%% v20.2 (August 22, 2026): THE TWENTY-TWO-PAPER DISCLOSURE.
%%   Every reference to a companion paper now carries that companion's own
%%   permanent page -- e.g. https://preprints.arisaka-gg.org/a3-bi6/ -- in place
%%   of the single program address every reference used to share.  Papers that
%%   are not cohort members keep their own locators: the 1985 Kamiokande paper,
%%   the three PsyArXiv works, and the CKM paper still in preparation are
%%   untouched.
%%   Sixty-four companion references carried no locator at all and now carry
%%   one, in this paper's own format.
%%   A11-Origin is posted with the rest, so the twenty-one papers that called it
%%   "in preparation" no longer do, and fifteen printed statements about its
%%   status -- footnotes, table rows, captions and the staged-submission
%%   paragraph -- were corrected to match.  The author's own reservation about
%%   its upper layers is kept verbatim; only its posting status changed.
%%   Two malformed sentences were repaired: A3's "UCLA preprint,." and P2's
%%   title running into the identifier.
%%   No claim, value, figure, table or grade differs from the previous version.
%% ===================================================================
%% ===================================================================
%% v15.9 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 21 of this paper's 21 cohort bibitems named a companion by a title
%% that companion does not carry.  Measured, not recalled: the truth is
%% each paper's printed cover, read from its deployed PDF; the markup is
%% each paper's own \GGTitlePage arguments, so the title sets in the same
%% faces the cover does.  Cohort-wide the pass touched 233 sites in 23
%% papers, and afterwards all 477 companion bibitems agree with their
%% covers.
%%
%% TWO CLASSES, and the difference is worth recording.
%%   RENAME PROPAGATION -- a paper was retitled and its citers were never
%%   updated.  The signature is the same wrong string in many papers:
%%   A6-GRIP's retired subtitle "Gi-4 Pillars (TGG, CCI, GFF, and GRIP) as
%%   the Geometric Backbone of GDOS" stood in seventeen papers, and
%%   C2-Hubble's retired "Two Hubble Constants as the Two Ends of One
%%   Six-Dimensional Geometric Corridor" in seventeen more.  W1-Symmetry,
%%   W2-Holonomy, P5-CPT and P6-Proton were all renamed and none of their
%%   citers followed.
%%   HOUSE SHORTHAND -- P5-CPT, W1-Symmetry and W2-Holonomy each wrote
%%   their own descriptive short form for every companion ("GDOS: the
%%   Boundary Law", "The Kaxion", "The Cosmological Inventory of the
%%   Descent").  Readable in prose and useless in a reference list,
%%   because none of those strings finds the paper.
%%
%% ONLY TITLE TEXT CHANGED.  Author blocks, years, availability lines,
%% Relevance notes, keys, labels and \emph/\textit wrappers are asserted
%% unchanged in count; the changelog comments are asserted BYTE-IDENTICAL.
%% ====================================================================
%% ===================================================================
%% v15.8 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY PROMISED A PAPER THAT WILL NOT BE
%%     THERE.  It read "Preprint, available at https://arisaka-gg.org/",
%%     which becomes false the day the homepage goes up carrying
%%     twenty-one papers.  It now reads "In preparation.  To be posted
%%     ... on completion; no result of the present paper depends on it."
%% (2) THE A11 TITLE WAS TRUNCATED.  W1 cited it as "The Geometric
%%     Inevitability of Cosmos, Life, and Mind" -- correct as far as it
%%     goes, but the subtitle "The Self-Referential Closure of the
%%     GG-Theory Program" was missing.  Restored, for the same reason as
%%     (1): the entry now says in preparation, so the title is the whole
%%     of what a reader has to go on.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  2 printed sites; \label / \ref identifiers and changelog
%%     comments protected.
%%     THE FIRST RUN OF THIS SWEEP WAS WRONG AND THE GATE PASSED IT.  The
%%     suffix alternation read (e|ed|es|ing), and Python's | is
%%     first-match-wins, so "catalogued" matched the bare "e" and left the
%%     "d" stranded: the sweep wrote "catalogd".  No spelling audit has
%%     that string in its pair list, so nothing reported it; it was found
%%     by reading a rendered page.  Alternation reordered longest-first,
%%     the suffix map asserted exhaustive, and the loosened check re-run
%%     against text carrying the known defect to confirm it still fires.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
%% v15.6 (August 17, 2026) -- THE SHOWCASE HEADLINE COMES BACK TO SANS.
%%
%% This paper set its showcase headline in SERIF, and thirteen of the
%% cohort set theirs in sans-serif bold.  Measured across all
%% twenty-four after the v11.5 promotion, that was the ONLY one of the
%% four surviving divergences a reader turning pages could see -- the
%% trailing skip is invisible on a page centered by \vfill, and the one
%% size difference is A4's.
%%
%% IT WAS DRIFT, NOT A DECISION.  This paper and P6, W1, W2 and the
%% other study were the LAST of the showcase pass; the sans form is the
%% one the author read and approved on A1-Constitution and on the
%% twelve that followed it.
%%
%% ONE LINE CHANGES: \GGShowcaseSeriftrue becomes
%% \GGShowcaseCenteredtrue, and style file v11.6 puts \sffamily in
%% that branch.  THE SHAPE DOES NOT CHANGE -- this page still centers
%% its headline with \begin{center} and still takes 8 pt beneath it,
%% because both are measured fits on this page and because three of the
%% five call the macro with no \centering around it, so the
%% \begin{center} is what centers them.
%%
%% ONE PAGE MOVES, and it is the showcase page.  Everything else in
%% this paper is asserted identical to v15.5, page for page.
%% ===================================================================
%% ===================================================================
% v15.5 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the centered serif form
%%   and says so in one line.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.4 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.4 (August 17, 2026) -- THE PREFACE OFF THE PAGE NUMBER, AND
%% FIFTEEN BRITISH SPELLINGS NO GATE COULD SEE.
%%
%% 1. THE BOTTOM MARGIN.  Found by the cohort-wide bottom_scan.py sweep,
%% run after the author found the same failure on page 2 of W2-Holonomy
%% by reading it.  Two pages in twenty-four papers put body text past
%% the bottom of the text block; page 16 of this paper was one, with the
%% Preface's last line 30.14 pt below the block and printing through the
%% folio.  Page 17 then carried ONE line and nothing else -- so the
%% collision AND the near-empty page it was invented to prevent were
%% both shipping.
%%
%% The cause was \enlargethispage{2\baselineskip} before "If you read
%% only fifteen pages", and the v15.0 changelog at that site had already
%% written "the author's 2 is already at the limit" without an
%% instrument that could say which side of the limit.  FOURTH occurrence
%% of this failure in the run: P5, W1, W2, W1 again.
%%
%% The enlargement goes and three printed lines come out of the
%% Preface's own repetitions -- the appendices' starting level said
%% three times, "a reader at the level of" echoing "A word on the level"
%% one clause earlier, "that appears in this paper", and "the appendices
%% are where the work is done" where the appendices are the subject.  NO
%% CLAIM IS WITHDRAWN AND THE READING GUIDE IS NOT TOUCHED.
%%
%% 2. stabiliser -> stabilizer, FIFTEEN SITES.  Charter 1.1 names
%% "stabilize" in its pair list; spelling_audit.py's PAIRS list does not
%% carry it, so this paper has reported MIXED PAIRS 0 and ALWAYS-BRITISH
%% 0 at every gate since migration while printing fifteen of them.  A
%% cohort scan on the whole -ise/-iser/-isation family, hand-classified
%% site by site, finds sixty-one real British spellings in twelve live
%% papers; the inventory is in Report 406.
%%
%% Substitution is on PRINTED TEXT ONLY: changelog comments are the
%% historical record and \label/\ref/\cite arguments are keys, so both
%% are masked.  Dirac's published 1931 title "Quantised singularities in
%% the electromagnetic field" is asserted present and unchanged.
%% ===================================================================
%% ===================================================================
%% v15.3 (August 17, 2026) -- THE LADDER LEAVES THE FRONT MATTER.
%%
%% The author's instruction: remove Figure 1 from the Executive Summary,
%% trim the Executive Summary by a few lines, and bring the keywords up
%% onto page 4.
%%
%% THE FIGURE IS MOVED, NOT DELETED.  Figure 1 carries \label{fig:ladder}
%% and the label is read in Part II: \S ssec:promise opens "Figure~\ref
%% {fig:ladder} draws the claim the rest of the paper is required to
%% earn."  Deleting the float would have left that sentence pointing at
%% nothing.  The float now sits immediately after the quoted claim it
%% draws, which is where it belonged -- beside its own claim instead of
%% three pages ahead of it.  Placement stays [H]; it remains a savebox
%% with two uses, the body copy and the showcase copy.
%%
%% THE TRIM IS OF MY OWN PROSE, NOT THE AUTHOR'S.  Three edits, all
%% inside the closing remarks written at v15.2, and each removes
%% something already said rather than compressing something said once:
%% the fourth paragraph's restatement of the Executive Summary's own
%% "What is claimed, and what is not"; "which is unsurprising" before a
%% clause that says why; and "actually" in "the world we actually
%% inhabit".  Not one word of the original Executive Summary is touched.
%%
%% THE KEYWORDS now set at the foot of page 4, under the closing
%% remarks, with no page of their own.
%%
%% FIGURE NUMBERS MOVE, and the count is in the report.  No cohort paper
%% cites this one by figure number.
%% ===================================================================
%% ===================================================================
%% v15.2 (August 17, 2026) -- THE SHOWCASE PAGE AND THE CLOSING REMARKS.
%%
%% SHOWCASE: fig:ladder above, fig:closed-open below, under the author's
%% own sentence from the abstract -- "conservation laws grow harder to
%% break as they say less about motion".  The house pairing this pass
%% settled on is the machinery, then what it decides: the ladder IS the
%% machinery of this review, and the two panels of Part III are what it
%% decides -- admissibility on the left, exclusion on the right, "not
%% inconsistent but unaffordable".  Both plates are SAVEBOXES, filled
%% once inside \setstretch{1} with \global\sbox and used twice, at their
%% BODY widths (0.574 and 0.750) and not a point above.
%%
%% CLOSING REMARKS, on the author's commission, for a broader audience:
%% four paragraphs closing the Executive Summary.  They restate what the
%% review already argues, in language a reader outside the field can
%% follow -- that the century tracked a bookkeeping identity wearing a
%% symmetry as its visible feature; that what survives in a universe
%% whose symmetries are spent is what can be counted, because nothing
%% gradual can move a whole number; that breaking is the productive step
%% and not the damage; and that beauty is evidence of nothing, a standard
%% turned here on this program's own premises.  NO NEW CLAIM: every
%% sentence has a home in the abstract, the Executive Summary or
%% fig:ladder's caption.
%%
%% NEW PROSE IN THIS SHIP: the four closing paragraphs, the showcase
%% headline and the showcase caption.  Nothing in the body is rewritten;
%% no figure, table, equation or number moves.
%% ===================================================================
%% ===================================================================
%%  W1-Symmetry v1.6  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  13 figures re-placed so their node text lands at the house size,
%%  7.79pt: what GG-P3-Electron Figure 3 renders (\footnotesize at
%%  0.865 of natural size).  For a picture of measured width W drawn
%%  at a base font of B points the target is f = (7.79/B) * W / 159.2
%%  of \textwidth, W being measured by compiling each picture
%%  standalone and reading its ink bounding box.  Nothing inside any
%%  picture changes: no font command, no dimension, no text, no
%%  relative position.  The cohort map and the central-claim funnel
%%  are excluded, both being at the house size already.
%%  Per-figure table and recipe: [v13-support]/allsweep.py and
%%  [v13-support]/figure_census.json.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v1.5  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.4 -> v1.5), author-directed: Part III gains an
%%  overview chapter.
%%    * NEW Section 11, "The open-system turn: what changes, and the
%%      four moves that make it", in seven subsections.  Part III
%%      previously opened on AX-2 two paragraphs after the Part title,
%%      and named its third and fourth moves nine and fourteen pages
%%      later.  The new chapter states the proposal before any of it
%%      is argued.
%%    * It asserts no new physics.  Both of its central answers exist
%%      already in G2 -- ssec:closed-world and ssec:exclusion -- and
%%      are previewed at claim level with the graded treatments left
%%      where they are and forward-referenced.
%%    * NEW figure: the closed frame and the open frame, ending at
%%      admissibility and at exclusion.
%%    * NEW figure: the twenty-two-paper cohort map, on its own page,
%%      cohort-standard form.  This paper had no cohort figure at all
%%      and becomes the twenty-first carrier.  Do not localise it.
%%    * NEW table: the four moves by the question each answers and the
%%      paper that owns it -- orientation, forward-referencing the
%%      G-table, which remains the audit.
%%    * ssec:frame moved out of G1 into the new chapter (it describes
%%      all four chapters, not the first); label unchanged.
%%    * Six cohort papers had no bibitem here: P1-KK, P3-Electron,
%%      P4-Real, C1-Kaxion, C2-Hubble, C3-BH.  Added with Relevance
%%      notes, so the bibliography knows every paper the map draws.
%%    * Part III subtitle amended; cover date August 4 -> August 5, to
%%      agree with \PaperDate.
%%  Plan and diagnosis: [v13-support]/GG-Report-205.
%%  Script: [v13-support]/w1_v15_sec11.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v1.2  (August 5, 2026)
%%  CHANGELOG (v1.3 -> v1.4): the A6 paper identifier.
%%    ucla-arisaka-gg-11.sty v11.1 fixes \ASix, which had expanded to
%%    the retired GG-A6-Gi4 since the May 2026 rename.  Twenty papers
%%    mask that with a local \renewcommand; this one does not, so it
%%    was printing the retired name in the GG-A6-GRIP bibitem's
%%    Relevance note -- directly under a title line already reading
%%    Arisaka-GG-A6-GRIP.  One rendered site; source unchanged apart
%%    from this note and the version stamp.  Page count unchanged.
%% ===================================================================
%% ===================================================================
%%  CHANGELOG (v1.2 -> v1.3): P7-Inflow residue.  Ship 4 re-pointed
%%    this paper's eight prose mentions but left the macro that named
%%    the retired paper, \providecommand{\PaperPSeven}, standing in
%%    the preamble with no remaining use.  Removed.  One site, no
%%    rendered output changes.
%% ===================================================================
%% ===================================================================
%%  CHANGELOG (v1.1 -> v1.2): P2/P7 merger ripple, ship 4 of 4.
%%    * P7-Inflow is retired into P2-N3 Appendix H (P2 v15.2).  This
%%      paper carried the largest ripple: eight prose mentions, now
%%      pointing at that appendix.  P7 was described here as "the
%%      member paper", "the inflow paper" and "a member paper" --
%%      none of which is true after the merge -- so the prose is
%%      rewritten, not merely re-keyed.  The cohort-inheritance list
%%      loses a bullet (the pincer, the capacity and the pushforward
%%      now come from one paper).  The ArisakaP2N3 relevance note
%%      inherits the two-sided-audit paragraph the P7 note carried.
%%      No claim, number, grade, figure or table value changes.
%%  Script: ship4_repoint.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v1.1  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.0 -> v1.1), author-directed: an open-handed polish.
%%  The paper argues against a large and serious research programme,
%%  and these four changes are all in the direction of arguing with it
%%  rather than past it.
%%
%%    * ssec:steelman gains the landscape's case in the LANDSCAPE'S
%%      OWN VOICE, at full strength, before it is answered: that the
%%      many vacua were found rather than wanted, by people who would
%%      have preferred one; that AdS/CFT and the extremal
%%      black-hole microstate count survive independently of the
%%      landscape reading; and that Weinberg 1987 predicted the order
%%      of magnitude of Lambda from selection considerations a decade
%%      before it was measured, closer than any first-principles
%%      calculation of the day.  A programme asking to be judged by
%%      dated forward commitments cannot dismiss its opponent's one
%%      dated forward commitment that landed.  The disagreement is
%%      then restated as what it is: not about competence but about
%%      what an explanation IS -- permitted-and-unsurprising against
%%      compelled -- with the note that which standard is right is a
%%      real question, not settled by either side asserting it.
%%      NEW bibitem Weinberg1987.
%%
%%    * ssec:exposure opens by owning the ladder as a CONSTRUCTION:
%%      a reviewer chose these seven rungs and this order, and the
%%      selection is not innocent of the conclusion it reaches.  A
%%      different reviewer could put spontaneous symmetry breaking,
%%      the renormalisation group, or Wigner's classification on it
%%      and arrive elsewhere.  Two defences offered, neither a proof;
%%      and the explicit concession that a reader who rejects the
%%      ordering criterion should reject the ladder, Part II's
%%      chapters surviving intact.
%%
%%    * The trials factor, previously an admission, becomes a DATED
%%      METHODOLOGICAL COMMITMENT in the shape the physical
%%      falsifiers use -- what, kills, by whom, window, status if
%%      unmet -- with the alphabet and target list to be fixed and
%%      published BEFORE the computation is run.  The paper's own
%%      method applied to the paper's own weakest point.
%%
%%    * Superlatives thinned where they cluster (Parts V and VI).
%%      Deflationary and opponent-generous superlatives kept; the
%%      maximal self-assessments softened, including one of the two
%%      "in this review's judgement, the deepest" constructions.
%%  Script: w1_v11_polish.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v1.0  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  PROMOTION v0.64 -> v1.0, author-directed.
%%
%%  The paper is complete.  It carries the seven-rung ladder of
%%  Part II, the open-system turn of Part III, the ledger of Part IV,
%%  fifteen technical appendices with eighty-two worked problems, one
%%  hundred and two questions answered, and one hundred and seven
%%  references each with a Relevance paragraph and a first-citation
%%  link.  All eight criteria of the final read (GG-Report-189, 190)
%%  are closed.
%%
%%  Changes at promotion, both mechanical:
%%    * \pdfoutput=1 added before \documentclass -- the arXiv flag
%%      that forces pdflatex.  W2-Holonomy carries the same line; W1
%%      did not, and would have been processed by the wrong engine.
%%    * Version stamp v0.64 -> v1.0.
%%  No text, number, figure, table or reference changed.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.64  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.63 -> v0.64), author-directed: a List of Tables, and
%%  four near-empty pages closed.
%%
%%  LIST OF TABLES
%%    * New \GGListOfTables in ucla-arisaka-gg-11.sty, a clone of
%%      \GGListOfFigures reading the .lot file.  ADDITIVE: no existing
%%      macro changed, so every other cohort paper is unaffected.
%%      Always starts its own page -- a paper long enough to want one
%%      has too many tables to share the front-matter page.
%%    * Called immediately after \GGListOfFigures.  35 entries, one
%%      page.
%%    * Six tables had a \caption with no short form, so the LoT
%%      printed their whole caption and ran to three pages.  Each now
%%      carries \caption[short]{long}: cosmos-sheet, comparison,
%%      falsifiers, same-integers, L-triality, M-chain.
%%
%%  NEAR-EMPTY PAGES  (measured by ink coverage; a full page reads
%%  ~10.3% at 20 dpi, the median page 9.0%)
%%    * Preface spilled four lines.  The grade-vocabulary itemize
%%      becomes a run-on paragraph (the author's suggestion), the
%%      reader-assumptions and fifteen-pages paragraphs are tightened,
%%      and the page is enlarged by two lines.  The Preface is now one
%%      page.
%%    * The timeline picture was preceded by \clearpage, which
%%      reserved a whole page and left one line stranded.  It is now a
%%      proper float, so the text closes up behind it and the
%%      check-yourself box that used to sit alone joins the page
%%      before.
%%    * The Part IV closing seam ran twelve lines past the page.  Its
%%      two paragraphs become one, and the page takes four more lines.
%%    * The Part V closing seam ran one line past.  Tightened, and the
%%      page takes two more lines.
%%    * Net: 351 -> 347 pages, four near-empty pages gone.
%%      Two sparse pages remain and are correct as they stand: the
%%      Part V banner page (every Part banner looks like that) and the
%%      thirteen-line page closing the Conclusions.
%%  Scripts: w1_v64_layout.py, w1_v64_trim.py (+ two inline passes).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.63  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.62 -> v0.63), author-directed: the remaining medium
%%  and low findings of GG-Report-189, 119 anchored patches plus one
%%  block move.
%%
%%  REFERENCES (criterion 4) -- 35 patches
%%    * 17 new Section-B bibitems, each alphabetically filed with a
%%      house-format Relevance note and cited at the point of
%%      attribution: Atiyah-Singer, Bennett, Casas-Ibarra,
%%      Coleman-Weinberg, Curie, Davidson-Ibarra, Gavela et al.,
%%      Goldstone, Greenberg, Harvey-Turner, Jarlskog, Jost, Landauer,
%%      Nambu, Peccei-Quinn, Pilaftsis-Underwood.  Curie's principle
%%      and the CPT theorem had been quoted and dated with no source.
%%    * Four "First cited in" notes repointed to their true first use.
%%
%%  FLOATS (criterion 5) -- 30 patches
%%    * The six bare center+tabular blocks that were really tables now
%%      carry \small, a house-style caption, a label and a number, and
%%      are cited inline: the abelian ledger, the centre-triality and
%%      n-ality censuses, the alphabet audit, the two-fibre table and
%%      the Sakharov anatomy.
%%    * Table standard: all five \footnotesize tables to \small (three
%%      needed one column widened by 0.01\textwidth, measured not
%%      guessed); the three longtables to \small; the verification
%%      harness gains the caption and label it lacked.
%%    * fig:seam gains a local pointer -- its only citation was 6,300
%%      lines away.
%%
%%  CONSISTENCY AND ACCESSIBILITY (criteria 1, 8) -- 51 patches
%%    * cos(3pi/5) is -1/(2phi), not the golden ratio; the kaon CPT
%%      bound is 4.4 orders below the K_S width, not six; the theta-bar
%%      leakage is seven orders, not eight (found in review, in two
%%      places); the horizon problem is not dated by Guth's paper.
%%    * "Primitive", "off shell", "characteristic class", the grade
%%      D^*, and the n of Delta B = 3n are now defined at first use.
%%    * A washout threshold that could not be reproduced from the
%%      printed formula; a neutrino-mass sum stated two ways; a
%%      charged-fermion count; two article errors; a subsection title
%%      repeated as its own first line.
%%
%%  STRUCTURE -- 3 patches + one block move
%%    * Part III promised a grade box at the end of every chapter and
%%      delivered two of four.  G2 and G3 now have theirs.
%%    * Chapter G1 ran its Steps 1, 3, 1(cont), 2 -- out of order in
%%      the chapter that had just announced the order.  Now 1, 1(cont),
%%      2, 3, 3(cont), 3(cont).  No text changed; the block moved.
%%    * The Acknowledgments name the century they walk through --
%%      Noether, Bianchi, Wilson, Ward and Takahashi, Green and
%%      Schwarz, Hopkins and Singer, and the experimenters: Wu's team,
%%      and Christenson, Cronin, Fitch and Turlay.
%%  Scripts: patches_refs.py, patches_floats.py, patches_consist.py,
%%           patches_struct.py, apply_patches.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.62  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.61 -> v0.62), author-directed (GG-Report-189, batch 2
%%  of 2: cohort alignment and the remaining corrections).
%%    * Appendix F's problems were numbered D.1-D.6 and Appendix G's
%%      E.1-E.6, colliding with the real D and E problem sets.  Now
%%      F.1-F.6 and G.1-G.6.
%%    * Hopkins-Singer pushforward degree 2k-d -> 4k-d (2k-d gives -2
%%      where the same paragraph concludes H^2).
%%    * K.5: sin(1.144) = 0.91030, not 0.90930.
%%    * NEW paragraph closing the GRIP chapter: which fibre does which
%%      job.  DGI's equivalences are the circle's group; the
%%      GSSB->GA descent is the interval's semigroup; GSSB sits on
%%      the seam.  The custody sentence, as shipped in A6 v13.16 and
%%      W2 v1.2.
%%    * The Wilson chapter's protection claim gains its limiting
%%      clause: the circle protects two different things two
%%      different ways -- a winding cannot move at all; a Wilson-line
%%      phase is continuous and does move, and what non-locality buys
%%      there is finiteness, not fixity.
%%    * "Nineteen coordinated papers" -> "The coordinated papers of
%%      the cohort" (stale count).
%%  Script: w1_v62_batch3.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.61  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.60 -> v0.61), author-directed (GG-Report-189, batch 1
%%  of 2: correctness and structure).
%%
%%  CORRECTNESS
%%    * Cover date: \GGTitlePage said August 1, \PaperDate said
%%      August 4.  Two dates on one page.  Now both August 4.
%%    * fig:ladder caption counted rows in one sentence and rungs in
%%      the next; S4 carries an integer and still cannot fail.  Recast
%%      in rows, with S4 named explicitly.
%%    * fig:ladder dated the Wilson rung 1983 (that is Hosotani);
%%      Wilson's loop is 1974, as Figure 2 and the bibliography say.
%%    * The plain-English overview and the timeline box both said the
%%      Ward identity "survives quantisation", full stop, while the
%%      body's own subsection is "What survives quantisation, and what
%%      does not".  S5 is the first rung that CAN fail; both summaries
%%      now say so.
%%    * S3's forward pointer described S6 (deliberate breaking), not
%%      S4.  Stale from before the Wilson chapter was inserted.
%%    * "the whole of Chapter 3 in miniature" -> "of this chapter":
%%      Noether is Chapter 4.
%%    * The flavour-graph count said (N-1)^2 and then read off 0,0,1,
%%      which is (N-1)(N-2)/2.  Both counts now stated, with unitarity
%%      as the reason they differ.
%%    * GRIP written as the cohort writes it: the four-move descent
%%      DGI -> (GFF) -> GSSB -> GA.  GFF is named in the chain and
%%      parenthesised, because it is the terrain rather than a stage
%%      with its own critical point -- W1's own reason, kept.
%%    * The age ledger printed 140.24 = 36.1 + 91.6 + 12.5, which sums
%%      to 140.2.  Canonical two-decimal summands restored.
%%    * Landauer's 0.0179 eV was called "seven percent" of a molecule's
%%      thermal energy; it is a little under half of (3/2)kT.
%%
%%  STRUCTURE (criterion 7): seven Part seams closed
%%    * Parts I, III, IV, V, VII and VIII gain a closing
%%      "Part N in one paragraph, and what comes next"; Part VI gains
%%      its handoff to the back matter at the end of sec:established,
%%      before the personal remark, so the ending is not spoilt.
%%      Only Parts II and IX already closed correctly.
%%
%%  FLOATS (criterion 5)
%%    * Ten numbered floats were never cited.  All now are, including
%%      fig:ladder -- the reader's map, which had zero \ref in 23,391
%%      lines -- and tab:failures, tab:g-table, tab:law-identity,
%%      tab:two-ends, tab:E-vector-axial, tab:I-closed-open,
%%      tab:H-dictionary.
%%  Script: w1_v61_batch1.py, w1_v61_batch2.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.60  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.59 -> v0.60), author-directed:
%%  Part VIII expanded from 66 to 102 questions, to the standard of
%%  W2-Holonomy's Part X.
%%    * NEW opening cluster, twenty questions, 'For the curious
%%      reader' -- basic in form, philosophically deep in substance,
%%      placed ahead of every specialist cluster as W2 places its
%%      twenty-two.
%%    * NEW specialist cluster, ten questions, on Part IV (the
%%      ledger): the parameter count, the running, CPT, strong CP,
%%      the Kaxion window, the baryon asymmetry, Sakharov's third
%%      condition, proton stability, and the FLAGGED age ledger.
%%      Part IV had no cluster at all.
%%    * Six further entries filling thin clusters: two on Bianchi
%%      (the gravitational twin; whether a monopole could still be
%%      found), two on the turn (what a semigroup is; what this adds
%%      to decoherence), one on Green--Schwarz (what BI-6 adds to
%%      hypercharge assignment), one on Hopkins--Singer (necessary
%%      or ornamental).
%%    * All sixteen clusters now open with an orienting paragraph
%%      naming the hardest question in the cluster.
%%    * All questions renumbered contiguously Q-W1-1..102.
%%  Script: w1_v60_assemble.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.59  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.58 -> v0.59), author-directed.  See GG-Report-183.
%%    PART VII (Technical Appendices) completed.  The Part carried eight
%%    appendices covering the Ladder of Part II and NOTHING from the
%%    open-system turn (Part III) or the ledger (Part IV).  Seven new
%%    ones close that gap, inserted before the dictionary so A-G keep
%%    their letters and their problem numbering:
%%      H  app:bianchi   the Bianchi identity, derived and placed (S3)
%%      I  app:opensys   density matrices, channels, the semigroup
%%      J  app:params    the parameter count; 'zero free parameters'
%%      K  app:discrete  C, P, T and the CPT theorem
%%      L  app:confine   confinement, colour triangle, strong CP
%%      M  app:lepto     baryogenesis and leptogenesis
%%      N  app:time      chronology, entropy, the arrow of time
%%      O  app:dictionary  (was H; unchanged, now last)
%%    Each in the house four movements: Step 1 the standard material
%%    derived from its textbook starting point at length; Step 2 what
%%    that treatment assumes and leaves open; Step 3 the body's
%%    reading; then six problems with full worked solutions.  42 new
%%    problems, 90 in the Part.
%%    Part opener and prerequisites rewritten: 'Eight' -> 'Fifteen',
%%    with a map of how the fifteen are arranged and how to use them.
%%  Scripts: w1_v59_assemble.py.  Sources: newapp_H..N.tex.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  v0.58  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.57 -> v0.58), author-directed.  See GG-Report-182.
%%    * ACKNOWLEDGMENTS relocated from between the Q&A Part and the
%%      References Part (p. 215) to the seam between the Conclusions
%%      and the Technical Appendices -- cohort convention (b), the one
%%      adopted for A9-SM and W2-Holonomy and now recorded in the sty's
%%      placement note.  One tailored paragraph added, on what a review
%%      in particular owes.
%%    * PART IX (References) rebuilt to the cohort A/B standard:
%%      - Section A: 13 cohort entries, canonical order, each note
%%        replaced by a full Relevance paragraph.
%%      - Section B: 75 external entries, RE-SORTED ALPHABETICALLY by
%%        first author (the B.1-B.9 rung headers retired; that ordering
%%        now lives in the 'Further reading, by level' block at the end
%%        of the section, where it is more useful).  Every note replaced
%%        by a full paragraph.
%%      - All 88 entries end with a first-appearance cross-reference,
%%        carried over verbatim from v0.57 so no link was re-guessed.
%%      - Part IX opening rewritten to announce the standard and to say
%%        why the rung grouping moved.
%%  Scripts: w1_v58_a1.py (acknowledgments), a2.py (Section A),
%%           b1-b3.py (Section B notes), b4.py (assemble + sort),
%%           b5.py (opening + stamp).  Harness: w1_refs_lib.py.
%% ===================================================================
%% ===================================================================
%%  W1-Symmetry v0.42  (August 1, 2026)  -- Q&A doubled and categorised.
%%  The thirty existing entries are kept verbatim and reorganised into
%%  eight topic sections (A-series style) by inserting section headers
%%  at the Part-tag boundaries. Thirty-six new entries (Q-W1-31..66) are
%%  appended in six new topic sections: Wilson & Ward (the two steps the
%%  short list skipped); breaking, time & existence; the multiverse and
%%  the landscape; integers & numerology; method, honesty & the cohort;
%%  necessity, mind & why anything. Several borrowed from A4/A9/A10/A11
%%  per author request. Total 30 -> 66. Subtitle/intro updated. Version
%%  bumped v0.41 -> v0.42.
%% ===================================================================
%% ===================================================================
%%  W1-Symmetry v0.41  (August 1, 2026)  -- R6: the design register.
%%  The word 'design' is retired from every site where the paper
%%  asserts it in its own voice, and the Concluding remark is re-aimed
%%  from 'design' to 'forced', consistent with A11 v13.61's
%%  "Not contingent. Not lucky. Not designed. Forced."
%%   (1) Abstract + Part I overview + Part III: "the mechanism by which
%%       the design selects a world" -> "the mechanism that selects a
%%       world" (assertive 'design' removed; the denial "not the loss
%%       of a design" is kept -- it denies a framing, asserts nothing).
%%   (2) Part III ssec:gssb disclaimer refreshed: A10 v13.20 Sec. 64 now
%%       uses 'productive' for baryogenesis, so the cohort carries the
%%       word at one site; recorded honestly.
%%   (3) Part V + the v0.40 three-horizon paragraph: "the design step"
%%       -> "the productive step" (A10's own wording).
%%   (4) Sec. title -> "Concluding remark: designed, or forced?";
%%       "mark of a design" -> "mark of a necessity" with the
%%       autobiography kept; "the design is symmetry" -> "the necessity
%%       is symmetry"; "if there is a design" -> "whatever is forced
%%       here"; "piece of engineering" -> "condition"; "a world tuned
%%       to the narrow strip" -> "driven into"; closing contrast
%%       rebuilt on 'forced' with the A11 four words cited.
%%  UNTOUCHED (correctly): the anthropic paragraph's "apparent design
%%  is a selection effect" and "the constants look tuned" (rival's
%%  voice); "A tuned universe would have to tune each region" (argument
%%  against tuning); ssec:twelve's "brought in from outside and tuned"
%%  (the closure claim, which Report 121 leaves standing).
%% ===================================================================
%% ===================================================================
%%  W1-Symmetry v0.40  (August 1, 2026)  -- S3: the three-horizon ladder.
%%  One paragraph added to the Discussions Part, directly after the
%%  design-step reconciliation paragraph: the H1/H2/H3 ladder (same
%%  conditions / same things / same choices), owned by A10 (v13.20
%%  Sec. 64); baryogenesis as the design step that threads the
%%  gauged-vs-global needle (A10 Sec. 32.13). Pointers only -- no new
%%  physics in W1. Version bumped v0.39 -> v0.40.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.39 -- R5: what the twenty counts.
%%
%%  Table 15 has twenty rows but eighteen distinct quantities:
%%  sin^2 theta_W appears under both N and L, zeta_Lambda under both K
%%  and L, because each closed form draws on two of the four integers
%%  at once.  "Twenty jobs" counts entries and the caption said so, but
%%  a sceptic counting the table finds the softness before the caption
%%  addresses it.  Both numbers are now printed, with the point that a
%%  quantity consuming two integers at once is a tighter constraint
%%  than one consuming a single integer -- re-use of a second kind, not
%%  a double count.  Both counts are recomputed from the tabular body
%%  by the harness; neither is carried in prose.
%%
%%  Part VI takes the short form of the z* correction, on the author's
%%  second ruling: "a topological marker in the recombination era",
%%  matching the Executive Summary, with one clause keeping the
%%  distinction explicit and the two-quantity discussion left where it
%%  already lives (S21.2, S17.3).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.38 -- R4: twenty jobs, not twenty-one.
%%
%%  Table 15 listed "the number of families" as one of the jobs done by
%%  the ledger integer N.  It is not one.  N is the colour index N_c,
%%  fixed by a chain in which no colour count appears until it is
%%  concluded; the family count N_g is not a ledger member and is
%%  reached by the P2-N3 pincer, sharing no premise with that chain.
%%  The row was the last site in W1 carrying the retired one-quantity
%%  reading, and it sat in the one table a sceptic reads closely.
%%  Removing it takes the count 21 -> 20 at five sites: the abstract,
%%  the Executive Summary, the S21.2 title, the caption's lead and the
%%  caption's closing clause.  The row count is recomputed from the
%%  tabular body before and after, not remembered.
%%
%%  The absence is stated as a claim rather than left as a gap: two
%%  derivations sharing no premise and reaching the same small number
%%  is a stronger fact than one integer spent twice, and belongs where
%%  it is earned.
%%
%%  Also R4: the sweep for residual one-quantity language found one
%%  site, in the first-person Conclusions -- "J = 7, fixing a coupling
%%  measured at a collider and the epoch at which the first atoms
%%  formed".  S21.2 warns against exactly that four pages earlier.
%%  On the author's ruling the passage keeps the atoms and names both
%%  quantities: e^J gives the topological switch z_sw, not the atomic
%%  z_dec, and the two sit 0.4% apart in e-folds.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.37 -- the three corrections the author
%%  ruled on 30 July, applied together and shipped under their own
%%  version number.  (R1 and R2 were each built and deployed as v0.36;
%%  three distinct files carrying one version string is the defect that
%%  bit A2 this week, so this ship bumps.)
%%
%%  R1 -- the units.  Newton's constant is not the anchor of the
%%  theory; it is the energy-unit conversion for the scale dimension,
%%  co-equal with c for the Lorentzian directions and hbar for the
%%  compact phase.  All three equal unity in natural units, so the
%%  dimensionful content is a unit system and not a parameter.  Three
%%  sites: the Executive Summary price list, the Physics-intuition note
%%  of the twelve-quantities section, and the claim block.
%%
%%  R2 -- how the ledger is determined.  W1 said (1,3,5;7,35) was forced
%%  *given* two integers supplied upstream, checked by exhaustive search
%%  over a natural box with the ordering M < N < L load-bearing.  Every
%%  clause of that is retired.  M = 1 by dimensional minimality; L = 5
%%  by single-tensor Green--Schwarz consistency with no colour count
%%  entering it (T-BI6-9.0), which is what makes the chain
%%  non-circular; N = 3 by substituting the AX-4 clause J = 2N+1 into
%%  the dual lock, giving N = L +/- sqrt(L - M^2), roots {3,7} at L = 5,
%%  the block embedding N < L selecting 3 (T-BI6-10.2a); J and K by
%%  arithmetic.  Uniqueness is by construction -- the roots always sum
%%  to 2L -- not by search, and the discarded root (1,7,5;15,75) closes
%%  the identities too and is now said so plainly.
%%
%%  R3 -- N_c, N_g, N_f.  The cohort now distinguishes three integers
%%  that all equal three.  W1 wrote a bare N for the colour index and
%%  the family count alike.  Renamed to N_g where this paper speaks in
%%  its own voice about families: the toll ladder and solvency
%%  condition, the three-generation pincer, the fourth-family exercise,
%%  and two grade-ledger rows (24 sites, math mode only).  NOT renamed
%%  in the Standard-Model sheet, which reproduces A9-SM's closed forms
%%  and is that paper's to relabel; a note at the sheet says so.  Two
%%  notation paragraphs added.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.34 -- Stage 6: the close.  The subtitle
%%  names the ledger rather than the ladder alone; the abstract and the
%%  Executive Summary carry the payout and the price; and the concluding
%%  remark, written before Part IV existed, is finished -- the design
%%  question cashed against the sheets, the keystone paper's credo
%%  quoted verbatim and cited, and a signed closing statement of the
%%  position that has no other home in the corpus.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.33 -- Stage 5 of the ledger surgery: the
%%  Discussion revisits Parts I-III now that the ledger exists.  A new
%%  chapter cashes the ladder against the sheets -- four integers,
%%  twenty-one unrelated jobs, none re-chosen -- states the reading of
%%  symmetry breaking as the source of structure rather than damage to
%%  it (marked as this paper's, not the cohort's), answers the anthropic
%%  move on four counts, and puts the family number in proportion: one
%%  row of forty-nine, derived-conditional, consuming an empirical input.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.32 -- Stage 4 of the ledger surgery: the
%%  comparison, and the dated falsifiers.  The comparison is redrawn:
%%  the count-versus-count contrast the cohort's own papers make is a
%%  category error and does not survive, so the opponent's position is
%%  stated at its strongest first and the contrast that holds is about
%%  where the freedom goes once the dials are gone -- with the
%%  programme's own discrete set conceded unproved in the same section.
%%  The falsifiers are printed as a table with instruments and dates.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.31 -- Stage 3 of the ledger surgery: the
%%  cosmological sheet.  Twenty-one entries, of which four are anchored
%%  and say so; the two-redshift discipline kept intact; the expansion
%%  rate by two routes and the one concession attached to the third; the
%%  epoch stated as the price of the three absolute fractions; and the
%%  finding that no fully derived entry in the inventory is compared
%%  against a measured number.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.30 -- Stage 2 of the ledger surgery: the
%%  Standard-Model sheet.  Twenty-eight rows with closed form,
%%  prediction, measurement, deviation and the owning paper's own status
%%  letter in the same table; the four entries a reader checks first;
%%  charge and CP; and the section that says what "derived" means when
%%  six of twenty-eight are derived and one is a theorem.  Every
%%  numerically checkable row was recomputed before it was typeset.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.29 -- Stage 1 of the ledger surgery.
%%  A new Part IV opens: the accounting discipline that admits no
%%  continuous freedom, the twelve quantities the construction is
%%  allowed, the two constants the ledger pays out at once, and --- in
%%  the same chapter, not a footnote --- what the word zero does not
%%  mean.  The three member papers that own the quantitative results
%%  enter the bibliography, which until now did not carry them.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.28 -- phase R8: the front matter brought
%%  into line with a seven-rung ladder and a Part III of four moves.
%%  The Executive Summary's walk down the steps now includes the two
%%  that were added; its statement of the thesis is the one the steps
%%  support, that failure becomes discrete rather than impossible; and
%%  its closing paragraph fences the fourth move.  Keywords extended.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.26 -- phases R5 and R6: Part III as four
%%  audited moves.  Each is stated in one fixed three-step frame with a
%%  grade box, five subsections are added (the frame, the Wilson line
%%  inside the equivalence, anomaly inflow, the open Ward identity, and
%%  the split between an attractor and the ground attractor), a fourth
%%  chapter states and fences the second group of pillars, and the Part
%%  closes with a table that can be read against Part II's arc table.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.25 -- phase R4: the two new rungs made
%%  load-bearing rather than inserted.  Eleven passages across Parts I
%%  and II now point forward to them or read back from them; the
%%  anomaly chapter's opening is rewritten to name the identity that
%%  fails and its global case becomes a derivation with the colour
%%  count worked; the Hopkins-Singer chapter gains a subsection placing
%%  the Wilson line and the Dirac flux at the two ends of one exact
%%  sequence, and answers what moves the middle datum of its cocycle.
%% ===================================================================
%% ===================================================================
%%  v0.24 stage 7 -- the five-to-seven sweep: the subtitle, the
%%  Executive Summary, the Part II and Part IV blurbs, the chapter
%%  title, three table captions and the Part III opening.  The arc
%%  table's last column head follows its cells from dynamical content
%%  to the integer reading, and the caption sentence that read the
%%  column is rewritten with it.
%% ===================================================================
%% ===================================================================
%%  v0.24 stage 5 -- the three rung tables gain their S4 and S5 rows,
%%  and their last column changes from dynamical content to the
%%  integer reading, which is the one the seven-rung ladder supports.
%%  The dictionary's OGN-5 entry is brought into line with the owning
%%  paper, which derives J from the sum of squares and no longer
%%  carries J = 2N+1 as a member of the closure system.
%% ===================================================================
%% ===================================================================
%%  v0.24 stage 4 -- ten external references in two new rung groups:
%%  holonomy and compact dimensions, and the Ward identity with its
%%  non-abelian form.  Each carries a Relevance note and a
%%  first-citation pointer.
%% ===================================================================
%% ===================================================================
%%  v0.24 stage 3 -- the two new rungs.  Chapter 6 is S4, the Wilson
%%  line: what the Bianchi identity leaves out.  Chapter 7 is S5, the
%%  Ward identity: what it becomes when the theory is quantised, and
%%  the first statement on the ladder that can fail -- by an integer.
%%  Both sit between the Bianchi and Green-Schwarz chapters, so the
%%  anomaly chapter now opens by breaking an identity already given.
%% ===================================================================
%% ===================================================================
%%  v0.24 stage 2 -- two new technical appendices at the established
%%  depth: Wilson lines and the one-loop phase potential, and Ward
%%  identities derived from the path integral.  The appendix letters
%%  become A forms, B Noether, C connections, D Wilson, E Ward, F
%%  anomalies, G differential cohomology, H dictionary; the two
%%  incumbent label namespaces are re-prefixed D->F and E->G.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.24 -- stage 1 of the seven-rung revision:
%%  the Green-Schwarz identity moves from S4 to S6 and Hopkins-Singer
%%  from S5 to S7, freeing S4 for the Wilson line and S5 for the Ward
%%  identity.  Figure 1 is redrawn to eight rungs, and its two arrows
%%  are relabelled: the right-hand arrow now reads how much of the
%%  conservation statement is carried by an integer, which is the
%%  reading the seven-rung ladder supports and the one Part I asks for.
%% ===================================================================
%% ===================================================================
%%  Front matter: Figure 1 moved ahead of the Keywords so that
%%  \GGKeywords's own \vfill puts the Keywords on the last line of the
%%  Executive Summary's final page, as in the A-series, instead of
%%  leaving a gap in the middle of it.
%% ===================================================================
% ===================================================================
%  Second fitting pass on the abstract: 1622 normalised characters, at
%  the low end of the cohort band, with the riser above it reduced to
%  zero so the whole abstract sits on the front page.
% ===================================================================
% ===================================================================
%  Abstract rewritten to fit page 1: 1735 normalised characters, inside
%  the band of the cohort abstracts that fit their front page.  All
%  five steps, the counterweight and the declared-grade placement are
%  retained; the cuts are verbal only.
% ===================================================================
% ===================================================================
%  Abstract condensed to the cohort ceiling: four verbal condensations
%  in the abstract only, taking it from 1944 to 1916 normalised
%  characters against the ceiling of 1920.  No claim, hedge, grade or
%  attribution is altered.
% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.23  (July 30, 2026) -- FRONT MATTER
%%  rebuilt to the A-series structure, on the author's eight
%%  instructions.  Order now identical to A3-BI6 and A9-SM: title page
%%  (third argument the DATE, as there), abstract, Executive Summary,
%%  keywords, figures, \GGfrontmatteronepagetrue + Overview + List of
%%  Figures on one page, table of contents, THEN the Preface, then
%%  Part I.  New title and subtitle; the descriptive paragraph removed
%%  from the title page; abstract rewritten in plain English; Executive
%%  Summary made less technical and given inline citations to eight
%%  cohort papers and eight external works.  Series rule colour changed
%%  from the WP green to orange!85!black -- a soft orange, brighter than
%%  the A-series orange!50!black -- overridden in the paper preamble and
%%  NOT in the shared style file.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.22  (July 30, 2026) -- Part V expanded to
%%  roughly double length on the author's instruction: broader-audience
%%  physics intuition, and greater philosophical depth.
%%  The charter's ruling C7 forces the division: Ch. 14 takes the
%%  METHODOLOGICAL depth and Ch. 15 the METAPHYSICAL, since Part V
%%  carries the author's view "as the concluding remark only".
%%    Ch. 14: new 14.3, what kind of claim each Part makes -- review
%%      result / proposal at declared grades / wager, why most disputes
%%      about such programmes are really disputes about which is on
%%      offer, and why agreement is not fit.  New 14.4, the whole
%%      argument in four everyday pictures (the till and the ledger;
%%      dials and counts; the budget; the shadow).  Intuition paragraphs
%%      added to 14.1 and 14.2.  No first person anywhere.
%%    Ch. 15: rewritten whole so the new movements interleave.  Added:
%%      how strange the rigidity is (a consistent theory need not have
%%      integers); the conviction reported AS a report, not dressed as
%%      an argument; the anthropic/landscape alternative stated at full
%%      force, conceded three points, and answered on KIND rather than
%%      value; Wigner's question and why the invented-language answer
%%      does not hold, with Bianchi and q(x) as the two cases; the
%%      refinement claimed modestly and defended on CHECKABILITY; beauty
%%      taken personally ("I would have been wrong about parity"); four
%%      concrete things that would change the author's mind; and a close
%%      that declines the theological question rather than answering it.
%%  A9-SM cited and added to Group A; all pointers recomputed.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.21  (July 30, 2026) -- PHASE 5 (b): the
%%  verification harness recorded in the paper, per the charter's
%%  requirement that it be green "and its record printed".  New
%%  Appendix F.4 inventories the harness's blocks and states the number
%%  of passing checks -- a number this script reads from the harness's
%%  own output rather than accepting by hand, so it cannot go stale.
%%  The closing paragraph repeats the three notes that record places
%%  where the text was deliberately weakened against a first draft.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.20  (July 30, 2026) -- PHASE 5 (a): the
%%  reference apparatus completed.
%%  (1) Eighteen citation gaps closed.  The shipped text named Sakharov's
%%      conditions, Gleason, Lindblad/GKS, Berry, Milnor-Stasheff,
%%      Fujikawa, Wess-Zumino, Adler-Bardeen, Bogomolny, Weyl,
%%      Georgi-Glashow, Wilson, Fradkin-Shenker, 't Hooft (twice),
%%      Guralnik-Hagen-Kibble, Garwin, Byers, Brading,
%%      Alvarez-Gaume-Witten and Freed WITHOUT citing any of them.
%%      All twenty-two new entries were independently verified before
%%      typesetting; three corrections adopted (Bogomolny's title has no
%%      "The"; Freed is 2002, Surveys in Diff. Geom. 7, 129-194; Byers
%%      is cited without a page range, which could not be confirmed).
%%  (2) Group B regrouped BY RUNG per the charter, in nine groups with
%%      bold headers, and grown from 35 external entries to 57.
%%  (3) A further-reading-by-level block closes the references, with a
%%      second paragraph giving the cheapest order for the programme's
%%      own papers.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.19  (July 30, 2026) -- PHASE 4 (e):
%%  appendix expansion toward the A9/A10 depth benchmark, on the
%%  author's "the longer, the better" instruction.  One substantial new
%%  section in each of A-E, each closing a gap the body leans on:
%%    A.5  the Hodge star, and the formal statement of Ch. 5's
%%         law/identity split -- exactly one of Maxwell's two halves
%%         needs a metric; plus self-duality in 2m dimensions, which is
%%         what the six-dimensional tensor sector uses.
%%    B.6  the Belinfante improvement, and the constrained-Hamiltonian
%%         count done properly: 8 phase-space functions, two first-class
%%         constraints, 4 removed, 2 polarisations.
%%    C.6  bundles, transition functions, the cocycle condition and the
%%         clutching construction -- pi_1(U(1)) = Z is where Dirac's
%%         integer actually lives; plus the SU(2) instanton analogue and
%%         the Bogomolnyi bound the cascade tolls instantiate.
%%    D.5  the Wess-Zumino consistency condition, and why it FORCES the
%%         descent: uniqueness up to counterterms, additivity over
%%         species (hence Diophantine cancellation), and the
%%         availability of the Green-Schwarz counterterm.
%%    E.7  Cheeger-Simons characters written out, shown to BE the
%%         holonomy with Stokes mod Z; and the pushforward as
%%         integration over the fibre, with the chain Wu class -> a ->
%%         well-defined pushforward -> line bundle stated as one chain.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.18  (July 30, 2026) -- PHASE 4 (d):
%%  Part VII (thirty questions and answers) and the nine textbook
%%  bibitems Part VI cites.
%%  The Q&A uses the A9/A10 markup verbatim in form -- the same local
%%  environment, the same bold question in quotes, the same "Response."
%%  opener -- with the Part-of-origin as the tag where A9/A10 use a
%%  referee persona, since these are a reader's questions rather than a
%%  referee's.  The thirty questions are the charter's own list.
%%  Register: the BODY's level, not Part VI's, so that a reader who has
%%  read the chapters but not the appendices finds every answer
%%  self-contained.
%%  All first-citation pointers recomputed from document order.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.17  (July 30, 2026) -- PHASE 4 (c):
%%  Appendices E and F.
%%    E: the three data assembled as a differential cocycle with the
%%       compatibility condition written out; BOTH exact sequences, with
%%       the flat sector and the topologically trivial sector identified
%%       and the conclusion drawn explicitly (only the integer class
%%       cannot be deformed); Dirac recovered as the S^2 case;
%%       quadratic refinements PROVED to have odd characteristic element
%%       (integrality is the whole condition); Wu's formula with the
%%       vanishing 2j > n; the HS typing (k,i) = (2,2) solved and the
%%       n = 8 contrast worked as a problem; and the shift restated as
%%       lambda/2 = p_1/4 with the common misquote called out.
%%    F: the dictionary as a three-column longtable (term, definition,
%%       home) in the A9/A10 schema -- including the GRIP entry that
%%       records GFF as a SIBLING pillar; the full grade ledger with a
%%       fourth column the body's table does not carry ("if the premise
%%       fails"); and a flat statement that Part III derives NOTHING.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.16  (July 30, 2026) -- PHASE 4 (b):
%%  Appendices C and D.
%%    C: parallel transport on the sphere with holonomy = enclosed solid
%%       angle (harness block P4, including the octant giving exactly
%%       pi/2); the covariant derivative and Christoffel symbols; the
%%       curvature as a COMMUTATOR in both the gravitational and gauge
%%       cases, so that GGEP's collapse of two rungs is visible as an
%%       equation rather than asserted; the two Bianchi identities
%%       derived and contracted to nabla_mu G^{mu nu} = 0;
%%       holonomy, the Wilson loop, and the theorem that curvature does
%%       NOT determine holonomy where loops fail to bound;
%%       Aharonov-Bohm as the experiment that settles it.
%%    D: the anomaly THREE ways -- Landau-level counting carried out
%%       from the degeneracy eB/2pi and the drift eE/2pi; the triangle
%%       with the Adler-Bardeen exactness argument; Fujikawa's measure
%%       in words -- with the agreement of three unrelated derivations
%%       made the evidential point; the descent equations by degree
%%       ((D+2) -> (D+1) -> D, harness P5), the factorisation condition,
%%       and the BI-6 arithmetic in four hand-checkable lines including
%%       what the trace ratio does NOT fix.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.15  (July 30, 2026) -- PHASE 4 (a): the
%%  appendix apparatus, and Appendices A and B.
%%  AUTHOR RULING this phase: Part VI is pitched at first-year graduate
%%  students / senior undergraduates, from the standard textbooks, and
%%  follows the structure and depth of the A9-SM and A10-Cosmos
%%  appendices.  This supersedes the charter's "all at undergraduate
%%  level" for Part VI ONLY, and the Preface is amended to say so (a
%%  substantive edit to shipped text, reported in the ship record).
%%  House style adopted from A9/A10: \GGteachbox opening, \GGreadalong
%%  textbook line, the three-step spine (Step 1 as a graduate student
%%  meets it / Step 2 what it assumes and cannot explain / Step 3 what
%%  this paper does with it), \paragraph{Statement.} in place of theorem
%%  environments, every display labelled, "Physics intuition" closers,
%%  a problem set with full stepped solutions, and a \GGtakeaway close
%%  with A10's four slots including "Continue in".
%%  Part VI now opens with A10's prerequisites-and-textbooks paragraph.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.14  (July 30, 2026) -- PHASE 3 (c):
%%  Figure 2 redrawn and Figure 3 tidied, both from reading the pages.
%%  Fig. 2 had four collisions at once -- the rotated seam label running
%%  through the crossing arrows, the data labels printed on the bulk
%%  box's border, the fate notes printed on the boundary box's border,
%%  and the in-box text running into the frames.  Redrawn with the boxes
%%  moved apart, the crossing labels set above and below their own
%%  arrows in the open channel, and the seam label horizontal above the
%%  seam.  Fig. 3: landscape label moved clear of the DGI marker and the
%%  ledger column moved right so the drawing sits across the text block.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.13  (July 30, 2026) -- PHASE 3 (b):
%%  Chapters 10 and 11, and the Part III apparatus.
%%    Ch. 10 is written to A6's OWN structure, which differs from this
%%      thread's shorthand: GRIP is the pipeline DGI -> GSSB -> GA
%%      descending the GFF landscape, and GFF is a SIBLING Gi-4 pillar,
%%      not a stage of GRIP.  Sec. 10 opens by saying so.
%%      10.3 marks "breaking is productive" as THIS PAPER's reading and
%%      records that the owning paper does not use the word -- a reader
%%      who drops it loses no result.
%%      10.5 derives the toll ladder as the sum of pi*k, notes that it
%%      IS pi*q(N) with q the refinement of Sec. 7.5, and states the
%%      solvency condition in the per-family form (no budget window is
%%      quoted -- see harness note 6).
%%    Ch. 11: N = 3 with all THREE fences printed as a numbered list;
%%      the kappa-bridge told as two descriptions with no shared working
%%      parts agreeing, with both of the owning paper's cautions; an
%%      open-items list that includes the arrow of time as ASSUMED; and
%%      11.4, the single fenced forward-pointing paragraph, which cites
%%      A7 once and defends nothing.
%%    Check-yourself block for Part III; A7 bibitem added; every
%%    first-citation pointer recomputed from document order.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.12  (July 30, 2026) -- PHASE 3 (a):
%%  Figures 2 and 3 drawn, and Chapters 8 and 9 drafted.
%%    Fig. 2 (the seam): bulk, boundary, the projection, and the three
%%      data crossing -- with the ASYMMETRY as the point (the integer
%%      crosses intact, the form may vary, the holonomy is read on the
%%      boundary loop, not transported to it).
%%    Fig. 3 (the cascade): drawn TO SCALE for N = 3 -- Morse indices
%%      4, 2, 0; tolls 2pi and pi; total 3pi exactly filling the ledger
%%      column.  Every number checked in harness block M.
%%    Ch. 8: GGEP as AX-2 of the constitutional tower (AX-1 LC is prior,
%%      per A1), what the S2/S3 collapse does and does not buy, the bulk
%%      in physical language, and a grade box whose last line says this
%%      chapter's contribution is NONE.
%%    Ch. 9: the pivot.  Every rung re-read as a closed-system
%%      statement; OGGEP as AX-3 and explicitly an AXIOM; GDOS staged
%%      honestly (CPTP standing, memory kernel exact, GKLS a Markov-limit
%%      COROLLARY); the arrow of time recorded as CONSTITUTIONAL, NOT
%%      derived (A5's own words); admissibility-vs-exclusion attributed
%%      to P2-N3, which is where that framing actually lives, NOT to A5;
%%      and 9.6, the direction of dependence, written as a displayed
%%      six-link chain with "nothing in this paper founds quantum
%%      mechanics" stated flatly.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.11  (July 30, 2026) -- PHASE 2 (g): two
%%  defects found by reading the expanded Part II.  (1) Sec. 5.5: the
%%  law-against-identity table had been appended AFTER the section's
%%  closing paragraph, so the chapter's punchline and its hand-off to
%%  Ch. 6 no longer came last; the closing paragraph is moved to after
%%  the table.  (2) Table 5: column 1 widened (a code plus a name plus a
%%  section pointer needs about 0.20\textwidth -- this defect class has
%%  now appeared three times and that allowance is the standing rule).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v1.0-draft  (July 30, 2026) -- PHASE 2 (f):
%%  Part II expansion, second pass.  Five substantial additions:
%%    4.3  Elitzur on one site: the theta-integral factorising to give
%%         EXACTLY zero, contrasted with the global case where it does
%%         not factorise and breaking is real.
%%    5.5  Law-against-identity contrast table, six rows; its last row
%%         ("its premise stops applying") is what Ch. 6 exploits.
%%    6.2  How tight anomaly cancellation is: one Weyl fermion fails;
%%         a vector-like pair passes but is not chiral; exhaustive
%%         search finds no chiral solution below FIVE fermions, the
%%         smallest being {-9,-5,-1,7,8}.  Search scope stated in the
%%         text (harness block L1 and its note).
%%    7.1  Dirac's two-cap derivation with the transition function, so
%%         that the integer is identified as a WINDING NUMBER.
%%    7.7  Part II summary table: rung, statement, cost of violation,
%%         status, toolkit appendix.
%%  Part II: 18 pp (v0.8) -> 24 pp (v0.9) -> this build.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.9  (July 30, 2026) -- PHASE 2 (e): Part II
%%  brought up to the charter's apparatus budget, on the author's
%%  ruling that this be spent now rather than deferred to Phase 5.
%%  SEVEN new worked examples, every one checked in verify_w1.py block K
%%  before being typeset:
%%    3.2  energy as the time-translation charge, on the oscillator
%%         (E = qdot dL/dqdot - L turns T - V into T + V)
%%    3.3  the same complex scalar run globally and locally, side by side
%%    4.1  the bare derivative failing and D_mu repairing it, explicitly
%%    4.2  F invariant and dF = 0 shown to be the SAME fact
%%    4.5  the rank count: rank(SM) = 4 = rank SU(5) -- why five
%%    5.1  the four index triples of dF = 0, named
%%    5.2  the flux integral, and the argument FAILING when a monopole
%%         is enclosed (the two-cap picture, forward to Dirac)
%%    6.4  the Standard Model anomaly table, evaluated -- the cubes
%%         48, -1536, 192, -432, 1728 summing to zero
%%    6.5  the BI-6 ledger by hand, four lines
%%    7.2  the Aharonov-Bohm phase and its period h/q, with numbers
%%    7.5  the functional equation at x = 3, y = -5, and a = 2 failing
%%  Plus a physics-intuition paragraph closing each technical movement
%%  (fourteen added), per the charter's one-per-section budget.
%%  No existing text altered; all additions are appends.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.8  (July 30, 2026) -- PHASE 2 (d): the
%%  Check-yourself environment made unbreakable.  Found by reading the
%%  rendered pages: the Part II block split across a page break, leaving
%%  item 3 and the closing rule orphaned on the next page, so the block
%%  read as two fragments.  Body wrapped in a full-width minipage.
%%  No text changes.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.7  (July 30, 2026) -- PHASE 2 (c):
%%  apparatus.  Fifteen external bibitems added for Part II (Yang-Mills,
%%  Englert-Brout, Higgs, Voss, Bianchi, Aharonov-Bohm, Adler,
%%  Bell-Jackiw, Wu-Yang, Nielsen-Ninomiya, Dirac, Cheeger-Simons,
%%  Witten, Hopkins-Singer), each with a Relevance note.  EVERY
%%  first-citation pointer in the list -- including the ones written in
%%  Phase 1 -- is recomputed from document order, so none goes stale as
%%  earlier chapters acquire citations.  Check-yourself blocks added at
%%  the end of Parts I and II (new environment), closing the apparatus
%%  gap left open in Phase 1; every answer they assert is checked in
%%  verify_w1.py block J.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.6  (July 30, 2026) -- PHASE 2 (b): the
%%  top two rungs.  Chapters 6 and 7 drafted.
%%    Ch. 6: the anomaly derived by Landau-level counting (no diagrams,
%%      no path integrals), with the coefficient e^2 E.B / 4 pi^2 worked
%%      explicitly; why a gauge anomaly is fatal and a global one is a
%%      prediction; dH_3 = X_4 as a deliberate failure; "conservation as
%%      currency" with the Standard Model's own four integer anomaly
%%      identities written out; BI-6 as the six-dimensional instance,
%%      with BOTH what "locked" means and what it does not (the (5k,3k)
%%      family qualification, stated in the body, not a footnote); and
%%      the chapter grade box.
%%    Ch. 7: Dirac 1931 with the two-hemispheres argument; the third
%%      datum (holonomy) with Aharonov-Bohm and Wu-Yang; the
%%      Cheeger-Simons differential cocycle; the Witten shift stated as
%%      lambda/2 = p_1/4 (NOT p_1/2 -- harness block I4); q(x) =
%%      x(x-1)/2 with the odd-vs-even increment as the elementary form
%%      of the half-shift; the six-dimensional Wu count and why it makes
%%      the pushforward land on a LINE BUNDLE; and the top of the
%%      ladder, with the two features Part III consumes.
%%    Margin note corrected and expanded: THREE Wus, not two --
%%      Wen-Tsun (classes), Chien-Shiung (parity), Tai Tsun (phase
%%      factors).
%%  The 496 statement is worded to the harness caution: necessary, not
%%  sufficient; "the two NON-ABELIAN solutions", never "496 singles out".
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.5  (July 30, 2026) -- PHASE 2 (a): the
%%  first three rungs.  Chapters 3, 4 and 5 drafted.
%%    Ch. 3: the action principle with the free particle worked as the
%%      chapter in miniature; the first theorem with the complex-scalar
%%      current and the on-shell/off-shell distinction made explicit;
%%      the SECOND theorem with d_mu d_nu F^{mu nu} = 0; what stopping
%%      at the first theorem costs; the Goettingen 1918 box (Noether was
%%      answering Hilbert's energy problem, and the famous theorem was
%%      the by-product).
%%    Ch. 4: the phase made local, the connection FORCED, the curvature
%%      and the two free Maxwell equations; Elitzur; the Higgs
%%      misnomer with the state count done explicitly; SU(N) and the
%%      SU(5) branching 8+3+1+6+6 = 24; the locality/unitarity
%%      intuition for why redundancy is not a defect.
%%    Ch. 5: dF = ddA = 0; magnetic charge restated as the OBSTRUCTION
%%      to a global potential rather than as "forbidden"; the contracted
%%      second identity forcing the form of Einstein's equations; the
%%      priority box (Voss 1880, Ricci ~1889 via Padova, Bianchi 1902 --
%%      the name records the first proof, not the first discovery); and
%%      "an identity is not a law".
%%  All toolkits deferred to Appendices A, B, C as the charter requires.
%%  Chapters 6 and 7: Phase 2 (b).  Apparatus: Phase 2 (c).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.4  (July 30, 2026) -- PHASE 1 (c): table
%%  geometry, entirely from reading the rendered pages.  Column widths
%%  of the two arc tables retuned so rung labels and the "what
%%  conserves" header no longer wrap; the S3 cell of the earned table
%%  shortened so its formula does not break mid-expression; and every
%%  narrow prose column in all four Phase 1 tables made ragged-right
%%  (new column type P) to remove stretched interword spacing.
%%  No text changes elsewhere.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.3  (July 30, 2026) -- PHASE 1 (b): FRONT
%%  MATTER AND APPARATUS.  Abstract (to the 1920-character ceiling),
%%  Executive Summary (the whole arc, readable alone), and Preface (the
%%  three conventions, the four-grade vocabulary, what is assumed of the
%%  reader, and the fifteen-page path).  Group A grows to eight cohort
%%  entries and Group B to eleven external ones, every entry carrying a
%%  Relevance note.  First-citation pointers are COMPUTED from document
%%  order by the edit script, not written by hand.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry v0.2  (July 30, 2026) -- PHASE 1 (a): THE ARC.
%%  Parts I, IV and V drafted: Chapters 1, 2, 12, 13, 14, 15.
%%    Ch. 13 (the honest record) written first, so that the standard it
%%    sets is in place before any claim of the programme is stated:
%%    parity 1956-57, CP 1964, naturalness and the LHC null result, the
%%    read-off-the-data vs imposed-for-beauty pattern, and the
%%    programme's own exposure (the minimality axiom named as the
%%    load-bearing aesthetic premise).
%%    Ch. 12 (the arc earned) adds the cost-of-violation column.
%%    Chs. 1-2 state conservation and symmetry operationally, with the
%%    boundary clause planted early, and follow the integer thread.
%%    Ch. 14 separates the review result from the proposal and prints
%%    the grade table.  Ch. 15 is the first-person concluding remark,
%%    drafted for the author's revision, fenced at its head.
%%  Four new tables (failures, arc-promised, arc-earned, grades).
%%  Front matter and the new bibitems: Phase 1 (b).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W1-Symmetry  --  PHASE 0 SKELETON  (July 30, 2026)
%%  The White Paper series, member W1: "The Books Must Balance".
%%  This file is the labelled shell only: every Part, every chapter
%%  stub, every \label, the three figure slots, and the A/B reference
%%  apparatus.  NO PROSE.  Prose begins at Phase 1, after sign-off.
%%  Harness: verify_w1.py (43 OK, 0 FAIL) -- run before every build.
%% ===================================================================
\pdfoutput=1
%% ---- vv15.10 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 1 showcase page in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}

\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
\renewcommand{\LaTeX}{LaTeX}% cohort-standard shim

%% Cohort-standard Part structure: one-line Part entries in the TOC.
\makeatletter
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \addcontentsline{toc}{part}{\protect\textbf{Part \thepart}. #2}%
  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}%
  \thispagestyle{plain}%
  \begin{center}
    \vspace*{1.2em}
    {\Large\bfseries Part \thepart}\\[0.45em]
    {\huge\bfseries #2}\\[0.35em]
    {\large #3}
  \end{center}
  \vspace{0.5em}
  \GGThickLine
  \vspace{0.8em}
}
\makeatother

\PaperSeries{WP}
\PaperStatus{White Paper}
%% Series rule colour: the sty gives the WP series green!40!black.
%% The author's ruling is a soft orange, brighter than the A-series
%% brown (orange!50!black).  Overridden here, never in the .sty.
\renewcommand{\GGSeriesColour}{orange!85!black}
\PaperNumber{Arisaka-GG-W1-Symmetry}
%% ------------------------------------------------------------
%% v1.6 -> v15.0, August 15, 2026.  STAGE 1: MECHANICAL.
%%   The nine-stage v15 pipeline, ninth paper conversion.
%%   W1 IS A CONFIRMED SECTION 3.1 STALE ARTIFACT AND THE
%%   REGISTER'S STRONGEST CASE -- it shows BOTH signatures at
%%   once.  The deployed title page carries the pre-v11.3
%%   four-line author block AND PRINTS A COVER DATE, "August 5,
%%   2026", which the live v11.4 style accepts and no longer
%%   sets.  That printed date DISAGREES with the version stamp on
%%   the same page, which reads August 6: the deployed artifact
%%   showed a reader two different dates for one paper.
%%   Deployed 356 pages, rebuild 357, 350 of 356 differing.
%%   sty_check passes -- the style matches the live copy byte for
%%   byte -- so section 14.3 does not fire; section 3.1 does, and
%%   THE REBUILD IS THE BASELINE.
%%   THIS CLOSES THE STALE-ARTIFACT REGISTER.  Confirmed and now
%%   resolved: A9-SM, P1-KK, P2-N3, P4-Real, P6-Proton,
%%   W1-Symmetry.  Cleared as false: A11-Origin, C1-Kaxion.  The
%%   screen was six for eight.
%%   911 PRINTED BRITISH SPELLINGS -> 0 over 124 distinct words:
%%   THE LARGEST IN THE COHORT, past P5-CPT's 880 and P6's 543.
%%   2 IN THE PDF METADATA: pdfsubject `programme', pdfkeywords
%%   `quantisation'.  One British form sat inside a caption SHORT
%%   FORM (`two travellers'), which prints in the List of Figures
%%   and which a body-only sweep would have left standing.
%%   \pdfoutput=1 was ALREADY PRESENT -- the first paper in this
%%   run to arrive with section 12 item 10 satisfied.
%%   THE \GGTitlePage COVER-DATE ARGUMENT IS LEFT IN PLACE.
%%   Section 7: style v11.3+ accepts but does not print it, so it
%%   is inert, and A6-GRIP uses the same slot for a descriptive
%%   line rather than a date.  It is the reason the deployed
%%   artifact printed a stale date, and migrating to the live
%%   style is what fixes that.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  Table captions moved above the grid.
%%   Section 8's float standard ends "caption above" and section
%%   11 lists caption position among the Stage 1 mechanical items.
%%   W1 set ALL THIRTY-THREE table captions BELOW the grid, with
%%   none above -- the whole paper on the wrong side, where P5-CPT
%%   was 22 of 32 and inconsistent with itself.  MEASURED, not
%%   assumed: every shipped v15 paper checked is 100 %% above
%%   (A5-GDOS 32/0, A2-GG6 21/0, P4-Real 5/0, P6-Proton 29/0).
%%   Caption and label move together to just after \centering,
%%   which is where every paper that has them above puts them.
%%   Caption
%%   text, short form, label and grid are copied byte for byte;
%%   only their order inside the float changes.
%%   FIGURES ARE NOT TOUCHED.  Every figure in the cohort captions
%%   below the picture, P4-Real included, and float_audit does not
%%   test position; moving them would be a house change, not a
%%   repair, and section 14.2 owns house changes.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  Thirteen thin captions extended.
%%   EVERY EXISTING SENTENCE KEPT VERBATIM; section 14.2 reserves
%%   rewriting a caption's claim and none is rewritten.
%%   W1's captions already did the harder half of the job: nearly
%%   every one of the thirteen already pointed at the row the
%%   argument turns on.  What they left implicit was the column
%%   inventory, and that is what is added -- nothing that makes a
%%   claim the table does not already carry.
%%   NEW PROSE: thirteen caption extensions.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  The bibliography split into A and B.
%%   W1 held ONE thebibliography carrying 113 entries -- 21
%%   Arisaka cohort papers followed by 92 external works, ALL of
%%   them under a heading reading "A. Papers by Katsushi Arisaka
%%   (the GG-Theory cohort)".  Adler, Bianchi, Dirac, Noether,
%%   Wilson and eighty-seven others printed under that heading.
%%   THE PAPER ALREADY PROMISED THE SPLIT.  The prose immediately
%%   above the bibliography reads: "Section A lists the cohort
%%   papers this review draws on, in the program's canonical
%%   order; Section B the external literature, alphabetically by
%%   first author."  Section B had no heading and no environment.
%%   The paper described a structure it did not have.
%%   SECOND OCCURRENCE.  P2-N3 had 54 external works under the
%%   same heading.  Section 12 item 10 requires the two
%%   environments in every arXiv package, so this was a v20
%%   blocker as well as a reader-facing defect.
%%   Nothing is reordered, no entry is edited, no citation key
%%   changes.  The boundary is where the entries themselves put
%%   it: after ArisakaW2Holonomy, before Adler1969.  Asserted
%%   afterwards: 21 in A, 92 in B.
%%   NO PROSE WAS WRITTEN IN THIS STAGE -- the Section B heading
%%   is the wording the paper's own prose already used.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 4, same day.  Palette: 6 violations -> 0.
%%   palette_fix cleared all six across eleven pictures: four bare
%%   `draw' node borders given hue shades, two fills above the !18
%%   ceiling compressed (!22->!13, !25->!16), and every named TikZ
%%   weight resolved to an explicit 0.60/0.80/1.10 pt.  No hand fix
%%   was needed -- the first paper in this run where the tool alone
%%   took the count to zero.
%%   THREE PATH STROKES ABOVE 1.1 pt ARE LEFT ALONE and printed by
%%   the tool rather than silently clamped: inline \draw marks in
%%   fig:oggep-bulk, fig:seam and fig:z3-winding, not node borders.
%%   Clamping inline strokes destroyed P3-Electron's own prediction
%%   line, which was set heavier than its foils on purpose.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B RIDER, same day.  Four caption extensions corrected.
%%   FOUR OF THE THIRTEEN EXTENSIONS I WROTE ENUMERATED COLUMNS AND
%%   ROWS I HAD NOT READ.  They were written from the header line
%%   alone, and where the header is bare -- `& a law & an identity'
%%   -- I filled the row names in from what the table sounded like it
%%   ought to contain:
%%     tab:law-identity    invented `what survives if the theory
%%       changes underneath it'.  The six rows are example, where it
%%       comes from, could it be otherwise, what tests it, what it
%%       constrains, how it fails.
%%     tab:partii-summary  listed `the equation it turns on' as a
%%       separate column; it is inside the statement column.
%%     tab:B-two-theorems  gave three vague items for six named rows.
%%     tab:I-closed-open   listed five items of which two are not
%%       rows at all; there are seven, ending at `a conservation law
%%       says'.
%%   All four corrected against the grids themselves.  Section 5 says
%%   measure, never estimate, and it does not stop applying inside my
%%   own prose -- the second time in three papers, after the invented
%%   `two places' in P4-Real's Part II introduction.
%% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.0 -> v15.1, August 15, 2026.  THE FOLIO RIDER.
%   folio_scan.py measures, on every page of every deployed v15
%   paper, the gap between the bottom of the lowest text word and
%   the top of the page number.  The modal gap is 15.7 pt (20.3 in
%   the four papers with the taller foot).  FOUR PAGES IN THREE
%   PAPERS CAME OUT NEGATIVE -- the folio inside the last line's
%   band -- and all four are visible at 150 dpi and invisible at 55:
%     W1-Symmetry p153  -8.9   the folio printed through a word
%     P5-CPT      p13   -6.4   the Contents' last line on the folio
%     P5-CPT      p6    -6.1   the List of Figures, likewise
%     A10-Cosmos  p211  -4.3   a page of body text, tight
%   TWO OF THE FOUR WERE MINE, both introduced by an
%   \enlargethispage placed to close a one- or two-line orphan and
%   both reported as clean fixes.  US Letter is 0.69 inch shorter
%   than A4 with the margins unchanged, so the room below the text
%   block is smaller on every page of every converted paper, and an
%   enlargement that was safe on A4 is not safe here.
%   W1's one: my \looseness=-1 removed, the author's own
%%   enlargement kept, and the two-line orphan it was placed
%%   to close accepted instead.
%   NO PROSE WAS WRITTEN IN THIS RIDER.
% ------------------------------------------------------------
\PaperVersion{v20.2}
\PaperDate{August 22, 2026}

%% ---- paper-local macros -------------------------------------------
\providecommand{\tr}{\operatorname{tr}}
\providecommand{\TR}{\operatorname{Tr}}
\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}
\providecommand{\GGA}{\ifmmode\mathsf{GGA}\else\textsf{GGA}\fi}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}
\providecommand{\EGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\EPl}{\ensuremath{E_{\mathrm{Pl}}}}
\providecommand{\Mlock}{\ensuremath{M_{\mathrm{lock}}}}
\providecommand{\Jq}{\ensuremath{J_Q}}
\providecommand{\Ja}{\ensuremath{J_A}}
\providecommand{\bZ}{\ensuremath{\mathbb{Z}}}
\providecommand{\dCKM}{\ensuremath{\delta_{\mathrm{CKM}}}}
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}
\providecommand{\HS}{\textsf{Hopkins--Singer}}
\providecommand{\GS}{\textsf{Green--Schwarz}}

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\providecommand{\etaB}{\ensuremath{\eta_B}}
\providecommand{\gstar}{\ensuremath{g_{*}}}
\providecommand{\dPMNS}{\ensuremath{\delta_{\mathrm{PMNS}}}}
\providecommand{\JPMNS}{\ensuremath{J_{\mathrm{PMNS}}}}
\providecommand{\mbb}{\ensuremath{m_{\beta\beta}}}

%% ---- Appendix apparatus, in the form used by A9-SM and A10-Cosmos --
%% \GGteachbox opens an appendix; \GGreadalong names its textbooks;
%% \GGtakeaway closes it.  Plain \fbox{\parbox{...}}, no tcolorbox.
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  \par\medskip\noindent
  \fbox{\parbox{0.955\textwidth}{\small\textbf{What this appendix
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  \par\noindent{\small\itshape Read alongside:} {\small #1}\par\medskip}
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  \par\medskip\noindent
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%% Ragged-right p-column: kills the loose interword spacing that
%% justified narrow table columns produce (cohort table standard).
\newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}}

%% Check-yourself block: two or three questions at the end of each
%% Part, at \small, set off by rules (charter apparatus budget).
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  {\end{enumerate}\endgroup
   \par\vspace{0.2em}\noindent
   {\color{\GGSeriesColour}\rule{\textwidth}{0.9pt}}%
   \end{minipage}\par\bigskip}

%% Phase-0 stub marker.  Every stub is visible in the preview, so an
%% empty heading cannot hide.  REMOVED as each Phase fills its text.
\newcommand{\GGstub}[1]{\par\noindent
  \textcolor{gray!60!black}{\small\itshape[\,#1\,]}\par\vspace{0.3em}}

\hypersetup{
  pdftitle  = {The Books Must Balance: Symmetry, Conservation, and the Geometry of What Cannot Change (GG-W1-Symmetry)},
  pdfauthor = {Katsushi Arisaka},
  pdfsubject = {GG-Theory program -- White Paper series member W1: a review of symmetry and conservation from Noether to Hopkins-Singer, and the open-system turn},
  pdfkeywords = {symmetry, conservation laws, Noether's theorems, gauge invariance, Bianchi identity, Green-Schwarz mechanism, Hopkins-Singer quantization, differential cohomology, open systems, GG-Theory}
}

%% \GGShowcaseHead -- defined per paper rather than in the style file,
%% because promoting it to ucla-arisaka-gg-11 would mean rebuilding and
%% reshipping twenty-two arXiv packages for one macro.  Flagged, not
%% done.
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
\GGShowcaseCenteredtrue
\renewcommand{\GGShowcaseHeadSkip}{\vspace{8pt}}

\newsavebox{\GGplateLadder}
\newsavebox{\GGplateFrames}
\begin{document}

%% ---- showcase saveboxes, FILLED AT \setstretch{1} ----
%% PROBED INSIDE BOTH ENVIRONMENTS, not one line above
%% them: this paper runs BSTR 1.15 in the front matter
%% and BSTR 1 inside fig:ladder and fig:closed-open.
%% \global\sbox and not \sbox -- a box assignment inside
%% \begingroup is LOCAL, and the group that supplies the
%% correct stretch would throw the filled box away on its
%% way out.  That cost a build on P6.
\begingroup\setstretch{1}\renewcommand{\arraystretch}{1}%
\global\sbox{\GGplateLadder}{\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
  \definecolor{rungcol}{HTML}{B26A16}
  \foreach \i/\name/\what in {%
     0/{S1\ \ Noether I (1918)}/{conserves \textbf{a current} --- no integer anywhere},
     1/{S1$'$\ \ Noether II (1918)}/{yields \textbf{an identity} --- off-shell, no new charge},
     2/{S2\ \ local gauge invariance}/{encodes \textbf{a redundancy} --- bookkeeping},
     3/{S3\ \ the Bianchi identity}/{conserves \textbf{a flux} --- a geometric identity},
     4/{S4\ \ the Wilson line (1974)}/{the integer is \textbf{the frame} --- one phase per cycle},
     5/{S5\ \ the Ward identity (1950)}/{the integer is \textbf{the failure} --- it breaks discretely},
     6/{S6\ \ \GS{} (1984)}/{the integer is \textbf{the trade} --- two failures cancel},
     7/{S7\ \ \HS{} (2005)}/{the integer is \textbf{the whole content} --- pure topology}}
  {
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  }
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  \draw[-latex,line width=1.1pt,rungcol]
     (10.3,0.05) -- (10.3,7.95)
     node[midway,rotate=-90,anchor=south,font=\small\bfseries]
     {how much of the statement is an integer};
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     (-0.5,7.95) -- (-0.5,0.05)
     node[midway,rotate=90,anchor=south,font=\small\bfseries]
     {the statement's grip on motion};
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  \definecolor{clsd}{HTML}{6B6B6B}
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  %% Vertical budget: a \scriptsize line is about 0.36 in these units, so
  %% every multi-line node below is allotted 0.36 per line plus padding,
  %% and the two verdict bars are aligned across the panels.
  %% ---------------- left panel: the closed frame -----------------
  \fill[clsd!6] (0,0.35) rectangle (4.2,6.9);
  \draw[clsd,line width=0.9pt] (0,0.35) rectangle (4.2,6.9);
  \node[clsd,anchor=north,font=\scriptsize\bfseries,align=center]
    at (2.1,6.78) {The closed frame\\ (Part~II)};
  \draw[clsd,line width=0.8pt] (1.55,4.15) -- (1.55,5.65);
  \draw[clsd,line width=0.8pt] (2.65,4.15) -- (2.65,5.65);
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    {\draw[clsd,line width=0.7pt] (1.55,\y) -- (2.65,\y);}
  \node[anchor=east,font=\scriptsize,text=clsd] at (1.45,5.65) {S7};
  \node[anchor=east,font=\scriptsize,text=clsd] at (1.45,4.15) {S1};
  \node[anchor=west,font=\scriptsize,text=black!60,align=left,
        text width=1.15cm] at (2.78,4.90) {seven\\ rungs};
  \node[anchor=north,font=\scriptsize,text=black!62,align=center,
        text width=3.7cm] at (2.1,4.05)
    {a complete configuration,\\ a total anomaly,\\
     a manifold with no outside};
  \draw[clsd!45,line width=0.5pt] (0.75,2.90) -- (3.45,2.90);
  \node[anchor=north,font=\scriptsize,text=black!62,align=center,
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    {no account, and so no cost:\\ exchange is outside\\ the formalism};
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  \fill[clsd!14] (0.15,0.70) rectangle (4.05,1.50);
  \draw[clsd,line width=0.8pt] (0.15,0.70) rectangle (4.05,1.50);
  \node[align=center,font=\scriptsize\bfseries,text=clsd,
        text width=3.8cm] at (2.1,1.10)
    {ADMISSIBILITY\\ consistent \textbar{} inconsistent};
  %% ---------------- the turn -------------------------------------
  \draw[-latex,black!55,line width=1.1pt] (4.45,4.95) -- (7.95,4.95);
  \node[anchor=south,font=\scriptsize\bfseries,text=black!60,align=center]
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  \node[anchor=north,font=\scriptsize,text=black!55,align=center,
        text width=3.4cm] at (6.20,4.80)
    {a measurement is\\ a boundary transaction\\ (\S\ref{ssec:why-open})};
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  \draw[opn,line width=0.9pt] (8.10,0.35) rectangle (15.30,6.9);
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    at (9.975,3.05) {\textbf{G3 \GRIP{}}\\ a descent that\\ pays to cross};
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  \draw[fnc,line width=0.8pt,dash pattern=on 2.5pt off 2pt]
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  \node[align=center,font=\scriptsize,text=fnc,text width=2.8cm]
    at (13.425,3.05) {\textbf{G4 \QiFour{}}\\ stated, fenced,\\ not defended};
  \node[anchor=north,font=\scriptsize,text=black!58,align=center,
        text width=3.4cm] at (13.425,2.32)
    {nothing to its left\\ depends on it (\S\ref{ssec:qi4-fence})};
  \draw[-latex,opn,line width=0.9pt] (9.975,2.40) -- (9.975,1.55);
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  \node[align=center,font=\scriptsize\bfseries,text=opn] at (11.70,1.10)
    {EXCLUSION \quad affordable \textbar{} unreachable};
\end{tikzpicture}}%
\endgroup

% ---- v0.57 (August 2, 2026) ----
% New section 'What is time? The chronology of the ledger'
% (sec:what-is-time, new 22; comparison shifts to 23 etc.): the
% essence of P5 v3.48 Part IX in the review's voice --- the four
% endowments located on the anatomy, the tick/arrow split, the
% boxed slope law, the mirrored chronology figure (after
% GG-P5-CPT), the age ledger 140.24 = 36.1+91.6+12.5, lock at
% r=K, ln(1+z_sw)=J, the flagged 4K candidate, the clock ladder,
% grades + derivation-home pointer. New Discussion subsection
% 'Time, read instead of assumed' after the symmetry-reading
% capstone: the three significances and the closing judgement
% (four questions, one answer). Numbers inherited from the
% verified session computation (GG-Report-153).

% ---- v0.56 (August 2, 2026) ----
% Capstone Discussions subsection 'Symmetry breaking as symmetry
% reading: the two fibres' (ssec:symmetry-reading), after the
% exhibit: Curie's principle and the three standard escapes; the
% two-fibre table (holonomy requires closed loops; an interval has
% none -- the 5th forgets, the 6th remembers); the three stacked
% determinism mechanisms (dial not hat / unique floor / cannot
% re-choose); Curie dissolved (symmetric dynamics, asymmetric
% arithmetic; symmetry breaking = symmetry reading; the groove);
% Sakharov's conditions as the anatomy of the bulk (table); the
% residual mirror bit at its honest grade; uniqueness as four
% structural contrasts; the fences. Grounding note:
% GG-Note-Deterministic-Symmetry-Breaking.

% ---- v0.55 (August 2, 2026) ----
% New chapter 'Leptogenesis: the matter the mirrors made'
% (sec:leptogenesis, new section 20; cosmological sheet shifts to
% 21): the observed eta_B; Sakharov's three conditions and who owns
% them (the 3Z door / the pentagon + Majorana parity / the lock
% crossing with zero dilution); the audited chain to eta_B =
% 5.87e-10 vs 6.12e-10 with the throughput identity eps*kappa =
% e^{-K/2} (chain flowchart figure, after GG-P5-CPT); the alphabet
% audit (which mirror made the matter; alphabet-circle figure,
% after GG-P5-CPT); 'Was the asymmetry tuned?'; grades + exposure.
% One paragraph appended to the Discussion exhibit (the pair of
% facts becomes a triple). Macros etaB/gstar/dPMNS/JPMNS/mbb
% added. Companion ship: P5 v3.47 (GG-Report-149).

% ---- v0.54 (August 2, 2026) ----
% Section 19 rebuilt as a full chapter: new 19.1 'The circle's
% holonomy, and the abelian ledger' (holonomy = U(1) gauge
% invariance cashing Chapter 7; the four abelian objects of GG-6;
% the centre of SU(3); the Z6 global form made geometric); 19.2
% 'The lock' promoted from paragraph, with the centre identity
% B = t/3 and the fragment table; new 19.3 'The singlet rule'
% (closure chain, the zoo table, why a triangle, the EM shadow
% Q+B in Z, fences); 19.4 pentagon/triangle unchanged; 19.5 'Why
% theta-bar vanishes' promoted from paragraph. Discussion: new
% subsection 'The exhibit: confinement and the immortal proton'
% after ssec:retreat. Macros UoneB/UoneQi/UoneY added. All new
% arithmetic verified programmatically (Z6 triviality, Q+t/3 in
% Z, zoo trialities).

% ---- v0.53 (August 2, 2026) ----
% Language: removed six reader-facing third-person 'the author'
% self-references (the author = Arisaka). S19 rephrased to the body's
% impersonal voice; the Conclusions header and the single-authorship
% Q&A response use the first person; coherence-audit note -> 'for
% review'. No physics, numbers, or structure changed.

% ---- v0.52 (August 2, 2026) ----
% New section 'The colour triangle: confinement, and why theta-bar
% vanishes' (sec:triangle) in Part IV, after S18 the discrete
% symmetries and before the cosmological sheet (which shifts to S20).
% Imports P6-Proton Figure 11 (fig:z3-winding, 'the lock') as native
% TikZ, reframed as the colour triangle. Answers the pentagon-vs-
% triangle question: two holonomies of one S^1_theta; CP violation =
% loop left open, confinement = loop forced shut; the same closure
% forces theta-bar = 0 (strong CP), extending S18.1. Macros PaperPSix,
% Stheta added.

% ---- v0.51 (August 2, 2026) ----
% Reorg: relocated the discrete-symmetry section from end of Part V
% to a new section in Part IV (after the SM sheet), retitled 'The
% discrete symmetries: C, P, T, CP, and CPT' (sec:discrete). Added a
% parity paragraph; folded old 17.4 CP numbers (delta_CKM, Jarlskog,
% strong CP) into its CP subsection; slimmed 17.4 to charge + pointer;
% repointed the SM-sheet grade box.

% ---- v0.50 (August 1, 2026) ----
% New section 'The discrete mirrors: C, P, T, and CPT' (+ subsection
% on CP violation / flavour pentagon) at end of Part V Discussions;
% imports three native-TikZ figures from GG-P5-CPT (fig:cpt-reversals,
% fig:cpt-flavour-graph, fig:cpt-pentagon). Macros Ja/Jq/bZ/dCKM/PaperPFive.

% ---- v0.49 (August 1, 2026) ----
% Imported A10-Cosmos Figure 13 (fig:oggep-bulk here; native TikZ,
% 6D bulk + OGGEP projection to the 4D boundary) at the head of
% Part III sec:oggep. Boundary slab widened 9.6->12.6 cm; enlarged
% ~10%% (full \textwidth vs A10's 0.92); \EGUT/\EPl/\Mlock added.

% ---- v0.48 (August 1, 2026) ----
% Imported A8-QM Figure 5 (fig:bulk-projection here; native TikZ,
% the 6D-bulk bundle projection pi: M6 -> M4) into Part III sec:oggep,
% ssec:oggep 'OGGEP: observation as a projection'. Enlarged ~15%%
% via \scalebox; caption copied from A8; figref hook in the intro.

% ---- v0.47 (August 1, 2026) ----
% Figure 2 (fig:ladder-timeline): fonts enlarged ~20% and all year
% labels bolded; clock lane pulled left so the wider text does not
% balloon the figure. TikZ block regenerated and swapped in place.

% ---- v0.46 (August 1, 2026) ----
% Part II head: new overview Section 3 ('The seven steps, in plain
% English', broad-audience, ~1 page) + dedicated full-page timeline
% figure fig:ladder-timeline (native TikZ): logical ladder S1->S7
% left, historical clock 1910-2030 right, \clearpage-isolated.

% ---- v0.45 (August 1, 2026) ----
% Enlarged Figure 3 (fig:grip-engine) fonts ~10-20%: within the
% figure's TikZ block, base \large->\Large and \small->\normalsize;
% box widths unchanged so the resizebox renders text larger.

% ---- v0.44 (August 1, 2026) ----
% Imported A11-Origin Figure 3 (the GRIP engine, native TikZ) as a
% full-page figure fig:grip-engine at the head of Section 12 (G3 ---
% GRIP). Added \GRIPin/\GRIPout providecommands; caption cross-ref
% repointed from A11 Part III to this chapter's step-by-step walk.

% ---- v0.43 (August 1, 2026) ----
% Imported A10-Cosmos Figure 5 (the cosmic GRIP descent; external
% asset a10_cosmic_descent.pdf) as a full-page figure fig:cosmic-
% descent in Part IV (The Ledger), sec:cosmos-sheet. Caption copied
% from A10 with \EGUT/\EPl expanded and the chronology reference
% pointed to A10 by citation; figref hook in the section intro.


%% ===================================================================
%%  COVER PAGE
%% ===================================================================
\GGTitlePage%
  {Symmetry, Conservation, and Geometry}%
  {Seven Steps from Noether, Bianchi, to \HS{},\\
   and the Ledger the Open-System Turn Pays Out}%
  {August 5, 2026}
\thispagestyle{empty}

\vspace{0pt}
\begin{abstract}
\noindent
Physicists have long believed that things are conserved because the world
is symmetric. This review follows that belief through the seven results
that built it, in the order they were found: at each step it gives way,
and what replaces it is more interesting.

Noether's first theorem ties a conserved quantity to a symmetry: a law
that constrains motion. Her second theorem, in the same 1918 paper, shows
that a symmetry holding independently at every point delivers no
conserved quantity --- only an identity, true whether or not anything obeys the equations, so
gauge invariance is a redundancy in our description, not a feature of
nature. The Bianchi identity conserves
magnetic flux for no dynamical reason: a curvature is a curvature. Where
a loop does not close, a phase survives that no curvature sees.
Quantized, the identity can fail at last --- and only by a whole number.
Green--Schwarz then lets two failures pay for each other, and
Hopkins--Singer arrives where the conserved thing is an integer, safe
because integers have nowhere to move.

The sequence has a direction: conservation laws grow harder to break as
they say less about motion. Parity, $CP$ and naturalness were
believed partly for their beauty, and each failed.

\GGTheory{} is then placed at the end of that sequence, at declared
grades: a six-dimensional geometry with exact books, read off by an open
boundary. Its payout is set beside the measurements ---
twenty-eight Standard-Model numbers and twenty-one cosmological ones,
each graded, from four integers doing twenty unrelated jobs. Open books say not what cannot leave but what must be shown when
something does --- so the breaking of symmetry becomes not the loss of a
design but the step that selects a world.
\end{abstract}

%% ===================================================================
%%  EXECUTIVE SUMMARY
%% ===================================================================
\begin{ExecutiveSummary}
\noindent
\textbf{The question.} Two ideas organize theoretical physics: that some
quantities never change, and that some changes make no difference.
\citet{Noether1918} connected them, and the connection has been so
productive that most physicists carry it as a single conviction ---
\emph{symmetry causes conservation}. \citet{Wigner1960} called the
effectiveness of symmetry principles unreasonable;
\citet{Yang1980} summarized the century in three words, \emph{symmetry
dictates interaction}. This paper takes the seven decisive steps in the
history of that conviction and shows that the connection loosens at each
one, while the conservation statements themselves grow harder to break.

\medskip\noindent
\textbf{The seven steps, and where they turn.} The abstract walks the
sequence; what matters here is where it turns. Noether's second theorem
--- same 1918 paper, rarely taught --- gives no new charge but a relation
among the equations, so gauge symmetry is a redundancy of bookkeeping,
and \citet{Elitzur1975} proved the consequence: there is nothing there to
break.

The sequence turns twice. First, curvature does not determine the field:
where a loop fails to bound, a gauge-invariant phase survives with the
curvature zero everywhere, and \citet{Hosotani1983} showed that its
value, not the Lagrangian, decides which gauge symmetry the low-energy
world displays. Second, the identity of \citet{Ward1950} and
\citet{Takahashi1957} is the first on the ladder that \emph{can} fail,
and it fails only by a whole number. That is what makes the last two
steps possible: \citet{GreenSchwarz1984} let one sector's failure cancel
another's, available only because each failure is known exactly; and
\citet{HopkinsSinger2005}, building on \citet{Dirac1931}, reach a level
where what is conserved is an integer, protected not by any symmetry but
by the impossibility of moving continuously through the whole numbers.

\medskip\noindent
\textbf{The pattern.} Two things happen together along that sequence:
symmetry retreats as the explanation, and what a statement says about
motion gives way to what it says about a whole number. The thesis, in
the form the seven steps support, is that \emph{the sequence is not one
of statements that become unbreakable, but one in which failure becomes
discrete}: four steps carry no integer and cannot fail at all, and the
rest carry one as a frame, as a failure, as a trade, and finally as
their whole content. Two consequences follow. Nothing small can touch an integer, so the
quantities still known exactly in a universe whose symmetries are mostly
gone are, without exception, the counting statements --- and that is the
universe we have. And an identity that cannot fail is of less use than
one that fails only by a whole number: the first says nothing when
something goes wrong, the second says exactly how much. Ledgers are
built on the second kind.

\medskip\noindent
\textbf{The honest record.} A review that celebrates symmetry without
printing its failures has not made an argument. Parity was assumed until
\citet{LeeYang1956} noticed nobody had checked, and is violated
maximally \citep{Wu1957}. $CP$ was the repair, and broke in 1964
\citep{Christenson1964}. Naturalness --- the most quantitative
symmetry-driven prediction of the last forty years --- pointed to
superpartners near a TeV, and the region it preferred has been excluded
\citep{ATLASsusy2026}. The lesson drawn here is sharper than ``symmetry
reasoning is unreliable'': it does well when the symmetry is read off the
data and badly when it is imposed because the alternative would be ugly.
That standard is then turned on this program's own premises, and the
one most exposed to it is named rather than left to be found.

\medskip\noindent
\textbf{The turn.} Every one of the seven steps describes a \emph{closed}
world. \GGTheory{} proposes that ours is not: what we observe is a projection
from a six-dimensional bulk onto a four-dimensional boundary
\citep{ArisakaA1Constitution}, whose dynamics is consequently open and
irreversible \citep{ArisakaA5GDOS}. The
consequence for this paper's theme is exact. In a closed system a
conservation law says what cannot leave; in an open one it says what the
books must show when something does. That is why closed-system methods
deliver \emph{admissibility} and never \emph{exclusion}, and why an upper
bound on a count needs an open-system argument \citep{ArisakaP2N3}.

\medskip\noindent
\textbf{Breaking as the mechanism.} In the usual story symmetry is the
ideal and breaking is what happens to it. Here breaking sits in the
middle of the process as the productive stage: a relaxation cascade
descends a landscape, pays a toll at each step from a fixed budget, and
settles \citep{ArisakaA6Gi4}. The conservation laws survive that descent not as symmetries of the
state it ends in --- that state has very little symmetry left --- but as
ledger identities of the whole descent. One consequence --- that the
number of families is a solvency question whose arithmetic runs out
after three --- is the program's most quoted result and, in
proportion, one row of forty-nine. The six-dimensional identity whose
coefficients are locked by a single ratio \citep{ArisakaA3BI6} and an agreement between the bulk's topology and the boundary's quantum mechanics that no parameter was adjusted to produce \citep[Appendix~H]{ArisakaP2N3} are reported in outline.

\medskip\noindent
\textbf{What it pays out, and at what price.} A program that removes
the dials must be judged by what it then produces, so the numbers are
set out in full: twenty-eight Standard-Model parameters
\citep{ArisakaA9SM} and twenty-one cosmological ones
\citep{ArisakaA10Cosmos}, each with its closed form, its measured value,
its deviation and the owning paper's own grade. The honest reading is
printed with them. Of the twenty-eight, one is a theorem end to end;
four of the twenty-one are anchored, and sit at zero deviation by
construction rather than by agreement. What is striking is not any one row but the re-use: four integers do
twenty unrelated jobs, and $J = 7$ fixes both a coupling measured at
a collider and a topological marker in the recombination era. The price is stated too --- and one item long counted as a price
is not one. The dimensionful content is a unit system: the co-equal
trio $\{c,\, G_N,\, \hbar\}$, one conversion factor for each kind of
dimension the bulk has, all equal to unity in natural units and no one
of them the anchor \citep{ArisakaA4Euclid}; expressing them in
kilograms and meters is a choice of units, not a dial. What remains a
price: the discrete completion class is asserted rather than proved;
and the statistic that would bound the whole impression --- the global
trials factor --- has not been computed, which the owning paper says
makes the closures an honest ansatz and not a proof. The commitments carry instruments and dates.

\medskip\noindent
\textbf{What is claimed, and what is not.} Part~II is a review: its
results are standard and the contribution is the reading. Part~III is a
proposal in four moves, each set out in one frame --- what is assumed,
what follows, what it buys and what it leaves open --- so that any one of
them may be rejected without disturbing the others. Part~IV reports the
numbers at the grades their owning papers give them and recomputes every
one of them; Part~V reads the ladder against those numbers and answers
the anthropic alternative at its strongest. Every claim carries a grade, axiom through open, with the member paper
that owns it. Nothing here founds quantum mechanics; the Born rule
enters at a declared grade, with its premises and its one residue named
\citep{ArisakaA8QM}. The fourth move, an extension beyond physics, is
stated at grade \emph{open}, checked nowhere here, and depended on by
one marked clause elsewhere. And nothing here
asks a reader to accept the proposal in order to accept the review: the
ladder can be taken and the rest left.

\medskip\noindent \textbf{A closing word, for a reader who is not a
specialist.} The conviction this review follows is one of the most
productive ideas in the history of physics, and the argument here is not
that it was wrong. It is that it was a good approximation to something
more durable. A century of physicists looked for symmetry and found it
everywhere: where a symmetry exists it is the easiest thing about a
theory to see. What was being tracked at each of
the seven steps was a bookkeeping identity that wore a symmetry as its
most visible feature. When the symmetries were later found to be broken
--- and the important ones were --- the books did not go with them.

\medskip\noindent That observation can be stated without any mathematics
at all. A quantity protected by a symmetry is protected only for as long
as the symmetry holds, and symmetries have turned out to be exactly the
kind of thing that stops holding. A quantity that is a whole number is in
a different situation entirely: to change it you would have to pass
through the values in between, and there are none. Nothing gradual can
touch it. So when one looks at a universe whose symmetries have mostly
been spent and asks what is still known exactly, the answer is not the
beautiful things. It is the things that can be counted.

\medskip\noindent The third remark took the longest to state honestly,
and it inverts the picture most of us were taught. In the usual telling
symmetry is the ideal and breaking is the damage. Here breaking is the
productive step. A perfectly symmetric world is one in which nothing has
happened yet --- no direction picked out, no scale, no particular value
of anything. Every specific fact about the world we inhabit is the
residue of a symmetry that was available and was spent. The breaking
of a symmetry is not the loss of a design. It is the step that selects a
world.

\medskip\noindent None of that is offered as proof, and the review is
careful to say where the confidence ends. Parity, $CP$ and naturalness
were each believed partly because they were beautiful, and each failed;
beauty is evidence of nothing, and the same standard is turned here on
this program's own premises, with the most exposed of them named rather
than left to be found. Which leaves the question a general reader is
entitled to ask: how would we know if this were wrong? It is why the
numbers of Part~IV are printed with their deviations rather than
described. A framework whose content is integers
cannot be adjusted. There is no dial to turn when a measurement moves,
because the quantities that would have been dials are whole numbers
doing several unrelated jobs at once --- and a whole number that is
wrong in one job is wrong in all of them. That is an uncomfortable
position to occupy and it is the correct one: it is what distinguishes
a framework that could fail from a framework that can always be
repaired. The commitments carry instruments and dates. If the integers
are wrong, the experiments that will say so are already running.


\end{ExecutiveSummary}



\GGKeywords{symmetry; conservation laws; Noether's theorems; gauge
invariance as redundancy; Bianchi identity; Wilson lines and holonomy;
the Hosotani mechanism; Ward identities; anomalies; \GS{} mechanism;
\HS{} quantization; differential cohomology; quadratic refinement; open
systems; relaxation cascade; \GGTheory{}}

\clearpage%% Self-filling showcase frame (Report 462): this page carried no
%% frame at all, which is why it printed hard against the top.  The
%% glue is page-level -- a longtable cannot live inside a minipage.
\null\vspace*{\stretch{1}}

\GGShowcaseHead{Conservation Laws Grow Harder to Break ---\\[3pt]
as They Say Less About Motion}

\par\vspace*{\stretch{2}}
\begin{center}
\resizebox{0.574\textwidth}{!}{\usebox{\GGplateLadder}}

\vspace{14pt}

\resizebox{0.750\textwidth}{!}{\usebox{\GGplateFrames}}
\end{center}

\par\vspace*{\stretch{2}}
\captionof*{figure}{\textbf{The whole review on one page.} Above: the ladder. Seven steps in the order they were found, and reading upward each one says less about how things move and carries more of its content in a whole number. The lower four carry none and cannot fail at all; the Ward identity is the first that \emph{can}, and it fails only by an integer; Green--Schwarz lets one sector's failure pay for another's, which is possible only because each failure is known exactly; and Hopkins--Singer arrives where the conserved thing \emph{is} an integer, safe not by any symmetry but because there is nothing between the whole numbers to move through. Below: what the open-system turn changes. Every step on the ladder describes a closed world, and the strongest closed-system tools --- anomaly cancellation, integrality --- sort candidate theories into consistent and inconsistent and then stop. They are silent about which consistent option obtains, because sorting by cost requires an account and an account requires a boundary. Open the system and the verdict changes in kind: not \emph{inconsistent} but \emph{unaffordable}. In full: Figure~\ref{fig:ladder}, p.~\pageref{fig:ladder}, and Figure~\ref{fig:closed-open}, p.~\pageref{fig:closed-open}.}
\par\vspace*{\stretch{1}}
\clearpage



%% ===================================================================
%%  TABLES OF CONTENTS  (Overview and List of Figures share one page)
%% ===================================================================
\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

%% ===================================================================
%%  PREFACE  (after the contents, before Part I -- A-series order)
%% ===================================================================
\GGPreface
\noindent
This paper is a review with a proposal attached, and the two are
deliberately easy to separate. It is written to be audited rather than
admired. Three conventions govern every page.

\looseness=-1
\emph{First, every fence is printed.} Where a step is a theorem, it says
so. Where it rests on a measurement, the measurement is named. Where a
premise is assumed rather than derived, it is called an axiom in the
place where it is used, and the most vulnerable of them is identified by
name in \S\ref{ssec:exposure} rather than left for a hostile reader to
find. Chapter~\ref{sec:honest-record} exists for the same reason: it
prints the occasions on which reasoning of the kind this paper admires
has failed.

\looseness=-1
\emph{Second, the physics is stated beside the mathematics.} Every
technical movement closes with a paragraph in plain terms. A reader who
wants the argument without the machinery can follow those paragraphs
alone and lose nothing essential. Where a piece of mathematics is needed
that a physics undergraduate will not have seen --- differential forms,
connections and holonomy, anomalies, differential cohomology --- it is
developed in an appendix that begins at textbook level, and the body
never uses a term whose appendix has not been written.

\looseness=-1
The appendices are written for a senior undergraduate
or first-year graduate student in physics. Each names the
textbooks it runs alongside, then follows a three-step plan: the
construction as a graduate student meets it; what that construction must
assume and cannot explain; and what the argument of this paper does with
it. Each closes with problems and full solutions. A reader of the body
alone may skip Part~\ref{part:appendices} entirely and lose no argument; a reader who wants
to check the body's claims will find the work is done there.

\emph{Third, the arithmetic is reproducible.} Every integer, coefficient
and identity was checked by an independent
computation before it was typeset, and none of it requires more than a
few lines of arithmetic that a reader can redo.

\bigskip
\noindent
\looseness=-1
\textbf{The grade vocabulary.} Claims in Part~III are labeled, and the
four labels mean exactly this:

\looseness=-1
\textbf{axiom} --- assumed, not derived, and it could be false;
\textbf{theorem} --- proved from what precedes it, with the proof in a
named member paper; \textbf{derived-conditional} --- proved subject to a
clause stated whenever the result is used, the clause being an empirical
input, an unproved premise, or an outstanding residue; and
\textbf{open} --- unsettled, and labeled as such rather than quietly
omitted.

\noindent
\looseness=-1
The grade boxes closing the later chapters of Part~II use the same
vocabulary, splitting \emph{theorem} into \emph{standard} (established in
the literature) and \emph{cohort} (proved in a member paper), and adding
\emph{model-dependent}, for what follows only once a field content is
chosen, and \emph{not claimed}.

\noindent
The complete ledger is Appendix~\ref{app:dictionary}; the summary is
Table~\ref{tab:grades}.

\bigskip
\noindent
\looseness=-1
\looseness=-1
\textbf{What is assumed of the reader.} Multivariable calculus, linear
algebra, differential equations, introductory quantum mechanics, and
electromagnetism through Maxwell's equations in vector form; special
relativity helps and is not required. Readers from chemistry and the life
sciences need only orbitals and thermodynamic reasoning, and
\S\ref{ssec:chemistry} and \S\ref{ssec:chirality} are written for them.

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\bigskip
\noindent
\textbf{If you read only fifteen pages.} Read the Executive Summary,
then Chapter~\ref{sec:two-ideas} for the question,
Chapter~\ref{sec:oggep} for the pivot the proposal turns on, and
Chapter~\ref{sec:one-argument} for the argument assembled. With a
sixteenth, add Chapter~\ref{sec:concluding} --- the only passage in the
first person, and the only place a view is expressed that physics cannot
adjudicate.

%% ===================================================================
%%  PART I --- THE QUESTION
%% ===================================================================
\GGPart[The question]{The Question}{What a conservation law is, what a
symmetry is, why everyone believes the two are the same thing --- and
why the question reaches past physics into chemistry and the life
sciences}
\label{part:question}

\section{Two ideas that turned out to be one}
\label{sec:two-ideas}
This paper is about two ideas that most physicists, most of the time,
treat as one: that certain quantities in nature do not change, and that
certain changes in nature make no difference. The connection between them
is the most celebrated structural result in theoretical physics. It is
also, as the paper will argue, weaker at every step than its reputation
suggests --- and what replaces it at each step is more interesting than
what it replaces.

Before any of that, the two ideas need stating in a form that can be
checked.


\subsection{What a conservation law is, operationally}
\label{ssec:cons-law}
Operationally, a conservation law is a promise about a measurement made
twice.

Pick a quantity. Measure it now. Wait. Measure it again. If the two
numbers agree, and if they keep agreeing under every circumstance
anybody has arranged, the quantity is conserved. Energy, momentum,
angular momentum, electric charge and baryon number are the standard
examples, and each of them is in daily use in exactly this form: a
chemist balances charge across a reaction; an engineer balances energy
across a turbine; a particle physicist balances momentum across a
collision and infers a neutrino from what is missing.

The last example already shows the clause that this paper turns on. The
momentum did not balance, and the physicist did not conclude that
momentum is not conserved. She concluded that something had left the
apparatus carrying the difference. Every conservation law in practice is
stated with a boundary: \emph{this quantity does not change, provided
nothing crosses the surface of the region being watched, and if
something does, the books must show it.} For a closed system the proviso
is invisible, and it is easy to forget that it is there at all. For an
open system it is the entire content of the law. Part~III is about what
happens when one takes that seriously as a matter of principle rather
than of laboratory practice.

Note also what a conservation law is \emph{not}. It is not a statement
that nothing happens. A conserved quantity is precisely a quantity whose
constancy survives everything else changing, which is what makes it
useful: it is a fixed point to reason from in a situation where almost
nothing else is fixed.


\subsection{What a symmetry is, operationally}
\label{ssec:symmetry}
Operationally, a symmetry is a change you can make that no experiment
detects.

Rotate the entire apparatus and repeat the experiment: the same result.
Move it across the room: the same result. Perform it tomorrow instead of
today: the same result. Each of these is a symmetry of the physics, and
each is a claim about the world that could have failed and does not.

Two rather different things travel under this name, and the difference
matters more than any textbook treatment suggests. The first is a change
made to the \emph{system}: the apparatus really is rotated, and it really
is in a different orientation than before, and the claim is that nature
does not care. The second is a change made only to the
\emph{description}: nothing is moved, and what changes is a convention in
the bookkeeping --- the zero of potential energy, the choice of
coordinates, the overall phase of a quantum state. Nobody expects a
change of convention to be detectable, so the second kind of statement
seems to say nothing at all.

It says a great deal, and Chapter~\ref{sec:gauge} is about how much.
When a redundancy in a description is required to hold \emph{locally},
independently at every point, the requirement is severe enough to
determine the interactions. That is the strangest fact in this paper's
first half: the most powerful organizing principle in the Standard Model
is a statement about bookkeeping.


\subsection{The naive link, and why everyone believes it}
\label{ssec:naive-link}
The link between the two ideas is Noether's first theorem \citep{Noether1918}: to every
continuous symmetry of the action there corresponds a conserved current.
Time translation gives energy; spatial translation gives momentum;
rotation gives angular momentum; the phase of a complex field gives
electric charge. It is derived in \S\ref{ssec:noether-i} and in full in
Appendix~\ref{app:noether}, and it is one of the genuinely beautiful
results in mathematical physics: a general theorem, proved once, that
delivers the conservation laws of any theory whose action one can write
down.

Its reputation is deserved and its influence is difficult to overstate.
\citet{Wigner1960} called the effectiveness of symmetry principles
unreasonable and argued that symmetries are laws about laws --- a tier of
structure above the equations of motion. \citet{Yang1980} summarized the
century's lesson in three words: symmetry dictates interaction. Generations
of physicists, including the present author, learned the subject inside
that conviction, and it produced general relativity, the Standard Model,
and every unification program since.

So the naive link is believed for the best of reasons: it works. The
question this paper asks is not whether it works but how far up it
reaches --- and the answer, developed one rung at a time in Part~II, is
that it stops earlier than almost anyone is taught.


\subsection{The promise: the arc, stated as a claim to be earned}
\label{ssec:promise}
Figure~\ref{fig:ladder} draws the claim the rest of the paper is
required to earn.

\begin{quote}
Physics has climbed a ladder on which conservation laws become
progressively more rigid and progressively less dynamical --- from
Noether's currents, through the identities of gauge redundancy and the
geometry of the Bianchi identity, to the tradeable ledger of the
\GS{} mechanism and finally to integers protected by topology.
\end{quote}

%% v15.3: FIGURE 1 MOVED HERE FROM THE EXECUTIVE SUMMARY, on the
%% author's instruction.  It is the plate the sentence above reads
%% by name, and it now sits beside the claim it draws instead of
%% three pages ahead of it.  Still [H], still a savebox with two
%% uses -- this one and the showcase page.
\begin{figure}[H]
\centering
\resizebox{0.574\textwidth}{!}{%
\usebox{\GGplateLadder}}
\caption[The ladder: seven steps from Noether to \HS{}.]{\textbf{The
ladder.} Reading upward, each step says less about how things move and
carries more of its content in an integer. The lower four rows carry
none: they are algebra, convention and geometry. S4 carries one as a
frame, and still cannot fail. From S5 up, something can --- and what it
can fail by is a whole number.}
\label{fig:ladder}
\end{figure}

Table~\ref{tab:arc-promise} sets out the seven rungs in the order Part~II
climbs them. \emph{Eight rows, seven numbers:} Noether's two theorems
share the first, and the prime on S1$'$ marks the second of them, which
is why the ladder is seven steps long and the table is eight lines deep.
At this stage the table is an assertion. Every rung but S1$'$ has a
chapter of its own, and the table returns in \S\ref{ssec:arc-earned}
with a column added that only the intervening chapters can supply.

\begin{table}[H]
\centering
\caption[The arc, promised.]{\textbf{The arc, promised.} The claim of
Part~II, stated in advance so that the reader can hold the chapters to
it. The same table returns as Table~\ref{tab:arc-earned} once it has
been earned. Four columns per rung: what conserves at that step, why it conserves, and how much of the answer has hardened into an integer. The fourth column is the one to watch down the page, because the arc the Part claims is a trend in that column and nowhere else.}
\label{tab:arc-promise}\small
\begin{tabular}{@{}P{0.235\textwidth}P{0.17\textwidth}P{0.275\textwidth}P{0.21\textwidth}@{}}
\toprule
\textbf{rung} & \textbf{what conserves} & \textbf{why} &
\textbf{how much is an integer} \\
\midrule
S1 Noether I (\S\ref{sec:noether}) & a current & a continuous global
symmetry of the action & none --- it constrains motion \\
S1$'$ Noether II (\S\ref{ssec:noether-ii}) & nothing new & a local
symmetry gives identities & none --- it holds off shell, without
the field equations \\
S2 gauge (\S\ref{sec:gauge}) & a redundancy & gauge-equivalent means the
same state & none --- bookkeeping \\
S3 Bianchi (\S\ref{sec:bianchi}) & flux & a curvature is a curvature &
none --- a geometric identity \\
S4 Wilson (\S\ref{sec:wilson}) & a holonomy & a loop that does not bound
& the \emph{frame} --- one phase per cycle \\
S5 Ward (\S\ref{sec:ward}) & a current, in every correlator & the
measure is invariant & the \emph{failure} --- it breaks discretely \\
S6 \GS{} (\S\ref{sec:gs}) & the \emph{net} ledger & failures cancel
failures & the \emph{trade} --- two integers cancel \\
S7 \HS{} (\S\ref{sec:hs}) & an integer class & integers cannot move &
the \emph{whole content} --- pure topology \\
\bottomrule
\end{tabular}
\end{table}

Two further things are owed. Part~III asks what stands above the top
rung, and answers with a proposal rather than a review: a six-dimensional
geometry whose bookkeeping is exact, projected onto a four-dimensional
boundary on which the breaking of symmetry is not the loss of the design
but the mechanism that selects a world. Every claim in
that Part carries a grade, and none of them is asked to lean on the
review that precedes it. And Chapter~\ref{sec:honest-record} prints the
record of where symmetry-first reasoning has failed, because a paper that
argues from the successes of an idea and hides its failures has not
argued at all.


\section{Why the question reaches beyond physics}
\label{sec:beyond}
The reader of a review in theoretical physics is entitled to ask why the
question should matter to a chemist or a biologist. The answer is that
both fields already run on symmetry arguments, in most cases without
naming them, and that both contain the sharpest available illustration
of this paper's second theme --- that what one actually observes is
usually the \emph{breaking}.


\subsection{Chemistry: orbital shapes and selection rules}
\label{ssec:chemistry}
The shape of the periodic table is a symmetry statement.

An electron in an isolated atom moves in a potential that depends only on
the distance from the nucleus. That potential is invariant under
rotations, and rotational invariance is what forces the solutions to
organize themselves by angular momentum: for each value of the orbital
quantum number $\ell$ there are $2\ell + 1$ orientations, and with the
two spin states this gives $2(2\ell + 1)$ states in a shell --- $2$ for
$s$, $6$ for $p$, $10$ for $d$, $14$ for $f$. Those four numbers are the
widths of the four blocks of the periodic table. A chemist who has never
written down a rotation operator uses the consequence of one every time
she reads across a row.

Selection rules are the same statement in its more useful form. A photon
carries one unit of angular momentum, so an electric-dipole transition
must change $\ell$ by exactly one. Transitions that would violate this
are not slow; to the extent that the symmetry is exact, they are absent.
A spectroscopist reading a spectrum is reading, line by line, which
symmetries the molecule in the sample possesses.

Now the second half. Put the same atom into a crystal or a complex, and
the environment is no longer spherically symmetric. The five $d$ states,
degenerate in the free ion because rotations related them, split into
groups; the splitting is small, it lies in the visible range, and it is
the reason transition-metal compounds are colored. The symmetry gave the
degeneracy; the \emph{breaking} of the symmetry gave the color. Nothing
in this paper is more characteristic of its argument than that: the
symmetry organizes the possibilities, and the departure from it is what
one sees.


\subsection{Life science: molecular chirality, and the sterility of a perfectly symmetric universe}
\label{ssec:chirality}
Most molecules of biological interest are handed: they cannot be
superposed on their mirror images. Living systems do not use the two
hands impartially. Proteins are built almost exclusively from
L-amino acids and nucleic acids from D-sugars, and the exceptions are
few enough to be individually notable. The machinery could in principle
have been built the other way round throughout --- a complete mirror
biochemistry is not obviously forbidden --- but it was not, and the same
choice appears across all known life.

This is worth dwelling on, because it is a case where the physics and the
biology sit at wildly different scales. Of the known fundamental
interactions, only the weak interaction distinguishes left from right,
and the energy difference it induces between two mirror-image molecules
is fantastically small --- many orders of magnitude below the thermal
energy available in any environment where chemistry proceeds. Whether
that tiny bias is amplified into the observed uniformity, or whether the
uniformity arose from an amplification mechanism indifferent to which
hand it started with, remains an open question in origins-of-life
research \citep{Blackmond2010}. This paper takes no position on it. What
matters here is the structural point: a reflection-symmetric universe
would contain both hands in equal measure and no basis for the
chemistry that living systems actually run, and the biochemistry we have
is downstream of a broken symmetry.

Life scientists therefore have a stake in the second half of this paper
even if they never open the first. Chapter~\ref{sec:grip} is about a
framework in which symmetry breaking is a productive stage rather than a
defect, and biology is the largest existing body of evidence that
broken symmetry is where the interesting structure lives.


\subsection{The thread this paper follows: why is anything an integer?}
\label{ssec:integers}
Here is the thread this paper follows from beginning to end.

Most of the numbers in physics are approximate. Masses are measured to
some number of digits; coupling constants run with energy; the age of the
universe carries an error bar. A small and remarkable set of numbers are
not like this at all. They are integers, they are exactly what they are,
and no measurement has ever found a fraction of one.

Electric charge is the cleanest case. Every isolated charge ever measured
is an exact multiple of one unit. The electron and the proton, which
share no other property, carry charges equal in magnitude to a precision
that experiments on the neutrality of bulk matter place at the level of
$10^{-21}$ of that unit \citep{MarinelliMorpurgo1984}. Nothing in the
Standard Model as ordinarily presented explains this. The theory is
perfectly consistent with a proton whose charge differs from the
electron's in the twenty-second decimal place; the world declines to
provide one.

There are exactly three colors of quark. There are exactly three
families of matter, and the fourth has been searched for with some
determination and is not there. Anomaly-cancellation conditions in the
Standard Model are equations between integers, and they balance.

The question this paper follows is: \emph{why is anything an integer?}
That is a question about rigidity, not about symmetry, and it turns out
to have a good answer --- an answer which occupies the top rung of the
ladder and has no symmetry in it at all. A quantity that takes values in
the integers cannot change by a little. It can only jump, and in a
continuous world nothing jumps. Chapter~\ref{sec:hs} makes this precise;
Chapter~\ref{sec:payoffs} is about what a physical theory can do with it.

A reader who wants one sentence to carry through the whole paper might
take this one: the deepest things in physics are not conserved because
something is symmetric, but because there is nowhere for them to go.



\subsection{Part~I in one paragraph, and what comes next}
\label{ssec:part-i-close}
Part~I asked one question and refused two easy answers to it. A
conservation law is not the statement that a quantity never changes; it is
a statement about a boundary --- what crosses it, and over what interval
--- and the clause is there even when nobody writes it down. A symmetry is
not a fact about the world but about a description of it, and the two stay
different in kind even where a theorem connects them. The chemistry and
the biology were not decoration: the symmetry supplies the possibilities,
and the \emph{breaking} supplies everything one observes --- the color of
a transition-metal salt, the single hand of every protein in every cell.
Underneath sits the thread the paper follows. A small number of quantities
in physics are known exactly; every one is a count; and nothing in the
standard account says why.

Part~II answers that one rung at a time, climbing the century from Noether
in 1918 to \HS{} in 2005 and asking of each result the same two things:
what exactly is conserved, and what it would cost to violate it. Read them
in order, because the order is the argument --- the answer waiting at the
top has no symmetry in it at all.

%% ===================================================================
%%  PART II --- THE LADDER
%% ===================================================================
\begin{checkyourself}{Part~I}
\item \S\ref{ssec:cons-law} claims that every conservation law in
  practice carries a boundary clause. Write the clause out for electric
  charge in a laboratory that is not sealed, and then describe a
  measurement whose interpretation would depend on it.
\item \S\ref{ssec:chemistry} obtains the block widths $2, 6, 10, 14$
  from $2(2\ell+1)$. Suppose the electron had three spin states rather
  than two. What would the four widths become, and which of the two
  factors in $2(2\ell+1)$ is doing the work in each case?
\item \S\ref{ssec:integers} claims that the exactly known quantities in
  physics are counting statements. Name a quantity you believe is known
  exactly, decide whether it is a count, and if it is not, say what
  protects it.
\end{checkyourself}

\GGPart[The ladder]{The Ladder}{Seven steps, one chapter each: Noether's
two theorems, local gauge invariance, the Bianchi identity, the Wilson
line, the Ward identity, the \GS{}-modified identity, and \HS{}
quantization --- with every toolkit deferred to the appendices}
\label{part:ladder}

\section{The seven steps, in plain English}
\label{sec:ladder-overview}

Part~II climbs a ladder. Each rung is about a \emph{conservation law} ---
a rule that some quantity never changes, no matter what else happens ---
and each holds that rule to a stricter standard than the one below. At the
bottom the law is the familiar kind a physicist proves in an afternoon; at
the top it has hardened into a whole number that could not have been
otherwise. Here is the whole climb once, in plain words.

\textbf{S1 --- Noether (1918).} Symmetry and conservation are two names for
one fact. If the laws do not care \emph{when} you run an experiment, energy
is conserved; if they do not care \emph{where}, momentum is. Noether made
that a theorem; a second theorem of hers, about symmetries chosen freshly
at every point, sets up everything above.

\textbf{S2 --- Local gauge invariance.} Take a quantity no instrument can
read --- the ``phase'' of a quantum field --- and \emph{insist} that it be
adjustable independently at every point of space. That cannot be met for
free: a new field must appear to reconcile the choices, and that field is
electromagnetism. Requiring a local freedom \emph{forces} a force into
being --- the most consequential idea in twentieth-century physics.

\textbf{S3 --- The Bianchi identity.} Some conservation laws need no
symmetry at all. Write a field as a curvature, and a certain combination of
its derivatives is exactly zero, always --- conservation for free, and
unbreakable even in principle.

\textbf{S4 --- The Wilson line.} A field records less than the world
contains. Where space carries a loop that cannot be shrunk to a point,
there is information read only by following a quantity \emph{around} the
loop. The deepest bookkeeping is about shape, not force.

\textbf{S5 --- The Ward identity (1950).} Do these tidy classical
identities survive the passage to quantum theory, where everything
fluctuates? For this family, yes --- and this is the first rung on which
the answer could have been no. The identity can fail; it fails only when
the quantum measure is not invariant; and when it fails, it fails by a
whole number.

\textbf{S6 --- The \GS{}-modified identity (1984).} Here a conservation law
is broken \emph{on purpose}, locally, and an equal and opposite breaking
elsewhere cancels it, so the total is still conserved. The violation is the
mechanism; conservation becomes sacred only in the grand sum.

\textbf{S7 --- \HS{} quantization (2005).} At the top the conserved object
is an \emph{integer} --- a count, with no symmetry behind it and no
dynamics left to strip away. It cannot drift, the way the number of holes
in a surface cannot --- conservation in its most rigid form.

\textbf{Above the top rung: \GGTheory{} (2026).} The seven rungs were
built by different people across a century, yet read in order they are one
argument --- symmetry not chosen but \emph{forced}, down to an integer
(Figure~\ref{fig:ladder-timeline}).

\begin{figure}[p]
\centering
\resizebox{0.877\textwidth}{!}{%
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\node[anchor=south] at (3.4,0.4) {\normalsize\bfseries The logical ladder (S1$\to$S7)};
\node[anchor=south] at (10.35,0.4) {\normalsize\bfseries A century of milestones};
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\end{tikzpicture}}
\par\medskip
\caption[The seven-step ladder set against a century of physics.]{\textbf{The
ladder of Part~II, set against real time.} \emph{Left:} the seven steps in
their logical order --- S1 at the top, \GGTheory{} at the foot --- each
tagged with the year of the idea it rests on. \emph{Right:} a linear clock
from 1910 to 2030 carrying the wider milestones the ladder draws on:
general relativity, quantum mechanics, non-Abelian gauge theory, parity
violation, the electroweak Standard Model, and more. Dashed lines join each
dated rung to its year; they cross where the logical order departs from the
historical one --- the Ward identity (1950) sits logically \emph{after} the
Wilson line (1974), and the Green--Schwarz and Hopkins--Singer rungs rest on
ideas that arrived only decades after Noether. The ladder is a
reconstruction, not a chronology.}
\label{fig:ladder-timeline}
\end{figure}

\section{S1 --- Noether (1918), both theorems}
\label{sec:noether}
Emmy Noether published two theorems in 1918 \citep{Noether1918}. The
first is one of the most quoted results in physics. The second is barely
taught. This chapter derives both, in that order, and argues that the
second is the more general and the more consequential --- and that
leaving it out is what makes the rest of the ladder look like a series
of unrelated curiosities instead of one argument.

The variational machinery is assumed only at the level of a first course
and is developed from scratch in Appendix~\ref{app:noether}; what is
given here is the physics and the two statements.

One promise is worth making explicitly, because the question of this
section is the one the whole ladder answers. Each rung will be asked how
much of its conservation statement is carried by an integer, and the
answers run in one direction: none at all through
Chapter~\ref{sec:bianchi}, where the statements are algebra, convention
and geometry; then the integer as a \emph{frame}
(\S\ref{ssec:flat}), as a \emph{failure} (\S\ref{ssec:ward-cost}), as a
\emph{trade} (\S\ref{ssec:let-it-fail}), and finally as the whole content
(\S\ref{ssec:top-of-ladder}). The anomaly-cancellation conditions quoted
above are a case in point: they are equations between integers because
they are the conditions for a set of identities --- named in
Chapter~\ref{sec:ward} --- to survive quantization.


\subsection{The action principle in one page}
\label{ssec:action}
A mechanical system is specified by a Lagrangian $L$, and the path it
actually follows between two fixed endpoints is the one that makes the
action
\begin{equation}
S = \int_{t_1}^{t_2} L\bigl(q(t), \dot q(t), t\bigr)\, \dd t
\label{eq:action}
\end{equation}
stationary. Setting the first variation to zero gives the
Euler--Lagrange equation,
\begin{equation}
\frac{\dd}{\dd t}\frac{\partial L}{\partial \dot q}
  - \frac{\partial L}{\partial q} = 0 .
\label{eq:EL}
\end{equation}

\paragraph{Worked example (the whole of this chapter in miniature).} Take a
free particle in one dimension, $L = \tfrac12 m \dot q^2$. Then
$\partial L/\partial \dot q = m\dot q$, which is the momentum, and
$\partial L/\partial q = 0$, because $q$ does not appear in $L$ at all.
Equation~\eqref{eq:EL} therefore reads
\begin{equation}
\frac{\dd}{\dd t}\,(m \dot q) = 0 .
\end{equation}

Look at what just happened. The equation of motion \emph{is} the
conservation law. They are not two facts, one following from the other;
they are one line of algebra read from two directions. And the reason
momentum is conserved is visible in the derivation: $q$ was absent from
$L$, which is to say that translating the particle in $q$ changes
nothing. A symmetry of the Lagrangian became a conserved quantity, in
one step, with no further input.

That is the whole content of the first theorem. Everything that follows
is this observation made general.



\emph{Physics intuition.} The action principle is often introduced as
though nature were solving an optimization problem, which invites
questions about how the particle knows the future. It is better read the
other way round. The variational statement is a compact way of writing
a \emph{local} condition --- the Euler--Lagrange equation --- and its
value is not metaphysical but bookkeeping: it packages the whole dynamics
into a single scalar $L$, so that a statement about $L$ becomes a
statement about every equation of motion at once. That is why a theorem
about symmetries of $L$ can be a theorem about all conservation laws
simultaneously, which is what Noether proved.
\subsection{The first theorem, and the three classical cases}
\label{ssec:noether-i}
\begin{quote}\itshape
\textbf{Noether's first theorem.} If the action is invariant under a
continuous group of transformations depending on constant parameters,
then to each parameter there corresponds a current
$j^\mu$ satisfying $\partial_\mu j^\mu = 0$ whenever the field equations
hold.
\end{quote}

The proof is a short calculation and is carried out in full, with the
boundary terms kept, in Appendix~\ref{app:noether}. The essential move
is this: vary the fields by the symmetry transformation. The action does
not change, because the transformation is a symmetry. But the action
also does not change under \emph{any} variation, once the field
equations hold, because that is what the field equations say. Subtract
the two statements and what survives is a total divergence set to zero.

\paragraph{Worked example: the phase of a complex field.} Let $\phi$ be
a complex scalar with Lagrangian
$\mathcal{L} = \partial_\mu \phi^{*}\partial^\mu \phi - \mu^2 \phi^{*}\phi$.
Rotating the phase, $\phi \to e^{\ii\alpha}\phi$ with $\alpha$ a
constant, leaves $\mathcal{L}$ alone, since $\phi$ and $\phi^{*}$ appear
in matched pairs. The associated current is
\begin{equation}
j^\mu = \ii\left(\phi^{*}\partial^\mu \phi - \phi\,\partial^\mu \phi^{*}\right),
\qquad
\partial_\mu j^\mu
 = \ii\left(\phi^{*}\Box\phi - \phi\,\Box\phi^{*}\right).
\label{eq:scalar-current}
\end{equation}
The second equality is an identity: it holds for any $\phi$ whatsoever.
The divergence vanishes only when the Klein--Gordon equation
$\Box\phi = -\mu^2\phi$ holds, at which point the two terms become
$-\mu^2\phi^{*}\phi$ and cancel. This is worth pausing on, because it is
the distinction the next section turns on. The current is conserved
\emph{on shell} --- that is, on solutions of the field equations --- and
not otherwise.

\paragraph{The three classical cases.} Table~\ref{tab:noether-cases}
collects them. Each is the free-particle computation of
\S\ref{ssec:action} with a different symmetry substituted, and all three
are worked explicitly in Appendix~\ref{app:noether}.

\begin{table}[H]
\centering
\caption[The classical Noether correspondences.]{The classical
correspondences. The first three are symmetries of the arena; the fourth
is already a symmetry of the \emph{description}, and
Chapter~\ref{sec:gauge} is about what happens when that fourth kind is
made local. Three columns: the symmetry of the action, the quantity it conserves, and what that conservation says physically. The rows are ordered so that the shift from symmetries of the arena to symmetries of the description happens once, at the last row, rather than being spread across the table.}
\label{tab:noether-cases}\small
\begin{tabular}{@{}P{0.27\textwidth}P{0.25\textwidth}P{0.34\textwidth}@{}}
\toprule
\textbf{symmetry of the action} & \textbf{conserved quantity} &
\textbf{what it says physically} \\
\midrule
translation in time & energy & the laws do not depend on when you run
the experiment \\
translation in space & momentum & they do not depend on where \\
rotation & angular momentum & they do not depend on which way the
apparatus faces \\
phase rotation of a complex field & electric charge & the overall phase
of the field is not measurable \\
\bottomrule
\end{tabular}
\end{table}



\paragraph{Worked example: where energy comes from.} Time translation is
the case worth doing explicitly, because the answer is not the
Lagrangian. If $L$ has no explicit $t$-dependence, the conserved charge
is
\begin{equation}
E = \dot q\,\frac{\partial L}{\partial \dot q} - L .
\label{eq:energy-noether}
\end{equation}
Take the harmonic oscillator, $L = \tfrac12 m\dot q^2 - \tfrac12 m\omega^2 q^2$.
Then $\partial L/\partial \dot q = m\dot q$, and
\begin{equation}
E = m\dot q^2 - \left(\tfrac12 m\dot q^2 - \tfrac12 m\omega^2 q^2\right)
  = \tfrac12 m\dot q^2 + \tfrac12 m\omega^2 q^2 = T + V .
\end{equation}
The minus sign in $L = T - V$ has become a plus. Differentiating
Eq.~\eqref{eq:energy-noether} in time gives $\dot q$ times the
Euler--Lagrange expression, so $E$ is constant precisely on solutions ---
on shell again, and for the same reason.

\emph{Physics intuition.} It is worth noticing what the theorem does not
say. It does not say that a symmetric world has more conserved
quantities in some loose sense; it says that each independent parameter
of the symmetry group supplies exactly one current, and it tells you
which one. That specificity is what makes it useful: given a Lagrangian
and a symmetry, one turns a crank and a formula comes out. The
converse --- given a conserved quantity, find the symmetry --- is not
part of the theorem, and Chapter~\ref{sec:bianchi} exhibits a conserved
quantity for which no symmetry exists to be found.
\subsection{The second theorem: local symmetry gives identities, not charges}
\label{ssec:noether-ii}
Now change one word. Let the parameter of the transformation be not a
constant but an arbitrary function of position --- a \emph{local}
symmetry. What does the theorem give?

\begin{quote}\itshape
\textbf{Noether's second theorem.} If the action is invariant under a
group of transformations depending on arbitrary functions and their
derivatives, then there are identities among the Euler--Lagrange
expressions --- relations that hold identically, whether or not the
field equations are satisfied.
\end{quote}

No new charge. Identities.

\paragraph{Worked example: electromagnetism.} The Maxwell action is
invariant under $A_\mu \to A_\mu + \partial_\mu \Lambda$ for an arbitrary
function $\Lambda(x)$. The Euler--Lagrange expression is
$\partial_\nu F^{\mu\nu}$, and the identity the second theorem delivers
is
\begin{equation}
\partial_\mu \partial_\nu F^{\mu\nu} \equiv 0 .
\label{eq:noether-ii}
\end{equation}
This requires no proof beyond noticing that $\partial_\mu\partial_\nu$ is
symmetric in its two indices while $F^{\mu\nu}$ is antisymmetric, so the
contraction vanishes term by term. It holds for \emph{any} $A_\mu$
whatsoever --- any configuration, physical or not, solution or not. It
is off shell.

Compare the two results directly, because the comparison is the point of
the chapter.

\begin{itemize}[leftmargin=1.4em,itemsep=0.3em]
\item \emph{Global} phase rotation gives
  Eq.~\eqref{eq:scalar-current}: a current, conserved on shell, whose
  integral is a number you can measure --- the charge.
\item \emph{Local} phase rotation gives Eq.~\eqref{eq:noether-ii}: an
  identity, true off shell, whose content is that one combination of the
  field equations was never independent in the first place.
\end{itemize}

The second is not a weaker version of the first. It is a different kind
of statement. The first constrains the world; the second constrains the
\emph{equations}, telling you that you have written down fewer of them
than you thought. In electromagnetism that is why $A_\mu$ has four
components but only two propagate: one is removed by the identity and
one by the freedom it reflects.

And notice what Eq.~\eqref{eq:noether-ii} costs to violate: nothing can
violate it. There is no configuration of the electromagnetic field, and
no modification of the dynamics, for which $\partial_\mu\partial_\nu
F^{\mu\nu}$ is anything other than zero. It is the first appearance in
this paper of a conserved statement with no dynamical content at all,
and it appears on the ladder's first rung, in Noether's own paper.

Two chapters later this identity acquires a quantum form, and the
quantum form is not so safe: Chapter~\ref{sec:ward} shows that the same
local symmetry, asked what it does inside a correlation function, gives
a statement that \emph{can} fail --- and fails only by an integer.
Eq.~\eqref{eq:noether-ii} is one half of that statement;
\S\ref{ssec:two-faces} sets the two halves side by side.



\paragraph{Worked example: the same field, twice.} The clearest way to
see the difference is to run both computations on one theory. Take the
complex scalar coupled to electromagnetism.

\emph{Global.} Rotate $\phi \to e^{\ii\alpha}\phi$ with $\alpha$ constant.
The Lagrangian is invariant. Noether's first theorem returns the current
of Eq.~\eqref{eq:scalar-current}, its divergence vanishes on shell, and
$Q = \int j^0 \dd^3x$ is a number. Measure it: it is the electric charge
in the region, and it is the same number tomorrow.

\emph{Local.} Rotate $\phi \to e^{\ii\alpha(x)}\phi$ and simultaneously
shift $A_\mu$ as Chapter~\ref{sec:gauge} requires. The Lagrangian is
again invariant --- more strongly invariant, in fact, since $\alpha$ is
now arbitrary. Noether's second theorem returns
Eq.~\eqref{eq:noether-ii}. There is no new number. What there is instead
is the statement that the time-derivative part of the field equations
was never independent of the rest, which is why one cannot specify
initial data for $A_\mu$ freely.

Same field, same Lagrangian, two symmetries differing by one word, and
two outputs of entirely different type: a measurable number, and a
constraint on what one is allowed to write down.
\subsection{Why textbooks stop at the first theorem, and what that costs}
\label{ssec:textbook-stop}
Almost every undergraduate treatment presents the first theorem and
stops. The reason is understandable: the first theorem is what one uses.
It produces the conservation laws that appear in problem sets, and the
second theorem produces nothing one can compute a number with.

The cost of stopping there is that the rest of this paper becomes
unintelligible. A reader who knows only the first theorem carries the
rule \emph{symmetry gives conservation} and then meets, in sequence:
a gauge ``symmetry'' that yields no charge (Chapter~\ref{sec:gauge}); a
conservation law with no symmetry behind it at all
(Chapter~\ref{sec:bianchi}); a deliberate violation of a conservation
law used as a repair (Chapter~\ref{sec:gs}); and finally a conserved
quantity produced by an argument in which no symmetry is mentioned
(Chapter~\ref{sec:hs}). Each of these looks like an anomaly in the rule.
Read with the second theorem in hand, they look like what they are: the
successive consequences of a distinction Noether had already drawn.

There is a second cost, subtler and worth naming. A reader who has only
the first theorem will hear ``gauge symmetry'' and supply the wrong
mental picture --- states related by a transformation --- and will then
find the standard account of the Higgs mechanism genuinely confusing,
because it is describing something else. \S\ref{ssec:higgs} returns to
this.



\emph{Physics intuition.} There is a diagnostic worth carrying through
the rest of Part~II. When someone says that a symmetry is responsible
for a conservation law, ask whether the symmetry's parameter is a
constant or a function. If a constant, expect a charge, expect it to be
conserved only on shell, and expect it to fail if the dynamics is
modified. If a function, expect no charge at all, expect an identity
instead, and expect it to be immune to any modification of the dynamics
whatsoever. Every remaining rung of this ladder is a variation on the
second case.
\subsection{Historical box: G\"ottingen 1918 and the energy problem}
\label{ssec:goettingen}
\begin{quote}\small
\textbf{Historical box: G\"ottingen, 1918.}
Noether did not set out to write a general theorem about symmetry. She
was answering a specific and embarrassing technical problem that Hilbert
and Klein had run into in the new general theory of relativity: energy
conservation there does not take the form it takes everywhere else. In a
generally covariant theory the quantity one wants to call the energy of
the gravitational field is not a tensor, and the conservation law
degenerates into an identity. Hilbert found this so striking that he
described it as a characteristic feature of general relativity, and
asked Noether to sort it out \citep{Byers1998}.

Her answer was that this is not a peculiarity of gravitation at all. It
is what \emph{always} happens when the invariance group depends on
arbitrary functions rather than constants --- and she proved it in
general, which is the second theorem. The first theorem, the one that
made her famous, is in the same paper largely as the contrasting case.
The episode is reconstructed in detail by \citet{Brading2002}, who traces
what Weyl and Noether each took the conservation of electric charge to
follow from.

The history is worth knowing because it inverts the usual pedagogical
order. The celebrated result was the by-product; the neglected one was
the answer to the question actually asked. Noether was at the time
lecturing at G\"ottingen under Hilbert's name, the university having
declined to appoint her.
\end{quote}


\subsection{Physics intuition}
\label{ssec:noether-intuition}
\emph{Physics intuition.} A global symmetry is a statement about the
world: the apparatus can be moved and nothing changes. Because it is a
statement about the world, it can be false, and when it is true it buys
you something the world did not have to give --- a quantity that stays
put while everything else moves.

A local symmetry is a statement about your description: you have used
more variables than there are physical facts. Because it is a statement
about your description, it cannot be false; you can always check it by
inspecting your own notation. And what it buys you is correspondingly
different. Not a number that stays put, but a guarantee that your
equations are not all independent --- that some of what you wrote down
was never a physical claim at all.

The first rung of the ladder therefore already contains its whole shape
in miniature. On one side, a conservation law with real dynamical
content, contingent on the world being a certain way. On the other, an
identity with no dynamical content, and no way of being wrong. Every
step from here to Chapter~\ref{sec:hs} moves further toward the second
kind.


\section{S2 --- Local gauge invariance}
\label{sec:gauge}
The second rung is the most important idea in twentieth-century physics
and the most widely misdescribed. This chapter builds it in the standard
way --- take a phase nobody can measure, insist that it be adjustable
independently at every point, and watch the electromagnetic field appear
as the price --- and then says plainly what the construction has and has
not established.

The geometry behind it (parallel transport, connections, curvature,
holonomy) is developed from scratch in
Appendix~\ref{app:connections}. This chapter uses only the physics.


\subsection{The phase nobody can measure, made local}
\label{ssec:phase}
A quantum state is described by a complex wavefunction, and the overall
phase of that wavefunction is not measurable: $\psi$ and $e^{\ii\alpha}\psi$
predict identical results for every experiment. This is the fourth row
of Table~\ref{tab:noether-cases}, and by the first theorem it gives
electric charge conservation.

Now ask a question that looks pedantic and is not. The statement ``the
phase is unmeasurable'' fixes one number for the whole universe. But if
the phase at a point is genuinely without physical content, why should
the choice made here have to agree with the choice made in the next
laboratory? Why should there be a single global convention for a
quantity that nobody can measure anywhere?

Suppose there is not. Let $\alpha$ become a function of position,
$\psi(x) \to e^{\ii\alpha(x)}\psi(x)$. Almost everything survives
unchanged --- $|\psi|^2$, for instance, is untouched. One thing does not:
the derivative. Differentiating picks up an extra term from the
$x$-dependence of $\alpha$, so $\partial_\mu \psi$ does not transform the
way $\psi$ does, and any Lagrangian built from derivatives fails to be
invariant.

Here is where the construction earns its reputation. There is exactly
one minimal repair. Introduce a field $A_\mu(x)$, and replace every
derivative by the combination
\begin{equation}
D_\mu = \partial_\mu - \ii q A_\mu ,
\label{eq:covderiv}
\end{equation}
with the new field required to shift as
$A_\mu \to A_\mu + q^{-1}\partial_\mu\alpha$ whenever the phase is
rotated. The two extra terms then cancel, $D_\mu\psi$ transforms exactly
as $\psi$ does, and invariance is restored.

The logic is worth stating in the order it actually runs, because it is
the reverse of the historical order. We did not postulate an
electromagnetic field and then discover it couples to charge. We
demanded that an unmeasurable convention be adjustable locally, and a
field with precisely the couplings of electromagnetism was
\emph{forced}. That is the sense in which, as \citet{Yang1980} put it,
symmetry dictates interaction. The construction is Weyl's
\citep{Weyl1929}, and it is the origin of the word ``gauge''.



\paragraph{Worked example: why the repair is forced.} The two lines of
algebra are worth seeing, because they show that nothing was chosen.
Under $\psi \to e^{\ii\alpha(x)}\psi$ the bare derivative gives
\begin{equation}
\partial_\mu\!\left(e^{\ii\alpha}\psi\right)
 = e^{\ii\alpha}\,\partial_\mu\psi
 + \ii\,(\partial_\mu\alpha)\,e^{\ii\alpha}\psi ,
\end{equation}
so it fails by exactly the unwanted term $\ii(\partial_\mu\alpha)$ times
the field. Now use $D_\mu$ of Eq.~\eqref{eq:covderiv} and let $A_\mu$
shift by $q^{-1}\partial_\mu\alpha$:
\begin{equation}
D_\mu\!\left(e^{\ii\alpha}\psi\right)
 = e^{\ii\alpha}\partial_\mu\psi + \ii(\partial_\mu\alpha)e^{\ii\alpha}\psi
   - \ii q\!\left(A_\mu + \tfrac{1}{q}\partial_\mu\alpha\right)\!
     e^{\ii\alpha}\psi
 = e^{\ii\alpha}\,D_\mu\psi .
\end{equation}
The two unwanted terms cancel identically, for arbitrary $\alpha(x)$.
Note the shape of the argument: the shift of $A_\mu$ was not postulated
and then found to work --- it is the unique shift that cancels the term,
and the coupling constant $q$ appears in $D_\mu$ precisely so that the
cancellation is exact.

\emph{Physics intuition.} What has been bought is a way of comparing the
phase of $\psi$ at two nearby points when there is no agreed convention
at either. $A_\mu$ is the exchange rate between the two local
conventions, and $D_\mu\psi$ is the difference of $\psi$ measured in a
consistent currency. Appendix~\ref{app:connections} says the same thing
in the language of parallel transport, where $A_\mu$ is a connection and
the ``exchange rate'' reading is exact.
\subsection{Curvature appears, and half of Maxwell comes free}
\label{ssec:curvature-free}
The field $A_\mu$ is not observable --- it shifts when the convention
shifts. What is observable is the failure of the correction to be
consistent: the extent to which comparing phases along one route differs
from comparing them along another. That failure is measured by
\begin{equation}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu ,
\label{eq:F}
\end{equation}
which is invariant under the shift, because the extra term is a
gradient and gradients have symmetric second derivatives. In the language
of Appendix~\ref{app:connections}, $A_\mu$ is a connection and
$F_{\mu\nu}$ is its curvature. In the language of a laboratory,
$F_{\mu\nu}$ is the electric and magnetic field.

Something is now free that was not paid for. Because $F$ was
\emph{defined} as the antisymmetrized derivative of $A$, it satisfies
\begin{equation}
\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda}
  + \partial_\nu F_{\lambda\mu} = 0
\label{eq:bianchi-flat}
\end{equation}
identically, for every $A_\mu$ whatsoever. Written in three-vector
notation, Eq.~\eqref{eq:bianchi-flat} is
\begin{equation}
\nabla \cdot \mathbf{B} = 0,
\qquad
\nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} = 0 ,
\end{equation}
which is to say: two of Maxwell's four equations. They did not come from
a Lagrangian, they do not follow from the field equations, and no choice
of dynamics can alter them. They are the price of having written $F$ as
a curvature, and they are the subject of Chapter~\ref{sec:bianchi}.

The other two equations --- Gauss's law and the Amp\`ere--Maxwell law ---
are of an entirely different character. They come from varying an action,
they involve the sources, and a different action would give different
ones. Maxwell's equations, which a first course presents as four members
of one family, are in fact two identities and two laws.



\paragraph{Worked example: two invariances, one reason.} Check
directly that Eq.~\eqref{eq:F} is untouched by the shift:
\begin{equation}
\partial_\mu\!\left(A_\nu + \partial_\nu\Lambda\right)
 - \partial_\nu\!\left(A_\mu + \partial_\mu\Lambda\right)
 = F_{\mu\nu} + \left(\partial_\mu\partial_\nu
   - \partial_\nu\partial_\mu\right)\Lambda
 = F_{\mu\nu} ,
\end{equation}
because mixed partial derivatives commute. Now notice that
Eq.~\eqref{eq:bianchi-flat} follows from the \emph{same} fact: expand
$\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} +
\partial_\nu F_{\lambda\mu}$ using $F = \partial A - \partial A$ and the
six terms cancel in pairs, again because second derivatives commute.

One elementary property of partial differentiation is therefore doing
two apparently unrelated jobs: it makes the field strength observable,
and it makes two of Maxwell's equations free. That is not a coincidence.
Both are statements that $\dd\dd = 0$, and Chapter~\ref{sec:bianchi} is
about the second one.

\emph{Physics intuition.} A useful picture: $A_\mu$ is like a choice of
sea level, arbitrary and unmeasurable, while $F_{\mu\nu}$ is like a
slope, which is not. Changing the reference level everywhere by a
function changes no slope. The two free Maxwell equations then say that
slopes computed from a height function cannot be arbitrary --- they must
be consistent around any closed circuit --- which is a constraint on the
slopes and not a physical law about what the world does.
\subsection{The uncomfortable truth: redundancy, not symmetry (Elitzur)}
\label{ssec:elitzur}
Now the correction. Everything above is standard, and every word of it
is usually narrated with the term ``gauge symmetry.'' The term is
misleading, and the misunderstanding it produces is not harmless.

A symmetry, in the sense of \S\ref{ssec:symmetry}, is a change one can
make to a system that takes it to a different but
experimentally indistinguishable state. A gauge transformation does not
do that. The configurations $A_\mu$ and $A_\mu + \partial_\mu\Lambda$ are
not two states related by a transformation; they are two \emph{names}
for one state. The physical configuration space is the set of gauge
orbits, and a gauge transformation moves you within a single point of
it.

This is why Noether's second theorem returned an identity rather than a
charge (\S\ref{ssec:noether-ii}): there was no second state for a charge
to distinguish. It is also why the following theorem is true.

\begin{quote}\itshape
\textbf{Elitzur's theorem} \citep{Elitzur1975}. A local gauge symmetry
cannot be spontaneously broken. The expectation value of any
gauge-non-invariant local quantity vanishes identically, at every
temperature and coupling.
\end{quote}

The proof idea is simple enough to state. Spontaneous breaking requires
that arbitrarily weak long-range order pick a direction out of a
degenerate set. A gauge transformation can be performed independently at
each point and costs no energy at all, so there is no long-range
correlation for a weak field to align; the average over the local
freedom kills any non-invariant quantity exactly.

Elitzur's theorem is not a curiosity about lattice gauge theory. It is
the sharpest statement of what the object in question actually is. One
cannot break what is not there.



\emph{Physics intuition.} Elitzur's theorem sounds destructive and is
not. What it destroys is a picture, not a calculation: every prediction
made using the language of spontaneous gauge breaking remains correct,
because the calculations were always gauge-fixed computations of
gauge-invariant quantities. What the theorem supplies is the reason a
gauge-fixed computation is legitimate at all --- there is nothing being
chosen between, so fixing the gauge cannot bias an answer.

There is also a practical payoff. Because a local gauge redundancy
cannot break, any genuine order parameter must be gauge invariant, and
looking for one is a good way to find out what a phase transition is
actually about --- and in the lattice gauge-Higgs system the answer is
that the confining and Higgs regions are not separated by any sharp
transition at all \citep{FradkinShenker1979}. In the electroweak case the honest order parameter is
not the Higgs field's value but the spectrum: which vectors are massive,
and how many scalars remain. That is what \S\ref{ssec:higgs} counts.

\paragraph{Worked example: Elitzur on one site.} The theorem is easier to
believe after doing the smallest case, in the lattice regularization of
\citet{Wilson1974}. Put a complex variable $\phi_i$ on
each site $i$ of a lattice, with a gauge freedom that multiplies
$\phi_i$ by an independent phase $e^{\ii\theta_i}$ at each site
separately, and an action that depends only on gauge-invariant
combinations. Ask for the expectation value of the non-invariant
quantity $\phi_i$ itself.

Because $\theta_i$ appears nowhere in the action, the integration over it
factorizes completely:
\begin{equation}
\langle \phi_i \rangle
 \;\propto\; \int_0^{2\pi}\!\!\dd\theta_i\, e^{\ii\theta_i}
 \;=\; 0 .
\end{equation}
Exactly zero, at every temperature, at every coupling, on a lattice of
any size --- and, crucially, even in the presence of an external field
that tries to align $\phi$, provided that field respects the gauge
freedom.

Contrast the \emph{global} case, where a single $\theta$ multiplies every
site at once. Then the integral does not factorize: the action does
depend on the relative phases, an arbitrarily weak aligning field is
amplified by the long-range order, and $\langle\phi\rangle$ survives the
limit. That is spontaneous symmetry breaking, and it is a real
phenomenon --- ferromagnets, superfluids, the chiral condensate. The difference between the two cases is one word, exactly as in
\S\ref{ssec:noether-ii}, and again the local case is the one that turns
out to be empty of the thing one was looking for.

The theorem leaves a question that the next chapter but one answers. If
a local symmetry cannot break, what is happening in the many places
where physicists say one has? \S\ref{ssec:higgs} gives one answer. There
is a second, and it is cleaner, because its order parameter is
gauge invariant before any gauge is fixed: \S\ref{ssec:centre}.
\subsection{What the Higgs mechanism actually does}
\label{ssec:higgs}
Which raises the obvious question, since every account of the Standard
Model says that the Higgs field breaks the electroweak gauge symmetry,
and \S\ref{ssec:elitzur} has just said that this is impossible.

Both statements are defensible; the phrase is the problem. What
physically occurs is this. A scalar field acquires a non-zero value in
the vacuum. That value is not gauge invariant, so choosing it amounts to
\emph{fixing a gauge} --- selecting one name from the orbit --- rather
than choosing between physical states. What is genuinely broken is the
\emph{global} part of the symmetry, which is a real symmetry with real
consequences. And the physically meaningful result, which does not
depend on any of this language, is a change in the spectrum: three of
the four electroweak vector fields acquire mass, one does not, and one
scalar degree of freedom remains.

The counting is worth doing, because the counting is the content.
Before: the electroweak group has $3 + 1 = 4$ generators, and the scalar
doublet has $4$ real components. After: one generator remains manifest
--- the photon --- so $4 - 1 = 3$ do not; each of those three
corresponds to a vector field that has become massive, and a massive
vector has one more polarization state than a massless one. Three of the
doublet's four components supply those extra polarizations. The
remaining $4 - 3 = 1$ is the physical Higgs scalar. Every number in that
paragraph is a count of states, and the counts balance exactly.
\citet{Englert1964}, \citet{Higgs1964} and
\citet{GuralnikHagenKibble1964} are the original papers, and none of
them uses the phrase that later became standard. That the resulting
theory is renormalizable --- the fact that made the construction
usable --- was shown by \citet{tHooft1971}.

\emph{Why the misnomer survives.} Because the mathematics is easiest to
carry out in the gauge that the vacuum value selects, and in that gauge
the calculation looks exactly like spontaneous breaking of a symmetry.
The formalism is unimpeachable; only the sentence describing it is
wrong. This paper is careful about it for one reason: the distinction
between a symmetry of nature and a redundancy of description is the
distinction the whole ladder is built on, and this is the place where
physics conventionally blurs it.



\emph{Physics intuition.} The cleanest way to hold the whole episode in
mind is that nothing was broken and nothing was created --- the states
were re-sorted. Before, one counts eight vector polarizations that
propagate ($2 \times 4$ for four massless vectors) plus four real
scalars, which is twelve. After, one counts nine states in the massive
sector, two for the photon and one physical Higgs boson --- twelve
again. A massless vector has two states, a
massive one has three, and the extra states came from the scalars.
``The gauge boson eats the Goldstone boson'' is the usual phrase for
this, and it is a better phrase than the one about breaking a gauge
symmetry, because eating is a redistribution and breaking is a loss.

One feature of this construction is worth marking, because
Chapter~\ref{sec:wilson} will offer an alternative that does not share
it. The order parameter here, $\langle\phi\rangle$, is not gauge
invariant: it is a vector in an internal space, and its direction is a
choice of gauge rather than a fact about the world. What is invariant is
the spectrum that results. A Wilson line is invariant from the start ---
which is why \S\ref{ssec:centre} can call it an order parameter without
any of the care this section has needed.
\subsection{From U(1) to SU(N): the Standard-Model group, and why SU(5) has 24 generators}
\label{ssec:sun}
Nothing above used the fact that a phase is a single number. Replace the
phase $e^{\ii\alpha}$ by a matrix $U(x)$ from a group $G$, let $\psi$ be a
column of fields, and every step goes through: the derivative fails to
transform correctly, a connection $A_\mu$ valued in the Lie algebra of
$G$ is forced, and its curvature acquires an extra term because matrices
do not commute,
\begin{equation}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
  - \ii g\,[A_\mu, A_\nu] .
\label{eq:H-nonabelian}
\end{equation}
That commutator is the entire difference between electromagnetism and
the strong and weak interactions: it makes the gauge field charged under
its own gauge group, so that gluons interact with gluons in a way that
photons do not with photons \citep{YangMills1954}.

\paragraph{Worked example: counting the gauge fields.} The number of
independent gauge fields is the dimension of the group.
For $\SU(n)$ that is $n^2 - 1$: an $n \times n$ complex matrix has $2n^2$
real parameters, unitarity removes $n^2$, and unit determinant removes
one more. So the Standard Model group
$\SU(3) \times \SU(2) \times \U(1)$ carries
\begin{equation}
8 + 3 + 1 = 12
\end{equation}
gauge fields: eight gluons, the three that become $W^{\pm}$ and $Z$ after
the reshuffling of \S\ref{ssec:higgs}, and the photon.

Now the number that matters for Part~III. For $\SUfive{}$ the count is
$5^2 - 1 = 24$, and the way those $24$ divide is the whole reason the
group is interesting:
\begin{equation}
\mathbf{24} \;\longrightarrow\; \underbrace{8}_{\text{gluons}}
 + \underbrace{3}_{W} + \underbrace{1}_{B}
 + \underbrace{6 + 6}_{X,\,Y} = 24 .
\label{eq:su5-branching}
\end{equation}
Twelve of the twenty-four are exactly the Standard Model's twelve. The
other twelve carry both color and weak charge, and would mediate
transitions between quarks and leptons. The arithmetic is not a
coincidence in the sense that matters: $\SU(3)\times\SU(2)\times\U(1)$
fits inside $\SUfive{}$ with nothing left over except one complete
multiplet, and it is the smallest simple group of which that is true
\citep{GeorgiGlashow1974}.
Chapter~\ref{sec:gs} will meet the number $5$ again from an entirely
different direction, as the denominator of a trace ratio.



\paragraph{Worked example: why five.} Suppose one wants the smallest
simple group containing $\SU(3)\times\SU(2)\times\U(1)$. A necessary
condition is available without any representation theory: the rank ---
the number of simultaneously diagonalizable generators, which is the
number of independent conserved charges one can assign at once. The
Standard Model group has rank $2 + 1 + 1 = 4$. So does $\SUfive{}$. The
next simple candidate down, $\SU(4)$, has rank $3$ and cannot contain it;
the next up, $\SO(10)$, has rank $5$ and can, at the cost of one extra
diagonal charge that must then be explained away.

Rank $4$ is therefore the tight fit, and $\SUfive{}$ is the group that
realizes it. The reader should notice how weak this argument is as a
piece of physics --- it is a counting statement about a group, with no
dynamics anywhere --- and how much it nevertheless determines. That
combination is the theme of Part~II arriving in a very concrete place.

\emph{Physics intuition.} The twelve extra generators of
Eq.~\eqref{eq:su5-branching} are the price of the fit, and they are also
the prediction: fields carrying both color and weak charge can turn a
quark into a lepton, so the proton need not be stable. Whether it is
stable at the level searched for is an experimental question and not one
this paper takes up. What matters here is the structural point ---
enlarging a symmetry is never free, and the surplus generators always
have to mean something.

How a group of this size is cut down to the observed one is a separate
question, and the standard answer in a geometry with a compact direction
is the subject of Chapter~\ref{sec:wilson}: a holonomy around a loop
that does not bound leaves massless only the generators that commute
with it. Where the compactification is not simply connected the
admissible holonomies are discrete, which is the mechanism
\citet{Witten1985} uses.
\subsection{Physics intuition: redundancy reconciles locality with unitarity}
\label{ssec:gauge-intuition}
\emph{Physics intuition.} It is natural to feel cheated by this chapter.
If gauge invariance is only bookkeeping, how can it predict anything?

The answer is that the bookkeeping is not free. Quantum field theory has
to satisfy two requirements that pull against each other: locality ---
nothing propagates faster than light, and fields interact where they
meet --- and unitarity --- probabilities add to one, with no negative
probabilities anywhere. For a massless spin-one particle these two are
very hard to satisfy at once. A vector field has four components; a
massless spin-one particle has two states; and the two extra components
carry, if taken seriously, negative probability.

Redundancy is the resolution. By describing the particle with more
variables than it has states, and declaring the surplus to be
convention, one obtains a description that is manifestly local, and the
surplus is precisely what would otherwise have violated unitarity. The
severity of the constraint --- the requirement to hold independently at
every point --- is what leaves so few consistent theories that the
interactions are essentially determined.

So ``symmetry dictates interaction'' survives, in a sharpened form:
\emph{the demand that a redundancy be consistent, locally and to all
orders, dictates interaction}. That is a stronger statement than the
original, because a redundancy cannot fail to be present, whereas a
symmetry can simply turn out to be false --- as
Chapter~\ref{sec:honest-record} records that two famous ones did.


\section{S3 --- The Bianchi identity}
\label{sec:bianchi}
The third rung is among the shortest chapters in Part~II, and the one
that
changes the reader's mind. Its subject is a conservation law with no
symmetry behind it, no Lagrangian behind it, and no possibility of being
false.

The language of differential forms makes the argument fit on one line,
and that language is built from scratch in Appendix~\ref{app:forms}.
Only two facts are needed here: $\dd$ is an operator that takes a
$k$-form to a $(k+1)$-form and generalizes grad, curl and div at once;
and $\dd\dd = 0$.


\subsection{\texorpdfstring{$F = \dd A$ implies $\dd F = 0$}{F = dA implies dF = 0}}
\label{ssec:dF}
Write the gauge potential of Chapter~\ref{sec:gauge} as a one-form
$A$ and its curvature as the two-form $F = \dd A$. Then
\begin{equation}
\dd F = \dd\dd A = 0 ,
\label{eq:dF}
\end{equation}
and that is the derivation in its entirety. Component by component it is
Eq.~\eqref{eq:bianchi-flat}, all four independent components of it, and
Appendix~\ref{app:forms} checks that the two statements agree.

Everything interesting about Eq.~\eqref{eq:dF} is in what is missing
from it. No action was varied. No symmetry was invoked. No field
equation was used. Nothing was assumed about the dynamics, about the
sources, or about the dimension of spacetime. The only input is that $F$
was obtained by differentiating something --- that $F$ is a curvature ---
and $\dd\dd = 0$ does the rest.

For the non-abelian case of Eq.~\eqref{eq:H-nonabelian} the same
statement holds with the covariant exterior derivative in place of
$\dd$, and the proof is the same two lines with a commutator carried
along.



\paragraph{Worked example: counting the identity.} How many equations is
Eq.~\eqref{eq:dF} in four dimensions? A three-form in four dimensions has
$\binom{4}{3} = 4$ independent components, indexed by the four ways of
choosing three distinct directions:
\begin{equation}
(1,2,3), \quad (0,1,2), \quad (0,1,3), \quad (0,2,3) .
\end{equation}
The purely spatial triple gives $\nabla\cdot\mathbf{B} = 0$. The three
containing the time direction give the three components of
$\nabla\times\mathbf{E} + \partial_t\mathbf{B} = 0$. Four components,
four familiar statements, and no dynamics used anywhere. The other four
Maxwell equations come from $\dd\star F = \star J$, which does require an
action and does involve the sources.

\emph{Physics intuition.} A one-line derivation with no inputs should
provoke suspicion, so it is worth asking what work is actually being
done. The answer is that all of it was done earlier, when one decided to
write the field as the derivative of a potential. That decision was not
innocent: it encodes the assumption that phases can be compared
consistently, and Chapter~\ref{sec:hs} shows what happens when they
cannot. $\dd F = 0$ is the receipt for an assumption made two chapters
ago.
\subsection{Reading it as physics: no monopoles, flux through a closed surface}
\label{ssec:no-monopoles}
Read Eq.~\eqref{eq:dF} as physics and it says that magnetic flux is
conserved. Integrate $F$ over a closed two-dimensional surface: by
Stokes' theorem the answer is the integral of $\dd F$ over the enclosed
volume, which is zero. No net magnetic flux emerges from any closed
surface. There is no magnetic charge.

It is worth being exact about what has and has not been shown, because
this is a place where a fast reading goes wrong. What has been shown is
that if $F = \dd A$ everywhere, then there are no monopoles. The
converse route is the interesting one, and Chapter~\ref{sec:hs} takes
it: if a monopole \emph{is} present, then $F$ cannot be written as
$\dd A$ on the whole space, and one is forced instead to use one
potential here and another there, agreeing up to a gauge transformation
where they overlap. That structure is not a failure of the theory. It is
a topological classification, and the integer that labels it is Dirac's.

So the honest statement of this rung is not ``monopoles are forbidden.''
It is: \emph{magnetic charge is exactly the obstruction to writing the
field as a curvature of a globally defined potential}. Nothing forbids
it; if it exists, the bookkeeping changes character, and the changed
bookkeeping is quantized.



\paragraph{Worked example: the flux that cannot escape.} Take any closed
surface $\Sigma$ --- a sphere, say --- in a region where $F = \dd A$.
Then
\begin{equation}
\oint_{\Sigma} F = \int_{V} \dd F = 0 ,
\end{equation}
by Stokes' theorem applied to the enclosed volume $V$. Physically: the
magnetic flux entering a closed surface equals the flux leaving it,
always, with no source term available. Compare the electric case, where
the corresponding integral gives the enclosed charge and the difference
is precisely that Gauss's law comes from an action while this one comes
from a definition.

The instructive step is to see the argument fail. If a monopole sits
inside $\Sigma$, then no single $A$ exists on all of $\Sigma$; the best
one can do is use one potential on the northern cap and another on the
southern, related by a gauge transformation on the overlap. Stokes'
theorem then applies to each cap separately, and the two answers differ
by the winding of the transition function around the equator --- an
integer. That integer is Dirac's, and \S\ref{ssec:dirac} derives it.
\subsection{The gravitational second identity, and the form Einstein's equations had to take}
\label{ssec:einstein}
Gravitation has an identity of the same kind, and its consequence is
larger.

The Riemann curvature tensor satisfies
\begin{equation}
\nabla_{[\lambda} R_{\mu\nu]\rho\sigma} = 0 ,
\label{eq:bianchi-2}
\end{equation}
for the same structural reason as Eq.~\eqref{eq:dF}: the curvature is
built by differentiating a connection, and the antisymmetrized second
derivative collapses. Contracting Eq.~\eqref{eq:bianchi-2} twice gives
\begin{equation}
\nabla_\mu G^{\mu\nu} = 0,
\qquad
G^{\mu\nu} \equiv R^{\mu\nu} - \tfrac12 g^{\mu\nu} R ,
\label{eq:div-G}
\end{equation}
identically, for every metric whatsoever. (Appendix~\ref{app:connections}
carries out the contraction; the identity is also checked directly on a
cosmological metric in the paper's verification harness, where it holds
for an arbitrary scale factor.)

Now consider what Einstein needed. He wanted field equations of the form
$(\text{geometry})^{\mu\nu} = \kappa\, T^{\mu\nu}$, with $T^{\mu\nu}$ the
energy--momentum tensor. Local energy--momentum conservation requires
$\nabla_\mu T^{\mu\nu} = 0$. So the geometrical side must have
identically vanishing divergence --- otherwise the field equations would
impose a condition on matter over and above conservation, and would be
inconsistent for generic sources.

Equation~\eqref{eq:div-G} says that $G^{\mu\nu}$ does. And among tensors
built from the metric and its first two derivatives, linear in the
second derivatives, the divergence-free combinations are
$G^{\mu\nu}$ and $g^{\mu\nu}$ --- the latter being the cosmological term.
The form of Einstein's equations is therefore not a choice made on
grounds of elegance. It is very nearly forced by an identity that has
nothing to do with gravitation and everything to do with what a
curvature is.

This is the most consequential appearance of the third rung in physics,
and it fits the pattern exactly: an identity constrains what the
\emph{laws} may look like, without itself being a law.



\emph{Physics intuition.} It is worth stating what the identity does
\emph{not} do. It does not derive general relativity; it does not fix
Newton's constant, and it permits the cosmological term. What it does is
eliminate almost every alternative. Any theory in which the geometrical
side of the field equations has non-vanishing divergence must impose
extra conditions on matter beyond energy--momentum conservation, and
those conditions are generically inconsistent with sources one is
entitled to consider --- an arbitrary distribution of dust, for instance.

So Eq.~\eqref{eq:div-G} works like a filter. It rules out candidates
without selecting a winner. That is characteristic of the whole rung,
and characteristic of the higher rungs too: identities are the strongest
tool physics has for narrowing a search, and the weakest for finishing
one.

The argument just given has an electromagnetic twin, and it is the more
famous of the two, though it is usually met in a different part of the
subject. There, an off-shell identity on the geometry side transmits
itself through the field equation to the matter side and forces the
current to be conserved; the quantum form of that transmission is the
subject of Chapter~\ref{sec:ward}, and the two are set out together in
Table~\ref{tab:two-faces}. A reader who has followed this section has
already followed the harder case.
\subsection{Historical box: Bianchi 1902 and the contested priority}
\label{ssec:bianchi-history}
\begin{quote}\small
\textbf{Historical box: whose identity?}
The name is Luigi Bianchi's, from a paper of 1902 in which he proved the
identities by direct calculation \citep{Bianchi1902}. He was not the
first to have them. A contracted form had been published by Aurel Voss in
1880 \citep{Voss1880}, in a paper on the transformation of quadratic
differential forms that had nothing to do with physics. Around 1889 Ricci
obtained the identities independently; he communicated them verbally to
Padova, who announced them without proof, and Ricci then appears to have
forgotten them --- they are absent from his later work with Levi-Civita.

So the attribution records the first published proof rather than the
first discovery, and the two earlier episodes are nine years apart and
should not be run together. What is striking is less the priority
question than the timing: the identity that would later force the shape
of Einstein's field equations was written down in the 1880s, in pure
differential geometry, and then mislaid twice by people who had no
reason to think it mattered.
\end{quote}


\subsection{The deep point: an identity is not a law}
\label{ssec:not-a-law}
Here is the point of the chapter, stated as plainly as it can be.

\begin{quote}
\emph{An identity is not a law. Nothing can violate it, and nothing is
explained by it.}
\end{quote}

A law of nature is a claim that could have been otherwise. Newton's law
of gravitation might have had a different exponent; the electron's
charge might have had a different value; the weak interaction might have
respected parity, and everyone believed it did until 1956. These are
contingent facts about our universe, discovered by measurement, and each
of them is falsifiable in the ordinary way.

$\dd F = 0$ is not like this. It is a consequence of writing $F = \dd A$,
which is to say a consequence of what one means by a field strength. No
experiment can test it, because no experimental result could conflict
with it. If tomorrow a magnetic charge were found, this identity would
not be refuted; the description would change --- one would stop writing
$F$ as the curvature of a single global potential --- and the identity
would continue to hold, in each patch, exactly as before.

There is a tempting conclusion here that should be resisted: that
identities are therefore empty, mere tautologies dressed as physics.
They are not, and Eq.~\eqref{eq:div-G} shows why. That identity is
content-free as a statement about the world and enormously constraining
as a statement about \emph{admissible theories}. It rules out almost
every field equation one might write down for gravity. Identities do not
tell you what happens. They tell you what you are allowed to say ---
which, for a theorist choosing among candidate laws, is the more
valuable information. Table~\ref{tab:law-identity} sets the two side by
side, because the distinction is the one the third rung turns on and it is
easy to lose.




The distinction can be put in a table, and it is worth having in front of
the reader for the two chapters that follow.

\begin{table}[H]
\centering
\caption[Law against identity.]{The distinction the third rung turns on.
The last row is the one Chapter~\ref{sec:gs} exploits: the
\GS{} mechanism does not violate an identity, it redefines the object so
that the identity's premise no longer holds of it. Two columns, one question per row, asked of a law and of an identity in turn: what each is derived from, what it would take to break it, and what survives if the theory changes underneath it. The distinction is elementary and it is the one most often elided, which is why it gets a table rather than a sentence.}
\label{tab:law-identity}\small
\begin{tabular}{@{}P{0.21\textwidth}P{0.33\textwidth}P{0.33\textwidth}@{}}
\toprule
 & \textbf{a law} & \textbf{an identity} \\
\midrule
example & Gauss's law, $\dd\star F = \star J$ &
$\dd F = 0$; $\nabla_\mu G^{\mu\nu} = 0$ \\
where it comes from & varying an action &
the definition of the object \\
could it be otherwise? & yes --- a different action gives a different law
& no --- not without changing what the symbols mean \\
what tests it & experiment & nothing; no measurement can conflict
with it \\
what it constrains & what the world does & what a theorist is allowed to
write down \\
how it fails & it is found false & it does not fail; its
\emph{premise} stops applying \\
\bottomrule
\end{tabular}
\end{table}

That is the third rung: conservation with no dynamics in it, purchased
entirely by a definition, and more binding than either rung below it.
The next rung does not break this structure. It asks what the structure
fails to record --- because an identity that constrains $F$ says nothing
at all about the part of $A$ that $F$ cannot see.

\section{S4 --- The Wilson line: what the identity leaves out}
\label{sec:wilson}

The third rung said that a curvature obeys $\dd F = 0$ because it is a
curvature. This chapter is about what that sentence does not say.

An identity constrains $F$. It says nothing about $A$ beyond what $F$
records --- and the map from $A$ to $F$ loses information. Wherever the
space carries a loop that does not bound a surface, there is a
gauge-invariant, physically measurable quantity that survives with the
field strength equal to zero everywhere. It is a phase. It is not a
label but a field, with a potential that can be computed; and its value
decides which gauge symmetry the world displays.

\subsection{\texorpdfstring{$F$}{F} does not determine \texorpdfstring{$A$}{A}: the flat connection on a circle}
\label{ssec:flat}
Take spacetime to be four ordinary dimensions and one more that closes
on itself, a circle of circumference $L$. Put on it the least
interesting gauge field that is not zero: every component vanishes
except the one pointing around the circle, and that one is constant,
\begin{equation}
A_y = \frac{\theta}{g L} .
\label{eq:flatA}
\end{equation}

Three facts, each checked in Appendix~\ref{app:wilson}.

\begin{itemize}[leftmargin=1.4em,itemsep=0.3em]
\item \emph{The field strength vanishes.} Every component of $F_{MN}$ is
  zero, at every point, identically. The connection is \emph{flat}.
\item \emph{The holonomy does not.} The phase acquired on transport once
  around the circle is
  \begin{equation}
  W = \exp\!\left(\ii g \oint A_y \,\dd y\right) = e^{\ii\theta} ,
  \label{eq:wilsonline}
  \end{equation}
  which is not $1$ unless $\theta$ is a multiple of $2\pi$. The coupling
  and the circumference cancel: what is left is an angle.
\item \emph{It cannot be gauged away.} The function that would remove
  Eq.~\eqref{eq:flatA} is $\Omega = e^{-\ii\theta y/L}$, and it returns
  to itself after one lap only when $e^{-\ii\theta} = 1$. For any other
  $\theta$ it is not single-valued, so it is not a gauge transformation
  at all.
\end{itemize}

So the curvature vanishes everywhere and the configuration is not pure
gauge. What is left over is Eq.~\eqref{eq:wilsonline}, and it is
invariant under every legitimate gauge transformation, because a
single-valued gauge function integrates to zero around a closed loop.

This is \S\ref{ssec:third-datum} arriving early. That section will say
that the complete gauge-invariant description of a field is the set of
its holonomies, and that curvature determines holonomy only where loops
bound. Here is the case where they do not, and here is what survives.

\emph{Physics intuition.} A gauge transformation is a change of
bookkeeping, and bookkeeping has to close. The function that would
remove Eq.~\eqref{eq:flatA} fails to close by $e^{-\ii\theta}$ after one
lap, and the failure is a property of the space rather than a defect of
the function. Everything in this chapter follows from that one sentence.

\subsection{The phase you can measure, twice}
\label{ssec:measurable}
A phase no experiment can see would be worth nothing. This one is seen
twice, in two different laboratories.

\paragraph{First: the interferometer.} The Aharonov--Bohm effect
\citep{AharonovBohm1959} is Eq.~\eqref{eq:wilsonline} in a region where
$\mathbf{B} = 0$: a beam split around a solenoid recombines with a
relative phase fixed by the flux \emph{enclosed}, not the flux crossed.
The curvature is zero along every path the electron takes, and the
effect is there. \S\ref{ssec:third-datum} returns to it.

\paragraph{Second: the spectrum.} A field of charge $q$ on the circle has
its momentum around the circle quantized, and the phase shifts the
quantization:
\begin{equation}
m_k \;=\; \frac{\left| 2\pi k + q\theta \right|}{L} ,
\qquad k \in \mathbb{Z} .
\label{eq:kktower}
\end{equation}
The whole tower of masses moves when $\theta$ moves. That is not a
gauge artifact; it is a spectrum, and spectra are measured. Two features
of Eq.~\eqref{eq:kktower} matter later. Sending
$\theta \to \theta + 2\pi$ maps every level onto a level with the mode
number relabeled $k \to k+q$: the set is unchanged, so the physics is
$2\pi$ periodic and $\theta$ genuinely lives on a circle. And the
spectrum is even under $\theta \to -\theta$, so the physical range is
half of it, $0 \le \theta \le \pi$, with the two endpoints $W = +1$ and
$W = -1$. Two kinds of thing therefore live on this one circle, and they
are not the same kind: the mode numbers are integers and cannot move at
all, while $\theta$ is a continuous coordinate inside the frame they
set, and it moves. The rest of this chapter is about $\theta$;
\S\ref{ssec:centre} is about the integer.

\subsection{Hosotani: the phase is a field, and it has a potential}
\label{ssec:hosotani}
Nothing so far makes $\theta$ dynamical. Classically it is a free label:
by Eq.~\eqref{eq:flatA} the field strength vanishes for every $\theta$,
so every value costs the same action and the family is exactly
degenerate.

Quantum mechanically it is not degenerate, because the tower of
Eq.~\eqref{eq:kktower} has a zero-point energy and that energy depends
on $\theta$. Summing it gives, per degree of freedom in five dimensions,
\begin{equation}
V(\theta) \;=\; \mp\,\frac{3}{4\pi^2 L^4}
 \sum_{w=1}^{\infty} \frac{\cos\!\left(w q\theta\right)}{w^5} ,
\label{eq:hosotani-potential}
\end{equation}
the upper sign for a boson and the lower for a fermion periodic around
the circle. This is the mechanism of \citet{Manton1979} and
\citet{Hosotani1983}, treated systematically in \citet{Hosotani1989}.

Three properties of Eq.~\eqref{eq:hosotani-potential} carry the rung.

\emph{It is finite.} A five-dimensional gauge theory is not
renormalizable, and yet the $\theta$-dependent part of the vacuum energy
needs no regulator, no counterterm and no renormalization condition. The
reason is that the divergence is $\theta$-independent and cancels; the
$\theta$-dependence enters only through terms suppressed by
$e^{-2\pi w x}$, the cost of a mode traveling $w$ times around the
circle. Appendix~\ref{app:wilson} does the resummation by hand.

\emph{It has a shape, and the shape is not a convention.} The overall
coefficient and the degeneracy factors depend on the field content and
the boundary conditions and must be read off a model. The weight
$w^{-5}$, the $2\pi$ periodicity, the evenness, and the absolute
convergence --- bounded by $\zeta(5) = 1.0369$ --- do not.

\emph{Its minimum is decided by what is in the loop.} With the bosonic
sign the minimum is at $\theta = 0$ and the holonomy is trivial. With
the sign of a periodic adjoint fermion it is at $\theta = \pi$ and the
holonomy is not. Which one wins is a competition of signs among the
degrees of freedom circulating in the loop.

\subsection{Which symmetry is realized is an output, not an input}
\label{ssec:output}
The four-dimensional theory left over after compactification keeps
massless the gauge fields that commute with the holonomy $W$, and gives a
mass to the rest --- by Eq.~\eqref{eq:kktower}, anything carrying
non-zero effective charge acquires a mass even in its lowest mode: at
least $|q\theta|/L$ while $|q\theta| \le \pi$, and in general the
distance from $q\theta$ to the nearest multiple of $2\pi$, divided by
$L$. When $W$ is trivial the commutant is the whole group. When
it is not, the commutant is a proper subgroup, and the pattern of
massless vectors is smaller.

Neither outcome was written into the Lagrangian. The Lagrangian contains
a group, a compactification, and a list of matter fields. The symmetry
that is actually realized is the output of
Eq.~\eqref{eq:hosotani-potential}.

\begin{quote}\itshape
Which gauge symmetry is realized is a dynamical consequence of the
matter content, not an input to the Lagrangian.
\end{quote}

There is no Higgs field here, and no scalar potential written by hand.
The order parameter is a holonomy; its value is decided by a one-loop
computation; and the symmetry pattern follows.

\subsection{Elitzur revisited: a gauge-invariant order parameter, and the center}
\label{ssec:centre}
Section~\ref{ssec:elitzur} established that a local gauge symmetry cannot
break, and that talk of ``breaking a gauge symmetry'' is a picture rather
than a fact. A Wilson line is nevertheless described everywhere as an
order parameter for symmetry breaking. Both statements are correct, and
the reconciliation is the most useful thing this rung contributes to the
argument of Chapter~\ref{sec:gauge}.

It does not break the local symmetry. The order parameter,
Eq.~\eqref{eq:wilsonline}, is gauge invariant from the start --- which is
exactly what \S\ref{ssec:elitzur} demanded of any honest order parameter,
and exactly what the Higgs expectation value of \S\ref{ssec:higgs} is not
without a gauge choice. Section~\ref{ssec:higgs} gave the reader one
mechanism compatible with Elitzur's theorem. This is a second, and a
cleaner one.

And there is a genuine symmetry here that genuinely breaks. The
\emph{center} of $\SU(N)$ --- the $N$-th roots of unity times the
identity --- acts on a Wilson line wrapping a non-contractible cycle by
multiplication,
\begin{equation}
\tr P \;\longrightarrow\; z^{-1} \tr P , \qquad z^N = 1 .
\label{eq:centre}
\end{equation}
The transformations that generate it are periodic only up to a center
element, so they are \emph{not} gauge transformations; the symmetry is
global, with one parameter for the whole system, and Elitzur's theorem
does not touch it. Its only fixed point is $\tr P = 0$, so a vanishing
loop is the symmetric phase and a non-vanishing one breaks the center.
For the thermal circle that breaking is deconfinement, in the sense of
\citet{Wilson1974}.

So the Wilson line is the object that converts a would-be statement
about gauge symmetry into a legitimate statement about a global one ---
and the global symmetry it delivers is discrete, labeled by an integer
$N$. That is the first appearance on this ladder of an integer that is
not a failure but a \emph{frame}.

\subsection{Why the potential is finite --- and what that says about naturalness}
\label{ssec:finite}
The finiteness of Eq.~\eqref{eq:hosotani-potential} has a structural
reason worth stating in the language of the next chapter.

A counterterm in a local field theory is the integral of a polynomial in
the fields and their derivatives at a point. The phase $\theta$ is not
the value of any such polynomial: every local gauge-invariant operator is
built from the gauge field is made of $F_{MN}$ and its covariant
derivatives, and all of those vanish; nor does anything local built from
the matter fields see $\theta$, for the same reason. Gauge invariance
therefore forbids a local counterterm from
generating or absorbing a potential for $\theta$. The divergences and
the $\theta$-dependence live in disjoint places because locality and
topology put them there. Chapter~\ref{sec:ward} gives that argument its
name.

That observation has a well-known consequence, and this paper states it
with its grade attached. If the four-dimensional Higgs field is
identified with the extra-dimensional component of a gauge field, then
its mass is protected by the argument above --- protection by topology
and non-locality, rather than by a symmetry that could break. Be exact
about what that buys: the calculability, not the value. The phase is a
continuous modulus, it moves, and the quantities built from it run; what
no local counterterm can do is generate or absorb its potential, so
whatever value it takes is finite and computable rather than a
subtraction. Set beside
\S\ref{ssec:failures}, where the symmetry-based answer to the same
question is recorded as excluded in the region it preferred, the contrast
is instructive.

\begin{quote}\small
\textbf{Grade.} The finiteness of the $\theta$-dependent potential is a
\emph{theorem}. That gauge--Higgs unification resolves the hierarchy
problem is \emph{open}: no realization with the observed spectrum is
established, and this paper takes no position on it. The program and
its difficulties are surveyed by \citet{HatanakaInamiLim1998}.
\end{quote}

\subsection{Historical box: a mechanism ahead of its evidence}
\label{ssec:hosotani-history}
\begin{quote}\small
\textbf{Historical box: Manton 1979, Hosotani 1983.}
\citet{Manton1979} asked whether the Higgs field of the electroweak
theory could be a component of a gauge field in a higher-dimensional
space, and found that the structure worked. \citet{Hosotani1983} supplied
what was missing: the extra-dimensional component is a Wilson-line phase,
it is not a free parameter, and the one-loop potential that fixes it is
finite and calculable. \citet{Hosotani1989} gave the systematic
treatment. The ingredients were older still --- the non-integrable phase
factor of \citet{WuYang1975}, the loop variable of
\citet{Wilson1974} --- and the mathematical fact that a flat connection on
a multiply connected space carries physical information was known to
geometers before it was used by physicists. What remains open, four
decades on, is not the mechanism but its application: no
extra-dimensional model built on it has been confirmed, and the searches
that excluded the natural region for superpartners are pushing its scale
up too. It is a mechanism in good standing and in want of evidence.
\end{quote}

\emph{Physics intuition.} The phase of a loop is not the field at a
point. Everything local --- every curvature, every counterterm, every
divergence --- is blind to $\theta$, and the one thing that sees it has
to go all the way around. That is why the rung sits where it does: it is
the first place on this ladder where the discreteness comes from the
shape of the space rather than from the dynamics, and the first where an
integer appears before anything has failed.

\subsection{Grade box}
\label{ssec:wilson-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Theorem, standard.} That a constant potential along a compact
direction has vanishing field strength and non-trivial holonomy
(Eqs.~\eqref{eq:flatA}--\eqref{eq:wilsonline}); that the removing gauge
function is single-valued exactly on the integers; that the
Kaluza--Klein tower depends on the phase, Eq.~\eqref{eq:kktower}; that
the $\theta$-dependent one-loop potential is finite, and has the shape
Eq.~\eqref{eq:hosotani-potential}; that the center symmetry is global and
may break, with a gauge-invariant order parameter.

\emph{Model-dependent.} The overall coefficient and the degeneracy
factors in Eq.~\eqref{eq:hosotani-potential}, which follow from a field
content and a choice of boundary conditions and not from anything in this
chapter.

\emph{Open.} That this mechanism resolves the hierarchy problem.

\emph{Not claimed.} That the Wilson line breaks a local gauge symmetry;
it does not, and \S\ref{ssec:centre} says what it does instead.
\end{quote}


\section{S5 --- The Ward identity (1950)}
\label{sec:ward}

The previous chapter asked what the Bianchi identity leaves out. This
one asks what it becomes.

Every identity so far on this ladder has been unbreakable. Noether's
second theorem gives $\partial_\mu\partial_\nu F^{\mu\nu} \equiv 0$,
which holds by antisymmetry and could not fail if the world tried. The
Bianchi identity holds because a curvature is a curvature. The Wilson
phase of Chapter~\ref{sec:wilson} exists whenever the space has a loop
that does not bound, and no dynamics can remove it.

This rung is different, and the difference is the hinge of the whole
ladder. Here is an identity with no dynamical content --- and it
\emph{can} fail. When it does, it fails by an integer.

\subsection{What survives quantization, and what does not}
\label{ssec:survives}
A classical conservation law is a statement about field configurations.
A quantum field theory does not deal in configurations; it deals in
correlation functions. So the question left open by
Chapter~\ref{sec:noether} is what becomes of $\partial_\mu j^\mu = 0$
when the theory is quantized.

The answer is obtained by a change of variables. Rotating the phase of
the charged fields by an arbitrary function of position cannot change the
value of a path integral, because renaming an integration variable never
does. Expanding that statement to first order and collecting what
survives gives, for any set of charged operators,
\begin{equation}
\partial_\mu \big\langle T\, J^\mu(x)\, \phi(x_1)\cdots\phi(x_n)
 \big\rangle
 \;=\; -\,\ii \sum_{k} q_k\, \delta^{4}(x - x_k)\,
 \big\langle T\, \phi(x_1)\cdots\phi(x_n)\big\rangle .
\label{eq:ward-master}
\end{equation}
This is the Ward identity, in the general form this chapter's argument
needs; Appendix~\ref{app:ward} derives it in full, keeping every term.

\subsection{The identity in one line, and what it costs}
\label{ssec:ward-cost}
Read Eq.~\eqref{eq:ward-master} twice.

\emph{Away from the insertion points} it says
$\partial_\mu \langle T J^\mu \cdots\rangle = 0$. The classical
conservation law survives quantization, exactly, to all orders, with no
correction.

\emph{At the insertion points} it says something the classical law could
not: the divergence is a sum of delta functions weighted by the
\emph{charges} of whatever operators happen to be sitting there. These
are the contact terms, and their content is the point of the rung.

\begin{quote}\itshape
The right-hand side of Eq.~\eqref{eq:ward-master} is pure kinematics. It
contains delta functions and charges. It contains no coupling constant,
no mass, no loop expansion, and no reference to the Lagrangian beyond
the fact that it was invariant.
\end{quote}

Change the dynamics --- add interactions, resum, go to strong coupling
--- and the right-hand side does not move. That is the same ``no
dynamical content'' property this paper has attributed to the identities
of Chapters~\ref{sec:noether}, \ref{sec:gauge} and \ref{sec:bianchi},
and it is why the Ward identity belongs on the ladder rather than beside
it.

\subsection{Worked example: Ward--Takahashi, and why every charge runs the same way}
\label{ssec:wt}
Specialize Eq.~\eqref{eq:ward-master} to one electron line and one
photon. The result, due to \citet{Ward1950} in its forward form and to
\citet{Takahashi1957} in general, is
\begin{equation}
q_\mu \Gamma^\mu(p+q,\,p) \;=\; S^{-1}(p+q) - S^{-1}(p) ,
\label{eq:wt}
\end{equation}
relating the vertex to the difference of two inverse propagators. It is
exact: nothing was assumed about the order of perturbation theory.

At tree level it is immediate --- $q_\mu\gamma^\mu = \slashed{q}$ on the
left, and $(\slashed{p} + \slashed{q} - m) - (\slashed{p} - m) =
\slashed{q}$ on the right, the masses canceling because they sit
identically at both ends of a difference. That cancellation is the whole
difference between this current and the axial one, whose corresponding
relation involves an anticommutator in which the same two mass terms add
instead.

\paragraph{The consequence.} Equation~\eqref{eq:wt} forces
$Z_1 = Z_2$: the vertex and the electron field are renormalized by the
same constant. So in
\begin{equation}
e \;=\; \frac{Z_2}{Z_1}\,\sqrt{Z_3}\; e_0
 \;=\; \sqrt{Z_3}\; e_0 ,
\label{eq:charge-universality}
\end{equation}
everything belonging to the \emph{matter} field has canceled, and what
is left, $Z_3$, belongs to the \emph{photon} --- one object shared by
every charged species there is. Therefore charge \emph{ratios} do not
run. An electron and a proton, which share no other property, carry
charges whose ratio is protected from all of that dynamics.

\paragraph{What this is not.} It is universality, and it is not
quantization. Equation~\eqref{eq:charge-universality} says that whatever
the ratio is, the dynamics will not change it. It does not say the ratio
is $-1$. A world in which the proton's charge exceeded the electron's by
one part in $10^{22}$ satisfies every equation of this chapter; the
measured bound is $10^{-21}$ \citep{MarinelliMorpurgo1984}, and the
explanation of that number is not here.

\begin{quote}\itshape
Ward protects the ratio from dynamics. Dirac protects it from
continuity.
\end{quote}

This is the last rung on which a \emph{dynamical} argument ---
renormalization --- still buys protection. Above it, everything is
topology. Chapter~\ref{sec:hs} supplies the other half.

\subsection{Transversality, and why the photon stays massless}
\label{ssec:transversality}
Apply Eq.~\eqref{eq:ward-master} with no charged operators inserted at
all. The right-hand side is empty, and what remains constrains the photon
self-energy to be transverse:
\begin{equation}
q_\mu \Pi^{\mu\nu}(q) = 0
\qquad\Longrightarrow\qquad
\Pi^{\mu\nu}(q) = \left(q^2 \eta^{\mu\nu} - q^\mu q^\nu\right)\Pi(q^2) .
\label{eq:transversality}
\end{equation}
Two form factors have become one, and the surviving structure already
carries a factor of $q^2$. A photon mass would require a term
$m_\gamma^2\,\eta^{\mu\nu}$ with no such factor, and
Eq.~\eqref{eq:transversality} forbids exactly that structure. The photon
stays massless, order by order and to all orders.

Notice what is doing the work. Gauge invariance is a redundancy of
description, as \S\ref{ssec:elitzur} insisted; there is no gauge charge
to measure and no vacuum that could fail to be invariant. And yet the
masslessness is protected. What protects it is an identity among
amplitudes, not an unbroken symmetry --- rigidity with no symmetry left
to break, which is the pattern \S\ref{ssec:retreat} follows up the whole
ladder.

\subsection{The identity's two faces: Bianchi and Ward}
\label{ssec:two-faces}
This section is the reason the rung sits where it does.

Apply Noether's second theorem to a theory of gauge field plus matter
with a local $\mathrm{U}(1)$. The theorem gives one off-shell identity,
$\partial_\mu\left(\delta S/\delta A_\mu\right) \equiv 0$. Split it into
its two pieces:
\begin{equation}
\underbrace{\partial_\mu \partial_\nu F^{\nu\mu}}_{\text{vanishes by antisymmetry alone}}
 \;+\;
\underbrace{\partial_\mu J^\mu}_{\text{vanishes on shell for the matter}}
 \;\equiv\; 0 .
\label{eq:two-faces}
\end{equation}
The first term is Eq.~\eqref{eq:noether-ii}, and it needs no field
equation and no dynamics whatever. Therefore the second must vanish too
--- on shell for the matter, off shell for the gauge field.

That asymmetry is the whole relation between this rung and the last.
Charge conservation is not an independent law. It is the \emph{matter
half} of a Bianchi-type identity, and the Ward identity is what promotes
it to a statement about correlation functions.
Table~\ref{tab:two-faces} sets the two faces against each other.

\begin{table}[H]
\centering
\caption[One identity, two faces.]{\textbf{One identity, two faces.} The
gravitational row is \S\ref{ssec:einstein}, which the reader has already
seen; the gauge row is the same argument, and it is the more famous of
the two. Two columns, the geometry half and the matter half, and two rows for the two theories in which the identity appears. Reading across shows what is forced in each case; reading down shows that the gravitational and gauge statements are one argument, differing in the objects and not in the reasoning.}
\label{tab:two-faces}\small
\begin{tabular}{@{}P{0.14\textwidth}P{0.37\textwidth}P{0.39\textwidth}@{}}
\toprule
 & \textbf{the geometry half (identically)} &
   \textbf{the matter half (forced)} \\
\midrule
gauge &
$\partial_\mu \partial_\nu F^{\nu\mu} \equiv 0$ &
$\partial_\mu J^\mu = 0$, and at $\hbar \ne 0$ the Ward identity
Eq.~\eqref{eq:ward-master} \\
gravity &
$\nabla_\mu G^{\mu\nu} \equiv 0$, Eq.~\eqref{eq:div-G} &
$\nabla_\mu T^{\mu\nu} = 0$ --- the form Einstein's equations had to take
\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Non-abelian: Slavnov--Taylor in one paragraph}
\label{ssec:slavnov}
For a non-abelian theory the identity needs ghosts, and the compact
statement is that the effective action is invariant under a single
fermionic operator $s$ with $s^2 = 0$: the Slavnov--Taylor identity
$s\Gamma = 0$ \citep{Slavnov1972,Taylor1971}. The nilpotency is not an
extra assumption --- it is the Jacobi identity of the Lie algebra in
different clothes, as Appendix~\ref{app:ward} shows in three lines.

One consequence is worth carrying forward now, because
Chapter~\ref{sec:gs} will need it. If the identity fails, and the failure
is written $s\Gamma = \mathcal{A}$, then applying $s$ again gives
\begin{equation}
s\,\mathcal{A} \;=\; 0 .
\label{eq:wz-cocycle}
\end{equation}
The anomaly is not free to be any functional whatever: it obeys an
identity of its own, and that identity is $s^2 = 0$ --- the same
algebraic fact as $\dd\dd = 0$, which is where this paper's third rung
came from. A Bianchi identity for the anomaly.

\subsection{Historical box: a one-paragraph paper}
\label{ssec:ward-history}
\begin{quote}\small
\textbf{Historical box: half a page in 1950.}
\citet{Ward1950} is one paragraph long. It states a relation between the
vertex function and the derivative of the electron self-energy, notes
that it holds to all orders, and stops. What it bought was the equality
$Z_1 = Z_2$, and with it the fact that the renormalized charge is the
same for every species --- the reason a proton and a positron have
charges that agree to twenty-one decimal places rather than merely to
the accuracy of a calculation. \citet{Takahashi1957} generalized it seven
years later to non-zero momentum transfer, which is the form in use
today, and it is Takahashi's version that Eq.~\eqref{eq:wt} states. The
identity is older than most of what this paper's later chapters
describe, and it is the one that quantum field theory uses most often.
\end{quote}

\subsection{Physics intuition: conservation as an audit}
\label{ssec:ward-intuition}
\emph{Physics intuition.} There is a temptation to read this ladder as a
sequence of statements that become progressively harder to break, ending
in one that cannot break at all. That reading is wrong, and this rung is
where it fails.

The Ward identity has no dynamical content --- and it can fail, because
the derivation assumed something the Lagrangian does not guarantee: that
the integration measure is invariant. When that assumption is false, the
identity acquires an extra term, and the extra term is the integral,
over a closed space, of a quantity built from the curvature whose total
is always a whole number --- a \emph{characteristic class}, which
Appendix~\ref{app:anomalies} builds. It is an integer.

That is a better property than unbreakability, not a worse one. An
identity that cannot fail says nothing when something goes wrong. An
identity that can fail only by a whole number tells you exactly how much
--- which is a quantity you can count, cancel against, and audit. The
next two chapters do exactly that: Chapter~\ref{sec:gs} arranges two such
failures to cancel, and Chapter~\ref{sec:hs} shows that what is left
when they do is an integer class.

\subsection{Grade box}
\label{ssec:ward-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Theorem, standard.} The master identity
Eq.~\eqref{eq:ward-master}, with its contact terms; the
Ward--Takahashi relation Eq.~\eqref{eq:wt} and $Z_1 = Z_2$; the
species-independence of the renormalized charge,
Eq.~\eqref{eq:charge-universality}; transversality and the masslessness
of the photon, Eq.~\eqref{eq:transversality}; the Slavnov--Taylor
identity and the consistency condition Eq.~\eqref{eq:wz-cocycle}.

\emph{Theorem, and the hinge of Part~II.} That the identity can fail,
that it fails only when the path-integral measure is not invariant, and
that the size of the failure is an integer.

\emph{Not claimed.} That the Ward identity explains why charges are
integer multiples of one another. It does not; it protects a ratio, it
does not choose one.
\end{quote}


\section{S6 --- The \GS{}-modified identity (1984)}
\label{sec:gs}
Chapter~\ref{sec:bianchi} argued that $\dd F = 0$ cannot be violated.
This chapter is about a construction in which the corresponding identity
for a different field is violated \emph{deliberately}, and in which the
violation is the mechanism rather than the problem. It is the rung at
which conservation stops being sacred entry by entry and becomes sacred
only in total.

The anomaly machinery is developed three separate ways in
Appendix~\ref{app:anomalies}; the argument used here is the one that
requires no path integrals at all.


\subsection{Anomalies as the failure of a Ward identity}
\label{ssec:anomalies}
Chapter~\ref{sec:ward} left one thing open. The Ward identity
Eq.~\eqref{eq:ward-master} was derived from a single assumption --- that
renaming the integration variable in a path integral does not change the
measure --- and that assumption is separate from the invariance of the
action. When it fails, the identity acquires an extra term.

That is what an anomaly is. Not a symmetry that mysteriously evaporates,
and not a defect of a calculation: a Ward identity with a term on the
right-hand side that the classical theory did not put there. The name is
unfortunate, since nothing irregular is happening, and it has outlived
the confusion that produced it.

The failure can be computed, and it can be computed by counting states,
with no diagrams and no functional integrals.

\paragraph{Worked example: the anomaly by counting.} Take a massless
fermion of one handedness and charge $e$, and switch on a uniform
magnetic field $\mathbf{B}$ along $z$. The motion transverse to
$\mathbf{B}$ is quantized into Landau levels, and the lowest of these is
special: it is one-dimensional, and it moves in only one direction along
$z$. Its transverse degeneracy is
\begin{equation}
\frac{\text{states}}{\text{area}} = \frac{eB}{2\pi} ,
\end{equation}
the familiar flux-quantum counting. Now switch on an electric field
$\mathbf{E}$ parallel to $\mathbf{B}$. Momenta drift, $\dd p/\dd t = eE$,
and states cross zero energy at a rate $eE/2\pi$ per unit length per
unit time. Multiplying,
\begin{equation}
\frac{\dd n}{\dd t}
 = \frac{eB}{2\pi} \cdot \frac{eE}{2\pi}
 = \frac{e^2}{4\pi^2}\, \mathbf{E}\cdot\mathbf{B} .
\label{eq:anomaly-rate}
\end{equation}

The number of fermions of that handedness is \emph{not} conserved. It
changes at a rate fixed by the fields, and the coefficient in
Eq.~\eqref{eq:anomaly-rate} is exactly the chiral anomaly first found by
\citet{Adler1969} and \citet{BellJackiw1969} from the triangle diagram.
The counting argument in this form is due to
\citet{NielsenNinomiya1983}. Nothing in it is subtle: states flow up out
of the filled sea, the sea has no bottom, and a conservation law that
looked exact is quietly leaking at a computable rate.

That is the first genuinely new phenomenon on the ladder. A symmetry
present in the classical Lagrangian --- and therefore, by
Chapter~\ref{sec:noether}, equipped with a Noether current --- delivers a
current whose divergence is not zero.



\emph{Physics intuition.} The step in the counting that does the damage
is the one that looks like an accounting convenience. A relativistic
theory of massless fermions has, in the Dirac picture, an infinitely deep
sea of filled negative-energy states, and infinity has the property that
one can remove a state from it forever without ever running out. Switch
on the fields and states flow steadily upward through zero energy; the
supply never fails, because there is no bottom. A quantity that is
conserved in every finite approximation is therefore not conserved in the
theory, and the rate at which it fails is finite and calculable.

This is why the anomaly is a quantum effect with a classical-looking
answer. Nothing about Eq.~\eqref{eq:anomaly-rate} involves $\hbar$
explicitly; what involves quantum mechanics is the existence of the sea.
\subsection{Why an anomaly in a gauge symmetry is fatal rather than interesting}
\label{ssec:fatal}
Whether this matters depends entirely on which kind of symmetry it
happens to.

If the anomalous symmetry is a \emph{global} one, the identity of
Chapter~\ref{sec:ward} constrains physical amplitudes, and a term added
to its right-hand side is therefore an amplitude that can be measured.
The result is a prediction.

\paragraph{Worked example: counting the colors.} The classic case is
the decay $\pi^0 \to \gamma\gamma$, which the naive symmetry argument
says should be strongly suppressed and which is observed at the rate the
anomaly predicts. The rate is worth quoting, because it is the cleanest
measurement in the range of this paper. With no free parameter beyond
the pion decay constant,
\begin{equation}
\Gamma(\pi^0 \to \gamma\gamma)
 = \frac{\alpha^2 m_{\pi^0}^3}{64\pi^3 f_\pi^2}
   \left(\frac{N_c}{3}\right)^{\!2} ,
\label{eq:pi0width}
\end{equation}
and with $N_c = 3$ and $f_\pi = 92.21$~MeV this is $7.761$~eV against
a measured $7.802 \pm 0.117$~eV \citep{PDG2024} --- agreement to half a
percent. With $N_c = 1$ it
is $0.862$~eV, low by a factor of exactly nine.

Read what has happened. An integer --- the number of colors --- has been
read off a decay rate, because the anomaly coefficient counts fermion
species rather than measuring a coupling. It is quadratic in that count,
so a wrong count is not slightly wrong but wrong by a factor. The
arithmetic is carried out in Appendix~\ref{app:ward}.

If the anomalous symmetry is a \emph{gauge} redundancy, the situation is
categorically different, and Chapter~\ref{sec:gauge} explains why. The
redundancy was not decorative: it was the device that reconciled
locality with unitarity by declaring surplus components of the field to
be convention. An anomaly says that quantum mechanics does not respect
that declaration. The surplus components stop being convention and start
propagating --- and they are the ones that carry negative probability.

So an anomalous gauge symmetry is not an interesting effect to be
measured. It is a theory that does not exist. This is the strongest
constraint in model building, and it is why the four integer identities
of \S\ref{ssec:currency} are not optional: a chiral gauge theory whose
anomalies do not cancel is not a bad description of nature, it is not a
description at all.



\paragraph{Worked example: how tight the constraint is.} Take the
simplest possible chiral gauge theory: a single left-handed Weyl fermion
of charge $q = 1$ under a $\U(1)$. The gauge anomaly is proportional to
$\sum_i q_i^3 = 1 \neq 0$, and a second condition, from the mixed
gauge--gravitational anomaly, requires $\sum_i q_i = 0$ and also fails.
The theory does not exist. Not ``is strongly constrained'': does not
exist.

Add a second fermion of charge $-1$ and both sums vanish. But this pair
is \emph{vector-like} --- the two charges are opposite, so a mass term is
allowed, and the two fermions can pair up and leave the low-energy
theory altogether. Nothing chiral has been achieved.

How hard is it to be genuinely chiral? An exhaustive search over charge
assignments with $|q| \le 9$ and at most six fermions finds no chiral,
anomaly-free, \emph{primitive} solution --- one whose charges share no
common factor, so that it is not a rescaling of a smaller one --- with
fewer than \emph{five} fermions,
and the smallest is
\begin{equation}
\{\,-9,\; -5,\; -1,\; +7,\; +8\,\},
\qquad \sum q_i = 0, \qquad \sum q_i^3 = 0 ,
\end{equation}
as one may check: $-729 - 125 - 1 + 343 + 512 = 0$. (The stated range is
the range searched; nothing here rules out solutions with larger
charges.) That is what a generic solution of these conditions looks like
--- five fields with unrelated-looking charges, arrived at by search.

Set it beside Table~\ref{tab:sm-anomaly}. Nature's solution to the same
kind of problem uses charges that organize into two doublets and three
singlets with a common denominator, and cancels four conditions at once
rather than two. That is the sense in which the Standard Model's
anomaly cancellation is not merely satisfied but conspicuously tidy ---
and it is the observation this chapter and Part~III both build on.
\subsection{The radical move: let the identity fail}
\label{ssec:let-it-fail}
By 1984 this constraint had come to look fatal for the chiral theories
in ten dimensions that string theory required. The anomaly was there, it
was large, and no arrangement of fermions canceled it.

\citet{GreenSchwarz1984} made a move that had not been tried. Ten
dimensions contains, besides the gauge and gravitational fields, an
antisymmetric tensor field $B$ with field strength $H_3$. Classically
$H_3 = \dd B$, and therefore $\dd H_3 = 0$ --- an identity of exactly the
kind Chapter~\ref{sec:bianchi} argued is unbreakable. Green and Schwarz
did not accept it. They took
\begin{equation}
\dd H_3 = X_4 ,
\label{eq:modified-bianchi}
\end{equation}
with $X_4$ a four-form built from the curvatures, which is to say: they
let the identity fail, by a prescribed amount.

The consequence is that $H_3$ is no longer gauge invariant. Under a
gauge transformation it must shift, in a way dictated by
Eq.~\eqref{eq:modified-bianchi}, and so any term in the action built
from $H_3$ acquires a variation that would otherwise be inexcusable. The
mechanism is to arrange that this new failure exactly cancels the
fermion loop's failure. Two things that are separately wrong are made to
be jointly right.

For this to work the total anomaly must factorize --- the twelve-form
that measures it must split as a product of the four-form in
Eq.~\eqref{eq:modified-bianchi} with an eight-form --- and factorization
is a strong arithmetic condition. In ten dimensions it requires, among
other things, that the gauge group have dimension $496$. Two non-abelian
groups meet the condition: $\SO(32)$, whose dimension is
$32 \cdot 31 / 2 = 496$, and $E_8 \times E_8$, whose dimension is
$2 \times 248 = 496$. It should be said precisely, because the result is
often overstated: dimension $496$ is necessary and not sufficient, and
abelian solutions of the same dimension exist. What is true, and was
startling in 1984, is that the two non-abelian solutions are exactly the
two groups that string theory could supply.



\emph{Physics intuition.} The move is easier to accept if one notices
that physics had done it before, at a lower level. A conserved current
can be modified by adding an improvement term; a Hamiltonian can be
shifted by a constant; a gauge potential can be changed by a gradient.
In each case the object one thought was fundamental turns out to be
defined only up to something, and the physics is in the equivalence
class. The \GS{} mechanism extends this to the identity itself: what is
physical is not $\dd H_3 = 0$ but the total anomaly, and $H_3$ is
allowed to be whatever it must be for that total to vanish.

The reason it feels radical is that the identity being modified is of
the type Chapter~\ref{sec:bianchi} argued cannot be violated. And that
argument was correct --- for a field strength that is the derivative of a
potential. What Green and Schwarz do is decline that premise: their
$H_3$ is not $\dd B$ but $\dd B$ plus Chern--Simons terms, and once it is
not the derivative of anything, $\dd H_3$ is under no obligation to
vanish. The identity was never broken. The object was redefined so that
the identity no longer applied to it.
\subsection{Physics intuition: conservation as currency}
\label{ssec:currency}
\emph{Physics intuition: conservation as currency.}

Every rung so far has treated a conservation law as something to be
protected. This one treats it as something to be \emph{spent}.

The picture that fits is double-entry bookkeeping. Before 1984 the rule
was that each account must balance separately: the fermion sector must
be anomaly-free, and the tensor sector must satisfy its identity. Green
and Schwarz observed that this is stricter than the physics requires.
What must hold is that the \emph{total} vanish. An overdraft in one
account is permissible if a matching credit sits in another, and the
theory is consistent precisely when the books close.

The shift in what is regarded as fundamental is worth stating. It is no
longer \emph{nothing is lost}. It is \emph{the books balance}. Those are
different principles, and the second is weaker locally and stronger
globally: it permits individual violations of any size, and it forbids
any net imbalance whatsoever.

That the Standard Model itself already obeys a principle of this kind is
easy to overlook. Its four anomaly-cancellation conditions are exact
identities among integers --- and, in the normalization in which
hypercharges are integers, they read
\begin{equation}
\sum Y = 0, \qquad
\sum Y^3 = 0, \qquad
\sum_{\text{doublets}} Y = 0, \qquad
\sum_{\text{triplets}} Y = 0 ,
\label{eq:sm-anomaly}
\end{equation}
with the sums over one generation. Written out, the first is
$6(2) + 3(-8) + 3(4) + 2(-6) + 12 = 0$ and the second is
$6(2)^3 + 3(-8)^3 + 3(4)^3 + 2(-6)^3 + (12)^3 = 0$. Nothing arranges
these; they are properties of the hypercharge assignments that were
determined by experiment. The quarks and the leptons of one generation,
which have no obvious business with each other, cancel each other's
anomalies exactly. That is the ``books balance'' principle already
operating in the theory we have, and it is one of the reasons for
believing that quarks and leptons belong to a single structure.



\paragraph{Worked example: the Standard Model's books, in full.} The
identities of Eq.~\eqref{eq:sm-anomaly} are worth seeing as a table
rather than a formula, because the point is that unrelated fields
cancel. One generation, written entirely in terms of left-handed Weyl
fields, with hypercharge in the normalization $Y_6 = 6Y$ that makes every
entry an integer:

\begin{table}[H]
\centering
\caption[The Standard Model's anomaly conditions as integer sums.]{The
first two of the four conditions of Eq.~\eqref{eq:sm-anomaly}, evaluated.
The remaining two are the doublet sum $3(+2) + 1(-6) = 0$ and the triplet
sum $2(+2) + (-8) + (+4) = 0$. Nothing arranges these; the hypercharges
were determined by experiment, one multiplet at a time, and the columns
add to zero.}
\label{tab:sm-anomaly}\small
\begin{tabular}{@{}P{0.20\textwidth}cccrr@{}}
\toprule
\textbf{multiplet} & \textbf{copies} & $Y_6$ & \textbf{doublets} &
$\sum Y_6$ & $\sum Y_6^3$ \\
\midrule
$Q\ (\mathbf{3},\mathbf{2})$        & 6 & $+2$ & 3 & $+12$ & $+48$ \\
$u^{c}\ (\bar{\mathbf{3}},\mathbf{1})$ & 3 & $-8$ & 0 & $-24$ & $-1536$ \\
$d^{c}\ (\bar{\mathbf{3}},\mathbf{1})$ & 3 & $+4$ & 0 & $+12$ & $+192$ \\
$L\ (\mathbf{1},\mathbf{2})$        & 2 & $-6$ & 1 & $-12$ & $-432$ \\
$e^{c}\ (\mathbf{1},\mathbf{1})$    & 1 & $+12$ & 0 & $+12$ & $+1728$ \\
\midrule
\textbf{total} & & & & $\mathbf{0}$ & $\mathbf{0}$ \\
\bottomrule
\end{tabular}
\end{table}

The last column is the striking one. The cube of a hypercharge has no
business being a physically meaningful quantity, and the numbers
involved --- $48$, $-1536$, $192$, $-432$, $1728$ --- are not small or
related in any obvious way. They cancel exactly. A reader who wants a
single piece of evidence that quarks and leptons are two halves of one
structure may take this table and need nothing else.
\subsection{\BIsix{} as the six-dimensional instance, and what ``locked'' means}
\label{ssec:bi6}
The cohort's own instance of Eq.~\eqref{eq:modified-bianchi} lives in
six dimensions rather than ten. Its content is the four-form on the
right-hand side \citep{ArisakaA3BI6}:
\begin{equation}
\dd H_3 = X_4
 = \tr R^2 - \TR F_5^2 - \tfrac{2}{5} F_B^2 - \tfrac{2}{5} F_{Qi}^2 .
\label{eq:bi6}
\end{equation}
Four terms. $R$ is the curvature and $\tr$ the trace in the fundamental;
$F_5$ is the field strength of the unified \SUfive{} block, with $\TR$
its trace in the adjoint; and $F_B$ and $F_{Qi}$ are the two abelian
field strengths --- one gravitational term, one non-abelian, two
abelian. The coefficient vector is $(1, -1, -\tfrac25, -\tfrac25)$.

\paragraph{What ``locked'' means.} The two abelian coefficients are not
fitted. They are the value of a ratio of traces, which for the ledger
integers $(M, N, L) = (1, 3, 5)$ of \citet{ArisakaA2GG6} reads
\begin{equation}
\gamma = \frac{L - N}{L} = \frac{5 - 3}{5} = \frac{2}{5} .
\label{eq:trace-ratio}
\end{equation}
Both abelian coefficients take this one value --- there is no
independent second parameter --- and the same integers satisfy the
closure identities
\begin{equation}
K = LJ = M^2 + N^2 + L^2 = 35, \qquad J = 2N + 1 = 7 .
\label{eq:ogn5}
\end{equation}
The arithmetic is checkable in a line: $5 \times 7 = 35$ and
$1 + 9 + 25 = 35$. The $5$ in the denominator of Eq.~\eqref{eq:trace-ratio}
is the same $5$ that gave $24$ generators in
Eq.~\eqref{eq:su5-branching}, which is the sense in which the two halves
of this paper are looking at one object.

\paragraph{And what ``locked'' does not mean.} Honesty requires the
qualification here rather than in a footnote. Equation~\eqref{eq:trace-ratio}
is satisfied not only by $(L, N) = (5, 3)$ but by the entire family
$(5k, 3k)$ for every positive integer $k$. The ratio alone does not
select $k = 1$; that selection is made by a minimality axiom of the
program, and \S\ref{ssec:exposure} identifies it as the premise most
exposed to the cautionary pattern of Chapter~\ref{sec:honest-record}.
What is locked by the ratio is the \emph{relationship} between the
coefficients. What fixes the scale is an assumption.



\paragraph{Worked example: the ledger by hand.} Every number in
Eqs.~\eqref{eq:trace-ratio} and \eqref{eq:ogn5} can be checked without a
calculator, and a reader should:
\begin{align}
L J &= 5 \times 7 = 35 , \\
M^2 + N^2 + L^2 &= 1 + 9 + 25 = 35 , \\
\frac{L - N}{L} &= \frac{5 - 3}{5} = \frac{2}{5} , \\
5 \times \tfrac{2}{5} &= 2 .
\end{align}
The last line is the one that matters for the ladder's top rung. Because
$5$ is the smallest positive integer that clears the denominators of the
coefficient vector $(1, -1, -\tfrac25, -\tfrac25)$, the class becomes
integral only after multiplication by $5$ --- and integrality is the
precondition for everything Chapter~\ref{sec:hs} does. The factor $5$ in
the minimal integral (primitive) lift and the $5$ of $\SUfive{}$ are the
same integer
reached by two routes.

\emph{Physics intuition.} It is fair to ask why a ratio of traces should
be the thing that fixes a physical coefficient. The reading the cohort
gives is an exchange-rate one: the abelian directions are not independent
currencies but are quoted against the non-abelian block they sit in, and
$(L-N)/L$ is the quoted rate. Whether that reading is right is a question
for the paper that owns it; what this chapter reports is that the
coefficients are the ratio's value and were not adjusted to anything.
\subsection{Grade box}
\label{ssec:gs-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Theorem, standard.} The chiral anomaly and its coefficient
(Eq.~\eqref{eq:anomaly-rate}); the fatality of a gauge anomaly; the
Green--Schwarz mechanism and the factorization condition in ten
dimensions; the Standard Model's four anomaly identities
(Eq.~\eqref{eq:sm-anomaly}).

\emph{Theorem, cohort.} The closure of the six-dimensional identity
Eq.~\eqref{eq:bi6}, the trace-ratio value $2/5$, the equality of the two
abelian coefficients, the closure identities Eq.~\eqref{eq:ogn5}, and the
primitivity of the resulting integral class. Proved in
\citet{ArisakaA3BI6}; not re-derived here.

\emph{Derived-conditional.} The selection of $k = 1$ --- equivalently, of
$\SUfive{}$ rather than $\SU(10)$ --- which follows from the program's
minimality axiom and not from Eq.~\eqref{eq:trace-ratio}.

\emph{Not claimed.} That dimension $496$ singles out $\SO(32)$ and
$E_8 \times E_8$; it does not. That the six-dimensional identity is
derived in this paper; it is imported.
\end{quote}


\section{S7 --- \HS{} quantization (2005)}
\label{sec:hs}
The top of the ladder. Here the conserved object is an integer, the
argument that produces it contains no symmetry, and there is no
dynamical content left to remove.

The mathematics --- differential cohomology, quadratic refinements ---
is built from the ground up in Appendix~\ref{app:diffcohom}. This
chapter tells the story with integers and phases, which is all it
actually needs.


\subsection{Dirac 1931: a phase that must close forces an integer}
\label{ssec:dirac}
Begin in 1931. \citet{Dirac1931} asked what would happen if a magnetic
monopole existed, and answered with an argument that is still the
cleanest quantization argument in physics.

Transport a charged particle's wavefunction around a closed loop in the
field of a monopole. The wavefunction picks up a phase equal to the
magnetic flux through the loop, times the charge. Now shrink the loop to
a point --- but note that a loop on a sphere surrounding the monopole can
be shrunk to a point in two ways, over the top or under the bottom, and
the two ways enclose fluxes differing by the total flux out of the
sphere. The wavefunction must come back to itself either way. So the two
phases can differ only by a multiple of $2\pi$, and
\begin{equation}
e g = \frac{n}{2}, \qquad n \in \mathbb{Z}
\label{eq:dirac}
\end{equation}
in units where $\hbar = c = 1$. Charge is quantized.

Read the argument again and notice what is not in it. There is no
Lagrangian. There is no action to vary. There is no symmetry, global or
local, anywhere in the reasoning. The only input is that a wavefunction
is single-valued --- that going around and coming back returns you to
where you started. A \emph{consistency} requirement, not a symmetry
requirement, has produced an integer.

The same structure appears in the laboratory without any monopole. The
magnetic flux through a superconducting ring is quantized in units of
$h/2e$, for exactly the reason above applied to the condensate's phase,
and the factor of two records that the carriers are pairs. What Dirac's
argument shows is that this is not a property of superconductors but of
phases and loops.



\emph{Physics intuition.} Dirac's argument has an odd logical shape that
is worth naming. It does not show that monopoles exist, and it does not
show that charge is quantized. It shows that the two statements are the
same statement: \emph{if} one monopole exists anywhere in the universe,
\emph{then} every charge everywhere is a multiple of a common unit. A
single object, once, would fix the arithmetic of all matter forever.

That is a strikingly cheap explanation of an otherwise mysterious fact,
and it is the reason the argument has never quite gone away despite
ninety years of unsuccessful searching. It is also the first example on
this ladder of a pattern Part~III makes central: a global consistency
requirement, imposed once, determining a local observable everywhere.

\paragraph{Worked example: the two caps.} The argument sketched above is
worth doing with the bookkeeping visible, because the integer appears in
a specific place.

Surround the monopole with a sphere. On the northern cap $F$ can be
written as $\dd A^{N}$ for a potential $A^{N}$ that is well behaved
there; on the southern cap, as $\dd A^{S}$. Neither works on the whole
sphere --- that is exactly the content of \S\ref{ssec:no-monopoles}. On
the overlap, a band around the equator, both are valid, and since they
give the same $F$ they can differ only by a gauge transformation:
\begin{equation}
A^{N} - A^{S} = \frac{1}{q}\,\dd\Lambda .
\end{equation}
Now integrate $F$ over the sphere, splitting the integral into the two
caps and using Stokes' theorem on each:
\begin{equation}
\oint_{S^2} F = \oint_{\text{equator}} \left(A^{N} - A^{S}\right)
 = \frac{1}{q}\oint_{\text{equator}} \dd\Lambda .
\end{equation}
The last integral is the total change in $\Lambda$ on going once around
the equator. The gauge transformation acts on the wavefunction as
$e^{\ii\Lambda}$, which must be single-valued, so $\Lambda$ may change
only by a multiple of $2\pi$. Therefore
\begin{equation}
\oint_{S^2} F = \frac{2\pi n}{q}, \qquad n \in \mathbb{Z} ,
\end{equation}
which is Eq.~\eqref{eq:dirac} with the integer identified: $n$ is the
\emph{winding number} of the transition function around the equator.

That identification is the whole reason this chapter exists. The
quantized quantity is not a field value, not an energy, and not the
eigenvalue of any operator. It is a winding --- a count of how many times
a map wraps around --- and a count cannot be a fraction.

\paragraph{What the previous chapter could not do.} It is worth being
exact about the division of labor here, because the two chapters are
often run together. \S\ref{ssec:wt} showed that charge \emph{ratios} do
not run: whatever the ratio of two charges is, no amount of dynamics
will change it. What it could not show is that the ratio is a whole
number. A world in which the proton's charge exceeded the electron's by
one part in $10^{22}$ satisfies every equation of Chapter~\ref{sec:ward}
exactly. The measured bound is $10^{-21}$
\citep{MarinelliMorpurgo1984}, and it is evidence for \emph{this}
chapter's statement, not the previous one's.

\begin{quote}\itshape
Ward protects the ratio from dynamics. Dirac protects it from
continuity. It takes both, and they are different arguments.
\end{quote}
\subsection{The third datum: charge, flux, holonomy}
\label{ssec:third-datum}
Dirac's argument points at something the standard presentation of a
gauge field leaves out.

A gauge field, as Chapter~\ref{sec:gauge} presented it, has a potential
$A$ (not observable) and a curvature $F$ (observable). That is two data.
There is a third: the \emph{holonomy}, the phase acquired on transport
around a closed loop. Where the loop bounds a surface, the holonomy is
determined by $F$ through Stokes' theorem, and one may reasonably ignore
it. Where the loop does not bound --- around a solenoid, around a hole,
around a compact extra dimension --- it is not determined by $F$ at all,
and it is physical: the Aharonov--Bohm effect
\citep{AharonovBohm1959} is the experiment, and it works in a region
where $F$ vanishes identically. \citet{WuYang1975} made the point
structurally: the complete gauge-invariant description of a gauge field
is the set of its non-integrable phase factors --- its holonomies ---
and neither $A$ nor $F$ alone is that description.

So a gauge field carries three data:
\begin{equation}
\underbrace{c}_{\text{an integer class}}, \qquad
\underbrace{\omega}_{\text{a curvature form}}, \qquad
\underbrace{h}_{\text{a holonomy phase}} ,
\end{equation}
and they are not independent: the curvature is the derivative of the
holonomy, and the integer is the obstruction to writing the whole thing
with a single potential.

Why this rung matters for Part~III can be stated now. A closed system
can be described by its curvature; the holonomy is a boundary datum, and
in a closed world there is no boundary to read it on. An open system
cannot ignore it. Chapter~\ref{sec:oggep} is about what follows from
taking that seriously.



\paragraph{Worked example: the phase you can measure.} Send an electron
beam around both sides of a long solenoid and recombine it. Outside the
solenoid $\mathbf{B} = 0$, so $F$ vanishes on every path the electron
takes. Nevertheless the two partial beams acquire a relative phase
\begin{equation}
\Delta\varphi = \frac{q}{\hbar}\oint A\cdot \dd\ell
 = \frac{q\,\Phi}{\hbar} ,
\end{equation}
with $\Phi$ the flux \emph{enclosed}, not the flux traversed. Shifting
$\Phi$ by $h/q$ shifts $\Delta\varphi$ by $2\pi$ and returns the
interference pattern to itself: the observable is periodic in the flux,
with period $h/q$. For $q = e$ that period is $4.1357 \times 10^{-15}$~Wb
and for a superconducting pair, $q = 2e$, it is half that.

The measurement is decisive for the argument of this chapter. The
curvature is zero everywhere the electron goes, and the effect is
nonetheless there and of the predicted size. So the curvature is not the
complete description, and the missing datum is the holonomy.

\emph{Physics intuition.} The three data stand in a definite relation:
the integer is the coarsest and the most robust, the form is the finest
and the most fragile, and the holonomy is the one that carries
information between them. It is also the only one of the three that is
measured on a \emph{loop} rather than at a point or in a region --- which
is why it is the datum an open system cannot dispense with, and why
Chapter~\ref{sec:oggep} begins where it does.

And it is not merely a datum. Chapter~\ref{sec:wilson} showed that where
the loop does not bound, the holonomy is a \emph{field}: it has a
potential, its value is settled by a one-loop competition among whatever
is circulating, and the symmetry the world displays follows from where
that potential is minimized. Of the three data, the holonomy is the one
with dynamics of its own.
\subsection{The differential cocycle $(c, h, \omega)$ in plain language}
\label{ssec:cocycle}
The object that carries all three data at once, without privileging any,
is a \emph{differential cocycle}: a triple $(c, h, \omega)$ with $c$ an
integral cochain, $\omega$ a differential form, and $h$ the phase that
interpolates between them, subject to one compatibility equation saying
that the curvature is the derivative of the holonomy and that both are
consistent with the integer.

The theory of these objects is due to \citet{CheegerSimons1985}, whose
differential characters are exactly the gauge-invariant holonomy
functions of the previous section organized into a group. The
construction is developed in Appendix~\ref{app:diffcohom}; two features
are all that this chapter needs.

First, it is a genuine \emph{refinement}, not a repackaging. Two gauge
fields can have identical curvature everywhere and differ physically ---
by a flat but topologically non-trivial holonomy, the Aharonov--Bohm
situation --- and the differential cocycle distinguishes them while the
curvature does not.

Second, it is where the rigidity lives. The form $\omega$ can be varied
continuously; the phase $h$ can be varied continuously; the class $c$
cannot be varied at all. It takes values in a discrete group, and a
continuous deformation of the field leaves it exactly where it was. That
is the entire content of the ladder's top rung, stated once: within a
single object, the differential cocycle separates what can move
continuously from what cannot, and the part that cannot is the part that
gets measured as an integer.



\emph{Physics intuition.} A comparison may help. A real number can be
written as an integer plus a fractional part, and neither piece alone
reconstructs it; one needs both, plus the rule for how they fit
together. A differential cocycle is the same arrangement one level up:
$c$ is the integer part, $\omega$ the continuous part, and $h$ the rule.
Splitting a gauge field this way is not a technical convenience --- it is
what makes it possible to say, precisely, which part of a physical
configuration can be deformed and which part cannot. Every quantization
statement in this paper is a statement about the integer part.

One question has been left hanging by the phrase ``can be varied
continuously,'' and Chapter~\ref{sec:wilson} has already answered it.
What varies $h$? On a space where every loop bounds, nothing does: $h$
is fixed by $\omega$ up to a choice with no physical content. Where a
loop fails to bound, $h$ carries a piece that $\omega$ cannot see, and
that piece is a field with a potential of its own,
Eq.~\eqref{eq:hosotani-potential}. It is settled dynamically, and what
it settles is which gauge symmetry the low-energy world displays.

So the three data are not three descriptions of one thing at three
resolutions. They are three physically distinct objects: $c$ frozen,
$\omega$ continuous and local, and $h$ continuous, non-local, and
\emph{dynamical}.
\subsection{Witten 1997: fermions shift the boson's lattice}
\label{ssec:witten-shift}
One further fact, and it is the one that makes the top rung more than a
tidy reformulation.

The naive expectation is that a field strength should have an integral
class: flux through a closed surface should be a whole number of quanta,
as in Eq.~\eqref{eq:dirac}. \citet{Witten1997} showed that when fermions
are present this can fail --- and fail in a specific, calculable way.
For the four-form flux of eleven-dimensional supergravity the correct
quantization law is not that $[G/2\pi]$ is integral but that
\begin{equation}
\left[\frac{G}{2\pi}\right] - \frac{\lambda}{2} \in H^4(X; \mathbb{Z}),
\qquad \lambda = \tfrac12 p_1 ,
\label{eq:witten-shift}
\end{equation}
so the flux lattice is displaced by half of a characteristic class.
(Since $\lambda$ is itself half the first Pontryagin class, the
displacement is $p_1/4$ --- worth stating carefully, because it is
routinely misquoted as $p_1/2$.)

Told physically: the fermions are not spectators to the boson's
quantization condition. Their determinant contributes a phase, and
consistency of the total requires the boson's lattice to sit half a unit
off where one would have put it. The pattern is exactly that of
Chapter~\ref{sec:gs} one level up --- one sector's inconsistency is
absorbed by shifting another sector's bookkeeping --- and it is the
reason the top rung needs the machinery of the next section rather than
just an integer.

There is a small arithmetic signature of the shift that requires no
topology at all, and \S\ref{ssec:qx} is about it: the difference between
a lattice built on $x^2$ and one built on $x(x-1)$.



\emph{Physics intuition.} The shift is easy to mis-hear as a small
correction. It is not: it is a change in what kind of object the flux
is. Before the shift, the set of allowed fluxes is a lattice containing
zero, and ``no flux'' is an allowed configuration. After, the allowed
set is a lattice displaced off zero, and the zero-flux configuration may
not be in it at all. Something that looked like an option --- switch the
field off --- can be excluded by an integrality condition, on a
background where the characteristic class does not vanish.

That is the strongest form of the ladder's top rung. It is not merely
that a quantity is quantized; it is that the quantization condition can
forbid the obvious configuration and force a non-zero one.
Chapter~\ref{sec:payoffs} reports a place where the cohort finds exactly
this: a pushforward whose minimal non-zero value is the object the
boundary independently constructs.
\subsection{Quadratic refinement using integers only: $q(x) = x(x-1)/2$}
\label{ssec:qx}
A shifted lattice needs a function that knows about the shift. The
right object is a \emph{quadratic refinement}: a function $q$ on the
lattice whose failure to be additive reproduces the pairing,
\begin{equation}
q(x + y) - q(x) - q(y) + q(0) = \langle x, y\rangle .
\label{eq:refinement}
\end{equation}
A quadratic form $\tfrac12 x^2$ satisfies this for the unit pairing but
is not integer-valued. The integer-valued solution is
\begin{equation}
q(x) = \frac{x(x-1)}{2} = \frac{x^2 - a x}{2}, \qquad a = 1 ,
\label{eq:qx}
\end{equation}
and one checks Eq.~\eqref{eq:refinement} directly:
$q(x+y) - q(x) - q(y) = xy$, with $q(0) = 0$.

The integer $a$ is the \emph{characteristic element}, and it is the
arithmetic image of the half-shift. On the integers the characteristic
elements are exactly the odd numbers, and $a = 1$ is the smallest
positive one --- which is why Eq.~\eqref{eq:qx}, and not some other
shift, is the one that appears.

\paragraph{Worked example: the shift, by increments.} Compare how the
two candidates step. For $x^2$ the increment from $x$ to $x+1$ is
$2x + 1$, always odd. For $x(x-1)$ it is $2x$, always even. That is the
half-shift in its most elementary form: the fermions move the boson's
lattice by exactly the amount that turns an odd step into an even one.
A reader who wants the topology stripped away entirely may take this as
the whole content of \S\ref{ssec:witten-shift}.

Equation~\eqref{eq:qx} will be met twice more. It is the quadratic
refinement of the cohort's six-dimensional tensor sector, at the same
value $a = 1$ \citep{ArisakaA3BI6}; and it is the toll ladder against
which the relaxation cascade of Chapter~\ref{sec:grip} pays. One
function, three appearances, and \S\ref{ssec:kappa} is about the fact
that this is not a coincidence.



\paragraph{Worked example: check the functional equation.} Take
$x = 3$ and $y = -5$. Then $q(3) = 3$, $q(-5) = 15$, and
$q(3 + (-5)) = q(-2) = 3$. So
\begin{equation}
q(x+y) - q(x) - q(y) = 3 - 3 - 15 = -15 = 3 \times (-5) = xy ,
\end{equation}
as Eq.~\eqref{eq:refinement} requires. Now try $a = 2$ instead, so that
$q(x) = x(x-2)/2$: at $x = 1$ this gives $-\tfrac12$, which is not an
integer. The even choice fails integrality at every odd argument, and
\S\ref{ssec:wu} explains why only odd characteristic elements are
available in the first place.

\emph{Physics intuition.} A quadratic refinement is the arithmetic
answer to a physical question: how does one add up phases that are
individually ambiguous by a sign, so that the total is not? The pairing
tells you how two contributions interfere; the refinement tells you what
a single one is worth on its own. That there is a canonical answer over
the integers, and that it is $x(x-1)/2$, is the reason the same function
turns up in three different corners of this program.
\subsection{Wu classes, and why six dimensions is special}
\label{ssec:wu}
Which characteristic element is the right one is not a matter of choice.
It is fixed by the topology of the space, through the \emph{Wu classes}
$\nu_j$, defined by the property that squaring operations on cohomology
are reproduced by cup product with $\nu_j$.

The fact that matters here is a dimension count. On a closed
$n$-manifold, $\nu_j = 0$ whenever $2j > n$. So on a six-manifold
$\nu_4$, $\nu_5$ and $\nu_6$ all vanish, and in particular
\begin{equation}
\nu_4 = 0 \quad\text{on } n = 6 .
\label{eq:wu-vanishing}
\end{equation}
A degree-four Wu structure --- the datum a quadratic refinement in that
degree requires --- therefore exists canonically. In eight dimensions it
does not: there $2 \cdot 4 = 8$ is not greater than $8$, $\nu_4$ need not
vanish, and the structure must be supplied by hand as extra data.

The same count controls where \citet{HopkinsSinger2005} applies. Their
construction takes a family of manifolds and pushes differential
cohomology forward along it, and the degrees have to fit: for a fiber
dimension of six the only consistent assignment puts the refinement on
degree-four data and lands the answer in degree two on the base --- a
line bundle. Not a number, not a higher gerbe: a line bundle. Six
dimensions is the case in which the top of the ladder produces exactly
the kind of object a boundary theory can use, and \S\ref{ssec:kappa}
reports what the cohort finds when it does the pushforward.

\begin{quote}\small
\textbf{Margin note: three Wus.} The Wu classes are named for Wen-Ts\"un
Wu (1919--2017), a topologist. The parity experiment of
\S\ref{ssec:failures} was performed by Chien-Shiung Wu (1912--1997), an
experimental physicist. The non-integrable phase factors of
\S\ref{ssec:third-datum} are due to Tai Tsun Wu (1933--2024), a
theorist. Three people, three different results, all central to this
paper, and no relation among them.
\end{quote}



\emph{Physics intuition.} Dimension is doing something unfamiliar here.
In most of physics the number of spacetime dimensions enters through
counting --- how many polarizations, how many components, how a
propagator falls off. In Eq.~\eqref{eq:wu-vanishing} it enters through
\emph{availability}: a structure either exists canonically or has to be
chosen. Six dimensions is on the favorable side of that line for
degree-four data, and eight is not.

A reader tracking the argument of this paper rather than its machinery
may take the following as the whole of \S\S\ref{ssec:cocycle}--\ref{ssec:wu}.
The top rung needs an integer; integers in this setting come from
quadratic refinements; refinements need a characteristic element;
characteristic elements are supplied by Wu classes; and in six dimensions
the relevant Wu class vanishes, so the element is $a = 1$ and the
refinement is $q(x) = x(x-1)/2$. Each link is a short step, and the chain
ends where Eq.~\eqref{eq:qx} began.
\subsection{The two ends of one sequence: Wilson lines and Dirac flux}
\label{ssec:two-ends}
The exact sequences of Appendix~\ref{app:diffcohom} say something this
chapter has so far read from one end only.

The first of them places the differential cohomology group between two
others: on the right the curvatures whose periods are integral, on the
left the objects whose curvature vanishes altogether. This chapter has
been reading the right-hand end. Dirac's argument, the flux quantum, the
winding number of \S\ref{ssec:dirac} --- all of them live there, in the
part of the group that records how much curvature is present and insists
that its total be a whole number. Table~\ref{tab:two-ends} sets the two
ends of the sequence against each other --- the first rung and the seventh,
what each conserves and what it would cost to break.

The left-hand end is the \emph{flat} part, and Chapter~\ref{sec:wilson}
has already been there. An object with vanishing curvature and
non-trivial holonomy is exactly an element of it. The two rungs are not
neighbors by accident: they are the two ends of one short exact
sequence.

The counting is instructive, because the two ends are independent and a
space may have either, both, or neither.

\begin{table}[H]
\centering
\caption[The two ends of the sequence.]{\textbf{The two ends of the
sequence.} The Wilson line of Chapter~\ref{sec:wilson} and the flux
quantum of \S\ref{ssec:dirac} are not competing descriptions. On the
circle there is a holonomy and no flux; on the sphere there is a flux
and no holonomy; on the torus there are both. What the differential
cocycle does is carry them in one object.}
\label{tab:two-ends}\small
\begin{tabular}{@{}P{0.20\textwidth}P{0.30\textwidth}P{0.34\textwidth}@{}}
\toprule
\textbf{space} & \textbf{holonomies (the flat end)} &
\textbf{flux integers (the curvature end)} \\
\midrule
the circle $S^1$ & one phase & none: there are no two-forms to
integrate \\
the sphere $S^2$ & none: every loop bounds & one, and it is Dirac's \\
the torus $T^2$ & two phases & one \\
\bottomrule
\end{tabular}
\end{table}

\emph{Physics intuition.} A field with no curvature anywhere and a phase
that will not go away, and a field whose curvature is spread out and
whose total is stuck at a whole number, look like unrelated
phenomena and are usually taught as such. They are the two ends of one
exact sequence, and the object in the middle is what
\S\ref{ssec:cocycle} calls a differential cocycle. That is the strongest
argument this paper can give for taking the refinement seriously: it is
not a convenience for bookkeeping, it is the only thing that holds both
ends at once.


\subsection{The top of the ladder: conservation as topological rigidity}
\label{ssec:top-of-ladder}
Part~II is finished, and the top rung can be stated in one sentence:
\emph{at the top of the ladder, what is conserved is an integer, and it
is conserved because integers have nowhere continuous to go.}

Set that beside the bottom rung for the last time. Noether's current is
conserved because the action has a symmetry --- a contingent fact about
the dynamics, removable by writing a different Lagrangian, and
constraining how things move. Dirac's integer is conserved because a
phase must close --- a consistency requirement on any description
whatsoever, removable by nothing, and constraining nothing about motion
at all. Between them lie three intermediate cases, and
Chapter~\ref{sec:one-argument} reads the whole sequence as one argument.

Two features of the top rung deserve to be carried into Part~III.

\emph{It is exact where everything else is approximate.} Every symmetry
in the actual world is broken somewhere: parity maximally, $CP$ weakly,
flavor by masses, scale invariance by almost everything. A conservation
law resting on a symmetry inherits the breaking and degrades with it. An
integer does not degrade. It is the only kind of statement that can
survive a universe as thoroughly broken as ours with its precision
intact, and \S\ref{ssec:counting-survives} argues that this is why the
exactly known quantities in physics are counting statements.

\emph{It requires a boundary to be read.} The third datum of
\S\ref{ssec:third-datum} is a holonomy, and a holonomy is a statement
about a loop --- something one evaluates on the edge of a region rather
than inside it. Every rung of this ladder, including this one, was
formulated for a closed system. The integer at the top is nonetheless
the kind of object that a boundary reads off a bulk. Part~III proposes
that this is not an accident of the formalism but the physical situation
we are actually in.



Table~\ref{tab:partii-summary} collects the Part in one place, with each
rung's defining equation and the appendix that carries its machinery.

\begin{table}[H]
\centering
\caption[Part II in one table.]{The seven rungs, with the equation each
turns on and the appendix that carries its machinery. The fourth column
is the one Chapter~\ref{sec:one-argument} reads as a single trend. Five columns per rung: the statement with the equation it turns on, what violating it would take, its status, and the toolkit appendix that carries its machinery. The table is here so that a reader can find any rung's toolkit in one step, and so that the whole Part can be checked against its own claims without rereading it.}
\label{tab:partii-summary}\small
\begin{tabular}{@{}P{0.20\textwidth}P{0.185\textwidth}P{0.235\textwidth}P{0.145\textwidth}P{0.105\textwidth}@{}}
\toprule
\textbf{rung} & \textbf{the statement} & \textbf{what violating it would
take} & \textbf{status} & \textbf{toolkit} \\
\midrule
S1 Noether I (\S\ref{sec:noether}) &
$\partial_\mu j^\mu = 0$, Eq.~\eqref{eq:scalar-current} &
a different action, or the field equations failing &
a law: contingent & App.~\ref{app:noether} \\
S1$'$ Noether II (\S\ref{ssec:noether-ii}) &
$\partial_\mu\partial_\nu F^{\mu\nu} \equiv 0$,
Eq.~\eqref{eq:noether-ii} &
nothing: it holds off shell & an identity &
App.~\ref{app:noether} \\
S2 gauge (\S\ref{sec:gauge}) &
$D_\mu = \partial_\mu - \ii q A_\mu$, Eq.~\eqref{eq:covderiv} &
nothing: there is no second state & a redundancy &
App.~\ref{app:connections} \\
S3 Bianchi (\S\ref{sec:bianchi}) &
$\dd F = 0$, Eq.~\eqref{eq:dF} &
declining to write $F$ as a curvature & an identity &
App.~\ref{app:forms} \\
S4 Wilson (\S\ref{sec:wilson}) &
$W = e^{\ii\theta}$, Eq.~\eqref{eq:wilsonline} &
a space in which every loop bounds & a holonomy &
App.~\ref{app:wilson} \\
S5 Ward (\S\ref{sec:ward}) &
Eq.~\eqref{eq:ward-master} &
a measure that is not invariant & an identity &
App.~\ref{app:ward} \\
S6 \GS{} (\S\ref{sec:gs}) &
$\dd H_3 = X_4$, Eq.~\eqref{eq:modified-bianchi} &
a net imbalance, not a local one & a trade &
App.~\ref{app:anomalies} \\
S7 \HS{} (\S\ref{sec:hs}) &
$q(x) = x(x-1)/2$ on a shifted lattice, Eq.~\eqref{eq:qx} &
a discontinuous jump & pure topology &
App.~\ref{app:diffcohom} \\
\bottomrule
\end{tabular}
\end{table}
%% ===================================================================
%%  PART III --- THE OPEN-SYSTEM TURN
%% ===================================================================
\begin{checkyourself}{Part~II}
\item \S\ref{ssec:noether-ii} states
  $\partial_\mu\partial_\nu F^{\mu\nu} \equiv 0$ as an off-shell
  identity. Establish it in one line. Then explain why the same style of
  argument does \emph{not} establish $\partial_\mu j^\mu = 0$ for the
  current of Eq.~\eqref{eq:scalar-current}, and identify precisely what
  extra ingredient that one needs.
\item Redo the counting of \S\ref{ssec:anomalies} for a fermion of
  charge $2e$ instead of $e$. By what factor does the rate in
  Eq.~\eqref{eq:anomaly-rate} change? The answer is not two; say which
  step contributes each factor.
\item Verify Eq.~\eqref{eq:refinement} for $q(x) = x(x-1)/2$ at
  $x = 3$, $y = -5$. Then set the characteristic element to $a = 2$
  instead of $a = 1$ and evaluate the resulting $q$ at $x = 1$. What
  property of Eq.~\eqref{eq:qx} has been lost, and how does
  \S\ref{ssec:wu} decide which value of $a$ is available?
\end{checkyourself}

\GGPart[The open-system turn]{The Open-System Turn}{An overview of the
whole proposal, and then four moves, each in one frame: \GGEP{},
\OGGEP{}/\GDOS{}, \GRIP{}, and an extension that is stated and fenced
--- the move that is not on the ladder, and the payoffs it is asked to
produce}
\label{part:turn}

\section{The open-system turn: what changes, and the four moves that
make it}
\label{sec:turn-overview}

Part~II is a review. Everything in it is standard, most of it is
decades old, and a reader who disagreed with any of it would be
disagreeing with the field rather than with this paper. Part~III is not
a review. From here the paper reports a \emph{proposal} --- one
program's answer to the question the ladder ends on --- and the
change of status should be visible on every page: each claim below is
imported from a member paper of the \GGTheory{} cohort, carries the
grade its owner gives it, and is not re-derived here.

Because the change is that sharp, this chapter comes before any of it.
It says what the proposal is before any part of it is argued, so that a
reader can see the shape of the thing he is being asked to audit.
Nothing in this chapter is established. Everything in it is claimed,
and every claim names the chapter that carries it.

The order below is the order of the questions a skeptical reader
actually asks. What did the ladder reach, and where did it stop
(\S\ref{ssec:what-part-ii-reached})? Why is the closed-system frame the
thing that has to give (\S\ref{ssec:why-open})? What becomes sayable
when it does, and at what price (\S\ref{ssec:what-changes})? What does
the whole turn stand on (\S\ref{ssec:two-axioms})? What are the four
moves (\S\ref{ssec:four-moves}), where do they live in the program
that owns them (\S\ref{ssec:cohort-map-w1}), and in what fixed shape
will each of them be set out (\S\ref{ssec:frame})?

\subsection{What Part~II established, and what it could not reach}
\label{ssec:what-part-ii-reached}
Seven rungs, and one trend. At the bottom, Noether's first theorem ties
a conserved current to a symmetry of the action: a contingent fact
about the dynamics, removable by writing a different Lagrangian, and
constraining how things move. In the middle, three of the field's most
quoted conservation statements turn out on inspection not to be laws at
all --- the gauge redundancy is a surplus in the description, the
Bianchi identity holds because a curvature is a curvature, and the Ward
identity is a statement about the measure. At the top,
\HS{}'s integer is conserved because integers have nowhere continuous
to go: exact, unbreakable by any deformation, and constraining nothing
about motion whatever. \S\ref{ssec:top-of-ladder} put the trend in one
line --- conservation laws grow harder to break as they say less about
motion --- and Table~\ref{tab:partii-summary} audits it rung by rung.

Two features of the top rung were flagged there, and they are the hinge
of this Part. It is \emph{exact where everything else is approximate}:
every symmetry in the actual world is broken somewhere, and a
conservation law resting on one inherits the breaking, while an integer
does not degrade. And it \emph{requires a boundary to be read}: a
holonomy is a statement about a loop, something evaluated on the edge
of a region rather than inside it.

Now the limit, and it is a limit of the whole ladder rather than of any
rung on it. Every one of the seven results was formulated for a
\emph{closed} system --- a complete field configuration, a total
anomaly, a manifold without an outside. Look again at
Table~\ref{tab:partii-summary}: it has four columns, and not one of
them is about exchange. That is not an oversight in the table. It is
that none of its rungs has anything to say about exchange, so there was
nothing to put in such a column.

This matters because the object at the top of the ladder is the one
object on it that a closed description cannot finish. It is read on an
edge, and a formalism with no outside has no edge to read it on. The
ladder therefore ends by pointing at something it cannot itself
describe --- which is either a curiosity of the formalism or a fact
about the physical situation, and Part~III proposes the second.

\subsection{Why an open system is essential}
\label{ssec:why-open}
Put one question to the ladder and take the answers in order: what would
each rung say about a system that exchanges something with an
environment? The rung-by-rung answers are given in full in
\S\ref{ssec:closed-world}; in preview they are these.

\emph{Noether} was derived from an action for a closed system. Applied
to an open one the conservation law becomes a balance equation with a
flux term, and the theorem says nothing whatever about what that term
is. \emph{Gauge redundancy} counts the states of a complete field
configuration and is silent about what a subsystem sees.
\emph{Bianchi} holds locally in any region and is indifferent to
boundaries --- which sounds like an advantage and is exactly the
problem, since it says nothing about the case where the surface is
where the physics is. \emph{\GS{}} requires the anomaly to cancel in
the total, and one is entitled to ask whose total. \emph{\HS{}}
constructs a statement about a manifold whose third datum, the
holonomy, is precisely the one a closed description may regard as
derived and an open one may not.

None of this is a defect. It is a \emph{scope}. Each rung is correct
within it, and the proposal of this Part is not to correct any of them
but to ask what a theory looks like when the scope is dropped.

Two further reasons make the dropping compulsory rather than optional,
and both are physical rather than formal.

The first is that no system we have ever measured has been closed. A
detector is read out; a laboratory is thermally coupled to a building;
the cosmological horizon is a boundary that recedes. The closed system
is an idealization of enormous utility that physics has never once
observed, and every fundamental law in the ladder was written for the
idealization rather than for the case.

The second is sharper, and it is the one the rest of this Part turns
on. \emph{A measurement is a boundary transaction.} Something crosses,
a record is left, and the record does not come back. A formalism with
no outside can define what is conserved, but it has no way to say what
is \emph{read}, because reading is exchange. That is why the object at
the top of the ladder --- the one that has to be read on an edge --- is
the object that forces the turn.

\subsection{What changes when the scope is dropped}
\label{ssec:what-changes}
Five things become sayable that were not sayable before, and one price
is paid for them. Figure~\ref{fig:closed-open} sets the two frames side
by side; the list below is what the right-hand panel buys.

\emph{First, a conservation law becomes an audit.} In a closed
description a conservation statement is a fact about the system: the
quantity does not change. In an open one it becomes a
\emph{bookkeeping} statement: what crossed the boundary is what the
interior lost, and the two columns have to agree. The difference is
that an audit can fail in a way a fact cannot, and can therefore be
checked. \S\ref{ssec:open-ward} writes the resulting identity out with
its source term.

\emph{Second, admissibility becomes exclusion.} This is the change the
program most needs and the one that is easiest to miss, so it is
worth stating in its sharpest form. Anomaly cancellation and
integrality --- the two most powerful selection tools the closed frame
has --- are \emph{admissibility} conditions: they partition candidate
theories into consistent and inconsistent and are silent about which of
the consistent ones obtains. A world with four families that canceled
its anomalies would pass every such test. To exclude a consistent
alternative one needs something that discriminates among admissible
options, and a cost will do it: if reaching a configuration requires
paying from a finite account, configurations that cannot be afforded
are excluded --- not as inconsistent, but as \emph{unreachable}. An
account is a boundary notion, because spending is exchange.
\S\ref{ssec:exclusion} makes the argument properly and attributes it
precisely.

\emph{Third, irreversibility acquires a price instead of an apology.}
In the closed frame the arrow of time is an embarrassment to be
explained away by coarse-graining or by initial conditions. In the open
frame it is the ordinary consequence of a one-way channel, and what
crossing it costs can be written down. \S\ref{ssec:solvency} turns that
cost into a solvency condition.

\emph{Fourth, parameters can become readings.} If the interior is a
structure and the boundary reads it, then a measured number need not be
an input at all: it can be what the reading returns.
Chapter~\ref{sec:sm-sheet} tests that against twenty-eight of them, and is
careful about which are readings and which are not.

\emph{Fifth, the top of the ladder finally has somewhere to stand.} The
integer needs an edge to be read on. An open system is a system with an
edge.

And the price, stated in the same breath, because a reader who takes
the five without it has been misled. \emph{Everything after this
chapter is a proposal.} Part~II's rungs are the field's; Part~III's
moves are one program's, they are imported at their owners' grades,
and this review derives none of them. Two of the four moves are axioms
--- assumptions, chosen, and attackable exactly where they are stated.
The fourth is an extension that this review reports and does not
defend. A reader who finds the five gains attractive should notice that
he is being asked to pay for them in assumptions, and
\S\ref{ssec:four-moves} prints the bill before the argument begins.

\begin{figure}[H]
\centering
\resizebox{0.75\textwidth}{!}{%
\usebox{\GGplateFrames}}
\caption[The two frames: admissibility on the left, exclusion on the
right.]{\textbf{What the turn changes.} The closed frame of Part~II
(left) reaches \emph{admissibility}: its strongest tools --- anomaly
cancellation and integrality --- sort candidate theories into consistent
and inconsistent, and stop. They are silent about which consistent
option obtains, because sorting by cost requires an account and an
account requires a boundary. The open frame of Part~III (right) is
built on two axioms, carries a third move that stands on them, and a
fourth that is stated and fenced; what it reaches is \emph{exclusion}
--- not ``inconsistent'' but ``unaffordable''. The whole of this Part is
an argument that the step between the panels is forced rather than
chosen, and Chapters~\ref{sec:ggep}--\ref{sec:qi4} pay for it in
assumptions, each named.}
\label{fig:closed-open}
\end{figure}

\subsection{The two axioms the turn stands on}
\label{ssec:two-axioms}
The program's constitution \citep{ArisakaA1Constitution} is a tower of
four axioms in a strict order, each presupposing the one before:
locality and causality (AX-1); the geometry--gauge equivalence
(\GGEP{}, AX-2); the operational bulk-to-boundary dictionary
(\OGGEP{}, AX-3); and a minimality principle (AX-4). The two in the
middle are the ones this Part is built on, and it is worth seeing them
beside each other once before either is argued.

\GGEP{} says that geometry and gauge are \emph{one} object. In the
standard account gravitation is the curvature of spacetime and the
gauge interactions are the curvature of an internal bundle sitting over
it: two structures, related by sharing a base and otherwise
independent, requiring separate postulates. \GGEP{} says they are one
structure seen in different directions. Chapter~\ref{sec:ggep} states
it precisely, and notes that one familiar object --- the Wilson line of
Chapter~\ref{sec:wilson} --- already satisfies it exactly.

\OGGEP{} says that observation is a \emph{projection}. A bulk theory is
not physically defined until it supplies the dictionary that maps bulk
structure to registered boundary observables and to the admissible
evolutions between them. Chapter~\ref{sec:oggep} states it precisely,
and the consequence it carries is the one \S\ref{ssec:what-changes}
listed second: a projection loses information, losing information
costs, and a cost excludes.

Two remarks about the pair, both of which a reader should hold onto.

They are \emph{axioms}, not results. Neither is derived here or
anywhere; each is assumed, named, and placed where an objection can be
aimed at it. That is the honest form for an assumption, and it is also
the vulnerable one, which is the point.

And they do different jobs. \GGEP{} is about \emph{what there is}: one
object rather than two. \OGGEP{} is about \emph{what we get to see}:
a boundary reading of it. A program could accept the first and refuse
the second, and would then have a unified geometry with no account of
measurement; it could accept the second and refuse the first, and would
have an open-system formalism with nothing definite in the bulk to
project. The two moves that follow --- \GRIP{} in
Chapter~\ref{sec:grip}, \QiFour{} in Chapter~\ref{sec:qi4} --- stand on
the pair rather than beside it, which is why they come after.

One term needs separating from another before it causes trouble. The
two axioms here are the \emph{constitutional} ones, AX-2 and AX-3. The
word \emph{pillar} is used later in a different and narrower sense, for
the four \QiFour{} admissibility conditions of
Chapter~\ref{sec:qi4} --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{} --- and
those four are not axioms of the constitution at all. Two axioms, four
pillars, and the two counts belong to different levels.

The fourth axiom, minimality (AX-4), is not used as a premise anywhere
in this Part, and \S\ref{ssec:exposure} names it as the program's
most exposed assumption. It is mentioned here only so that the tower is
complete in one place.

\subsection{The four moves at a glance}
\label{ssec:four-moves}
Part~III makes four moves, and Table~\ref{tab:four-moves} states each
in one line: the question it answers, the claim it makes, and the
cohort paper that owns it. It is deliberately not the audit.
Table~\ref{tab:g-table}, at the end of the Part, is the audit --- what
each move assumes, what follows, what it buys, and at what grade --- and
a reader who wants the ledger rather than the orientation should turn
there now and read the two tables together.

\begin{table}[H]
\centering
\caption[The four moves of Part~III, at a glance.]{\textbf{The four
moves, at a glance.} Orientation, not audit: the question each move
answers, the claim it makes, and the paper that carries the derivation.
Table~\ref{tab:g-table} gives the same four rows with their premises,
consequences and grades, and should be read against this one at the end
of the Part. Three of the four are the program's own; the fourth is an
extension this review states and does not defend, and nothing in the
first three depends on it (\S\ref{ssec:qi4-fence}).}
\label{tab:four-moves}\small
\renewcommand{\arraystretch}{1.18}
\begin{tabular}{@{}P{0.115\textwidth}P{0.175\textwidth}P{0.375\textwidth}P{0.155\textwidth}@{}}
\toprule
\textbf{move} & \textbf{the question it answers} & \textbf{the claim,
in one sentence} & \textbf{owner} \\
\midrule
G1 \GGEP{} (\S\ref{sec:ggep}) &
what is there? &
Geometry and gauge are one object, which forces a six-dimensional bulk
whose anomaly bookkeeping is exact rather than conventional. &
\ATwo{}, ledger \AThree{} \\
G2 \OGGEP{} (\S\ref{sec:oggep}) &
what do we get to see? &
What is observed is a boundary reading of that bulk, so the boundary
dynamics is open and irreversible, and a conservation law becomes an
audit with a source term. &
\AFive{} \\
G3 \GRIP{} (\S\ref{sec:grip}) &
why this world and not another admissible one? &
Reaching a configuration costs, the toll grows faster than the budget,
and what cannot be afforded is excluded. &
\ASix{} \\
G4 \QiFour{} (\S\ref{sec:qi4}) &
what would settle it in a laboratory? &
One abelian direction of the bulk carries information, with a fixed
conversion constant and a measurable consequence. &
\ASeven{} \\
\bottomrule
\end{tabular}
\end{table}

The shape of the four is worth one paragraph, because it is not a list.
G1 and G2 are the axioms of \S\ref{ssec:two-axioms}, and they are
independent of each other: one says what there is, the other says how
it is read. G3 stands on both --- there is no cost without something to
traverse and no account without a boundary --- and it is the move that
does the work the closed frame could not do, converting admissibility
into exclusion. G4 is different in kind. It is an extension, reported
here at the grade its owner gives it, fenced in \S\ref{ssec:qi4-fence},
and carrying the one thing the other three do not: a number that an
experiment could return tomorrow.

A reader deciding how much of this Part to take seriously can therefore
take it in pieces, and should. Rejecting G4 costs nothing above it.
Rejecting G3 leaves G1 and G2 standing as a unified geometry with an
open reading, and removes the exclusion argument. Rejecting either
axiom ends the Part, which is exactly what an axiom should do when it
is rejected.

\subsection{The cohort behind the moves}
\label{ssec:cohort-map-w1}
The four moves are not this review's. Each is derived in a member paper
of a twenty-two-paper program, and
Figure~\ref{fig:cohort-map-w1} is that program drawn once ---
the same plate every paper of the cohort carries. It is printed here,
in the middle of a review, for a specific reason: a reader who is about
to be handed four imported claims is entitled to see where they come
from, what else stands on the same foundations, and which papers would
have to be wrong if these four are.

Read against the four moves, the map sorts as follows.

\emph{The spine is the derivation.} At the top, in violet, the
constitutional root \AOne{} supplies the four axioms of
\S\ref{ssec:two-axioms} --- so AX-2 and AX-3, the two this Part stands
on, are issued by the box at the very top of the figure. Beneath it the
red row is \textbf{G1}: \ATwo{} realizes the six-dimensional bulk and
\AThree{} proves the anomaly ledger closes, which is why the G1 row of
Table~\ref{tab:four-moves} names two papers rather than one. The yellow
box below them is \textbf{G2}, and the map's own label for it is the
boundary law: \AFive{} is where \OGGEP{} becomes a dynamics. The two
green boxes beneath \emph{that} are \textbf{G3} and \textbf{G4} ---
\ASix{} and \ASeven{}, the geometric and the quantum-information
runtime, in that order and at that depth because both stand on the
boundary law rather than beside it.

\emph{The blue row is what the moves produce.} \AEight{}, \ANine{} and
\ATen{} are the readouts --- quantum mechanics, the Standard Model, the
cosmological inventory --- and they are the reason
Chapters~\ref{sec:zero-parameters}--\ref{sec:cosmos-sheet} of this
review have any numbers to quote. The gold capstone \AEleven{} below
them carries the cascade this review touches once, in
\S\ref{ssec:not-zero}, and does not endorse.

\emph{The two framed rows below are where the program can be wrong in
public.} The Predictions Branch in cyan --- \PaperPOne{}
\citep{ArisakaP1KK}, \PaperPTwo{}, \PaperPThree{}
\citep{ArisakaP3Electron} and \PaperPSix{} --- and the Cosmology \&
Astrophysics Branch in teal --- \PaperCOne{} \citep{ArisakaC1Kaxion},
\PaperCTwo{} \citep{ArisakaC2Hubble} and \PaperCThree{}
\citep{ArisakaC3BH} --- each state a number or a bound that an
experiment can retire. \S\ref{ssec:three-gen} of this Part takes its
worked payoff from the first of those rows, and
Chapter~\ref{sec:falsifiers} returns to them as a class.

\emph{The two flanking columns do opposite jobs, and this paper is one
of them.} On the left, in violet, the Foundations Branch feeds the
spine: \PaperPFour{} \citep{ArisakaP4Real} supplies the origin of the
$i$ that the quantum readout consumes, and \PaperPFive{} sits at the
interface between the Standard-Model and cosmological readouts. On the
right, in brown, the White-Paper Branch reads it --- and the upper of
its two boxes is the present paper. Every edge into that column is
drawn dashed, which in the figure's grammar means the deployment of a
result derived elsewhere rather than a step of derivation. That is the
status claim of Part~III, drawn: this review imports, at declared
grades, and derives nothing.

The single dashed feed into \PaperWOne{} comes from \AThree{}, and it
names the heaviest source rather than the only one --- the anomaly
paper is where the ladder of Part~II ends and where this Part's first
move begins. The solid arrow from this paper down to \PaperWTwo{} is
the one structural dependency between the two syntheses: the century
arrives at protection by topology, and the sibling paper takes that
structure across sixteen standing problems
\citep{ArisakaW2Holonomy}.

One caution about the figure, because a map invites a misreading it
should not get. It is a \emph{dependency} graph, not a reading order
and not a hierarchy of importance: an arrow means that the box below
uses what the box above proved, at the grade the box above declared.
Nothing in the figure is evidence for anything. It is the shape of the
program, and this Part's four moves are four of its boxes.

%% ===================================================================
%%  The twenty-two-paper cohort map, on its own page.  Verbatim copy
%%  of the master (A1-Constitution Figure 3) as carried by every other
%%  paper of the cohort: no self-reference marker anywhere, so the
%%  figure is identical in all twenty-one carriers and simple to
%%  replicate.  Only the internal node names MP4/MP5 differ from the
%%  master (collision avoidance).  Do not localise this block.
%% ===================================================================
\begin{figure}[p]
\centering
\resizebox{\textwidth}{!}{%
%% v22 type scaling, local to this group: the picture sets node text
%% in \footnotesize (the root in \small) and its edge and branch labels
%% in \scriptsize, so redefining the three commands here enlarges box
%% type by 15%% and everything outside a box by 20%% without touching a
%% single node.
\renewcommand{\footnotesize}{\fontsize{10.35}{12.4}\selectfont}%%
\renewcommand{\small}{\fontsize{11.5}{13.8}\selectfont}%%
\renewcommand{\scriptsize}{\fontsize{9.6}{11.5}\selectfont}%%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3.45pt},
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 minimum width=80.5mm, minimum height=16.1mm,
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 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
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%% --- Row 0: axiomatic root ---
\node[root] (A1) at (0, 0)
 {\AOne{} \\[1pt] \footnotesize 4 axioms + 7 postulates};

%% --- Row 1: GG6 Trio ---
\node[bulk] (A2) at (-50mm, -25mm)
 {\textbf{\ATwo{}} \\[-1pt] \footnotesize 6D bulk};
\node[bulk] (A3) at (0, -25mm)
 {\textbf{\AThree{}} \\[-1pt] \footnotesize \BIsix{} anomaly proof};
\node[hist] (A4) at (50mm, -25mm)
 {\textbf{\AFour{}} \\[-1pt] \footnotesize Historical lineage};

%% --- Row 2: A5-GDOS, the boundary law ---
\node[gdos] (A5) at (0, -48mm)
 {\textbf{\AFive{}} \\[-1pt] \footnotesize \GDOS{} boundary law};

%% --- Row 3: pillars ---
\node[pillars] (A6) at (-32mm, -69mm)
 {\textbf{\ASix{}} \\[-1pt] \footnotesize \GiFour{} pillars};
\node[pillars] (A7) at (32mm, -69mm)
 {\textbf{\ASeven{}} \\[-1pt] \footnotesize \QiFour{} pillars};

%% --- Row 4: 4D-Physics Trio ---
\node[qm] (A8) at (-58mm, -91mm)
 {\textbf{\AEight{}} \\[-1pt] \footnotesize 4D QM reformulation};
\node[qm] (A9) at (0, -91mm)
 {\textbf{\ANine{}} \\[-1pt] \footnotesize 28 SM parameters};
\node[qm] (A10) at (58mm, -91mm)
 {\textbf{\ATen{}} \\[-1pt] \footnotesize $\Omega_m = 1/\pi$};

%% --- Row 5: Closing capstone ---
\node[a11] (A11) at (0, -113mm)
 {\textbf{\AEleven{}} \\[-1pt] \footnotesize Closing capstone \\
 \footnotesize universal identity, GRIP cascade};

%% --- Row 6: Predictions Branch (P-series) ---
%% v22: four predictions on the row at 50mm pitch.
\node[pred] (P1) at (-75mm, -138mm)
 {\textbf{\PaperPOne{}} \\[-1pt] \footnotesize $\Mlock \approx 3.85$~\TeV{}};
\node[pred] (P2) at (-25mm, -138mm)
 {\textbf{\PaperPTwo{}} \\[-1pt] \footnotesize Three generations};
\node[pred] (P3) at (25mm, -138mm)
 {\textbf{\PaperPThree{}} \\[-1pt] \footnotesize Electron as ground state};
\node[pred] (P6) at (75mm, -138mm)
 {\textbf{\PaperPSix{}} \\[-1pt] \footnotesize The guarded proton};

%% --- Row 7: Cosmology & Astrophysics Branch (C-series) ---
\node[cosmo] (C1) at (-50mm, -160mm)
 {\textbf{\PaperCOne{}} \\[-1pt] \footnotesize Kaxion dark matter};
\node[cosmo] (C2) at (0, -160mm)
 {\textbf{\PaperCTwo{}} \\[-1pt] \footnotesize Two Hubble constants};
\node[cosmo] (C3) at (50mm, -160mm)
 {\textbf{\PaperCThree{}} \\[-1pt] \footnotesize BH entropy $A/4$};

%% --- Foundations Branch column (v1.3: MP5 raised so its centre
%%     height matches the A11 row). MP4 and MP5 form a single
%%     left-flank column at x = -96mm: MP4 at y = -64mm (unchanged),
%%     MP5 at y = -113mm, level with GG-A11-Origin, so the column
%%     brackets the A8 row with a compact gap.
\node[fnd] (MP4) at (-96mm, -64mm)
 {\textbf{\PaperPFour{}} \\[-1pt] \footnotesize The origin of $i$};
\node[fnd] (MP5) at (-96mm, -113mm)
 {\textbf{\PaperPFive{}} \\[-1pt] \footnotesize Mirrors \& matter (CPT)};

%% --- White-Paper Branch column (v22): the right flank, at the mirror
%%     positions of the Foundations column.  The left flank feeds the
%%     spine; this one reads it, so every edge into it is dashed.
\node[wht] (W1) at (96mm, -64mm)
 {\textbf{\PaperWOne{}} \\[-1pt] \footnotesize Conservation without symmetry};
\node[wht] (W2) at (96mm, -113mm)
 {\textbf{\PaperWTwo{}} \\[-1pt] \footnotesize Sixteen open problems};

%% --- Branch frames (background layer) ---
\begin{scope}[on background layer]
\node[branchframe, draw=cyan!60!black, fill=cyan!4,
 fit=(P1)(P2)(P3)(P6)] (PFRAME) {};
\node[branchframe, draw=teal!60!black, fill=teal!4,
 fit=(C1)(C2)(C3)] (CFRAME) {};
\end{scope}
\node[branchlabel, text=cyan!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west) {Predictions\\Branch\\(P-series)};
\node[branchlabel, text=teal!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west |- CFRAME) {Cosmology \&\\Astrophysics\\Branch\\(C-series)};
%% Foundations column label (v1.3): interior placement, in the gap
%% between MP4 and MP5, anchored east so it sits immediately LEFT of
%% the vertical MP4->MP5 edge; house style (small violet) kept. The
%% "shared i" edge label occupies the RIGHT side of the same edge.
\node[branchlabel, text=violet!40!black, anchor=east]
 at ($($(MP4.south)!0.5!(MP5.north)$)+(-1.5mm,0)$) {Foundations\\Branch};
%% v22: the mirror label, held to the RIGHT of the W1->W2 edge --
%% \ATen{}'s grown border reaches x = 78.7mm and an east-anchored
%% label would run into it.
\node[branchlabel, text=brown!45!black, anchor=west]
 at ($($(W1.south)!0.22!(W2.north)$)+(1.5mm,0)$) {White-Paper\\Branch\\(W-series)};

%% --- Edges: A1 -> GG6 Trio ---
\draw[arrowrt] (A1.south) -- node[edgelabel,pos=0.55] {\AX{2}} (A2.north);
\draw[arrowrt] (A1.south) -- node[edgelabel,pos=0.55] {\POS{3}} (A3.north);
\draw[arrowdash] (A1.south) -- (A4.north);

%% --- Edges: GG6 Trio -> A5 ---
\draw[arrow] (A2.south) -- (A5.north west);
\draw[arrow] (A3.south) -- node[edgelabel,pos=0.55] {\AX{3}} (A5.north);
\draw[arrow] (A4.south) -- (A5.north east);

%% --- Edges: A5 -> pillars ---
\draw[arrow] (A5.south) -- (A6.north);
\draw[arrow] (A5.south) -- (A7.north);

%% --- Edges: A5 + pillars -> 4D-Physics Trio ---
\draw[arrow] (A6.south) -- (A8.north);
\draw[arrow] (A5.south).. controls (0, -72mm) and (0, -80mm).. (A9.north);
\draw[arrow] (A7.south) -- (A10.north);

%% --- Edges: 4D-Physics Trio -> A11 ---
\draw[arrow] (A8.south) -- (A11.north west);
\draw[arrow] (A9.south) -- (A11.north);
\draw[arrow] (A10.south) -- (A11.north east);

%% --- Edges: parents -> branch frames ---
\draw[arrowdash] (A9.south west).. controls (-40mm, -105mm)
 and (-84mm, -122mm).. (PFRAME.north west);
\draw[arrowdash] (A9.south east).. controls (40mm, -105mm)
 and (84mm, -122mm).. (PFRAME.north east);
%% v22 RE-ROUTE.  The old line descended through what is now the
%% GG-W2-Holonomy node, and there is no corridor left inboard of it.
%% A polyline rather than a Bezier: a bounding box contains control
%% points, and at the clearance this edge needs those would have sat
%% 24mm beyond any ink, scaling the whole plate down for nothing.
\draw[arrowdash, rounded corners=7pt] (A10.east) -- (127mm,-91mm)
 -- (127mm,-160mm) -- (CFRAME.east);
%% sister import inside the Predictions frame
\draw[arrowdash, draw=cyan!50!black]
 (P2.east) -- node[edgelabel,pos=0.5] {$N{=}3$} (P3.west);
% v22: the second sister import.  \PaperPSix{} says it of
% \PaperPThree{} -- "the sister paper supplies the method this paper
% runs."  Three letters, because a white-filled label wide enough to
% say "method" masks the whole arrow it annotates.
\draw[arrowdash, draw=cyan!50!black]
 (P3.east) -- node[edgelabel,pos=0.5] {GGA} (P6.west);

%% --- Foundations Branch edges (rerouted v1.3 for the raised MP5) ---
%% A2 -> MP4 (solid; Thm 0 arena feed) -- unchanged from v1.0
\draw[arrow] (A2.west) .. controls (-88mm,-30mm) ..
 node[edgelabel,pos=0.55] {Thm~0} (MP4.north);
%% MP4 -> A8 (solid; the origin of i). Departs the east side (MP4.south
%% carries the column edge to MP5) and lands on A8's top border,
%% 6mm left of the A6->A8 arrowhead, staying left of A6 throughout.
\draw[arrow] ([yshift=-3mm]MP4.east) .. controls (-69mm,-74mm) ..
 node[edgelabel,pos=0.45] {origin of $i$} ([xshift=-6mm]A8.north);
%% MP4 -> A2 (dashed deployment: complex-structure machinery), nudged
%% 3mm up the east side so the two east-side departures stay distinct.
\draw[arrowdash] ([yshift=3mm]MP4.east) .. controls (-70mm,-52mm) .. (A2.south west);
%% MP4 -> MP5 (v1.3: SOLID, per the reading conventions -- forced
%% descent, labelled). A short straight vertical between the pair;
%% the "shared i" label sits on the right side of the edge, the
%% Foundations Branch label on the left.
\draw[arrow] (MP4.south) --
 node[edgelabel, pos=0.5, anchor=west, xshift=1mm] {shared $i$}
 (MP5.north);
%% A9 -> MP5 (solid; ledger phases). Leaves A9's west side, dips
%% through the corridor below the A8 row and enters MP5.east
%% horizontally at the A11 row height. Exactly one clean crossing
%% (the A8->A11 spine edge, near x = -40mm); stays above the dashed
%% A9->PFRAME.north-west edge throughout.
\draw[arrow] (A9.west) .. controls (-40mm,-97mm) and (-58mm,-113mm) ..
 node[edgelabel,pos=0.6] {$\delta,\ \alpha_{31}$} (MP5.east);
%% MP5 -> A10 (solid; eta_B). The one long edge: runs left-to-right
%% through the clear corridor BELOW the A11 node (above the
%% Predictions frame), then climbs into A10's bottom border right of
%% the A10->A11 arrowtail; it leaves MP5.south vertically. Controls
%% lifted 5mm with the raised MP5 so the dip still clears A11's
%% bottom border. One clean crossing with each of the two dashed
%% A9->Predictions-frame edges.
\draw[arrow] (MP5.south) .. controls (-96mm,-128mm) and (55mm,-130mm) ..
 node[edgelabel,pos=0.5] {$\eta_B$} ([xshift=8mm]A10.south);
%% MP5 -> A8 (dashed; Wick-quartet/statistics feed for T-A8.12): with
%% MP5 level with A11 this is now a short straight upward edge from
%% the column's top-right corner into A8's bottom border, 8mm left
%% of the A8->A11 arrowtail. No crossings.
\draw[arrowdash] (MP5.north east) -- ([xshift=-8mm]A8.south);
%% --- White-Paper Branch edges (v22) ---
%% A3 -> W1: \AThree{} is \PaperWOne{}'s heaviest spine source and its
%% thematic terminus -- the \BIsix{} anomaly proof is where its
%% Green--Schwarz / Hopkins--Singer ladder ends.
\draw[arrowdash] ([xshift=8mm]A3.south) .. controls (58mm,-43mm) ..
 node[edgelabel,pos=0.55] {\BIsix{}} (W1.west);
%% A10 -> W2: \ATen{} is \PaperWTwo{}'s heaviest source of any kind.
%% At +15%% those two boxes overlap in x (never in y), so this is drawn
%% off A10's south-east corner, clear of the W1->W2 arrowhead.
\draw[arrowdash] (A10.south east) .. controls (84mm,-101mm) ..
 ([xshift=-6mm]W2.north);
%% W1 -> W2, solid and labelled: the mirror of the shared-i edge on the
%% left flank.  \PaperWTwo{} states the dependency itself -- it takes
%% "the structure that the arc arrives at, protection by topology
%% rather than by symmetry".
\draw[arrow] (W1.south) --
 node[edgelabel, pos=0.80, anchor=west, xshift=1mm]
 {symmetry $\to$ topology} (W2.north);

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-w1}
\end{figure}

\subsection{How to audit Part~III: the three-step frame}
\label{ssec:frame}
Part~II was auditable because every rung carried the same four columns:
what it conserves, why, what violating it would take, and how much of it
is an integer. A reader could disagree with any cell without disturbing
the others.

Part~III makes claims of a different kind, and it would be easy to let
them run together into a story. They are set out instead in one fixed
shape, repeated four times.

\begin{quote}\itshape
\textbf{Step 1 --- what is assumed.} The move, stated as a proposition,
with its grade. Normally an axiom. Named, so that a reader who wants to
attack it knows exactly where to aim.

\textbf{Step 2 --- what follows.} The chain of consequence, each link
graded, with its conditioning printed where the link is used rather than
in a footnote.

\textbf{Step 3 --- what it buys, and what it leaves open.} The payoff ---
a number, a count, a structural collapse --- followed by the residue,
named.
\end{quote}

Each chapter closes with a grade box in the same vocabulary the Preface
fixed: \emph{axiom}, \emph{theorem}, \emph{derived-conditional},
\emph{open}. Table~\ref{tab:four-moves} above named the four
moves; Table~\ref{tab:g-table}, at the end of the Part, audits them,
so that Part~III can be read against Part~II's arc table as one ledger
against another.

That is the whole of the apparatus. The first move follows.

\section{G1 --- \GGEP{}, \GG{}, and \BIsix{}: one bulk with an exact ledger}
\label{sec:ggep}
The first of the four moves is an equivalence, and it is an axiom.

\subsection{Step 1 --- what is assumed: \GGEP{}, geometry and gauge as one}
\label{ssec:ggep-stated}
The program's constitution \citep{ArisakaA1Constitution} is the tower
of four axioms set out in \S\ref{ssec:two-axioms}. This chapter takes
the second of them; Chapter~\ref{sec:oggep} takes the third; the fourth is the
one \S\ref{ssec:exposure} names as the program's most exposed premise.

\GGEP{} asserts that there is a \emph{single primitive geometric
object}: a composite connection on a bundle over a six-dimensional bulk,
with all physical content carried by its curvature and holonomy
invariants, and with the local flattening property of the equivalence
principle holding along any timelike curve.

The claim to notice is the word ``single''. In the standard account,
gravitation is the curvature of spacetime and the gauge interactions are
the curvature of an internal bundle sitting over it --- two structures,
related by living on the same base but otherwise independent, and
requiring separate postulates. \GGEP{} says they are one structure seen
in different directions. Einstein's equivalence principle is thereby
extended: not merely that gravitation is locally removable, but that the
whole geometric-and-gauge complex is.


\subsection{Step 1 (continued) --- where the equivalence is a definition rather than a proposal}
\label{ssec:ggep-wilson}
The equivalence just stated is an axiom, and a reader is entitled to ask
whether anything in standard physics already satisfies it. One object
does, exactly, and Chapter~\ref{sec:wilson} has already introduced it.

A Wilson line is the holonomy of a connection around a loop. Read one
way it is a piece of \emph{geometry}: parallel transport, a rotation of a
frame carried around a closed path. Read the other way it is a piece of
\emph{gauge structure}: a phase, the exponential of a line integral of
the potential, and the only gauge-invariant thing a potential supplies
where its curvature vanishes. It is not that two descriptions happen to
agree. There is one object, and the two readings are two ways of
pronouncing it.

That is the strongest support the review side of this paper can give the
axiom of \S\ref{ssec:ggep-stated}. For the Wilson line, geometry and
gauge as one is not a proposal about how physics might be organized; it
is what the object is.

Two fences. First, this is an instance, not a proof: one object
satisfying an axiom does not establish it, and \S\ref{ssec:ggep-grade}
does not pretend otherwise. Second, the bundle-theoretic reading of gauge
fields is standard and is not this program's invention
\citep{WuYang1975}. What the program adds is narrower and should be
stated narrowly: not that the gauge sector is bundle geometry, but that
the abelian fiber is forced into a compact direction of the bulk itself.

\subsection{Step 2 --- what follows: a six-dimensional bulk, and its ledger}
\label{ssec:bulk}
The bulk of this program is six-dimensional \citep{ArisakaA2GG6}, and a
reader who has met extra dimensions before will want to know where they
are. That is the wrong question, and it is worth saying why.

In the Kaluza--Klein tradition the extra dimensions are small circles at
every point of space, and the physics one sees at low energy is the
average over them. In the present program the two additional
directions are not a hidden part of the room. One of them is a
\emph{scale} direction and the other a compact phase circle, and what
they do is carry structure that the four-dimensional description records
as internal: charge, generation, and phase.

The honest way for a reviewer to put this is that the question ``where
are they?'' presupposes that the six-dimensional space is a bigger
version of the space one walks around in, which under \GGEP{} it is not.
What the extra directions are \emph{for} is a better question, and the
answer is: they are what makes the bookkeeping of
Chapter~\ref{sec:gs} exact, and they are what a four-dimensional
observer reads off through the projection of Chapter~\ref{sec:oggep}.

\emph{Physics intuition.} A crude but honest analogy: a shadow on a wall
is not a smaller room next door. It is a projection, and the questions
one asks about a shadow --- why is it that shape, what does its outline
record --- are not questions about where the wall's extra dimension is
hiding. The program's claim is that the four-dimensional world stands
to the six-dimensional one roughly as the shadow does to the object, with
the crucial difference that the projection is specified exactly and the
information it discards is accounted for rather than lost.


\subsection{Step 3 --- what it buys: two rungs collapse into one}
\label{ssec:ggep-buys}
For the argument of this paper the immediate consequence is
structural. Chapter~\ref{sec:gauge} established that a gauge potential
is a connection and its field strength a curvature;
Chapter~\ref{sec:bianchi} established that the Bianchi identity is what a
curvature satisfies, in the gauge case and the gravitational case alike,
for the same reason. The two chapters were, in effect, telling one story
twice.

Under \GGEP{} that is not a pedagogical coincidence. The two rungs
\emph{are} one rung: there is a single connection, it has a single
curvature, and there is a single identity. What Part~II presented as
S2 followed by S3 collapses, and Table~\ref{tab:partii-summary}'s middle
rows become one row.

That is a genuine economy, and it is worth being clear about what kind
of economy it is. Nothing new is predicted by the collapse itself; two
axioms have become one. The value is in what it makes available further
up: because the gravitational and gauge sectors are one object, the
four-term identity of \S\ref{ssec:bi6} --- with its gravitational term
and its gauge terms locked to each other by a single ratio --- is a
statement about \emph{one} thing failing to close, rather than a
coincidence between two.


\subsection{Step 3 (continued) --- the bulk pays what the boundary cannot}
\label{ssec:inflow}
There is a second thing the arrangement buys, and it is the one that
connects this chapter to Chapter~\ref{sec:ward}.

A four-dimensional theory whose Ward identity fails is inconsistent on
its own. It is not inconsistent as the boundary of a five- or
six-dimensional theory whose gauge variation supplies exactly the missing
amount --- the mechanism of \citet{CallanHarvey1985}, usually called
anomaly inflow. The boundary's books do not balance; the bulk's do; the
seam is where the transfer is recorded.

Set that beside what this chapter has: a bulk carrying a modified
identity, $\dd H_3 = X_4$, whose right-hand side is a four-form built
from the curvatures. Read as inflow, $X_4$ is not merely a correction
term. It is the quantity the boundary is short by.

\begin{quote}\itshape
The boundary's Ward identity fails by exactly what the bulk's modified
Bianchi identity supplies.
\end{quote}

That sentence is the title of this paper, written as an equation rather
than a metaphor, and it is the reason Chapter~\ref{sec:ward} was placed
where it was. The cohort's development of the inflow reading is \citet[Appendix~H]{ArisakaP2N3}; what is used here is only the statement, and the mechanism itself is standard.

\emph{Grade.} The inflow mechanism is a \emph{theorem} of quantum field
theory. That the six-dimensional identity of this program is the
inflow partner of the observed four-dimensional theory is the
program's \emph{proposal}, and is graded with the rest of Part~III.

\subsection{Step 3 (continued) --- what it leaves open, and the grade box}
\label{ssec:ggep-grade}
\begin{quote}\small
\textbf{Grade box: Chapter~\ref{sec:ggep}.}

\emph{Axiom.} \GGEP{} itself. It is AX-2 of
\citet{ArisakaA1Constitution}, assumed and not derived, and it presupposes
AX-1 (locality and causality). It could be false, and if it is, every
result of Part~III fails with it.

\emph{Theorem, cohort.} The six-dimensional structure of the bulk and
the arithmetic of its ledger, in \citet{ArisakaA2GG6}. Imported, not
re-derived.

\emph{This paper's contribution.} None. Chapter~\ref{sec:ggep} is
exposition of an axiom, and the reader is entitled to treat it as a
hypothesis whose interest lies entirely in what the following chapters
do with it.
\end{quote}


\section{G2 --- \OGGEP{} and \GDOS{}: the boundary is open}
\label{sec:oggep}
This is the chapter the paper turns on. Everything before it, including
all seven rungs of Part~II, describes systems that are complete in
themselves. This chapter is about what changes when the system one is
describing has an outside. Figure~\ref{fig:oggep-bulk} shows the setup: the six-dimensional bulk, and the \OGGEP{} map that projects it onto the open four-dimensional boundary.


\begin{figure}[tbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
    >={Stealth[length=3.2mm,width=2.6mm]},
    font=\small,
    bulkfill/.style={top color=blue!10,bottom color=blue!4},
    edge/.style={line width=1.10pt},
    fiber/.style={line width=0.80pt,draw=teal!70!black},
    scaledot/.style={circle,fill=black,inner sep=1.4pt},
    boundaryslab/.style={top color=orange!18,bottom color=orange!28,draw=orange!60!black,line width=0.80pt},
    note/.style={font=\footnotesize\itshape,text=black!72},
  ]

  %--------------------------------------------------------------
  % Geometry constants for the throat (a tapering cutaway)
  %  top  = UV GUT edge (wide),  bottom = IR end (narrow), feeds the boundary
  %--------------------------------------------------------------
  \def\xL{0}      % left wall x at UV top
  \def\xR{8.6}    % right wall x at UV top
  \def\yTop{9.2}  % UV edge height (r=0)
  \def\yBot{1.7}  % IR end height
  \def\taper{2.55}% horizontal pull-in at the IR (per side)

  % corner coordinates of the throat
  \coordinate (TL) at (\xL,\yTop);
  \coordinate (TR) at (\xR,\yTop);
  \coordinate (BL) at (\xL+\taper,\yBot);
  \coordinate (BR) at (\xR-\taper,\yBot);

  %--------------------------------------------------------------
  % 1. The 6D bulk throat  M_4 x I_r x S^1_theta
  %--------------------------------------------------------------
  \begin{scope}[on background layer]
    \fill[bulkfill]
      (TL) -- (TR)
      .. controls ($(TR)+(-0.2,-2.6)$) and ($(BR)+(0.55,2.6)$) .. (BR)
      -- (BL)
      .. controls ($(BL)+(-0.55,2.6)$) and ($(TL)+(0.2,-2.6)$) .. (TL)
      -- cycle;
  \end{scope}
  % throat walls
  \draw[edge,blue!55!black]
      (TR) .. controls ($(TR)+(-0.2,-2.6)$) and ($(BR)+(0.55,2.6)$) .. (BR);
  \draw[edge,blue!55!black]
      (TL) .. controls ($(TL)+(0.2,-2.6)$) and ($(BL)+(-0.55,2.6)$) .. (BL);

  % bulk label
  \node[align=center,blue!45!black] at (4.3,7.3)
      {\textbf{6D bulk}\\[1pt]
       $\mathcal{M}_4 \times I_r \times S^1_\theta$};
  \node[note,align=center,text width=4.6cm] at (3.7,5.7)
      {a tapering ``throat'':\\ radius shrinks as energy\\ flows from UV to IR};

  %--------------------------------------------------------------
  % 2. UV GUT edge (top, r = 0)
  %--------------------------------------------------------------
  \draw[edge,red!65!black] (TL) -- (TR);
  \node[above=2pt of TR,anchor=south east,align=right,red!60!black,font=\small]
      {\textbf{UV GUT edge}\;($r=0$)\\[1pt]
       $\EGUT \sim 10^{16}\ \mathrm{GeV}$ --- the lawful beginning};

  %--------------------------------------------------------------
  % 3. Radial / RG scale axis  r = ln(E_GUT/mu)  on the left
  %--------------------------------------------------------------
  \def\axx{-1.7}
  % the directed scale axis itself (UV top -> IR bottom)
  \draw[->,line width=1.10pt,black!75] (\axx,\yTop+0.55) -- (\axx,\yBot-0.35);
  \node[rotate=90,anchor=south,font=\small,black!75] at (\axx-0.55,5.5)
        {RG / scale axis \ \ $r=\ln(\EGUT/\mu)$};
  % UV / IR end caps (placed clear of the chain labels)
  \node[right=2pt,font=\footnotesize,black!60] at (\axx,\yTop+0.5) {UV\,($r{=}0$)};
  \node[right=2pt,font=\footnotesize,black!60] at (\axx,\yBot-0.3)  {IR};

  % inherited scale chain  E_Pl -> E_GUT -> M_lock  (three distinct heights)
  \node[scaledot] (sPl)   at (\axx,\yTop+0.0)  {};
  \node[scaledot] (sGUT)  at (\axx,\yTop-2.6)  {};
  \node[scaledot] (sLock) at (\axx,\yBot+0.55) {};
  \node[right=4pt,font=\footnotesize] at (sPl.east)  {$\EPl$};
  \node[right=4pt,font=\footnotesize] at (sGUT.east) {$\EGUT$};
  \node[right=4pt,font=\footnotesize,align=left] at (sLock.east)
        {$\Mlock\sim 3.85~\mathrm{TeV}$\\[-1pt]\scriptsize(KK lock)};
  % "inherits" annotations alongside the axis
  \node[left=2pt,font=\scriptsize,black!55] at ($(sPl)!0.5!(sGUT)$)  {inherits};
  \node[left=2pt,font=\scriptsize,black!55] at ($(sGUT)!0.5!(sLock)$) {inherits};

  %--------------------------------------------------------------
  % 4. Compact angle S^1_theta : small fibre circle at a sample point
  %    (winding carries the Kaxion / dark sector)
  %--------------------------------------------------------------
  \coordinate (fpt) at (5.9,3.6);
  \node[scaledot] at (fpt) {};
  \draw[fiber] (fpt) ellipse [x radius=0.66,y radius=0.30];
  \draw[fiber,->] ($(fpt)+(0.66,0.0)$) arc [start angle=0,end angle=300,x radius=0.66,y radius=0.30];
  \node[above right=0.05cm and 0.05cm of fpt,align=left,text width=3.4cm,font=\footnotesize,teal!45!black]
      {compact phase $S^1_\theta$:\\
       winding carries the\\ \textbf{Kaxion / dark sector}};

  %--------------------------------------------------------------
  % 5. The 4D boundary slab  M_4  (FRW / LCDM cosmology)
  %--------------------------------------------------------------
  \node[boundaryslab,minimum width=12.6cm,minimum height=1.35cm,rounded corners=2pt]
      (bdy) at (4.3,0.05) {};
  \node[align=center] at (bdy.center)
      {\textbf{4D spacetime $\mathcal{M}_4$}\;:\; ordinary FRW\,/\,$\Lambda$CDM cosmology
       \ \ (the boundary)};

  %--------------------------------------------------------------
  % 6. OGGEP projection : bold bulk -> boundary arrow
  %--------------------------------------------------------------
  \draw[-{Stealth[length=5mm,width=4mm]},line width=2.2pt,red!70!black]
      ($(BL)!0.5!(BR)+(0,0.05)$) -- ($(bdy.north)+(0,0.02)$);
  \node[anchor=west,align=left,font=\footnotesize,red!55!black] at (6.7,1.55)
      {\textbf{\OGGEP{}} \,(bulk\,$\to$\,boundary):\\
       cosmological observables are the\\
       \emph{boundary shadows} of bulk structure};

  % GDOS boundary law tag under the slab
  \node[below=3pt of bdy.south,anchor=north,font=\footnotesize,orange!40!black]
      {boundary governed by the \GDOS{} boundary law};

  %--------------------------------------------------------------
  % 7. Caption note about the meaning of "warped"
  %--------------------------------------------------------------
  \node[draw=black!55,black!35,fill=black!3,rounded corners,inner sep=4pt,
        text width=5.0cm,note,anchor=north west] at (9.3,8.7)
      {``Warped'' here denotes \textbf{RG running along $r$}
       (the field-theory / holographic-RG reading), \emph{not}
       a Randall--Sundrum spatial dimension.};

\end{tikzpicture}%
}%
\par\smallskip
\caption[6D bulk and its \OGGEP{} projection to the 4D cosmological boundary.]%
{Schematic of the six-dimensional bulk
 $\mathcal{M}_4\times I_r\times S^1_\theta$ and its \OGGEP{} projection to the
 four-dimensional cosmological boundary. The radial axis $r=\ln(\EGUT/\mu)$ is
 the renormalization-group / scale axis: the ultraviolet GUT edge sits at the top
 ($r=0$, $\EGUT\sim10^{16}\,\mathrm{GeV}$, the lawful beginning) and low-energy
 physics lies toward the boundary, with the inherited scale chain
 $\EPl\!\to\!\EGUT\!\to\!\Mlock$ ($\Mlock\sim3.85\,\mathrm{TeV}$, the Kaluza--Klein
 lock scale) marked along it. The compact angle $S^1_\theta$ is the phase fiber
 whose winding carries the Kaxion / dark sector. The bold arrow is the \OGGEP{}
 bulk-to-boundary map: cosmological observables on $\mathcal{M}_4$ (ordinary
 FRW/$\Lambda$CDM) are boundary shadows of bulk structure, with the boundary
 governed by the \GDOS{} boundary law. ``Warped'' denotes RG running along $r$
 (the field-theory / holographic-RG reading), \emph{not} a Randall--Sundrum
 spatial dimension.}
\label{fig:oggep-bulk}
\end{figure}

\subsection{Step 1 --- what is assumed: every rung S1--S7 describes a closed world}
\label{ssec:closed-world}
Look back at the ladder with one question in mind: what would this rung
say about a system that exchanges something with an environment?

\emph{Noether.} The theorem is derived from an action for a closed
system. Applied to an open one, the conservation law becomes a balance
equation with a flux term, and the theorem itself says nothing about
what that term is.

\emph{Gauge redundancy.} The counting of states, and the argument that
the surplus is convention, is a statement about a complete field
configuration. It is silent about what a subsystem sees.

\emph{Bianchi.} $\dd F = 0$ holds locally, in any region, and is
indifferent to boundaries --- which sounds like an advantage and is in
fact the problem: it constrains flux through closed surfaces and says
nothing about the case where the surface is where the physics is.

\emph{\GS{}.} The anomaly must cancel in the total. Whose total? The
mechanism is formulated for a complete theory in a fixed number of
dimensions.

\emph{\HS{}.} The construction is a statement about a manifold, and the
third datum of \S\ref{ssec:third-datum} --- the holonomy --- is exactly
the one that a closed description can regard as derived and an open one
cannot.

None of this is a defect. It is a scope. Every rung is correct within
it, and the program of Part~III is not to correct any of them but to
ask what a theory looks like when the scope is dropped.


\subsection{Step 1 (continued) --- \OGGEP{}: observation as a projection}
\label{ssec:oggep}
The third axiom of the constitution \citep{ArisakaA1Constitution} makes
that move. \OGGEP{} (AX-3) states that a bulk theory is not physically
defined until it specifies an operational dictionary: a causal,
gauge-covariant map from bulk structure to registered boundary
observables and to the admissible open-system evolutions between them.
Concretely the map outputs a boundary Hilbert space, a state, a family of
measurement instruments, and the channels that carry the state from one
intervention to the next. Figure~\ref{fig:bulk-projection} draws that projection: over each boundary point sits the non-spatial fiber $(I_r,S^1_\theta)$ --- the scale interval and the phase circle --- and the map $\pi$ that erases it, leaving the boundary dynamics open.

Two features of this deserve emphasis before anything is built on it.

First, it is an \emph{axiom}, not a discovery. The projection is
postulated. What \citet{ArisakaA5GDOS} adds is a uniqueness result --- the
geometry of the bulk selects the reduction essentially uniquely, up to
bulk gauge --- and then the form of the dynamics that follows. But that
observation is a projection is assumed.

Second, the axiom inverts the usual order of business. In the standard
presentation one has a theory, and measurement is an extra topic bolted
on afterwards, generally with some embarrassment. Here the specification
of what is registered is part of what it means to have stated the theory
at all. The constitution puts it as: openness is constitutional, and
closure is its limiting case.


\begin{figure}[tbp]
\centering
\scalebox{1.15}{%
\resizebox{0.398\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, line width=0.80pt, font=\small]
  % --- bulk fibre over x ---
 \coordinate (x) at (0,0);
 \draw[->, red!70!black, line width=1pt] (x) -- (0,3.3);
 \node[red!70!black, right, align=left] at (0.2,3.25)
 {$I_r$ : \emph{scale} fiber\\[-1pt] \footnotesize RG flow --- irreversible, non-compact};
 \draw[violet!70!black, line width=1pt] (0,1.65) ellipse (0.55 and 0.25);
 \draw[->, violet!70!black] (0.55,1.67) arc (0:330:0.55 and 0.25);
 \node[violet!70!black, right, align=left] at (0.72,1.62)
 {$S^1_\theta$ : \emph{phase} fiber\\[-1pt] \footnotesize holonomy --- unitary, compact};
  % --- base stage M_4 ---
 \draw[fill=blue!7, draw=blue!55] (-2.3,-1.7) -- (2.3,-1.7) -- (3.5,-0.8) -- (-1.1,-0.8) -- cycle;
 \node[blue!55!black] at (2.55,-1.45) {$\mathcal{M}_4$};
 \node[blue!45!black, align=center] at (0.7,-2.05) {\footnotesize boundary stage (what the observer registers)};
 \fill (0,-1.1) coordinate (xb) circle (1.7pt);
 \node[below right] at (xb) {$x$};
 \draw[gray!60, dotted] (x) -- (xb);
  % projection pi
 \draw[->, dashed, gray!80!black, line width=0.9pt] (1.55,1.15).. controls (2.3,0.1) and (1.4,-0.7).. (0.2,-1.0);
 \node[gray!45!black, align=left] at (3.25,0.55)
 {$\pi$ : \emph{observation}\\[-1pt] \footnotesize erases $(r,\theta)$\\[-1pt] \footnotesize $\Rightarrow$ open / CPTP};
\end{tikzpicture}}}
\caption[Observation as the bundle projection $\pi:\mathcal{M}_6\to\mathcal{M}_4$.]{Observation as the bundle projection $\pi:\mathcal{M}_6\to\mathcal{M}_4$. Over each boundary point $x$ sits the non-spatial fiber $(I_r,S^1_\theta)$. The compact \emph{phase} circle $S^1_\theta$ carries the coherent, unitary content of quantum mechanics --- its holonomy is the Aharonov--Bohm/Berry phase \citep{AharonovBohm1959, Berry1984} --- while the non-compact \emph{scale} interval $I_r$ carries the irreversible, open content, its forward renormalization-group semigroup being the GKLS/\GDOS{} generator. Because both fiber coordinates are non-spatial, no boundary protocol resolves them; every registration marginalizes the fiber, so $\pi$ is non-invertible and the boundary dynamics is necessarily open. Unitary closure is the unreachable limit in which the fiber is not coarse-grained.}
\label{fig:bulk-projection}
\end{figure}

\subsection{Step 2 --- what follows: \GDOS{}, and a dynamics that does not run backwards}
\label{ssec:gdos}
Given the projection, what kind of dynamics does the boundary have?

\citet{ArisakaA5GDOS} answers in stages, and the staging matters because
it is where the honesty is. The reduction is completely positive and
trace preserving --- that much follows from the axiom. The exact
evolution is then a memory-kernel equation with no approximation
whatsoever. Only in a stated regime, where the environment's correlation
time is short compared with the system's, does the evolution take the
familiar semigroup form with a Lindblad generator
\citep{Lindblad1976,GoriniKossakowskiSudarshan1976}; and the paper is
explicit that this is a limiting corollary rather than a starting point.

The consequence for this paper is the loss of invertibility. A forward
semigroup has no inverse: information that has left is not recoverable
from the boundary, and the evolution has a direction.

Here the register must be exact, because it is a place where a
program of this kind could easily be read as claiming more than it
does. \emph{The arrow of time is not derived.} \citet{ArisakaA5GDOS}
states plainly that the arrow is constitutional rather than emergent ---
an axiom, not a theorem --- and locates it in AX-3. What is derived is
everything downstream: that the semigroup has no inverse, that a
monotone quantity increases, that the second law follows as information
loss rather than as a new postulate. A reader who was hoping to see the
arrow of time explained will not find it here, and should not be told
otherwise.

\emph{Physics intuition.} The picture is thermodynamic rather than
mechanical, and deliberately so. A closed system's state moves along a
trajectory that could be run backwards. A boundary state relaxes: it
loses distinctions, it forgets, and it settles. What Part~II's rungs
protect during that relaxation is the subject of the next two sections.


\subsection{Step 2 (continued) --- the pivot: what the books must show}
\label{ssec:pivot}
Now the pivot, and it can be stated in one sentence.

\begin{quote}
\emph{In a closed system, a conservation law says what cannot leave. In
an open system, it says what the books must show when something does.}
\end{quote}

Both are the same law. The difference is what one is entitled to do with
it.

For a closed system the law is a prohibition, and its use is to rule out
processes: this cannot happen, because the quantity would have changed.
For an open system the prohibition is empty --- anything can leave, it is
open --- and the law becomes an accounting requirement: whatever left
must appear on the other side of the ledger, in the environment, in the
bulk, in the record. Nothing is forbidden; everything is charged for.

This is the sense in which the title of this paper is not a metaphor.
The chapters of Part~II traced conservation from a prohibition on motion
(Noether) through a series of statements that prohibit less and less,
ending at an integer that prohibits nothing at all and merely cannot be
moved. Read that sequence again with the open-system reading in hand and
it has a destination: what survives at the top of the ladder is exactly
what a ledger identity looks like, and a ledger identity is what an open
system's conservation law \emph{is}.


\subsection{Step 2 (concluded) --- the open Ward identity: the books, written out}
\label{ssec:open-ward}
The previous section said what an open system's conservation law has to
do. It can be written down.

Chapter~\ref{sec:ward} gave the closed-system statement: the divergence
of a current, inside any correlation function, is a sum of contact terms
fixed by the charges of the operators present, and nothing else. Open the
system and one term is added:
\begin{equation}
\partial_\mu \big\langle J^\mu \big\rangle
 \;=\; \underbrace{(\text{contact terms})}_{\text{kinematics, as before}}
 \;+\; \underbrace{(\text{flux through the seam})}_{\text{what left}} .
\label{eq:open-ward}
\end{equation}
Nothing on the left has changed. The identity has not been repealed, and
it has not been weakened. What has changed is that the right-hand side is
no longer pure kinematics: it carries a term that records a transfer.
Figure~\ref{fig:seam} draws the seam that term is measured across, and
the three data of \S\ref{ssec:third-datum} making the crossing.

Read Eq.~\eqref{eq:open-ward} twice, as Chapter~\ref{sec:ward} read its
closed-system ancestor. In a closed system the second term is absent and
the equation says \emph{this quantity cannot leave}. In an open one it is
present and the equation says \emph{whatever leaves must appear here}.
The first is a prohibition. The second is an audit.

That is the whole content of the pivot, and it is why this paper is
called what it is. A member paper carries the corresponding statement for
the energy--momentum tensor of this program, in the same shape, with
the transfer terms named \citep{ArisakaA5GDOS}.

\emph{Grade.} Equation~\eqref{eq:open-ward} is a \emph{theorem} wherever
the seam is specified: it is Chapter~\ref{sec:ward}'s identity with a
source, and the derivation is unchanged. What is a \emph{proposal} is the
claim of \S\ref{ssec:oggep} that the world's observed sector is such a
boundary. The equation is not evidence for the proposal; it is what the
proposal would mean.

\subsection{Step 3 --- what it buys: admissibility becomes exclusion}
\label{ssec:exclusion}
The pivot has a consequence for counting, and it is the reason Part~III
exists rather than being an epilogue.

Consider what a closed-system method can establish about an integer such
as the number of matter families. Anomaly cancellation says: a theory
with this content is consistent. Integrality says: this class lives in
the right lattice. Both are \emph{admissibility} conditions --- they
partition candidate theories into the consistent and the inconsistent,
and they are silent about which of the consistent ones obtains. A
world with four families that canceled its anomalies would pass every
such test.

To exclude a consistent alternative one needs something that
distinguishes among admissible options, and a closed unitary description
has nothing to offer: its states are all equally admissible, and no
dynamics inside it prefers one to another. What can distinguish them is a
\emph{cost}. If reaching a configuration requires paying something from a
finite account, then configurations that cannot be afforded are excluded
--- not as inconsistent, but as unreachable.

That is an open-system argument, because an account is a boundary
notion: something must be spent, and spending is exchange.
\citet{ArisakaP2N3} makes the argument concretely, and it is worth
attributing precisely --- this framing is that paper's, not the boundary
paper's. Its observation is that a budget alone bounds nothing, since an
allowance growing linearly with the count excludes no count by itself;
what bounds is a budget set against a \emph{toll ladder} that grows
faster. Chapter~\ref{sec:grip} is about the ladder and
\S\ref{ssec:three-gen} about the resulting bound.

\emph{Physics intuition.} The distinction is the ordinary one between
what the law permits and what one can afford. A building code is an
admissibility condition; a budget is an exclusion. Physics has been very
good at the first kind of statement and has had almost nothing to say of
the second kind, and the claim here is that this is a consequence of
having formulated every fundamental law for closed systems.


\subsection{Step 3 (continued) --- where the Born rule sits}
\label{ssec:born}
One import needs its status stated with unusual care, because it is the
one a hostile reader will test first and the one most easily misread in
either direction.

The Born rule --- that the probability of an outcome is the squared
modulus of an amplitude --- is in this program a \emph{result} rather
than a postulate. \citet{ArisakaA8QM} is the home text. The status has
two parts and they carry different grades. The Born \emph{form} is
settled: given the operational dictionary of \S\ref{ssec:oggep}, a
Gleason-type argument \citep{Gleason1957} fixes the functional form of
any probability
assignment on it. The \emph{dynamical realization} --- that the descent
of Chapter~\ref{sec:grip} into a basin actually produces those weights
--- is closed at the relaxation level, and the Constitutional Born
Theorem that assembles it is conditional on four named premises: the
forward semigroup of \S\ref{ssec:gdos}, a Haar anchor with its recursion,
genericity of the descent, and the basin partition of
Chapter~\ref{sec:grip}. One analytic residue remains outstanding, named
in that paper as the field-grade representation step. There is a
falsifier attached, which is the mark of the register the cohort works
in: epochs before relaxation completed could leave relic non-Born
statistics in primordial correlations.

\paragraph{The direction of dependence.} This is the sentence the
chapter exists to state correctly. The dependence runs
\begin{equation}
\begin{aligned}
\text{axioms}
 &\;\longrightarrow\; \GDOS{}
  \;\longrightarrow\; \text{the Born result} \\
 &\;\longrightarrow\; \text{the adiabatic theorems of the family sector}
  \;\longrightarrow\; \text{the Berry bundle} \\
 &\;\longrightarrow\; \text{the bridge's boundary input,}
\end{aligned}
\label{eq:dependence}
\end{equation}
forward, in that order, and never backwards. Every arrow consumes what
is to its left. In particular the $\kappa$-bridge of
\S\ref{ssec:kappa} sits at the right-hand end of the chain and takes
boundary quantum mechanics as an input; it does not supply it.

\emph{Nothing in this paper founds quantum mechanics.} Not this chapter,
not Part~III, not the ladder of Part~II. A reader who wants to know
whether the program derives quantum mechanics should read
\citet{ArisakaA8QM} and judge that paper on its own terms; what is
imported here is a result at a declared grade, with its premises named
and its one residue named, and the review would be worthless if it
presented the matter any other way.


%% ---- FIGURE 2 ----
\begin{figure}[H]
\centering
\resizebox{0.741\textwidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
  \definecolor{bulkc}{HTML}{2F6F4E}
  \definecolor{bdyc}{HTML}{7A4E2D}
  %% ---- the two regions, set well apart --------------------------
  \fill[bulkc!7] (0,1.0) rectangle (4.5,4.6);
  \draw[bulkc,line width=0.9pt] (0,1.0) rectangle (4.5,4.6);
  \node[bulkc,anchor=north west,font=\scriptsize\bfseries,align=left,
        text width=3.9cm] at (0.3,4.4) {the bulk\\ six dimensions};
  \node[anchor=north west,text=black!65,font=\scriptsize,align=left,
        text width=3.9cm] at (0.3,3.55)
    {geometry and gauge are\\ one object (\GGEP{}, AX-2);\\
     the bookkeeping is exact};

  \fill[bdyc!8] (10.6,1.0) rectangle (15.1,4.6);
  \draw[bdyc,line width=0.9pt] (10.6,1.0) rectangle (15.1,4.6);
  \node[bdyc,anchor=north west,font=\scriptsize\bfseries,align=left,
        text width=3.9cm] at (10.9,4.4) {the boundary\\ four dimensions};
  \node[anchor=north west,text=black!65,font=\scriptsize,align=left,
        text width=3.9cm] at (10.9,3.55)
    {what is registered\\ (\OGGEP{}, AX-3);\\
     the dynamics is open (\GDOS{})};

  %% ---- the seam: drawn at the bulk's edge, so that it never
  %% ---- crosses a label; labelled horizontally, above ---------------
  \draw[black!45,line width=1.2pt,dash pattern=on 3pt off 2pt]
    (4.85,0.75) -- (4.85,4.85);
  \node[black!55,anchor=south west,font=\scriptsize\bfseries] at (4.9,4.95)
    {the seam: a projection, not a window};

  %% ---- the three data, in the open channel ------------------------
  \foreach \y/\name/\fate in {%
     3.55/{charge --- an integer class}/{crosses intact},
     2.45/{flux --- a curvature form}/{crosses, and may vary},
     1.35/{holonomy --- a phase on a loop}/{read \emph{on} the edge, not carried to it}}
  {
    \draw[-latex,black!60,line width=0.8pt] (5.15,\y) -- (10.35,\y);
    \node[anchor=south,font=\scriptsize] at (7.75,\y+0.09) {\name};
    \node[anchor=north,font=\scriptsize,text=black!58] at (7.75,\y-0.11)
      {\fate};
  }
  %% ---- the loop the holonomy is read on ---------------------------
  \draw[bdyc,line width=0.7pt] (14.35,1.35) circle (0.3);
  \draw[-latex,bdyc,line width=0.7pt] (14.65,1.35) arc (0:55:0.3);
\end{tikzpicture}}
\caption[The seam: bulk, boundary, and the three data crossing it.]{\textbf{The
seam.} The three data of \S\ref{ssec:third-datum} crossing from a
six-dimensional bulk to a four-dimensional boundary. The asymmetry is
the point: the integer class crosses intact, the curvature form crosses
and may vary, and the holonomy --- being a statement about a loop --- is
evaluated \emph{on} the boundary rather than transported to it. A closed
theory can ignore the third row. An open one cannot
(\S\ref{ssec:pivot}).}
\label{fig:seam}
\end{figure}


\subsection{Step 3 (continued) --- what it leaves open, and the grade box}
\label{ssec:oggep-grade}
\begin{quote}\small
\textbf{Grade box: Chapter~\ref{sec:oggep}.}

\emph{Axiom.} \OGGEP{} itself --- that what is observed is a boundary
reading of a bulk, and that the reading is one-way. The arrow of time
enters here, as an axiom and not a theorem, and \S\ref{ssec:not-a-law}'s
distinction applies to it: nothing in this chapter derives the
direction, it is assumed and then used.

\emph{Theorem, cohort.} That the reduced boundary evolution is
completely positive and trace-preserving, and that in the Markovian
limit its generator takes the GKLS form
\citep{ArisakaA5GDOS,GoriniKossakowskiSudarshan1976,Lindblad1976}.
Imported with its hypotheses, not re-derived.

\emph{Derived-conditional.} The Born rule, at the relaxation level, on
the four named premises of \S\ref{ssec:born}. One analytic residue --- the
field-grade representation of the channel --- remains open, and is
named as such in the parent paper \citep{ArisakaA8QM}.

\emph{This paper's contribution.} The placement: that this is the same
one-wayness the seventh rung of Part~II met as an integer that cannot
move, seen from the other side.
\end{quote}

\section{G3 --- \GRIP{}: the cascade \DGI{} $\to$ \GSSB{} $\to$ \GA{} on the \GFF{} landscape}
\label{sec:grip}
Chapter~\ref{sec:oggep} argued that exclusion requires a cost. This
chapter is about what does the spending.

A word on names first, because the literature's and this paper's differ.
The cohort groups four pillars together \citep{ArisakaA6Gi4}; two of
them matter here. The \GFF{} --- a geometric free-energy functional on
the boundary's moduli space --- is the \emph{landscape}. \GRIP{}, the
gauge-runtime invariance pipeline, is the \emph{descent} of that
landscape, in four moves, \DGI{} $\to$ (\GFF{}) $\to$ \GSSB{} $\to$ \GA{},
with \DGI{} at the top, \GSSB{} in the middle,
\GA{} at the bottom. So \GFF{} is not a stage of \GRIP{}; it is the
terrain \GRIP{} moves across. This chapter treats them together because
for a reader they are one mechanism, but the distinction is the owning
paper's and is worth keeping straight. Figure~\ref{fig:grip-engine} shows the whole engine at once --- the four-move descent, the two regimes, and the ground state where it halts --- and the rest of this chapter walks its moves in order.


\begin{figure}[p]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\Large,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 dgibox/.style={fill=red!13, draw=red!60!black, line width=0.7pt,
 minimum width=80mm, minimum height=18mm},
 gffbox/.style={fill=yellow!13, draw=olive!60!black, line width=0.7pt,
 minimum width=80mm, minimum height=18mm},
 gssbbox/.style={fill=green!14, draw=green!60!black, line width=0.7pt,
 minimum width=80mm, minimum height=18mm},
 ggabox/.style={fill=blue!14, draw=blue!60!black, line width=1.0pt,
 minimum width=80mm, minimum height=18mm},
 gabox/.style={fill=teal!12, draw=teal!60!black, line width=0.8pt,
 minimum width=80mm, minimum height=18mm},
 inbox/.style={fill=cyan!16, draw=cyan!60!black, line width=0.9pt,
 minimum width=60mm, minimum height=18mm},
 outbox/.style={fill=orange!16, draw=orange!70!black, line width=1.0pt,
 minimum width=60mm, minimum height=18mm},
 nextbox/.style={fill=red!8, draw=red!50!black, line width=0.65pt,
 minimum width=60mm, minimum height=16mm,
 dash pattern=on 2.5pt off 2pt},
 bigarrow/.style={-{Stealth[length=2.8mm]}, line width=0.7pt, draw=black!75},
 ribarrow/.style={-{Stealth[length=2.6mm]}, line width=0.85pt, draw=cyan!55!black},
 outarrow/.style={-{Stealth[length=2.8mm]}, line width=1.1pt, draw=orange!70!black},
 axlabel/.style={font=\normalsize\itshape, text=blue!50!black,
 fill=white, inner sep=2pt, align=right, anchor=east},
 qilabel/.style={font=\normalsize\itshape, text=green!50!black,
 fill=white, inner sep=2pt, align=left, anchor=west},
 arrowtext/.style={font=\normalsize\itshape, text=black!70,
 fill=white, inner sep=1pt, align=center},
]
\node[draw=none, inner sep=0pt] at (-95mm, 0) {};
\node[draw=none, inner sep=0pt] at (95mm, 0) {};
\node[dgibox] (DGI) at (0, 0) {\textbf{\DGI{} \quad (Layer $n$)}\\[-1pt]
 {\normalsize Dynamical Gauge Invariance --- the unbroken bulk substrate, carrying $J_{\rm Qi}$ on $\mathcal{M}_6$ under $U(1)_{\rm Qi}$}};
\node[gffbox] (GFF) at (0, -32mm) {\textbf{$\bm{(F_{\GFF{}})}$ \quad Geometric Free-Energy Functional}\\[-1pt]
 {\normalsize Lyapunov landscape on the 4D boundary, $\,dF_{\GFF{}}/dt \leq 0$}};
\node[gssbbox] (GSSB) at (0, -64mm) {\textbf{\GSSB{} \quad Geometric Spontaneous Symmetry Breaking}\\[-1pt]
 {\normalsize branch commitment ($\mathbb{Z}_2$-orbifold-fixed spectrum) $\to$ discrete admissible boundary observables}};
\node[gabox] (GA) at (0, -96mm) {\textbf{\GA{} \quad Geometric Attractor(s)}\\[-1pt]
 {\normalsize stable basins en route --- possibly way stations (a muon, an excited atom); \GRIPin{} continues through their exit channels}};
\node[ggabox] (GGA) at (0, -128mm) {\textbf{\GGA{} \quad Ground Geometric Attractor}\\[-1pt]
 {\normalsize layer's unique deepest stable Morse minimum on the four-pillar-admissible stratum --- \emph{where \GRIPin{} halts}}};
\draw[bigarrow] (DGI.south) -- (GFF.north);
\draw[bigarrow] (GFF.south) -- (GSSB.north);
\draw[bigarrow] (GSSB.south) -- (GA.north);
\draw[bigarrow] (GA.south) -- node[arrowtext, right=2pt]{exit channels: the descent continues} (GGA.north);
\node[axlabel] (AX2GGEP) at ($(DGI.west) + (-5mm, 0)$)
 {\textbf{\AX{2}~\GGEP{}}\\(GG-6 / \BIsix{})};
\node[axlabel] at ($(GFF.west) + (-5mm, 0)$)
 {\textbf{\AX{3}~\OGGEP{}}\\(\GDOS{})};
\node[axlabel] at ($(GSSB.west) + (-5mm, 0)$)
 {\textbf{\AX{4}~\MSMF{}}\\acts on \GSSB{}};
\node[qilabel] at ($(DGI.east) + (5mm, 0)$)
 {\textbf{\QiQUA{}}~(\QGL)\\Arisaka equation\\$J_E/\QB = J_e = J_B = J_{\rm Qi}$};
\node[qilabel] at ($(GFF.east) + (5mm, 0)$)
 {\textbf{\QiEP{}}\\anchored on $N_A\QB \approx 15$~kJ/mol\\drives gradient descent};
\node[qilabel] at ($(GSSB.east) + (5mm, 0)$)
 {\textbf{\QiCON{}}\\conformality diagnostic};
\node[qilabel] at ($(GA.east) + (5mm, 0)$)
 {way stations:\\$\mu$, $\tau$, excited states\\(non-ground \GA{}s)};
\node[qilabel] at ($(GGA.east) + (5mm, 0)$)
 {\textbf{\QiHAL{}}\\holographic attractor\\architecture};
\node[axlabel] at ($(GGA.west) + (-5mm, 0)$)
 {\textbf{where \GRIPin{} halts}\\(\textsf{T-GGA-No-Exit})};
\node[inbox] (GIN) at (-65mm, -166mm) {\textbf{\GRIPin{}}\\[-1pt]
 {\normalsize within-layer regime}\\
 {\normalsize loops through way-station \GA{}s until the \GGA{}}};
\node[outbox] (GOUT) at (65mm, -166mm) {\textbf{\GRIPout{}}\\[-1pt]
 {\normalsize between-layer regime}\\
 {\normalsize out-of-equilibrium shock}};
\draw[ribarrow] (GA.west) to[bend right=25]
 node[arrowtext, pos=0.55, below, sloped]{\GFF{} landscape unchanged}
 (GIN.north);
\draw[outarrow] (GGA.south) to[bend left=20]
 node[arrowtext, pos=0.55, below, sloped]{\GFF{} landscape reorganized}
 (GOUT.north);
\draw[ribarrow, dashed] (GIN.north) to[out=90, in=-90]
 node[arrowtext, pos=0.28, left=2pt, text=cyan!55!black, font=\normalsize\itshape]
 {within-layer\\closed loop}
 (AX2GGEP.south);
\node[nextbox] (NEXT) at (65mm, -202mm)
 {\textbf{\DGI{} \quad (Layer $n+1$)}\\[-1pt]
 {\normalsize next-layer cascade begins}};
\draw[outarrow] (GOUT.south) -- (NEXT.north)
 node[arrowtext, pos=0.5, right=3pt, text=orange!70!black, font=\normalsize\itshape]
 {ascent\\Layer $n \to n+1$};
\end{tikzpicture}}
\caption[The \GRIP{} engine: four moves, two regimes, one ground state (reproduced from \citealp{ArisakaA11Origin}).]{\textbf{The \GRIP{} engine: four moves, two regimes, one ground state.} The central column is the four-move descent \DGI{} $\to$ $(F_{\GFF{}})$ $\to$ \GSSB{} $\to$ attractor; the three constitutional axioms (left) force the moves and the four \QiFour{} pillars (right) filter the Morse minima. \GRIPin{} (cyan) loops within a layer to the unique ground state; \GRIPout{} (orange) reorganizes the landscape and opens the next floor. The four pillar names on the right (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) and the minimality axiom \MSMF{} on the left are stated in Chapter~\ref{sec:qi4}; nothing before that chapter uses them. The four moves are walked in order through the rest of this chapter (\S\S\ref{ssec:dgi}--\ref{ssec:grip-intuition}).}
\label{fig:grip-engine}
\end{figure}

\subsection{Step 1 --- what is assumed: \DGI{}, and what is preserved along the flow}
\label{ssec:dgi}
\DGI{} is the top of the descent: the unbroken configuration, at which
the full symmetry is intact and nothing has yet been committed. In the
Morse language of the next section it is the critical point of maximal
index --- the least stable place on the landscape, the one from which
every direction leads downward.

What the stage contributes to this paper's theme is the word
\emph{invariance} in its name. Gauge covariance is preserved at every
step of the descent; the pipeline is not permitted to break the
redundancy of Chapter~\ref{sec:gauge}, which by Elitzur's theorem it
could not do in any case. What gets broken along the way is something
else, and \S\ref{ssec:gssb} says what.


\subsection{Step 1 (continued) --- \GFF{}: the landscape being descended}
\label{ssec:gff}
The landscape is a functional on the boundary's moduli space, and the
structure that matters is that it is a \emph{Morse} function: its
critical points are isolated and each carries an index, the number of
independent downhill directions.

For the family sector the relevant stratum is the complex projective
space $\CP^{N_g-1}$, and its critical points have even index. For $N_g = 3$
that means three of them, of index $4$, $2$ and $0$ --- the top, one
intermediate saddle, and the minimum. This is Figure~\ref{fig:cascade},
and the reader may wish to look at it now.

\paragraph{Worked example: the two-dimensional cartoon.} Strip away the
projective geometry and keep the shape. Take a height function on an
ordinary surface with three critical points: a peak, a pass, and a
basin. A ball released at the peak rolls to the pass and then to the
basin, and the descent is completely determined by the landscape. Now
add the feature that makes this physics rather than mechanics: each
passage costs something, drawn from an account fixed in advance. The
question ``how many families?'' becomes the question ``how many passages
can be paid for?'', and it has an answer as soon as the tolls and the
account are known.


\subsection{Step 2 --- what follows: \GSSB{}, the breaking step}
\label{ssec:gssb}
The middle stage is where the symmetry goes.

The intermediate critical points are saddles, and passing one is a
partial breaking: a configuration that had been symmetric under some
subgroup ceases to be, and the descent commits to one basin rather than
another. In the family sector these saddles are not decoration --- the
cohort identifies the index-4 point with the third generation, the
index-2 saddle with the second, and the index-0 minimum with the first
\citep{ArisakaA6Gi4}, so that the generations \emph{are} the stages of
the descent rather than three copies of a thing.

\emph{This paper's reading, and it is a reading.} The thesis stated in
\S\ref{ssec:promise} is that in this architecture breaking is the
productive act rather than a defect --- that it is the mechanism that
selects a world. It should be said plainly where this formulation comes
from: the paper that owns \GSSB{} describes what it does without
characterizing it as productive, and the characterization is this
review's, though the cosmology paper now uses the same word for the one
breaking this review cares most about \citep{ArisakaA10Cosmos}. A reader
who dislikes the word may drop it and lose no result. What
survives without the word is the structural observation, which is not a
reading at all: the breaking sits in the \emph{middle} of the pipeline,
with stages before and after, and the count of the world's families is a
count of breakings. In the classical story breaking is what happens to a
theory at the end. Here it is a step in a process that continues past it.


\subsection{Step 2 (continued) --- \GA{}: attractors, and particles as ground states}
\label{ssec:ga}
The descent terminates at a minimum: no downhill direction remains, and
the configuration is stable. That endpoint is a \GA{}, a geometric
attractor, and where the stratum admits a unique deepest basin the
descent halts there.

The claim the cohort attaches to this is ontological, and it is worth
stating because it is the most unfamiliar thing in Part~III: a particle
is a ground state of the descent. Not an excitation carried on a
background, and not a point with attributes attached, but the endpoint
of a relaxation --- the configuration a boundary settles into when it has
finished paying.

\emph{Physics intuition.} The reading has an immediate consequence that
makes it testable in principle rather than merely evocative. If a
particle is an attractor, then its properties are properties of a basin,
and its stability is the stability of a minimum. Questions like ``why is
the electron stable?'' and ``why are there exactly this many kinds?''
stop being questions about a list and become questions about a
landscape, which is the sort of question that can have a derivation.
Whether the derivation succeeds is the subject of
Chapter~\ref{sec:payoffs}, and its conditioning is printed there.


\subsection{Step 2 (concluded) --- a \GA{}, and \emph{the} \GGA{}}
\label{ssec:gga-split}
Two statements are easy to run together here, and the difference between
them decides what the next chapter is for.

The first is that the cascade \emph{terminates}. Descending a landscape
by gradient flow, one arrives at a point from which every direction leads
upward: a local minimum, an attractor, a \GA{}. That statement needs
nothing but the cascade structure, and the owning paper marks it as the
one the structure alone supports \citep{ArisakaA6Gi4}.

The second is that a \emph{particular} attractor is selected --- the
deepest one on its stratum, the ground state, written \GGA{}. This is a
stronger claim, and it is not settled by the flow, because a landscape
with several basins does not tell a trajectory which basin to have
started in. Something has to select among the minima.

What the owning paper cites for that selection is the second group of
pillars, the subject of Chapter~\ref{sec:qi4}. The clause is explicit:
the \GGA{} is the attractor selected from among the \GA{}s by those
admissibility pillars.

\begin{quote}\itshape
Every load-bearing use of the cascade in this Part is a use of \emph{a}
\GA{}. The upgrade to \emph{the} \GGA{} imports Chapter~\ref{sec:qi4},
and is marked wherever it is made.
\end{quote}

That distinction is not pedantry. It is the seam along which this
review's fence runs, and it is the reason the fourth move is stated at
all rather than left as a footnote. A reader who stops at the end of
Chapter~\ref{sec:payoffs} keeps every counting result in this Part; what
that reader does not get is a claim that the attractor reached is the
deepest one available.

\subsection{Step 3 --- what it buys: a solvency condition, and a bound on families}
\label{ssec:solvency}
Now the accounting, which is where this chapter meets the paper's title.

Passing from a critical point of index $2k$ to one of index $2(k-1)$
costs an instanton action $\pi k$ \citep{ArisakaA6Gi4}. Summing the steps
down the whole cascade on $\CP^{N_g-1}$,
\begin{equation}
S_{\text{tot}}(N_g) \;=\; \sum_{k=1}^{N_g-1} \pi k
 \;=\; \frac{\pi N_g (N_g-1)}{2} ,
\label{eq:toll-ladder}
\end{equation}
which the reader will recognize: it is $\pi$ times the quadratic
refinement $q(N_g) = N_g(N_g-1)/2$ of \S\ref{ssec:qx}. The toll ladder of the
cascade and the integer model at the top of the ladder in Part~II are
the same function. That is the third of the ``one refinement, three
appearances'' noted in \S\ref{ssec:qx}, and \S\ref{ssec:kappa} is about
why it is not a coincidence.

Against this stands a capacity. In the sharper per-family form of
\citet{ArisakaP2N3}, the stratum earns $\pi$ of budget per family, so
that the solvency condition is
\begin{equation}
\frac{\pi N_g(N_g-1)}{2} \;\le\; \pi N_g .
\label{eq:solvency}
\end{equation}
Everything in \S\ref{ssec:three-gen} follows from comparing the two
sides, and the comparison is elementary.

The structural point is what this chapter was for. Conservation has
reappeared --- and not as a symmetry of the state the descent ends in,
which by construction has very little symmetry left. It has reappeared as
a condition on the \emph{whole descent}: the tolls paid must not exceed
the capacity earned. That is a ledger identity of a process, not an
invariance of a configuration, and it is the form conservation takes in
an open system, exactly as \S\ref{ssec:pivot} said it would be.


\subsection{Physics intuition: a landscape, a descent, and a bill}
\label{ssec:grip-intuition}
Three pictures, in the order the chapter used them.

\emph{The landscape} is fixed before anything moves. It is not adjusted
to fit the answer; it is a functional on a moduli space whose shape the
geometry determines, and its critical points are wherever they are.

\emph{The descent} is what a boundary does because it is open. It has a
direction for the reason given in \S\ref{ssec:gdos} --- the semigroup has
no inverse --- and it stops when there is nowhere lower to go.

\emph{The bill} is the part with no analogue in the closed-system
picture, and it is the part that does the physical work. A closed theory
can say that a configuration is consistent; only the bill can say that a
configuration is unaffordable. Every exclusion this program claims is
of the second kind.

One more observation, and it is the reason
Chapter~\ref{sec:one-argument} can treat Parts~II and III as one
argument rather than two. The tolls are quantized --- each is an integer
multiple of $\pi$ --- and the reason they are quantized is the reason
anything on the top rung of Part~II is: they are instanton actions, and
instanton actions are topological. The cascade does not merely have a
budget. It has a budget denominated in the currency of
Chapter~\ref{sec:hs}.


%% ---- FIGURE 3 ----
\begin{figure}[H]
\centering
\resizebox{0.755\textwidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
  \definecolor{casc}{HTML}{2F6F4E}
  \definecolor{ledg}{HTML}{7A4E2D}
  %% --- the landscape: three critical points, descending -------------
  \draw[casc,line width=1.1pt]
    (0,4.3) .. controls (1.1,4.3) and (1.4,3.1) .. (2.5,3.1)
            .. controls (3.6,3.1) and (3.9,1.5) .. (5.0,1.5)
            .. controls (6.1,1.5) and (6.4,1.5) .. (7.0,1.5);
  \foreach \x/\y/\lab/\idx in {%
     0.35/4.3/{\DGI{}}/{index 4},
     2.5/3.1/{\GSSB{}}/{index 2},
     5.9/1.5/{\GA{}}/{index 0}}
  {
    \fill[casc] (\x,\y) circle (2.2pt);
    \node[anchor=south,font=\scriptsize\bfseries,text=casc] at (\x,\y+0.12)
      {\lab};
    \node[anchor=north,font=\scriptsize,text=black!60] at (\x,\y-0.12)
      {\idx};
  }
  \node[anchor=south west,font=\scriptsize,text=black!65,align=left]
    at (-0.15,5.05) {the \GFF{} landscape, descended by \GRIP{}};
  %% --- the two tolls ------------------------------------------------
  \draw[-latex,ledg,line width=0.8pt] (1.45,4.05) -- (1.45,3.35);
  \node[anchor=west,font=\scriptsize,text=ledg] at (1.55,3.7) {toll $2\pi$};
  \draw[-latex,ledg,line width=0.8pt] (4.0,2.85) -- (4.0,1.75);
  \node[anchor=west,font=\scriptsize,text=ledg] at (4.1,2.3) {toll $\pi$};
  %% --- the ledger bar ----------------------------------------------
  \draw[ledg,line width=0.9pt] (10.6,1.3) rectangle (11.7,4.6);
  \fill[ledg!30] (10.6,1.3) rectangle (11.7,3.4);
  \fill[ledg!55] (10.6,3.4) rectangle (11.7,4.6);
  \node[anchor=south,font=\scriptsize\bfseries,text=ledg] at (11.15,5.05)
    {the budget: $3\pi$};
  \node[anchor=west,font=\scriptsize,text=black!70] at (11.85,4.0)
    {$2\pi$ spent at the first gate};
  \node[anchor=west,font=\scriptsize,text=black!70] at (11.85,2.35)
    {$\pi$ spent at the second};
  \node[anchor=north,font=\scriptsize,text=black!70,align=center]
    at (11.15,1.15) {nothing left over};
  \draw[black!45,line width=0.7pt,dash pattern=on 2pt off 2pt]
    (10.3,1.3) -- (12.0,1.3);
\end{tikzpicture}}
\caption[The cascade with its budget.]{\textbf{The cascade with its
budget.} The three-generation case, drawn to scale. The \GFF{} landscape
on the family stratum has critical points of Morse index $4$, $2$ and
$0$; the descent pays $\pi k$ to pass from index $2k$ to $2(k-1)$, so
$2\pi$ and then $\pi$; and the total $3\pi$ is exactly the budget. A
fourth family would need a further $3\pi$ and there is none
(\S\ref{ssec:three-gen}). Conservation reappears here not as a symmetry
of the endpoint but as the condition that the column on the right
balances.}
\label{fig:cascade}
\end{figure}


\paragraph{Which fiber does which job.} The cascade divides its labor
across the two fibers, and the division is worth stating once because it
is the same division \S\ref{ssec:breaking-source} draws for symmetry
breaking. The equivalences \DGI{} holds intact are the compact fiber's:
gauge structure lives on the circle, and its transports compose as a
group. The descent that executes \GSSB{} $\to$ \GA{} is the open fiber's:
it composes forward only, as a semigroup, which is why what it deposits
is final. \GSSB{} itself sits on the seam --- it reads circle data, runs
on the interval, and banks its result back into the classes the circle
protects. In one sentence: \emph{the circle states the alternatives, the
interval performs the selection, and the attractor is the circle's
receipt.} The runtime paper carries this as a proposition
\citep{ArisakaA6Gi4}; the fork it rests on is the subject of the
companion white paper \citep{ArisakaW2Holonomy}.


\subsection{Step 3 (continued) --- what it leaves open, and the grade box}
\label{ssec:grip-grade}
\begin{quote}\small
\textbf{Grade box: Chapter~\ref{sec:grip}.}

\emph{Axiom.} That the boundary carries a landscape at all --- a
functional whose critical points are where they are, fixed before
anything moves --- and that the descent across it is monotone.

\emph{Theorem, within its setting.} The toll ladder and the solvency
condition: that the cost of passing a gate of Morse index $2k$ is
$\pi k$, and that a cascade closes only if the tolls it pays do not
exceed the capacity it earns \citep{ArisakaA6Gi4}.

\emph{Import.} The budget itself, $3\pi$, from the capacity theorem of
\citet{ArisakaP2N3}, with that paper's coupling clause and integrality
lemma intact. Everything downstream of the budget inherits its
conditionality, including the exclusion of a fourth family.

\emph{Open.} Strict Morse positivity, and with it the upgrade from
\emph{a} attractor to \emph{the} attractor. The cascade is known to
stop; that it stops in one place is conditional.
\end{quote}

\section{G4 --- \QiFour{}: the extension, stated and fenced}
\label{sec:qi4}

The three moves so far are this program's account of physics. There is
a fourth, and it is of a different kind: an extension of the same
architecture to open systems that are not the subject of particle
physics at all --- chemical, living, cognitive.

This chapter states it. It does not defend it, and no argument anywhere
in this paper depends on it. It is here for one reason, and the reason is
specific: \S\ref{ssec:gga-split} showed that the cascade's terminus is
upgraded from \emph{a} minimum to \emph{the} minimum by a selection rule
that the owning paper attributes to this material. An import that
load-bearing should not be left in a footnote. Stating it, and fencing
it, is more honest than either defending it or omitting it.

\subsection{Step 1 --- what is assumed}
\label{ssec:qi4-assumed}
The owning paper is explicit that it adds no new axiom
\citep{ArisakaA7Qi4}. What it adds is a hypothesis about the bulk's
content: that among the abelian factors carried by the six-dimensional
geometry there is one that acts as the carrier of \emph{information} ---
that a bit is not a metaphor for a physical quantity but a charge under
a gauge factor, on the same footing as electric charge and baryon
number.

That is a single, sharp assumption, and its grade is stated in the
owning paper as plainly as one could ask. The four pillars, it says, are
theorems of the class of admissible theories in which that factor is
\emph{active} --- not of the constitution itself, which permits the
factor without requiring it. They are conditional theorems with an
experimental gate, and the paper prefers to say so.

\emph{Grade: axiom, conditional on an empirical question that is
open.} Nothing in Chapters~\ref{sec:ggep}--\ref{sec:grip} assumes it.

\subsection{Step 2 --- what follows: four pillars, in a chain}
\label{ssec:qi4-follows}
Granted the hypothesis, four statements follow in order, each forcing the
next rather than standing beside it. The owning paper names them
\QiQUA{}, \QiEP{}, \QiCON{} and \QiHAL{}, and describes their roles in a
sentence this review can quote without qualification: the first is the
bookkeeping layer, the second the selection layer, the third the
diagnostic layer, the fourth the architecture layer.

One feature of the first pillar can be checked from inside this paper,
and it is worth recording because it is the only point of contact.
The ratio it asserts among the four boundary currents is
$(1, -1, -2/5, -2/5)$ --- the coefficient vector of the modified identity
of \S\ref{ssec:bi6}, read on the boundary rather than in the bulk. The
claim is therefore not a new numerical input. It is the assertion that
the bulk's anomaly-cancellation ledger is also the boundary's
bookkeeping ledger, with information occupying the slot that the
identity already had.

Table~\ref{tab:qi4} lists the four pillars in order, with the grades
their owning paper gives them.

\begin{table}[H]
\centering
\caption[The second group of pillars.]{\textbf{The second group of
pillars, with the grades their owning paper gives them.} The grades
descend from left to right along the chain, and the owning paper says so
itself. This review reports them; it checks none of them. Four columns: the pillar, the layer it sits on, what it asserts, and the grade its owning paper gives it. The grade column is the one that matters here, and this review's discipline with it is stated in the caption rather than assumed --- the grades are reported as their owner declares them and none is checked.}
\label{tab:qi4}\small
\begin{tabular}{@{}P{0.13\textwidth}P{0.22\textwidth}P{0.36\textwidth}P{0.15\textwidth}@{}}
\toprule
\textbf{pillar} & \textbf{layer} & \textbf{what it asserts} &
\textbf{grade} \\
\midrule
\QiQUA{} & bookkeeping &
The four boundary currents --- energy, charge, baryon number,
information --- descend from a single one-dimensional source, so that
their ratios are fixed rather than independent &
theorem, on the hypothesis; its uniqueness step imported from
\citet{ArisakaA3BI6} \\
\QiEP{} & selection &
One projected variational principle, of which several familiar
boundary principles are gauge choices &
theorem, on a saddle-point reduction; the owning paper's strongest
reduction \\
\QiCON{} & diagnostic &
An intermediate scaling window with power-law correlators and
$1/f$-type spectra &
conditional; the owning paper calls the conformality
\emph{operational}, not mathematical \\
\QiHAL{} & architecture &
A memory-bearing open system factorizes into a fixed set of
coupled layers &
weakest of the four; downgraded by the owning paper from a necessity
claim to a canonical decomposition, and conditional on that paper's own
open conjectures \\
\bottomrule
\end{tabular}
\end{table}


\subsection{Step 3 --- what it buys, and what it leaves open}
\label{ssec:qi4-buys}
Three things, on the owning paper's account.

\emph{A constant.} The chain fixes a conversion factor between an
informational event and an energy, $Q_B = (2/5)\,E_{\mathrm{Ryd}}/K$ with
$K = 35$ the ledger integer of Eq.~\eqref{eq:ogn5}, numerically
about $0.16$~eV per bit at biological temperatures. Twice that, per mole,
is close to the free energy of hydrolysis of the cell's standard energy
currency --- an agreement of a few percent, on a quantity nobody adjusted.

\emph{A falsifier, and a sharp one.} The same statement is a prediction
about single events, and the owning paper states the kill criterion in
those terms: one elementary catalytic event --- a rotor step, a single
enzyme turnover, a single electron transfer --- whose four-way bookkeeping
disagrees by more than a stated tolerance refutes the first pillar.
Whatever else is true of this material, it is not unfalsifiable, and the
falsification class is a laboratory one.

\emph{A selection rule.} And, returning to why this chapter exists: the
pillars are what the owning papers cite when a cascade terminus is
upgraded from a minimum to the deepest minimum. That is the import
\S\ref{ssec:gga-split} marked.

What it leaves open is more than what it settles, and the owning paper
says so in a list. Whether the informational factor is realized at all
is open and is the experimental gate. The measurement protocols outside
particle physics do not yet exist. The functional form of the constant is
motivated rather than derived. The uniqueness of the architectural
factorization is open by that paper's own statement. The count of levels
in its hierarchy is an empirical reading rather than a theorem output.
And where the account reaches cognition it is restricted, explicitly, to
the availability and reportability of content --- the harder question is
recorded as a placeholder and not addressed.

\subsection{The fence}
\label{ssec:qi4-fence}
\begin{quote}\small
\textbf{What this review does and does not do with this chapter.}

\emph{It states it.} The four pillars, their chain, their grades as the
owning paper gives them, and the one point of contact with this paper's
own arithmetic.

\emph{It checks nothing.} No claim in this chapter has been verified
here, and none of the numbers quoted has been recomputed. Every grade in
Table~\ref{tab:qi4} is the owning paper's, reported.

\emph{Nothing above depends on it.} Parts~I and II are a review of
standard results and touch none of this. In Part~III, every load-bearing
use of the cascade is a use of \emph{a} \GA{}, in the sense of
\S\ref{ssec:gga-split}; the counting results of
Chapter~\ref{sec:payoffs} stand without this chapter. The one place the
program's own papers reach back for it is the upgrade to the deepest
attractor, and that upgrade is marked where it is made rather than
assumed.

\emph{A reader may stop before it.} A reader who finds the open-system
move of Chapter~\ref{sec:oggep} persuasive and this chapter unpersuasive
loses nothing in the rest of the paper. That was true when this material
was a single paragraph and it remains true now that it is a chapter; what
has changed is only that the import is visible.

\emph{And this review takes no position.} The extension is at a
different grade from anything else in this paper. It is reported because
a reader is entitled to know where the architecture's authors think it
leads, and because one clause of Chapter~\ref{sec:grip} cites it.
\end{quote}

\subsection{Grade box}
\label{ssec:qi4-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Established here.} Nothing. This chapter is a report.

\emph{Reported, at the owning paper's grade.} That the four pillars are
conditional theorems of the class of theories in which the informational
factor is active; that their chain runs in one direction; that the first
of them asserts the boundary ratio $(1, -1, -2/5, -2/5)$ already carried
by \S\ref{ssec:bi6}; that the constant and its falsifier are as stated.

\emph{Open, by the owning paper's own list.} Whether the factor is
realized; the measurement protocols; the functional form of the
constant; the uniqueness of the factorization; the level count; and the
whole of the question this review has been careful to call availability
rather than experience.

\emph{Not claimed anywhere in this paper.} That the extension is
correct, that it is supported, or that anything in
Chapters~\ref{sec:ggep}--\ref{sec:grip} would fail without it.
\end{quote}


\section{Three worked payoffs}
\label{sec:payoffs}
An architecture is judged by what it delivers and by how honestly the
delivery is described. This chapter takes three results, states each in
outline with its conditioning printed, and then names what is not
settled. None of it is derived here; all of it is imported, and the
member paper that owns each result is cited at the point of use.


\subsection{Why three generations: the pincer, with its conditioning printed}
\label{ssec:three-gen}
A word on the symbol first, because the cohort now distinguishes
three integers that all equal three. $N_c$ is the color index, which is
the \OGNfive{} integer $N$ of \S\ref{ssec:twelve}; $N_g$ is the number
of matter families, which is \emph{not} a member of that ledger and is
what this section is about; $N_f$ keeps its standard meaning, the number
of quark flavors in a beta function. That $N_c$ and $N_g$ are both $3$
is the conclusion of two derivations with no premise in common, and not
a premise of either \citep{ArisakaP2N3}.

The number of matter families is the cohort's sharpest claim, and it is a pincer: a lower bound and an upper bound arriving from unrelated
directions and meeting.

\emph{The lower bound.} A physical $CP$-odd phase in the quark mixing
matrix requires at least three families
\citep[the observation of][]{KobayashiMaskawa1973}; $CP$ violation is
observed; therefore $N_g \ge 3$. The input consumed is a single sign ---
that $CP$ violation exists --- rather than a fitted number. It is
nonetheless an empirical input, and \S\ref{ssec:failures} names the
debt: this leg runs on a \emph{broken} symmetry, and without the 1964
measurement there would be no bound at all.

\emph{The upper bound.} This is Eq.~\eqref{eq:solvency}. Canceling
$\pi N_g$ from both sides gives $N_g - 1 \le 2$, so
\begin{equation}
N_g \le 3 .
\end{equation}
Three lines of arithmetic, and the content is entirely in where the two
sides came from: the left from the Morse structure of the landscape, the
right from the capacity theorem of \citet{ArisakaP2N3}.

\emph{Together.} $N_g \ge 3$ and $N_g \le 3$, so $N_g = 3$. A third leg, a
constructive check against the measured charged-fermion masses, is
reported in the same paper and is a cross-check rather than a third
bound.

\paragraph{The conditioning, printed.} Three fences, and a reader
should hold the result at exactly this strength and no more.
\begin{enumerate}[leftmargin=1.5em,itemsep=0.25em]
\item \emph{An empirical input.} The lower leg consumes the observed
  non-vanishing of $CP$ violation.
\item \emph{A coupling clause.} The upper leg assumes that the inflow
  prices the family stratum through that stratum's own quantum geometry.
  This is a premise at the axiom level, not a theorem, and
  \S\ref{ssec:kappa} is about the member paper that closes the distance
  to it on a benchmark background.
\item \emph{A topological residue.} One integrality statement,
  registered by name, on which the lattice presentation of the bound
  depends.
\end{enumerate}
The program's own papers state all three, and this review would be
misrepresenting them if it reported $N_g = 3$ without them.


\subsection{The seam agreeing with itself: the $\kappa$-bridge in one page}
\label{ssec:kappa}
The second payoff is the one this paper is best placed to explain,
because Part~II built the machinery for it.

The question is whether the coupling clause of \S\ref{ssec:three-gen} is
merely assumed or is enforced by something. \citet[Appendix~H]{ArisakaP2N3} takes the six-dimensional differential data of the modified identity --- the object of \S\ref{ssec:bi6} --- and pushes it forward through the \HS{} quadratic functor of \S\ref{ssec:wu}, on a benchmark background of
the intended class. The degree count is the one \S\ref{ssec:wu} did: a
fiber dimension of six sends degree-four data to degree two on the base,
so the output is a line bundle on the family moduli space.

What is it? The boundary, quite independently, constructs a line bundle
on the same space: the Berry bundle \citep{Berry1984} of its own adiabatic
transport, which
is a piece of ordinary quantum mechanics on the four-dimensional side and
knows nothing about six dimensions. The result is that the two agree ---
the bulk, pushed forward, hands the boundary the very bundle the
boundary already had.

The reason this matters can be put without any of the machinery. Two
descriptions with no shared working parts, one topological and
six-dimensional, one quantum-mechanical and four-dimensional, are made
to compute the same object, and they return the same answer. Agreement
of that kind is evidence in a way that a successful fit is not, because
there was no dial to turn. And the phrase for what it achieves is exact:
a clause that had to be assumed becomes, on this background, something
enforced.

Two cautions, both from the owning paper. The result holds on a
\emph{benchmark} background, not in general; the residues that would be
needed off it are named there and remain open. And the direction of
dependence of \S\ref{ssec:born} applies here with full force: the bridge \emph{consumes} boundary quantum mechanics --- the Berry bundle is an input --- and cannot be read as founding it.


\subsection{What remains open, named}
\label{ssec:open}
The third payoff is a list of what is not settled. It belongs in the
body rather than in a footnote, and each entry is registered in the
member paper that owns it.

\begin{itemize}[leftmargin=1.4em,itemsep=0.3em]
\item \emph{The last analytic residue in the Born chain}: the
  field-grade representation step (\S\ref{ssec:born}). Until it closes,
  the Constitutional Born Theorem stands conditional on its four named
  premises.
\item \emph{The $\kappa$-bridge off its benchmark}: the residues that
  would extend \S\ref{ssec:kappa} to general backgrounds of the intended
  class are named and open.
\item \emph{The minimality axiom} (\S\ref{ssec:exposure}): not a residue
  but a premise, and the one this paper judges most exposed. It selects
  $k = 1$ in the family that the trace ratio leaves free.
\item \emph{Morse positivity} on the relevant strata, which
  \citet{ArisakaA5GDOS} registers as an open problem of its own and which
  the landscape of \S\ref{ssec:gff} assumes.
\item \emph{The arrow of time} (\S\ref{ssec:gdos}): not open, but
  \emph{assumed} --- constitutional rather than derived. A reader hoping
  for an explanation of temporal asymmetry should know that this
  program postulates it.
\end{itemize}

A reader who wants a single sentence for the state of the proposal may
take this one: the architecture is closed at the level of its
theorems, conditional at the level of four named premises and one
minimality axiom, and open in exactly the places listed above.


\subsection{The G-table: the moves of Part~III, audited}
\label{ssec:g-table}
Part~II ended with a table of its rungs. Table~\ref{tab:g-table} is the corresponding table
of this Part's moves, in the frame of \S\ref{ssec:frame}, so that the
two can be read side by side.

\begin{table}[H]
\centering
\caption[The moves of Part~III, audited.]{\textbf{The moves of
Part~III, audited.} Read against Table~\ref{tab:arc-earned}, the
contrast is the point of the paper's shape: every row of that table is a
standard result, and every row of this one is a proposal with its grade
attached. The fourth row is fenced in \S\ref{ssec:qi4-fence} and nothing
above it depends on the fourth row.}
\label{tab:g-table}\small
\begin{tabular}{@{}P{0.10\textwidth}P{0.20\textwidth}P{0.24\textwidth}P{0.20\textwidth}P{0.13\textwidth}@{}}
\toprule
\textbf{move} & \textbf{what is assumed} & \textbf{what follows} &
\textbf{what it buys} & \textbf{grade} \\
\midrule
G1 \GGEP{} (\S\ref{sec:ggep}) &
geometry and gauge are one object &
a six-dimensional bulk with an exact ledger, and the modified identity
$\dd H_3 = X_4$ &
two rungs collapse into one; the bulk supplies what the boundary's Ward
identity lacks &
axiom; the bulk a theorem of it \\
G2 \OGGEP{} (\S\ref{sec:oggep}) &
what is observed is a boundary of that bulk &
an open, irreversible boundary law, and the identity of
Eq.~\eqref{eq:open-ward} with a source &
a conservation law becomes an audit; admissibility becomes exclusion &
axiom; the boundary law derived-conditional \\
G3 \GRIP{} (\S\ref{sec:grip}) &
a landscape, and a descent that pays to cross &
the cascade \DGI{} $\to$ \GSSB{} $\to$ \GA{}, and a toll at each step &
a solvency condition, and a bound on the number of families &
theorem within its own setting; the budget an import \\
G4 \QiFour{} (\S\ref{sec:qi4}) &
one abelian factor of the bulk carries information &
four pillars in a chain, with a fixed conversion constant &
a laboratory falsifier; and the selection rule
\S\ref{ssec:gga-split} imports &
open throughout; reported, not checked \\
\bottomrule
\end{tabular}
\end{table}

\subsection{One paragraph pointing forward}
\label{ssec:forward}
\begin{quote}\small
\textbf{One paragraph pointing forward, and defending nothing.}
The program this paper has reviewed does not stop at physics, and
Chapter~\ref{sec:qi4} says where it goes: an exact bulk ledger, an open
boundary, and a relaxation cascade whose endpoint is what gets
registered, carried over to chemical, living and cognitive systems
through a second group of pillars \citep{ArisakaA7Qi4}. That chapter
states the extension and fences it. It checks nothing, it defends
nothing, and the only place the rest of the program reaches back for it
is the clause marked in \S\ref{ssec:gga-split}. A reader is entitled to
know where the architecture's authors think it leads, and equally
entitled to close the paper at the end of this Part and go no further.
This review goes no further.
\end{quote}


\begin{checkyourself}{Part~III}
\item \S\ref{ssec:pivot} claims that a conservation law says something
  different in an open system than in a closed one. Take electric charge
  in a region with current flowing across its boundary, and write both
  versions of the statement. Which one is the continuity equation?
\item \S\ref{ssec:exclusion} claims that anomaly cancellation is an
  admissibility condition and cannot exclude. Construct a hypothetical
  four-family world that satisfies every condition of
  Chapter~\ref{sec:gs}, and then say precisely what
  Eq.~\eqref{eq:solvency} adds that Chapter~\ref{sec:gs} did not.
\item Using Eq.~\eqref{eq:toll-ladder}, compute the total toll for
  $N_g = 4$ and compare it with the capacity $\pi N_g$. By how much is a
  fourth family short? Then work out whether the \emph{fixed}-budget
  reading $S_{\text{tot}} \le 3\pi$ and the per-family reading
  Eq.~\eqref{eq:solvency} ever disagree about which $N_g$ are admissible.
  They do not --- and the reason is worth a sentence: what does that
  coincidence say about the particular value $3\pi$?
\end{checkyourself}



\subsection{Part~III in one paragraph, and what comes next}
\label{ssec:part-iii-close}
Four moves, and not one of them is on the ladder. \GGEP{} makes geometry
and gauge one object and buys a six-dimensional bulk with an exact ledger.
\OGGEP{} says that what we observe is a boundary of that bulk, which turns
a conservation law into an audit and turns admissibility into exclusion.
\GRIP{} supplies the descent that does the spending, and with it a toll a
fourth family cannot pay. \QiFour{} is stated, fenced, and otherwise left
alone. Table~\ref{tab:g-table} carries the grades, and the grades are the
honest part of the Part: an axiom is called an axiom, a theorem is called a
theorem, and the premise this Part judges most exposed --- minimality ---
is named in \S\ref{ssec:open} rather than left for a reader to discover.

None of that is worth anything until it is made to pay. Part~IV presents
the bill. The twenty-eight numbers of the Standard Model and the twenty-one
of the cosmos are set out with a closed form, a measured value, a deviation
and a grade apiece; then come the chapters where the ledger is asked to do
something other than reproduce a number --- the mirrors, the color
triangle, the matter that survived, the chronology; and then, because a
theory with nothing left to adjust has nowhere to hide, the dated
falsifiers.

%% ===================================================================
%%  PART IV --- THE LEDGER
%% ===================================================================
\GGPart[The ledger]{The Ledger}{What the geometry pays out: the input it
is allowed, the twenty-eight numbers of the Standard Model, the
twenty-one of the cosmos, and what a theory with nothing left to adjust
is exposed to}
\label{part:ledger}

\section{Zero free parameters: the discipline, and what it does not mean}
\label{sec:zero-parameters}

Part~III set out a proposal in four moves. This Part asks the only
question that finally matters about a proposal of that kind: what does it
pay out, and at what price in assumptions?

The answer begins with an accounting discipline, and the discipline is
severe enough that it is worth stating before any result is quoted. The
program allows itself a fixed and very short list of primitives, and
forbids itself everything else. Nothing may be measured and inserted; no
continuous quantity may be adjusted after the fact to improve an
agreement. That is a strong claim, it is easy to overstate, and the
owning paper overstates it once itself. This chapter states it in the
form that survives audit.

\subsection{The discipline, stated}
\label{ssec:discipline}
The statement is a closure theorem \citep{ArisakaA4Euclid}:

\begin{quote}\itshape
Every dimensionless physical constant is one of the following: a member
of the integer ledger; derivable from that ledger by the arithmetic
cascade; or a ratio of the dimensionful conversion factors. Every
dimensionful quantity is a product of those conversion factors with
ledger integers and integer powers of $e$ and $\pi$. No third species of
primitive is admitted.
\end{quote}

Read the last sentence twice, because it is the whole content. The claim
is not that the list of inputs is \emph{short}. Short lists are common
and are a matter of taste. The claim is that the list is \emph{closed}:
that there is no place in the construction where a number may be brought
in from outside and tuned.

\subsection{The twelve admitted quantities}
\label{ssec:twelve}
The complete list is twelve items long:
\begin{equation}
\underbrace{\{\,c,\ G_N,\ \hbar\,;\ k_B,\ Q_B\,\}}_{\text{five conversion factors}}
\ \cup\
\underbrace{\{\,e,\ \pi\,\}}_{\text{two mathematical constants}}
\ \cup\
\underbrace{(M, N, L;\, J, K) = (1,3,5;\,7,35)}_{\text{five integers}} .
\label{eq:twelve}
\end{equation}
Three of the conversion factors belong to the bulk and two to the
boundary. The two mathematical constants are not physics. The five integers are not free. Two relations hold among them,
\begin{equation}
K \;=\; L\,J \;=\; M^2 + N^2 + L^2 ,
\qquad\text{together with}\qquad
J \;=\; 2N + 1 ,
\label{eq:closure}
\end{equation}
and the second is not a consequence of the first. It is a clause of the
constitution's fourth axiom: among admissible closure packets the charge
packet is the minimal one symmetric about a neutral slot. That is a
statement about the packet's \emph{shape}, and it mentions no value of
$N$ --- which is precisely what allows it to be substituted into the
first relation to determine $N$, rather than merely restating it
\citep{ArisakaA2GG6}.

The determination then runs in four steps, and no step consumes an
integer that a later step concludes.

\emph{$M = 1$}, by dimensional minimality: the bulk carries exactly one
compact direction, hence one Abelian holonomy ledger.

\emph{$L = 5$}, by single-tensor Green--Schwarz consistency: the
irreducible quartic and the cubic index of the charged core package both
equal $L - 5$ and both must vanish, with the minimality axiom selecting
the minimal chiral package \citep[T-BI6-9.0 of][]{ArisakaA3BI6}. This
step deserves a sentence of its own, because it is the one that makes
the chain non-circular: it fixes the block size \emph{with no color
count entering it at any stage}.

\emph{$N = 3$}, from the two relations of Eq.~\eqref{eq:closure} taken
together. Substituting $J = 2N+1$ into the dual lock gives a quadratic
in $N$,
\begin{equation}
N^2 - 2LN + \bigl(M^2 + L^2 - L\bigr) \;=\; 0 ,
\qquad
N \;=\; L \pm \sqrt{L - M^2} ,
\label{eq:ledger-quadratic}
\end{equation}
so at $M = 1$ integrality requires $L - 1$ to be a perfect square --- the
admissible block sizes are $2, 5, 10, 17, 26, \ldots$ --- and at the
$L = 5$ just fixed the two roots are $3$ and $7$. The colored block sits
inside the $L$-dimensional defining representation, so $N < L$, and the
root retained is $N = 3$ \citep[T-BI6-10.2a of][]{ArisakaA3BI6}.

\emph{$J = 7$ and $K = 35$} follow by arithmetic: $J = 2N + 1 = 7$, and
$K = L J = 35 = 1 + 9 + 25$.

Two features of that chain are easy to lose and worth marking. The
first is that uniqueness here is \emph{by construction} rather than by
search: the two roots of Eq.~\eqref{eq:ledger-quadratic} always sum to
$2L$, so exactly one of them lies below $L$ whenever the discriminant is
positive, and the embedding selects that one. No ordering among $M$, $N$
and $L$ has to be assumed, and no box has to be swept. The discarded
root closes the identities too --- $(1,7,5;15,75)$ satisfies
$LJ = 75 = 1 + 49 + 25$ --- and is excluded by the embedding and by
nothing else, which is worth saying plainly rather than leaving a reader
to find. The second is that the integer so determined is the
\emph{color} index: the size of the colored block inside the defining
representation. How many families the boundary carries is a different question, settled by a different argument in a different paper and reported at \S\ref{ssec:three-gen}; no step above bears on it.

\emph{Physics intuition.} Twelve items, of which two are mathematics and
five are whole numbers satisfying two equations, leaves five conversion
factors --- and all five are unit conventions. Three of them are the
bulk's, one for each kind of dimension it has: $c$ for the Lorentzian
directions, $G_N$ for the scale interval, $\hbar$ for the compact
phase. It is tempting to read the third of these as the one that sets
the scale, and the owning paper is explicit that it does not: in
natural units all three equal unity, and what sets a scale is the act
of expressing all three in human units, which every dimensionful theory
performs and none avoids \citep{ArisakaA4Euclid}. Everything else in the
construction is supposed to be arithmetic.

\subsection{What the ledger pays out at once}
\label{ssec:immediate}
Two consequences are available before any machinery is built, and they
are worth having early because they show what kind of statement the rest
of this Part contains --- the same kind that fills the two sheets this
Part will report, the particle sheet of \citet{ArisakaA9SM} and the
cosmological sheet of \citet{ArisakaA10Cosmos}.

\paragraph{The fine-structure constant.} In the infrared,
\begin{equation}
\alpha^{-1}(0) \;=\; 2^{J} + N^{2} \;=\; 128 + 9 \;=\; 137 ,
\label{eq:alpha137}
\end{equation}
against a measured $137.035999177$ --- a deviation of $-2.6\times10^{-4}$.
At the matching scale the same integer appears alone,
$\alpha^{-1}(M_Z) = 2^J = 128$, against $127.930(8)$, a deviation of
$+5.5\times10^{-4}$. Neither number was adjusted; both are readings of
$J = 7$ and $N = 3$, which were fixed by the anomaly identity of
\S\ref{ssec:bi6} before the fine-structure constant was consulted.

\paragraph{The elementary charge.} It follows that the charge on the
electron is not a primitive of the theory but an output of it:
\begin{equation}
e \;=\; \sqrt{\frac{4\pi}{2^{J}}} \;=\; 0.3133285 \ \ \text{at the
matching scale.}
\label{eq:echarge}
\end{equation}
A reader who has spent Part~II on the question of what makes a quantity
exact should pause here. The charge on the electron --- the quantity
whose universality \S\ref{ssec:wt} showed is protected from dynamics, and
whose \emph{value} \S\ref{ssec:dirac} showed is protected from
continuity --- is here claimed to be the square root of $4\pi$ over a
power of two.

\subsection{What ``zero'' does not mean}
\label{ssec:not-zero}
A claim of this shape invites a specific kind of misreading, and the
owning papers themselves are not uniformly careful about it. This review
will therefore never write the unqualified phrase. What is claimed, in
full, is:

\begin{quote}\small
\textbf{Zero free \emph{continuous} parameters.} No dial is adjusted
after the theory is fixed.

\textbf{One unit system, not one anchor.} The dimensionful content
of the bulk is the co-equal trio $\{c,\, G_N,\, \hbar\}$ --- one
conversion factor
for each kind of bulk dimension, all equal to unity in natural units,
none of them singled out \citep{ArisakaA4Euclid}. Fixing their values
against the world is the irreducible act of choosing units, and no
theory avoids it; it is a convention, not a parameter. The boundary's
two, $k_B$ and $Q_B$, are conversion factors of the same kind.

\textbf{A forced integer ledger}, in the sense of
\S\ref{ssec:twelve} --- and forced from the constitution rather than
given integers supplied upstream. $M$ and $L$ are fixed before any
color count enters the argument; $N$ then follows from the two
relations of Eq.~\eqref{eq:closure} together with the block embedding;
$J$ and $K$ are arithmetic.

\textbf{One frozen discrete completion class}, whose necessity the
owning papers assert rather than prove.

\textbf{Named open problems}, of which the largest is the sheet of
charged-fermion mass ratios --- structurally derived, not yet closed in
the strict sense.
\end{quote}

The owning paper says as much itself: a bare ``zero'' would be an
overclaim while that open problem stands, and its defense is that it
does not assert a bare zero but a closed audit plus a named, falsifiable
open item. What is not an overclaim is the
statement that the theory contains no continuous knob. Those are
different claims, and only the second is made here.

\emph{Why the distinction is worth the space.} Because the comparison this
Part draws is not between a theory that is finished and theories that
are not. It is between three postures toward
the same problem: one that adds continuous freedom, one that distributes
it across an ensemble, and one that removes it and thereby exposes
itself. A program that overstated its own completeness would forfeit
the third posture entirely.

It is worth recording that the program's own keystone paper takes the
same view of what is at stake. Its position is not that the constants
are beautiful, and not that a universe with observers must be selected
from an ensemble, but that once the constitution is granted nothing
further was free \citep{ArisakaA11Origin}. Part~\ref{part:conclusions}
returns to that in the first person, rather than in this review's voice.

\subsection{Grade box}
\label{ssec:zero-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Verified here.} The closure arithmetic of Eq.~\eqref{eq:closure}; the four-step determination of $(1,3,5;\,7,35)$ in \S\ref{ssec:twelve}, including the root formula of Eq.~\eqref{eq:ledger-quadratic}, the fact that its two roots sum to $2L$, and the closure of the discarded root; Eq.~\eqref{eq:alpha137} and its deviation; Eq.~\eqref{eq:echarge}. Each was recomputed independently before it was
typeset.

\emph{Reported, at the owning papers' grade.} The closure theorem
itself, and the claim that the twelve quantities of
Eq.~\eqref{eq:twelve} exhaust the admissible primitives.

\emph{Open, by those papers' own statement.} The charged-fermion mass
sheet; the necessity of the frozen completion class; and --- the item
this Part will return to twice --- the global trials factor of the whole
construction, which has not been computed. The sentence conceding what
that costs belongs to the paper that owns the Standard-Model codebook
rather than to this chapter's source, and is quoted where it is sourced,
in \S\ref{ssec:not-bought}.

\emph{Not claimed anywhere in this Part.} That the geometry is uniquely
forced. That is a conjecture in the owning papers and is marked as one.
\end{quote}


\section{The Standard-Model sheet: twenty-eight numbers}
\label{sec:sm-sheet}

The Standard Model is the most accurate theory in the history of
science, and it cannot explain a single one of its own constants. It has
twenty-eight of them. They are measured and written in by hand, and the
theory is silent about why they take the values they do.

This chapter reports what the program of Part~\ref{part:turn} claims to
do about that. The claim is not modest, so it is set out with its grades
attached, and the grades are printed in the same table as the numbers.

\subsection{The count: what twenty-eight means}
\label{ssec:count-28}
The number depends on how one counts, and the count used here is the
fullest:
\begin{equation}
\underbrace{5}_{\text{gauge, Higgs}}
+\underbrace{12}_{\text{fermion masses}}
+\underbrace{4}_{\text{CKM}}
+\underbrace{4}_{\text{PMNS}}
+\underbrace{2}_{\text{Majorana phases}}
+\underbrace{1}_{\bar\theta}
\;=\; 28 .
\label{eq:count28}
\end{equation}
The familiar figure of nineteen describes the theory as it stood before
1998, with massless neutrinos. Oscillations add seven, bringing it to
twenty-six; if the neutrinos are their own antiparticles, two further
physical phases appear, and the count reaches twenty-eight. The gap
between twenty-six and twenty-eight is precisely the Dirac-or-Majorana
question, and counting twenty-eight is the more honest bookkeeping
because it does not quietly assume the answer.

\subsection{The sheet}
\label{ssec:the-sheet}
Table~\ref{tab:sm-sheet} is the paper's central quantitative claim. Read
the last column first. Measured values are \citet{PDG2024}, as reported
by the owning paper.

\begin{table}[!t]
\centering
\caption[The Standard-Model sheet.]{\textbf{The Standard-Model sheet, at
first order, with grades.} Closed forms as the owning paper
\citep{ArisakaA9SM} states them; predictions recomputed independently
here from those forms and the ledger of \S\ref{ssec:twelve}; deviations
$(\text{predicted}-\text{measured})/|\text{measured}|$. \textbf{D} =
derived, D$^*$ = derived subject to one named lemma, A = asserted, in
that paper's own triage. Six D, fourteen D$^*$, eight A --- and of the
six, the weak mixing angle alone is a theorem end to end. Six rows have
no measured central value. The larger quark deviations are what a
second-order layer is for, and \S\ref{ssec:what-derived} says what that
layer costs.}
\label{tab:sm-sheet}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}P{0.13\textwidth}P{0.29\textwidth}P{0.14\textwidth}P{0.15\textwidth}P{0.13\textwidth}P{0.06\textwidth}@{}}
\toprule
\textbf{quantity} & \textbf{closed form} & \textbf{predicted} &
\textbf{measured} & \textbf{deviation} & \textbf{grade} \\
\midrule
\multicolumn{6}{@{}l}{\itshape Gauge and Higgs sector (five)} \\
$\alpha^{-1}(M_Z)$ & $2^{J}$ & $128$ & $127.930(8)$ & $+5.5\e{-4}$ & D$^*$ \\
$\sin^2\theta_W$ & $L^2/4N^3 = 25/108$ & $0.231481$ & $0.23155(4)$ & $-3.0\e{-4}$ & \textbf{D} \\
$\alpha_s(M_Z)$ & $1/\pi e$ & $0.117100$ & $0.1180(9)$ & $-7.6\e{-3}$ & D \\
$v$ [GeV] & $M_{\rm lock}(2/L)^{N}$ & $246.331$ & $246.21964$ & $+4.5\e{-4}$ & D$^*$ \\
$m_h$ [GeV] & $\sqrt{2(\alpha_s+\alpha_{\rm em})}\,v$ & $123.122$ & $125.25(17)$ & $-1.7\e{-2}$ & D \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Charged-lepton masses (three)} \\
$m_e$ [MeV] & $(v/\sqrt2)\,N/(L^3 2^{J+L+1})$ & $0.510300$ & $0.51099895$ & $-1.4\e{-3}$ & A \\
$m_\mu$ [MeV] & $(v/\sqrt2)\,L/2^{J+L+1}$ & $106.313$ & $105.658376$ & $+6.2\e{-3}$ & A \\
$m_\tau$ [GeV] & $(v/\sqrt2)\,NJ/2^{J+N+1}$ & $1.78605$ & $1.77693(9)$ & $+5.1\e{-3}$ & A \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Quark masses (six)} \\
$m_u$ [MeV] & $(v/\sqrt2)\,N e^{-4\pi}$ & $1.8223$ & $2.20(7)$ & $-1.7\e{-1}$ & A \\
$m_d$ [MeV] & $(v/\sqrt2)\,2N e^{-4\pi}$ & $3.6446$ & $4.69(5)$ & $-2.2\e{-1}$ & A \\
$m_s$ [MeV] & $(v/\sqrt2)\,2N e^{-3\pi}$ & $84.339$ & $92.74(54)$ & $-9.1\e{-2}$ & A \\
$m_c$ [GeV] & $(v/\sqrt2)\,N e^{-2\pi}$ & $0.97583$ & $1.275(9)$ & $-2.3\e{-1}$ & A \\
$m_b$ [GeV] & $(v/\sqrt2)\,(N/2^{N}) e^{-\pi}$ & $2.8227$ & $4.196(12)$ & $-3.3\e{-1}$ & A \\
$m_t$ [GeV] & $v/\sqrt2$, i.e.\ $y_t = 1$ & $174.183$ & $172.56(31)$ & $+9.4\e{-3}$ & D \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Neutrino masses (three); $m_\star = \pi v^2/E_{\rm GUT}$} \\
$m_1$ [meV] & $0$ & $0$ & not measured & --- & D$^*$ \\
$m_2$ [meV] & $m_\star$ & $8.361$ & not measured & --- & D$^*$ \\
$m_3$ [meV] & $2N\,m_\star$ & $50.167$ & not measured & --- & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Quark mixing (four)} \\
$\sin\theta_{12}$ & $\bigl|e^{-\pi/2}\!-\!e^{\ii 3\pi/5}e^{-\pi}\bigr|$ & $0.2250185$ & $0.225010(680)$ & $\mathbf{+3.8\e{-5}}$ & D$^*$ \\
$\sin\theta_{23}$ & $4\pi\sqrt3\,e^{-2\pi}$ & $0.0406460$ & $0.041830(740)$ & $-2.8\e{-2}$ & D$^*$ \\
$\sin\theta_{13}$ & $\sqrt3\,e^{-2\pi}$ & $0.0032345$ & $0.003732(88)$ & $-1.3\e{-1}$ & D$^*$ \\
$\delta_{\rm CKM}$ & $2\pi/5 = 72^\circ$ & $1.25664$ & $1.147(26)$ & $+9.6\e{-2}$ & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Lepton mixing (four)} \\
$\sin^2\theta_{12}$ & $L^2/3N^3 = 25/81$ & $0.308642$ & $0.304(16)$ & $+1.5\e{-2}$ & D$^*$ \\
$\sin^2\theta_{13}$ & $N/4K = 3/140$ & $0.0214286$ & $0.0220(6)$ & $-2.6\e{-2}$ & D$^*$ \\
$\sin^2\theta_{23}$ & $\tfrac12(1+\tfrac{1}{2N+1}) = 4/7$ & $0.571429$ & $0.560(18)$ & $+2.0\e{-2}$ & D$^*$ \\
$\delta_{\rm PMNS}$ & $\pi + 2\arcsin(1/7)$ & $196.43^\circ$ & $202^\circ(26)$ & $-2.8\e{-2}$ & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Majorana phases (two) and strong CP (one)} \\
$\alpha_{21}$ & $0$ & $0$ & not measured & --- & D \\
$\alpha_{31}$ & $\pi$ & $180^\circ$ & not measured & --- & D \\
$\bar\theta$ & $0$ & $0$ & $<10^{-10}$ & --- & D$^*$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{What a reader will check first}
\label{ssec:check-first}
Four entries carry most of the weight, and they are worth separating out
because they are of four different kinds.

\paragraph{A note on the symbol $N$ in this sheet.} The closed forms
are reproduced as the owning paper prints them \citep{ArisakaA9SM}, and
that paper writes a single $N$ for two quantities: the ledger formulas
read it as the color index $N_c$, while the Yukawa codebook reads it as
the family count $N_g$. Both equal $3$, so no number in the table moves
either way; which formula carries which is that paper's own outstanding
editorial item, and this review reports the sheet rather than deciding
it. Where this paper speaks in its own voice about the family count ---
the toll ladder of \S\ref{ssec:solvency} and the pincer of
\S\ref{ssec:three-gen} --- it writes $N_g$.

\paragraph{The weak mixing angle is a theorem.} $\sin^2\theta_W =
L^2/4N^3 = 25/108$, from the trace ratio of the identity of
\S\ref{ssec:bi6} and nothing else, with no empirical input anywhere in
the derivation. It agrees with the effective leptonic angle to $0.03\%$.
\emph{And it must be reported with its scheme}: against the
$\overline{\rm MS}$ angle $0.23122(4)$ the same prediction is a $+6.5\sigma$
tension. The owning paper prints that in the open and so does this
review. A closed form that agrees in one renormalization scheme and not
in another has not finished its work.

\paragraph{The top quark is a prediction, not a fit.} $y_t = 1$ exactly,
so $m_t = v/\sqrt2 = 174.18$ GeV against a measured $172.56$ ---
agreement at one percent for a quantity with no adjustable content
whatever. It is the cleanest parameter-free statement in the sheet.

\paragraph{The Cabibbo angle is the tightest row.}
$\sin\theta_{12} = \bigl|e^{-\pi/2} - e^{\ii 3\pi/5}e^{-\pi}\bigr|
= 0.2250185$ against a measured $0.225010$: a deviation of
$3.8\times10^{-5}$, from an expression containing two exponentials, an
angle of $108^\circ$, and nothing else.

\paragraph{The eight asserted mass rows are the weakest.} Of the nine
charged-fermion masses, eight are graded \emph{asserted} by the owning
paper --- every one but the top --- which says plainly that the choice
of which ledger integer plays which role is a working hypothesis rather
than a forced result. The first-order quark deviations run to $33\%$. A reader looking for the sheet's soft center will find it here,
and the owning paper points at it first.

\subsection{Charge as an output}
\label{ssec:charge-cp}
One result on the sheet earns its own paragraph because of what Part~II
said about it.

\paragraph{The electron's charge is an output.} From
Eq.~\eqref{eq:echarge}, $e = \sqrt{4\pi/2^J}$: the charge is a readout of
the integer $J$, not a primitive. Chapter~\ref{sec:ward} showed that no
dynamics can change the \emph{ratio} of two charges and
Chapter~\ref{sec:hs} showed that topology fixes it to an integer. Here
the integer itself is claimed to be arithmetic.

The sheet's two discrete-symmetry entries --- the $CP$-violating phase
$\dCKM$ and the strong-$CP$ angle $\bar\theta$ --- are set aside for their
own chapter: \S\ref{sec:discrete} draws the geometry behind both.

\subsection{What ``derived'' means here}
\label{ssec:what-derived}
This is the section that decides whether the sheet is worth anything, and
it is short.

\begin{quote}\itshape
Of the twenty-eight entries, six are derived, fourteen are derived
subject to one named lemma each, and eight are asserted. Of the six, one
--- the weak mixing angle --- is a theorem end to end. The others are a
cost-minimum selection in which the measured value enters the filter, a
curvature computed modulo an uncomputed lemma, a parameter-free
prediction, and two phases in a sector nobody has measured.
\end{quote}

So the sheet is not twenty-eight derivations. The formula the owning
paper itself uses is: \emph{twenty-eight reduced to five integers, plus
named lemmas, plus one measurement}. This review will not improve on
that phrasing, and will not use a shorter one.

Two further cautions belong here rather than in a footnote.

\emph{The second-order layer.} The larger first-order deviations are
closed by a second layer of small rational corrections. Its
\emph{denominators} are forced --- every one traces to the same ledger
--- but some of its numerators reach into content the geometry does not
supply, and where the first-order pull was largest the owning paper
concedes the correction is a bounded selection rather than a derivation.

\emph{The trials factor.} The largest open item is not a physics lemma
but a statistic. A small alphabet of integers and two mathematical
constants can express many numbers, and the honest question is how many
targets the alphabet had to hit. The owning paper states that the global
trials factor across the whole codebook \emph{has not been computed},
and that until it is, the closed forms are an honest ansatz and not a
proof. That sentence is the correct place for a skeptical reader to
begin, and this review will not obscure it.

What can be said against it is the ordering. The integers were fixed by
an anomaly identity before the constants were consulted; the same five
must then serve the gauge sector, the mass sheet, the mixing matrices and
--- in the next chapter --- the contents of the universe. A ledger that
must hit a dozen independent targets without being re-chosen is a
different object from a formula fitted to one.

\subsection{One number that is not on the sheet: the proton}
\label{ssec:proton}
Grand unification's characteristic prediction is that the proton decays.
The program of Part~\ref{part:turn} predicts the opposite, and predicts
it exactly: baryon number is carried by a massive gauge factor, the
selection rule forbids $\Delta B = 1$ and $\Delta B = 2$, and the
lifetime is infinite rather than long.

That is a stronger commitment than a number with an error bar. A single
observed proton decay --- one event, in any channel --- ends the
program, with nothing to renegotiate afterwards. The owning paper calls
it the cleanest falsifier the framework offers, and this review agrees.

\subsection{Grade box}
\label{ssec:sheet-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Verified here.} Every numerically checkable entry of
Table~\ref{tab:sm-sheet} was recomputed from its closed form and the
ledger of \S\ref{ssec:twelve}, and reproduces the owning paper's quoted
deviation. So did the Jarlskog invariant of \S\ref{sec:discrete}, the
neutrino scale, and the count of Eq.~\eqref{eq:count28}. What is verified
is the \emph{arithmetic}, not the derivations behind the closed forms.

\emph{Reported, at the owning paper's grade.} The closed forms
themselves; the six/fourteen/eight triage; the claim that the weak
mixing angle is a theorem; the exact stability of the proton.

\emph{Printed against the program's interest.} The
$\overline{\rm MS}$ scheme tension at $6.5\sigma$; the eight asserted
mass entries and their first-order deviations of up to $33\%$; the
second-order layer's selected numerators; and the uncomputed global
trials factor.

\emph{Not claimed anywhere in this chapter.} That twenty-eight
parameters have been derived. Six are, at that paper's own grade; one is
a theorem.
\end{quote}


\section{The discrete symmetries: \texorpdfstring{$C$, $P$, $T$, $CP$}{C, P, T, CP}, and \texorpdfstring{$CPT$}{CPT}}
\label{sec:discrete}
The Standard-Model sheet of \S\ref{sec:sm-sheet} listed the CKM phase $\dCKM{}$ and the strong-$CP$ angle among its twenty-eight numbers; what follows is the discrete-symmetry structure those two entries record.

The ladder of Part~\ref{part:ladder} and the open-system turn were about
\emph{continuous} symmetries and the identities they leave behind. Physics
carries three \emph{discrete} mirrors as well --- charge conjugation $C$,
parity $P$, and time reversal $T$ --- and every quantum-mechanics course
hands down the same table about them: $C$ and $P$ are represented by unitary
operators, $T$ by an antiunitary one, and the product $CPT$ is the
combination no local relativistic theory may break. That table is older
than the Standard Model and is normally \emph{assigned} --- justified after
the fact by Wigner's theorem and the demand that energies stay positive. The
cohort's discrete-symmetry paper, \PaperPFive{}, instead \emph{derives} it,
by reading each mirror as the reversal of one of the three real orientations
the six-dimensional bulk carries.

\paragraph{Three factors, three mirrors.} The bulk is real, oriented
geometry; there is no imaginary unit in it, and every $i$ that appears in
physics is the boundary's bookkeeping for a real rotation. The bulk has
exactly three orientable factors: the compact \emph{fiber circle} that
carries a field's phase, the \emph{spatial slice}, and the \emph{causal
clock}. Charge conjugation reverses the fiber circle; parity reverses the
spatial slice; time reversal reverses the clock --- one mirror per
orientation, no more and no fewer (Figure~\ref{fig:cpt-reversals}).

\begin{figure}[tbp]
\centering
\resizebox{0.604\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth]
 % C: circle
 \begin{scope}[shift={(0,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.75);
  \draw[->, line width=1.10pt, blue] (0.75,0) arc (0:120:0.75);
  \draw[->, line width=1.10pt, red, dashed] (0.75,0) arc (0:-120:0.75);
  \node at (0,-1.35) {\textbf{C}: fiber reversal};
  \node[font=\footnotesize] at (0,-1.8) {$\theta \mapsto -\theta$;\ \ $\Jq \mapsto \Jq$};
  \node[font=\footnotesize, blue] at (0,-2.25) {$J$-linear (unitary)};
 \end{scope}
 % P: space
 \begin{scope}[shift={(4.6,0)}]
  \draw[->, line width=1.10pt, blue] (-0.9,0) -- (0.9,0) node[right]{$\vec{x}$};
  \draw[->, line width=1.10pt, red, dashed] (0.55,-0.3) -- (-0.9,-0.3) node[left]{$-\vec{x}$};
  \draw[line width=0.80pt] (0,0.5) -- (0,-0.6);
  \node at (0,-1.35) {\textbf{P}: space reversal};
  \node[font=\footnotesize] at (0,-1.8) {touches neither source};
  \node[font=\footnotesize, blue] at (0,-2.25) {$J$-linear (unitary)};
 \end{scope}
 % T: clock
 \begin{scope}[shift={(9.2,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.75);
  \draw[line width=0.80pt] (0,0) -- (0,0.55);
  \draw[->, line width=1.10pt, blue] (0,0) -- (25:0.6);
  \draw[->, line width=1.10pt, red, dashed] (0,0) -- (155:0.6);
  \node at (0,-1.35) {\textbf{T}: clock reversal};
  \node[font=\footnotesize] at (0,-1.8) {$t \mapsto -t$;\ \ $\Ja \mapsto -\Ja$};
  \node[font=\footnotesize, red] at (0,-2.25) {$J$-antilinear (antiunitary)};
 \end{scope}
\end{tikzpicture}%
}
\caption[The three orientation reversals of T-P5-1.]{The three orientation reversals of T-P5-1. The fiber
reversal flips the rotation generator and the signed charge
together, leaving $\Jq = \partial_\theta/Q$ invariant; the space
reversal touches neither source of $i$; the clock reversal is the
one flip the imaginary unit is made of. The unitary/antiunitary
table is the map of which reversals touch the sources.~(after \PaperPFive{})}
\label{fig:cpt-reversals}
\end{figure}

\paragraph{Why time reversal is the odd one out.} The boundary builds its
quarter-turn --- the operation it calls ``multiply by $i$'' --- out of one
specific orientation, the clock's. A mirror is represented unitarily if it
leaves that quarter-turn alone and antiunitarily if it reverses it. Test the
three. Reverse space, and clock and fiber are untouched, so $i$ is untouched:
$P$ is unitary. Reverse the fiber, and the charge label flips together with
the rotation it generates, so $i$ survives the double negative: $C$ is
unitary. Reverse the clock, and ``a quarter-period ahead'' becomes ``a
quarter-period behind'' --- $i$ becomes $-i$, which is exactly antilinearity:
$T$ is antiunitary. Hence a discrete symmetry is antiunitary \emph{if and
only if it contains $T$}. Wigner's theorem offers the fork between unitary
and antiunitary; here which tine is taken is a one-line computation about
which mirror touches the source of $i$, not a convention. Two familiar facts
fall out for free: Kramers' degeneracy (a half-integer-spin state is at least
doubly degenerate, because $T^2=-1$), and the reason $T$ carries no conserved
quantum number (an antilinear operator has no rephasing-invariant eigenvalue
to carry).

\paragraph{Parity, the mirror that is unitary yet broken.} Of the three,
$P$ is the one nature treats most strangely. It is represented unitarily
--- reversing space leaves the clock and the fiber untouched, so $i$ is
untouched --- and yet the weak interaction does not respect it. Its real
content is the exchange of the two Weyl hands,
$(\tfrac12,0)\leftrightarrow(0,\tfrac12)$: a theory is $P$-symmetric exactly
when its two hands carry the same charges (vector-like) and $P$-violating
when they do not (chiral). The weak force chooses chirally, so $P$ is broken
--- the 1956--57 surprise of Lee and Yang's proposal and Wu's cobalt-60
experiment, a symmetry assumed exact until the first careful test. Among the
mirrors $P$ is thus the odd case in the opposite direction from $T$: exact as
a representation, broken as a dynamics.

\paragraph{$CPT$, and why it is the robust one.} The total reversal $CPT$
flips all three orientations at once. On a particle together with its
antiparticle the three flips compose to carry the pair back to itself, and
any coupling built from real geometric data --- magnitudes, which are even,
and fiber angles, which are merely relabeled --- is left invariant. So
$CPT$ is blind to the texture of the couplings: no texture, however
contrived, can break it, while the individual mirrors can. That asymmetry
--- the whole reversal exact, its three factors each breakable --- is the
engine of the next subsection and, downstream, of why the universe kept a
little matter. The reading is complementary to the textbook $CPT$ theorem
rather than a replacement for it: the classic result of
\citet{Jost1957} reaches
interacting fields through complex Lorentz transformations but says nothing
about \emph{what} the three mirrors are, while the orientation reading names
the anatomy at the price of proving the interacting case only conditionally.
Either way the experimental stakes are identical --- by Greenberg's theorem
\citep{Greenberg2002} a $CPT$-violating signal would take Lorentz
invariance down with it.

\subsection{\texorpdfstring{$CP$}{CP} violation: chirality on the flavor pentagon}
\label{ssec:cp-pentagon}
The mirror that matters for matter is $CP$ --- charge conjugation followed by
parity --- because it is the one that should exchange a particle for its
antiparticle. It is broken, very slightly, and the orientation picture says
exactly where the break lives: not in any single coupling, but in the
\emph{shape} of the couplings on the fiber circle. Write the quark Yukawa
couplings as a wiring diagram (Figure~\ref{fig:cpt-flavour-graph}): the
fields are terminals, each with a freely re-settable phase dial, and each
coupling is a wire carrying a twist. Turning a dial changes individual
wire-twists but cancels around any closed loop, so the only physical
quantities are the loop holonomies --- the flavor-space analogue of an
Aharonov--Bohm phase \citep{AharonovBohm1959}. $CP$ is a symmetry exactly
when every loop holonomy is $0$ or $\pi$; $CP$ violation is the statement
that some loop is \emph{chiral}, sitting off the real axis. The graph has
$(N-1)^2$ independent plaquettes, but unitarity ties them together, and
the count of \emph{physical} phases that survives is $(N-1)(N-2)/2$:
none for one, none for two, and precisely one for three --- which is why three
generations are the minimum for $CP$ violation, and why that one surviving
phase, the Jarlskog phase \citep{Jarlskog1985}, is a single number.

\begin{figure}[tbp]
\centering
\resizebox{0.609\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth,
 uvert/.style={draw=blue!60!black, line width=0.80pt, fill=blue!8,
   rounded corners=2pt, minimum width=13mm, minimum height=7.5mm},
 dvert/.style={draw=orange!75!black, line width=0.80pt, fill=orange!12,
   rounded corners=2pt, minimum width=13mm, minimum height=7.5mm}]
 \fill[red!10] (0.65,3.6) -- (5.35,3.6) -- (5.35,1.8) -- (0.65,1.8)
   -- cycle;
 \node[uvert] (u1) at (0,3.6) {$u$};
 \node[uvert] (u2) at (0,1.8) {$c$};
 \node[uvert] (u3) at (0,0.0) {$t$};
 \node[dvert] (d1) at (6,3.6) {$d$};
 \node[dvert] (d2) at (6,1.8) {$s$};
 \node[dvert] (d3) at (6,0.0) {$b$};
 \foreach \i in {1,2,3} \foreach \j in {1,2,3} {
   \draw[gray!45, line width=0.60pt] (u\i) -- (d\j);
 }
 \draw[line width=1.10pt, red!75!black] (u1) --
   node[above=0.5mm, pos=0.22, font=\footnotesize]{$\varphi_{ud}$} (d1);
 \draw[line width=1.10pt, red!75!black] (u1) --
   node[below=0.4mm, pos=0.16, font=\footnotesize]{$\varphi_{us}$} (d2);
 \draw[line width=1.10pt, red!75!black] (u2) --
   node[above=0.4mm, pos=0.16, font=\footnotesize]{$\varphi_{cd}$} (d1);
 \draw[line width=1.10pt, red!75!black] (u2) --
   node[below=0.5mm, pos=0.22, font=\footnotesize]{$\varphi_{cs}$} (d2);
 \foreach \v in {u1,u2,u3,d1,d2,d3} {
   \draw[line width=0.80pt] ([xshift=0mm,yshift=5.5mm]\v.center) circle (1.6mm);
   \draw[line width=0.80pt] ([xshift=0mm,yshift=5.5mm]\v.center) --
     ([xshift=1.0mm,yshift=6.6mm]\v.center);
 }
 \node[draw=black!55, rounded corners=2pt, fill=white, align=left,
   font=\footnotesize, anchor=north west] at (7.4,3.8)
  {\textbf{vertices}: fields, each with a\\ free phase dial
   (rephasing)\\[2pt]
   \textbf{edges}: Yukawa couplings\\ $|y_{ij}|\,R(\varphi_{ij})$
   --- a magnitude\\ and a fiber twist\\[2pt]
   \textbf{shaded loop}: the plaquette\\
   $\Phi_{\square} = \varphi_{ud} - \varphi_{us}
     + \varphi_{cs} - \varphi_{cd}$\\[2pt]
   dial turns cancel around any\\ closed loop:
   $\Phi_{\square}$ is physical};
\end{tikzpicture}%
}
\caption[The flavor graph for three generations.]{The flavor graph for three generations, drawn to the
cohort's wiring-diagram standard. Up-type and down-type fields
are vertices, each carrying a freely re-settable phase dial; the
nine Yukawa couplings are edges, each carrying a fiber twist
$\varphi_{ij}$. Turning any dial shifts every edge at that vertex
equally, so the shifts telescope to zero around a closed loop:
the shaded four-edge plaquette holonomy $\Phi_{\square}$ is
invariant --- a flavor-space Aharonov--Bohm phase
\citep{AharonovBohm1959}. For $N = 3$, unitarity ties all such
plaquettes to a single physical angle, the Jarlskog phase.~(after \PaperPFive{})}
\label{fig:cpt-flavour-graph}
\end{figure}

What the bulk adds is the \emph{value} that phase may take. With a single
compact circle and the block integer $L=5$ of the cohort's ledger, the
allowed first-order twists are multiples of $\pi/5$ --- a pentagonal alphabet
(Figure~\ref{fig:cpt-pentagon}). The Higgs sector keeps a fraction $2/L$ of
the unified block, which forces a branch phase of $\pi(1-2/L)=3\pi/5$: the
interior angle of a regular pentagon. Because $3\pi/5$ is not a multiple of
$\pi$, that pentagon is chiral, and $CP$ violation is therefore not an option
the theory could have declined --- there was no dial to turn it off. Removing
one unphysical family rephasing (one step of $\pi/5$) leaves the physical CKM phase at first order,
\begin{equation}
\dCKM \;=\; \frac{2\pi}{L} \;=\; \frac{2\pi}{5} \;=\; 72^\circ ,
\label{eq:delta-ckm}
\end{equation}
a count of pentagonal steps; and the same pentagon trigonometry ---
$\cos(3\pi/5) = -1/(2\varphi)$, the golden ratio where a pentagon puts
it
--- fixes the Cabibbo magnitude. One figure, two
of flavor physics' oldest numbers. The Jarlskog invariant that measures the
physical amount of $CP$ violation follows,
\begin{equation}
J^{(1)} \;=\; 2.739\times10^{-5} ,
\label{eq:jarlskog}
\end{equation}
against a measured $3.12\times10^{-5}$ --- a phase nobody fitted, landing the
invariant in the right decade at first order. The strong-$CP$ angle, the
twenty-eighth Standard-Model parameter, is a discrete-symmetry entry of the
same kind: the sheet gives it as $\bar\theta = 0$ exactly --- not small but
zero, by a determinant-phase cancellation graded not-yet-proved end to end
--- against an experimental bound below $10^{-10}$.

\begin{figure}[tbp]
\centering
\resizebox{0.373\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, scale=1.5, >=Stealth]
 % pentagon
 \foreach \k in {0,1,2,3,4} {
   \coordinate (P\k) at ({90+72*\k}:1.25);
 }
 \draw[line width=0.80pt] (P0)--(P1)--(P2)--(P3)--(P4)--cycle;
 \foreach \k in {0,1,2,3,4} \fill (P\k) circle (1.1pt);
 % lattice angles from center
 \draw[->, gray] (0,0) -- (P0);
 \draw[->, line width=1.10pt, blue] (0,0) -- (P1);
 \draw[->, line width=1.10pt, red] (0,0) -- ({90+72*1.5}:1.25);
 \node[font=\footnotesize] at (0.02,1.5) {$0$};
 \node[font=\footnotesize, blue] at ({90+72}:1.55) {$2\pi/5$ \ ($=\dCKM^{(1)}$)};
 \node[font=\footnotesize, red] at ({90+108}:1.62) {$3\pi/5$ \ ($=\varphi_{12}$)};
 % annotations
 \node[align=center, font=\footnotesize] at (0,-1.85)
  {lattice step $\pi/5$;\quad chiral because $3\pi/5 \notin \pi\bZ$\\
   golden ratio in $\cos(3\pi/5)$ $\Rightarrow$ the Cabibbo closed form};
\end{tikzpicture}}
\caption[The flavor pentagon.]{The flavor pentagon. With one compact circle and block
integer $L = 5$, first-order holonomies live on the lattice
$(\pi/5)\bZ$: the forced branch phase $3\pi/5$ (red) is not a
multiple of $\pi$, so CP violation must exist; one unphysical
lattice step leaves the physical phase $2\pi/5$ (blue); and the
pentagon's trigonometry --- the golden ratio --- sets the Cabibbo
magnitude. One object, two of flavor physics'
deepest numbers.~(after \PaperPFive{})}
\label{fig:cpt-pentagon}
\end{figure}

The honesty of the claim should travel with it. That $CP$ violation
\emph{exists}, and that it is the chirality of a texture on one circle, is
derived. That the first-order phase is forced to $2\pi/5$ follows once the
holonomies are pinned to the pentagonal lattice --- a step \PaperPFive{}
grades \emph{derived-conditional}, resting on the rigidity of the holonomy
moduli. The last stretch, from the raw $2\pi/5$ to the measured
$\dCKM \simeq 1.147$~rad, runs through a quantum-defect completion that the
parent paper carries at its own \emph{asserted} grade, and flags as such. The
direction that makes the mirror chiral is the same direction that let the
early universe keep six parts in ten billion of itself as matter: the mirrors
broke because the world has one.

\section{The color triangle: confinement, and why \texorpdfstring{$\bar\theta$}{theta-bar} vanishes}
\label{sec:triangle}
The sheet of \S\ref{sec:sm-sheet} carried two entries that are not
masses or mixing angles but multiply a topological density: the
$CP$-violating phase $\dCKM$ and the strong-$CP$ angle $\bar\theta$.
Section~\ref{sec:discrete} drew the geometry behind the first --- a
chiral texture on the pentagon. The second is different in kind.
$\bar\theta$ is not a misalignment between two family frames; it is a
\emph{vacuum-topology} angle, and its geometry is not the weak sector's
pentagon but the color sector's triangle. To read it we turn from
$\SUtwoL$ to $\SUthreeC$ --- and, as it happens, from the phase the
geometry leaves open to the phase it forces shut.

\subsection{The circle's holonomy, and the abelian ledger}
\label{ssec:abelian-ledger}
Before the twist is set, be precise about the tool, because the reader
of this review has already built it. Chapter~\ref{sec:wilson} climbed
the Wilson rung: the holonomy of a connection around a closed curve,
Eq.~\eqref{eq:wilsonline}, is the gauge-\emph{invariant} content of a
gauge field --- the loop integral that survives when every pointwise
phase is rotated away. For an abelian connection the path-ordering is
idle and the holonomy is a pure phase, $e^{i\Phi} \in \mathrm{U}(1)$,
measurable in interference \citep{AharonovBohm1959}. That is what
``local gauge invariance'' delivers when it is read around a loop:
nothing local survives the redundancy; the closed loop does. So the
sentence ``the holonomy of $\Stheta$'' and the sentence ``a
$\mathrm{U}(1)$ gauge phase on $\Stheta$'' name one object --- the
first names the geometry, the second the symmetry. What is
framework-specific is only the standing of the circle: $\Stheta$ is a
real, compact, non-spatial cycle of the bulk, so its holonomy is not
an analogy for a charge kept elsewhere. It is the charge, kept here.

That one circle serves several abelian masters, and the sheet's reader
deserves the ledger stated once. The bulk gauge group of the
constitution is $\SUfive \times \UoneB \times \UoneQi$ (with gravity);
around the compact cycle, four abelian objects can hold a phase, in
four different standings, and Table~\ref{tab:abelian-ledger} is that
ledger:

\begin{table}[H]
\centering
\caption[The four abelian circles on the compact cycle.]{\textbf{Four
things called $\mathrm{U}(1)$, none interchangeable.} Two are
fundamental factors of the bulk gauge group, one is a direction
\emph{inside} $\SUfive$, and one is what survives the descent. Only the
first row carries this section's subject: $\UoneB$ is anomalous and its
boson is Stueckelberg-massive, and what survives exactly is the rigid
charge, baryon number.}
\label{tab:abelian-ledger}\small
\begin{tabular}{@{}lp{0.30\textwidth}p{0.38\textwidth}@{}}
\toprule
circle & standing & fate and role \\
\midrule
$\UoneB$ & fundamental bulk factor & anomalous, paid by inflow; boson
Stueckelberg-massive; the \emph{rigid} charge survives exactly ---
baryon number, this section's subject \\
$\UoneQi$ & fundamental bulk factor & the information circle of the
dark sector; not part of the color story \\
$\UoneY$ & not a bulk factor: a direction \emph{inside} $\SUfive$ &
its Wilson line on $\Stheta$ is the dial the descent sets; its light
fluctuation is the Higgs \\
$\mathrm{U}(1)_{\rm EM}$ & not a bulk factor: the unbroken diagonal
$Q = T_3 + Y$ after the descent & a boundary descendant; carries the
triangle's electromagnetic shadow (\S\ref{ssec:singlet-rule}) \\
\bottomrule
\end{tabular}
\end{table}

\noindent Two constitutional factors, one direction inside the unified
group, one survivor of the breaking: four things called
``$\mathrm{U}(1)$,'' none interchangeable.

\paragraph{Where the abelian dial touches color.} Now the question
this chapter turns on. Confinement is $\SUthreeC$ physics ---
non-abelian, three charges, eight gluons; the cycle's holonomy is an
abelian phase. By what right does a $\mathrm{U}(1)$ speak to
$\mathrm{SU}(3)$ at all? At the \emph{center} --- the structure the
reader met in \S\ref{ssec:centre}, where the $N$-th roots of unity
act on a wrapped Wilson line, Eq.~\eqref{eq:centre}. For color,
\begin{equation}
Z(\SUthreeC) \;=\; \{\, \mathbf{1},\; \omega\mathbf{1},\;
\omega^{2}\mathbf{1} \,\}, \qquad \omega = e^{2\pi i/3} :
\label{eq:su3-centre-w1}
\end{equation}
three elements that belong to the non-abelian group yet commute with
all of it, carry no infinitesimal generator, and are invisible to the
Lie algebra. Turn the hypercharge Wilson line continuously and its
holonomy sweeps a circle of phases; at exactly three points that
holonomy is \emph{also} an element of the color center. Those three
points --- the cube roots of unity, at $0$, $\tfrac{2\pi}{3}$,
$\tfrac{4\pi}{3}$ --- are the vertices of an equilateral triangle
inscribed in $\Stheta$: the color triangle of this chapter's title.
An abelian dial with three detents, and each detent is a statement in
color.

\paragraph{The quotient that was always there.} The Standard Model
records this meeting in its own fine print. The group that acts
\emph{faithfully} on the Standard-Model fields is not the textbook
product but the quotient
\begin{equation}
G_{\rm SM} \;=\;
\bigl[\, \SUthreeC \times \SUtwoL \times \UoneY \,\bigr]
\big/\, \mathbb{Z}_6 ,
\label{eq:z6-quotient-w1}
\end{equation}
because one combination of a color-center rotation, a weak-center
sign, and a hypercharge phase --- $\omega^{t}\,(-1)^{2T_3}\,e^{2\pi i
Y}$, with $t$ the color triality --- acts as the identity on every
field the theory has. The check is finite and was run
programmatically for this review: all eight kinds of fragment return
the phase $1$. The $\mathbb{Z}_3$ part of the quotient is the lock of
triality to hypercharge, $B - 2Y \in \mathbb{Z}$ on every fragment.
In the Standard Model this quotient is a true, checkable, and inert
remark --- a footnote about which group one should have written down,
with no dynamical consequence. In the present framework the footnote
is load-bearing: the $\mathbb{Z}_3$ that color and hypercharge share
is realized as the holonomy of a real cycle, and it is exactly the
identity that parks the descent's twist at the center point, as the
next subsection shows.

\subsection{The lock: baryon number as a winding}
\label{ssec:lock-winding}
The descent breaks $\SUfive$ by
a holonomy on the same compact cycle $\Stheta$ the flavor sector used:
the Wilson--Hosotani mechanism, with the SM-preserving holonomies the
exponentials of hypercharge $W(c) = \exp(2\pi i\,c\,Y)$. Exactly one
quantized setting passes the spectrum's three demands at once --- break
$\SUfive$, lift the color triplet, keep the Higgs doublet light --- and
that setting is the center element
\begin{equation}
z \;=\; W(2) \;=\; e^{4\pi i Y} \;=\; \mathrm{diag}(\omega,\omega,\omega,1,1),
\qquad \omega = e^{2\pi i/3},
\label{eq:z3-centre}
\end{equation}
a cube root of unity on the circle (\PaperPSix{}, T-P6-2; \citealp{ArisakaP6Proton}). One twist,
four locks: the same third-of-a-turn breaks the unified group, removes
the proton-eating $X,Y$ bosons, splits the Higgs doublet from its
triplet, and --- with no further input --- writes baryon number onto the
matter fragments as a winding number.

The identity beneath that last clause deserves to be shown rather than
asserted, because it is arithmetic anyone can check. The center element
$z$ acts on any fragment of any representation by the fragment's
\emph{color triality} $t$ --- the phase $\omega^{t}$, with $t = +1$
for a color triplet, $-1$ for an antitriplet, $0$ for a singlet.
Run every fragment of the spectrum and compare $t/3$ with the
fragment's baryon number, as Table~\ref{tab:centre-triality} does:

\begin{table}[H]
\centering
\caption[Triality against baryon number, fragment by fragment.]{\textbf{Triality
against baryon number.} The center element $z$ of
Eq.~\eqref{eq:z3-centre} acts on every fragment of the spectrum by that
fragment's color triality $t$. The last two rows agree in every
column --- exactly, not modulo one --- which is the center identity,
Eq.~\eqref{eq:centre-identity-w1}: baryon number is the winding the
twist writes on the matter, not a quantum number assigned to it.}
\label{tab:centre-triality}\small
\begin{tabular}{@{}lcccccccc@{}}
\toprule
fragment & $Q$ & $u^c$ & $d^c$ & $L$ & $e^c$ &
$H_{\mathrm{doub}}$ & $H_{\mathrm{trip}}$ & $X,Y$ \\
\midrule
triality $t$ & $+1$ & $-1$ & $-1$ & $0$ & $0$ & $0$ & $+1$ & $+1$ \\
$t/3$ & $+\tfrac13$ & $-\tfrac13$ & $-\tfrac13$ & $0$ & $0$ & $0$ &
$+\tfrac13$ & $+\tfrac13$ \\
$B$ & $+\tfrac13$ & $-\tfrac13$ & $-\tfrac13$ & $0$ & $0$ & $0$ &
$+\tfrac13$ & $+\tfrac13$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent Equality in every column --- exactly, not modulo one:
\begin{equation}
B \;=\; \frac{t}{3}
\qquad\Longleftrightarrow\qquad
e^{2\pi i B} \;=\; z \ \ \text{on every fragment}
\label{eq:centre-identity-w1}
\end{equation}
(the \emph{center identity}; \PaperPSix{}). Baryon number was never
bookkeeping pasted onto the spectrum: its fractional content \emph{is}
the center of the color group, carried by the $\SUfive$ geometry
itself, in plain sight since 1974 \citep{GeorgiGlashow1974}. What the
lock adds is the promotion of this phase identity to an exact,
additive charge --- the integer lift of $t/3$ --- and phases live on
circles, so the lift of a phase is a winding. A quark mode winds the
cycle by a third; a baryon, three quarks, winds it once. \emph{Confinement is the
statement that only the closed loop is a state}: a fractional wrap
cannot close on itself, so it never appears in a boundary readout alone,
while a full wrap is a state --- and, because a closed loop cannot be
continuously unwound, a state forever. The immortality of the proton is
the impossibility of untying a knot in a closed loop
(Figure~\ref{fig:z3-winding}).

\subsection{The singlet rule: which states exist}
\label{ssec:singlet-rule}
Everything so far names what cannot exist --- the lone quark. The same
condition, run in the affirmative, is a census of what can, and the
census has been in every textbook since 1964 as \emph{the
color-singlet rule}: the hadrons nature exhibits are the
quark--antiquark pairs and the three-quark bundles, $q\bar q$ and
$qqq$. In the standard account that rule is an output of confinement
dynamics. Here it is one line of arithmetic per state.

The line: phases multiply, so triality adds. On a composite of
constituents $t_1, t_2, \ldots$ the center acts by
$\omega^{T}$ with $T = \sum_i t_i$ (mod 3), and the composite's
winding is $B = \sum_i t_i / 3$ --- an integer plus $T/3$. A completed
boundary readout closes on the cycle, and a curve closes when its
winding is an integer. So the whole rule is one chain:
\begin{equation}
\text{admissible readout}
\;\Longleftrightarrow\;
B \in \mathbb{Z}
\;\Longleftrightarrow\;
\omega^{T} = 1
\;\Longleftrightarrow\;
T \equiv 0 \ (\mathrm{mod}\ 3)
\;\Longleftrightarrow\;
\text{center-invariant}.
\label{eq:singlet-chain-w1}
\end{equation}
The right-hand end is the group theorist's ``color singlet under the
center''; the left-hand end is the boundary's ``a state that can be
handed over.'' They meet because they are one condition read at both
ends. Administered to the zoo, it gives
Table~\ref{tab:nality-census}:

\begin{table}[H]
\centering
\caption[The center's census of the hadron zoo.]{\textbf{The
color-singlet rule as a census of closure.} One question per
configuration --- does the total triality $T$ close the cycle? --- and
the answer decides whether the state can be handed to the boundary at
all. The exotic rows are where a closure law and a quark-counting law
part company: the tetraquark and the pentaquark pass here, and nature
exhibits both. The gluon passes the triality test too; its absence from
the free list is the mass gap's work, not the winding's.}
\label{tab:nality-census}\small
\begin{tabular}{@{}lccc p{0.335\textwidth}@{}}
\toprule
configuration & $T$ (mod 3) & $\omega^{T}$ & closes? & verdict \\
\midrule
quark $q$ & $1$ & $\omega$ & no & not a state of the boundary theory;
the sixteen-tonne string is the dynamics enforcing at finite energy
what the topology decrees at every energy \\
antiquark $\bar q$ & $2$ & $\omega^{2}$ & no & likewise, the arc run
the other way \\
diquark $qq$ & $2$ & $\omega^{2}$ & no & never free; a fragment
inside closed loops only \\
meson $q\bar q$ & $0$ & $1$ & yes & winding $0$: closed without
wrapping --- and mortal \\
baryon $qqq$ & $0$ & $1$ & yes & winding $1$: one full wrap; carries
$B = 1$ forever \\
tetraquark $qq\bar q\bar q$ & $0$ & $1$ & yes & permitted: the rule
counts triality, not quarks \\
pentaquark $qqqq\bar q$ & $0$ & $1$ & yes & permitted, and nature
obliges; rarity is dynamics, not grammar \\
gluon (adjoint) & $0$ & $1$ & --- & passes the triangle's test; its
absence from the free list is the mass gap's work, not the
winding's \\
\bottomrule
\end{tabular}
\end{table}

\noindent The table \emph{is} the color-singlet rule with its exact
boundary drawn, and its true name is the $N$-ality rule: it counts
winding thirds, not constituents. The exotic rows carry the point ---
a quark-counting law forbids the tetraquark and pentaquark; a closure
law permits them; nature sides with closure.

\paragraph{Why a triangle.} Once without algebra. Put the three
colors at the vertices of the equilateral triangle inscribed in the
cycle --- the cube roots of unity. A lone quark stands at one vertex,
its phase pointing a third of the way around, and nothing local can
point it home. A baryon puts one strand at each vertex: the three
phases multiply to $\omega^{3} = 1$; equivalently, the three vertex
vectors sum to the triangle's center. A meson pairs a vertex with its
conjugate. The triangle is a clock face with three hours: every quark
advances the hand one hour, every antiquark turns it back one, and the
boundary's door opens only when the hand reads twelve. Confinement,
in this picture, is not a wall around the quarks. It is the absence
of a door at one-third past.

\paragraph{The electromagnetic shadow.} One more run through the
fragment table, this time against electric charge, gives --- exactly
---
\begin{equation}
Q + \tfrac{t}{3} \,\in\, \mathbb{Z}
\quad\text{on every fragment,}
\qquad\text{hence}\qquad
Q + B \,\in\, \mathbb{Z}
\label{eq:em-shadow-w1}
\end{equation}
by the center identity ($u$: $\tfrac23 + \tfrac13$; $d$: $-\tfrac13 +
\tfrac13$; the electron: $-1 + 0$; the $X$ at $-\tfrac43 + \tfrac13$;
every row an integer --- verified programmatically across the
spectrum). Electric charge and baryon number share one fractional
part. A color singlet therefore has integer electric charge
automatically, and the century's three great never-statements
collapse into one: \emph{no experiment has ever seen a free quark; no
experiment has ever seen an isolated fractional electric charge; and
no boundary readout ever carries an open winding} --- one fact, in
three laboratory dialects. A fractional charge in isolation would
\emph{be} a free triality, which would \emph{be} an unclosed loop
presented as closed; a single confirmed one kills this reading
outright.

\paragraph{Two fences.} Triality zero is necessary, not sufficient:
the adjoint gluon passes the center's test, and its absence from the
free spectrum is the mass gap's work --- dynamics, carried by the
parent program at its own grade, not by the triangle. And this
subsection is the \emph{kinematic} half of confinement only --- which
states exist. Nothing here prices the string tension or touches the
Clay problem; the parent paper keeps that division of labor explicit,
and this review keeps it too.

\subsection{The pentagon and the triangle}
\label{ssec:pentagon-triangle}
Two figures now hang on the same nail. Section~\ref{sec:discrete}'s
pentagon and this section's triangle are not two circles but two
holonomies of the \emph{one} compact cycle $\Stheta$. The weak--flavor
sector reads it as a pentagon: block integer $L = 5$, holonomies on the
lattice $(\pi/5)\bZ$, a forced branch phase $3\pi/5$. The color--baryon
sector reads the same circle as a triangle: block integer $N = 3$, the
$\mathbb{Z}_3$ center, the three cube roots of unity. Five-fold and
three-fold, on one ring.

The contrast is exact, and it is the axis of this whole
chapter. \emph{The pentagon's loop does not close.} Its branch phase
$3\pi/5$ is not a multiple of $\pi$, so the residue $\dCKM = 2\pi/5$
survives every rephasing --- and that surviving phase, the holonomy the
loop failed to return to the identity, \emph{is} $CP$ violation.
\emph{The triangle's loop must close.} Three thirds make one full wrap,
and only the closed wrap is admitted as a state --- and that forced
closure \emph{is} confinement. Said once and plainly: $CP$ violation is
where the six-dimensional loop is left open and a phase remains;
confinement is where the same loop is forced shut and only integer
windings survive. Symmetry declining to close its circle, and rigidity
compelling it to: one continuous chiral phase at the weak end, one
discrete topological winding at the strong end, drawn on a single
$\Stheta$.

\subsection{Why \texorpdfstring{$\bar\theta$}{theta-bar} vanishes}
\label{ssec:theta-zero}
The strong-$CP$ angle is the phase
of the color loop:
\begin{equation}
\bar\theta \;=\; \theta_{\mathrm{top}} \;+\; \arg\det(M_u M_d).
\label{eq:theta-bar-w1}
\end{equation}
Read both pieces through the closed triangle. The first,
$\theta_{\mathrm{top}}$, is the coefficient of an independent seam
$\theta$-term --- and the four-dimensional seam is a projected image of
the six-dimensional bulk (\OGGEP{}), whose ledger licenses no free
continuous $\theta$-source. The color holonomy on $\Stheta$ is a
quantized center element, a cube root of unity by
Eq.~\eqref{eq:z3-centre}, not a dial that can be set to an arbitrary
value. There is no continuous angle available to be nonzero:
$\theta_{\mathrm{top}} = 0$. The second, $\arg\det(M_u M_d)$, is a sum
of quark-determinant phases --- relative holonomy phases carried around
the same closed loop; a loop that returns to the identity contributes no
net phase, so at first order the determinant phase cancels and
$\arg\det(M_u M_d) = 0$. The angle that the axion was invented to push
to zero is here zero by the very closure that confines: no
Peccei--Quinn symmetry \citep{PecceiQuinn1977}, no new light particle,
no tuning. In the
pentagon the loop is open and $CP$ violation is the residue it leaves;
in the triangle the loop is shut and there is simply no residue left to
\emph{be} $\bar\theta$.

So: does the strong-$CP$ problem still exist? It is best answered
by noting that its \emph{shape} changes. The
problem was: why is a free parameter, expected of order unity, smaller
than $10^{-10}$? In this geometry $\bar\theta$ is not free. It is the
phase of a closed color loop, and a closed loop has no phase to fine.
The question dissolves rather than being answered by an added device.

The honesty of the claim travels with it, at the grade
\S\ref{sec:discrete} already set for the sheet's entry. That the seam
admits no independent $\theta$-term is derived; that the quark
determinant phases cancel is graded \emph{derived-conditional} --- not
proved end to end, the codebook ``happening to be real'' at first order
--- so the standing prediction is not a hard zero but a tiny
quantum-defect leakage, $\bar\theta \lesssim 5\times10^{-18}$, seven
orders below the neutron electric-dipole bound of $10^{-10}$. A measured
neutron dipole above that envelope would be the falsifier, the same
standard the pentagon's numbers are held to. And the closure has one
last root worth naming: the reflection $\theta\to-\theta$ that shuts the
triangle acts, consistently, only in the conjugation-twisted form
$g\mapsto P\,\bar g\,P^{-1}$ --- which is precisely the identity
\emph{conjugation is orientation reversal} that \S\ref{sec:discrete}
used to build $CPT$. The geometry that makes the proton immortal and the
theorem that gives every particle its antiparticle rest on the same
turn of the same circle.

\begin{figure}[tbp]
\centering
\resizebox{0.798\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 lab/.style={font=\footnotesize, align=center},
 wind/.style={-{Stealth[length=2.6mm]}, line width=1.10pt},
]
%% ----- left: the circle with the Z3 marks -----
\begin{scope}
\draw[line width=1.2pt, draw=black!80] (0,0) circle (2.2);
\node[lab, anchor=south] at (0, 2.35) {$S^{1}_{\theta}$ --- the compact cycle\\ (phase; non-spatial)};
%% Z3 spokes
\foreach \a in {90, 210, 330}{
 \draw[draw=violet!60!black, line width=0.8pt] (0,0) -- (\a:2.2);
 \fill[violet!60!black] (\a:2.2) circle (2.0pt);
}
\node[lab, text=violet!60!black, anchor=north] at (0,-2.45) {the twist:\\ $z = W(2) = e^{4\pi i Y} = \mathrm{diag}(\omega,\omega,\omega,1,1)$};
\node[lab, text=violet!60!black, anchor=west] at (2.30, 1.1) {center marks at\\ $0,\ \tfrac{2\pi}{3},\ \tfrac{4\pi}{3}$:\\ $\omega = e^{2\pi i/3}$};
\end{scope}
%% ----- right: the two travellers -----
\begin{scope}[xshift=8.2cm]
%% quark: 1/3 wrap
\draw[line width=0.9pt, draw=black!50] (0,1.35) circle (1.05);
\draw[wind, draw=blue!60!black] (1.05,1.35) arc (0:120:1.05);
\fill[blue!60!black] (1.05,1.35) circle (1.8pt);
\node[lab, anchor=west, text=blue!60!black] at (1.45, 1.85)
 {a quark mode: winding $\tfrac13$\\ $B = t/3$ with $t = 1$\\ not closable --- not a state\\ (C-P6-1, \PaperPSix{})};
%% baryon: full wrap
\draw[line width=0.9pt, draw=black!50] (0,-1.6) circle (1.05);
\draw[wind, draw=black!80] (1.05,-1.6) arc (0:355:1.05);
\fill[black!80] (1.05,-1.6) circle (1.8pt);
\node[lab, anchor=west] at (1.45, -1.15)
 {a baryon: winding $1$\\ ($t = 3$: one full wrap)\\ closed --- a boundary state\\ carrying $B = 1$ forever};
\end{scope}
\end{tikzpicture}%
}
\caption[The color triangle: the $\mathbb{Z}_3$-lock, one twist, two travelers.]{\textbf{The
color triangle} --- the lock, drawn. \emph{Left:} the compact cycle $S^{1}_{\theta}$ with
the twist at the unique center point of the hypercharge line ---
$z = W(2) = e^{4\pi i Y} = \mathrm{diag}(\omega,\omega,\omega,1,1)$,
the sole quantized holonomy passing the three-demand scan
(\PaperPSix{}); its three marks at $0,\tfrac{2\pi}{3},\tfrac{4\pi}{3}$
are the cube roots of unity --- a three-fold lattice where the flavor
sector had a five-fold one. \emph{Right:} the two kinds of traveler. A
quark mode winds by one third --- an arc, not a loop: it cannot close,
which is why no boundary readout ever carries it alone. A baryon winds
once --- a closed loop whose winding number is baryon number itself,
$B = t/3$ by the center identity. A winding can no more be shed than a
knot untied in a closed loop. One twist, four locks: the same
third-of-a-turn breaks $\SUfive$, lifts the $X,Y$, splits the Higgs, and
writes $e^{2\pi i B} = z$ --- baryon number as the winding no operator can
change (T-P6-2); that sentence, made precise, is the stability theorem.
\ (after \PaperPSix{})}
\label{fig:z3-winding}
\end{figure}

\section{Leptogenesis: the matter the mirrors made}
\label{sec:leptogenesis}
Part~IV has so far read the ledger's output at a single instant: the
sheet of \S\ref{sec:sm-sheet}, the mirrors of \S\ref{sec:discrete},
the triangle of \S\ref{sec:triangle}. This chapter reads its deepest
entry across time. The observed universe contains roughly one baryon
for every $1.6$ billion photons ---
\begin{equation}
\etaB^{\mathrm{obs}} \;=\; (6.12 \pm 0.04)\times10^{-10}
\label{eq:etaB-obs}
\end{equation}
\citep{Planck2018}, a number certified two independent ways (the
acoustic peaks of the microwave background and the light-element
abundances of primordial nucleosynthesis agree on it) --- and it is
the most consequential small number in physics, because it is the
fraction of the early universe that survived the annihilation. Matter
and antimatter met and canceled, pair by pair, and for every ten
billion pairs that vanished into light, six particles of matter had
no partner and remained. Everything --- every galaxy, every atom,
every reader --- is that remainder. The question ``why is there
something rather than nothing?'' has, since Sakharov, a precise
physical core: \emph{where did the six come from?}

The Standard Model's answer is one of its sharpest failures, and the
failure has two independent halves. Sakharov showed in 1967
\citep{Sakharov1967} that any dynamical origin of the excess needs
three ingredients at once: a process that changes baryon number, $C$
and $CP$ violation to bias it, and a departure from thermal
equilibrium to keep the bias from relaxing away. The Standard Model
in fact owns a version of each --- and fails on two quantitatively.
Its electroweak transition, at the measured Higgs mass, is a smooth
crossover: no bubbles, no departure from equilibrium. And its $CP$
violation, the CKM phase of \S\ref{sec:discrete}, feeds the
inventory of matter at the level $\sim10^{-20}$ when run honestly
through the hot era (the Jarlskog invariant enters through quark
masses that are negligible at $10^{2}$~GeV temperatures, as Gavela,
Hern\'andez, Orloff and P\`ene showed) --- ten orders of magnitude
short \citep{Gavela1994}. The phase that makes $B$ mesons decay
asymmetrically never
touched the question of why matter exists. Any honest account must
therefore go through the one sector the Standard Model left open:
the neutrinos. That road is called \emph{leptogenesis} --- make the
excess first in the lepton sector, then let the hot electroweak era
convert it --- and in the present framework it is not one option on
a menu. It is forced, and every number in it is already on the
ledger. This chapter walks the whole account in the review's
register: the three conditions and who owns them, the chain and its
audit, the question of \emph{which} mirror did the work, and the
question the author of the parent cohort has carried for fifty
years --- whether the asymmetry was tuned.

\subsection{Sakharov's three conditions, and who owns them}
\label{ssec:sakharov}
The three conditions read, in this geometry, like three chapters of
this review meeting.

\paragraph{First condition: baryon number must change.} Chapter
\ref{sec:triangle} appears to forbid this outright: baryon number is
a winding, and no operator can unwind a closed loop. The escape is
exact and quantized, and it is the electroweak theory's own: the
weak group has topological sectors, and transitions between them ---
't~Hooft's anomaly vertex, the \emph{sphalerons} --- change $B$ and
$L$ together, in units of three ($\Delta B = \Delta L = 3n$, one
unit per generation), while conserving the difference $B{-}L$
exactly. The winding of \S\ref{ssec:lock-winding} is not
contradicted: the sphaleron is a bulk-topological event, not a
boundary operator, and it moves three thirds at once --- the door
respects the triangle. Two properties of this door decide
everything. Cold, it is welded shut: the transition is a tunneling
event priced at $e^{-4\pi/\alpha_W} \simeq e^{-372} \sim
10^{-162}$, a number for which ``suppressed'' is theater --- no
proton has decayed by it in the life of the universe, and none
will. Hot --- above the electroweak scale, where the barrier can be
walked over rather than tunneled through --- the same door swings
freely. So the early universe had exactly one lawful channel for
making baryon number, it counted in threes, it conserved $B{-}L$,
and it closed forever at the crossover. As the proton companion
puts it: \emph{the equation that forbids one proton's death is the
equation that permitted all protons' births}.

\paragraph{Second condition: the mirrors must be broken.} Chapter
\ref{sec:discrete} delivered this --- $CP$ violation exists because
the flavor pentagon is chiral, with no dial that could have
declined it --- but with the CKM phase disqualified by throughput,
the working question becomes: \emph{which} $CP$-violating structure
biases the lepton door? The answer is the subject of
\S\ref{ssec:alphabet-w1}, and it is the most instructive surprise
in this chapter: the work is done not by the celebrated phases but
by a Majorana \emph{parity} --- a lattice letter that the naive
criterion calls $CP$-conserving.

\paragraph{Third condition: a departure from equilibrium.} The
descent supplies it at a specific address: the lock crossing at
$\Mlock = \tfrac12 M_{\mathrm{Pl}}\,e^{-K} \simeq 3.85$~TeV, where
the seesaw partners of the neutrinos --- Majorana states $N_i$
parked at the lock with percent-level splittings $O(1/K)$ --- fall
out of equilibrium and decay. Two features distinguish this from
the generic story. The crossing is \emph{prompt}: no supercooling,
no secondary inflation, no late entropy release (the cosmology
companion establishes all three \citep{ArisakaA10Cosmos}), so an
asymmetry made at the lock survives \emph{undiluted} --- the coin
is struck once, and the CMB counts that same coin an age of the
universe later. And the address is \emph{reachable}: $3.85$~TeV is
collider-adjacent physics, not a $10^{9}$--$10^{15}$~GeV
never-never land --- a point \S\ref{ssec:lepto-grade} returns to,
because it converts the account from a story into a schedule.

\subsection{The chain, audited}
\label{ssec:lepto-chain}
The parent treatment (\PaperPFive{}, Part~VIII; \citealp{ArisakaP5CPT}) walks the chain in
nine steps with every grade displayed; this review compresses it to
its skeleton and keeps the audit. The heavy masses sit at the lock
($3.85$~TeV, splittings $O(1/K) \sim 110$~GeV --- derived). The
portal is unique: a uniqueness theorem of the parent paper shows the
weight-zero Majorana self-pairing is the only $\Delta L = 2$
structure the bulk admits, so $LHN$ is not chosen but forced. The
neutrino Yukawa is fixed by matching the seesaw to the derived
light masses: $Y_\nu^2 = \pi\Mlock/\EGUT \simeq 5.3\times10^{-13}$,
giving the tiny width $\Gamma_N = Y_\nu^2 M_N/8\pi \simeq
8\times10^{-11}$~GeV. A width that small would doom the textbook
hierarchical mechanism; what rescues it is the \emph{resonant}
regime: for near-degenerate Majorana states the $CP$ asymmetry in
decay is enhanced by the ratio of splitting to width, saturating
when the two are comparable --- the self-energy resonance of
\citet{PilaftsisUnderwood2004}, here with the percent-level lock
splittings as the near-degeneracy. \textbf{The on-resonance
condition is the chain's one named input}, and the parent papers
carry it in public as a named conjecture, sharpened to a partial
resonance $\Delta M/\Gamma^{\mathrm{eff}} \simeq 3\times10^{4}$ and
now backed by an identified candidate mechanism at candidate grade
--- the effective width is the rate at which the thermal bath
\emph{measures lepton flavor}, $\Gamma^{\mathrm{eff}} = c_\tau
y_\tau^2 T$, the only regulator in a sixteen-order landscape that
lands in the required decade (home: the cosmology companion's
Appendix~I \citep{ArisakaA10Cosmos}). On it, the flavored
asymmetry reaches $\epsilon^{\mathrm{res}} \simeq 2.5\times10^{-6}$.

Washout then takes its toll, and takes it in a way that should
delight anyone allergic to initial conditions. The equilibrium
threshold is \emph{ledger-free} --- $m_\star =
\tfrac{16\pi^{5/2}\sqrt{\gstar}}{3\sqrt5}\,(v/\sqrt2)^2/M_{\mathrm{Pl}}
\simeq 1.1\times10^{-3}$~eV, built from measured constants alone
--- and the derived neutrino masses sit far above it
($\widetilde m \sim m_3 \simeq 50$~meV), so the washout is
\emph{strong}: $K_{\mathrm{wash}} \simeq 50$, efficiency $\kappa
\simeq 10^{-2}$, and --- the standard blessing of the strong
regime --- all memory of initial conditions is erased. What
survives is the throughput
\begin{equation}
\epsilon^{\mathrm{res}}\,\kappa \;\simeq\;
(2.5\times10^{-6})(1.0\times10^{-2}) \;\simeq\; 2.5\times10^{-8}
\;=\; e^{-K/2},
\label{eq:throughput-w1}
\end{equation}
which reconstructs, to displayed precision, the ledger's throat
exponential --- the parent paper flags the exponent $K/2$ as a
postulate of the compression, geometrically suggestive as
$\sqrt{e^{-K}}$ but not yet derived, and the reconstruction as the
chain's central claim: an explicit seesaw-plus-washout transport
\emph{lands on} the ledger value rather than asserting it.
Sphalerons then convert the lepton excess to baryons at the fixed
one-doublet rate $28/79$ \citep{HarveyTurner1990}, the
entropy bath dilutes by $\gstar = 106.75$, and the pipeline closes:
\begin{equation}
\etaB \;\simeq\; 7.04\times\frac{28}{79}\times
\frac{1.25}{\gstar}\times e^{-K/2} \;\simeq\; 7.34\times10^{-10}
\qquad\text{against}\qquad
(6.12\pm0.04)\times10^{-10},
\label{eq:etaB-w1}
\end{equation}
$+19.9\%$ central, inside the published factor-$\sim3$ band of the
uncomputed $O(1)$'s --- with \emph{zero fitted parameters}. The factor
$1.25/\gstar$ is the triplet's equilibrium yield, $0.42$ per species
summed over three; the parent papers display it rather than absorbing it
silently into $1/\gstar$, and this review follows them.
Figure~\ref{fig:lepto-chain} draws the chain end to end, with the one
named input and the one postulated exponent marked. The
review repeats the parent's own warning: the percent-level
closeness must not be read as precision; the genuine claim is the
reconstruction \eqref{eq:throughput-w1} and the dial count. The
sheet of \S\ref{sec:cosmos-sheet} carries the result as one row;
this chapter is that row's story.

\begin{figure}[tbp]
\centering
\resizebox{0.58\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth,
 node distance=3.5mm,
 st/.style={draw=black!55, rounded corners=2pt, align=center, inner sep=4pt,
   minimum width=52mm}]
\node[st, fill=blue!8] (a) {ledger $K = 35$ $\Rightarrow$
 $\Mlock \simeq 3.85$~TeV; splittings $O(1/K)$};
\node[st, fill=blue!8, below=of a] (b) {unique Majorana door
 ($LHN$); seesaw $\Rightarrow$ $Y_\nu \simeq 7.3\times10^{-7}$};
\node[st, fill=orange!14, below=of b] (c) {\textbf{resonant decay}
 (the named input): flavored asymmetries, driver
 $\alpha_{31} = \pi$;\\ $\epsilon^{\mathrm{res}} \simeq
 2.5\times10^{-6}$};
\node[st, fill=orange!14, below=of c] (d) {strong washout:
 $\kappa \simeq 10^{-2}$;\quad throughput $\epsilon\kappa =
 e^{-K/2}$};
\node[st, fill=green!10, below=of d] (e) {sphalerons
 ($28/79$) $+$ entropy ($\gstar$) $\Rightarrow$
 $\etaB \simeq 7.34\times10^{-10}$};
\node[st, fill=green!10, below=of e] (f) {prompt lock crossing:
 zero dilution $\to$ the CMB reads $6.12\times10^{-10}$};
\draw[->, line width=0.80pt] (a) -- (b);
\draw[->, line width=0.80pt] (b) -- (c);
\draw[->, line width=0.80pt] (c) -- (d);
\draw[->, line width=0.80pt] (d) -- (e);
\draw[->, line width=0.80pt] (e) -- (f);
\end{tikzpicture}}
\caption[The leptogenesis chain as a flowchart.]{The leptogenesis
chain as a flowchart: blue = inherited and derived, orange = the
transport block carrying the named conjecture, green = standard
reprocessing and conservation. The one direction of the whole
diagram is the descent's. Every number is on the ledger; the one
named input (the on-resonance condition) and the postulated
exponent ($K/2$) are stated in the text at their parent
grades.~(after \PaperPFive{})}
\label{fig:lepto-chain}
\end{figure}

\subsection{The alphabet audit: which mirror made the matter}
\label{ssec:alphabet-w1}
Now the question this review is uniquely placed to ask, because it
has spent three chapters assembling the answer's vocabulary. The
flavor pentagon of \S\ref{sec:discrete} values every first-order
$CP$ phase on the lattice $(\pi/5)\bZ$ --- ten letters on one
circle, of which the two on the real axis, $\{0, \pi\}$, are
\emph{achiral} (a texture sitting there is indistinguishable from
its mirror) and the other eight are chiral in mirror pairs. Every
$CP$ quantity flavor physics owns is one of these letters. Run
the audit against the birth of matter, as
Table~\ref{tab:alphabet-audit} does:

\begin{table}[H]
\centering
\caption[The $CP$ alphabet audited against the birth of matter.]{\textbf{Which
letter made the matter.} Every $CP$ quantity flavor physics owns is
one of the ten letters on the lattice $(\pi/5)\bZ$; the third column
asks what each one does in the baryon asymmetry. The two phases the
experiments are chasing, $\dCKM$ and $\dPMNS$, do none of the work; the
driver is the achiral Majorana letter $\alpha_{31} = \pi$. The
magnitude is not a letter at all but the throughput $e^{-K/2}$ of
Eq.~\eqref{eq:etaB-w1}.}
\label{tab:alphabet-audit}\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}lp{2.5cm}p{6.8cm}@{}}
\toprule
$CP$ quantity & letter & job in the birth of matter \\
\midrule
$\dCKM = 2\pi/5$ & chiral & none --- throughput $\sim10^{-20}$,
ten orders short: the phase the $B$ factories measured is a
bystander \\
$\dPMNS = \pi + 2\arcsin\tfrac17 = 196.4^\circ$ & anchor $\pi$,
deviation $1/J$ & spectator --- the chain is $\JPMNS$-blind (the
Casas--Ibarra cancellation, \citealp{CasasIbarra2001}): the phase DUNE
will measure is not
the reason we exist \\
$\alpha_{21} = 0$ & achiral (parity $+$) & inert \\
$\alpha_{31} = \pi$ & achiral (parity $-$) & \textbf{the driver}:
sets the sign of the flavored asymmetries \\
throughput $= e^{-K/2}$ & not a letter & \textbf{the magnitude}:
Eq.~\eqref{eq:etaB-w1} \\
\bottomrule
\end{tabular}
\end{table}

\noindent The audit's surprise is printed in the middle rows: the
birth of matter is driven by the \emph{achiral} letters --- the
values the chirality criterion calls $CP$-conserving. The parent
paper resolves the apparent paradox in three steps, each worth
carrying in the review because each is a piece of physics in its
own right. \emph{First, a Majorana letter is read at half-angle}:
Majorana phases enter amplitudes as $e^{i\alpha/2}$, so the letter
$\alpha_{31} = \pi$ acts, wherever it appears in an interference,
as $e^{i\pi/2} = i$ --- a quarter turn, the maximally imaginary
insertion. Figure~\ref{fig:alphabet-w1} puts all ten letters on the
circle, with the four that do the work marked. What the Lagrangian's bookkeeping records as a parity,
the amplitudes experience as the strongest bias a discrete letter
can deliver. \emph{Second, at a resonance an amount becomes an
orientation}: on the resonance the asymmetry saturates and the
Yukawa magnitude cancels; the phases are no longer asked ``how
much?'', only ``which way?'' --- and a parity is precisely an
answer to ``which way?'' with no ``how much'' attached. Amounts
saturate; orientations survive. \emph{Third, something must still
be chiral, and it is the pentagon}: a parity can orient an
interference only if an interference exists, and that existence is
the branch phase $3\pi/5$ --- chiral because five is odd
(\S\ref{ssec:cp-pentagon}). Give the bulk an even block integer
and every letter lands on the real axis, the parities have nothing
to sign, and the universe annihilates to pure light. In one line
each: \emph{the pentagon writes the sentence; the parity chooses
its sign; the throat depth sets the print run.}

\begin{figure}[tbp]
\centering
\resizebox{0.663\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, scale=1.45,
 >=Stealth]
 \draw[line width=0.9pt] (0,0) circle (1.55);
 \foreach \k in {0,1,...,9}{
   \fill[black!55] ({\k*36}:1.55) circle (1.0pt);
 }
 \draw[dashed, black!45] (-1.55,0) -- (1.55,0);
 \node[black!60, font=\scriptsize, below] at (0.45,-0.06)
   {the achiral axis $\{0,\pi\}$};
 \draw[->, black!60, line width=0.80pt] (0,0) -- (0:1.55);
 \node[black!60, anchor=west, align=left] at (0:1.72)
   {$0$: \ $\alpha_{21}$\\ (parity $+$; inert)};
 \draw[->, blue!70!black, line width=1.10pt] (0,0) -- (72:1.55);
 \node[blue!70!black, anchor=south west, align=left]
   at (72:1.66) {$2\pi/5$: \ $\dCKM$ (chiral)\\
   $CP$ violation exists --- but\\ feeds no matter
   ($\sim\!10^{-20}$)};
 \draw[->, violet!80!black, line width=1.10pt] (0,0) -- (108:1.55);
 \node[violet!80!black, anchor=south east, align=right]
   at (108:1.66) {$3\pi/5$: the branch phase\\ the chirality
   no dial\\ could decline};
 \draw[->, red!75!black, line width=1.10pt] (0,0) -- (180:1.55);
 \node[red!75!black, anchor=east, align=right] at (180:1.72)
   {$\pi$: \ $\alpha_{31}$ --- the parity that\\
    signs the matter (half-angle $i$);\\
    and the anchor of $\dPMNS$\\
    ($196.4^{\circ} = \pi + 2\arcsin\tfrac17$)};
 \node[draw=black!55, rounded corners=2pt, align=center,
   font=\scriptsize\sffamily, inner sep=3.5pt] at (0,-2.35)
   {the magnitude is not on the circle:\quad
    $\epsilon\,\kappa = e^{-K/2}$, $K = 35$
    \ $\Rightarrow$ \ $\etaB \simeq 5.9\times10^{-10}$};
\end{tikzpicture}}
\caption[The alphabet audit, drawn.]{The alphabet audit, drawn:
the ten letters of the pentagonal lattice on the one compact
circle, with the four that decide the birth of matter marked. The
quark Dirac phase (blue) is chiral but feeds no matter; the branch
phase (violet) is the chirality the odd alphabet forced --- the
reason any interference exists to orient; the achiral letter $\pi$
(red) does double duty as the Majorana parity that signs the
matter and the anchor of the phase DUNE will measure; and the
magnitude lives off the circle entirely, in the ledger's throat
depth. Existence from the pentagon, sign from a parity, size from
the throat: three jobs, three structures, zero
dials.~(after \PaperPFive{})}
\label{fig:alphabet-w1}
\end{figure}

\subsection{Was the asymmetry tuned?}
\label{ssec:tuned}
The question deserves its own subsection because it is the modern
form of the oldest question, and because the ledger's answer is
structurally different from every answer the reader will have met.
In the Standard Model the CKM phase is a continuous parameter:
$\delta = 0$ --- no $CP$ violation, no matter, no us --- was
exactly as lawful as the measured value, and the theory offers no
comment. In supersymmetric and string constructions the phases are
moduli- and landscape-dependent: lawful values fill continua, and
the observed asymmetry selects among them anthropically or not at
all. Hierarchical leptogenesis, the field's default, inherits the
same economy: free heavy masses, a free $R$-matrix, free phases
--- the observed $\etaB$ becomes a constraint surface in a
many-dimensional parameter space, and the theory \emph{accommodates}
the number without \emph{producing} it.

Here the audit found no dial at any link. The alphabet is discrete
and its letters are forced: $CP$ violation exists because five is
odd, and no assignment of anything could have made the pentagon
achiral. The sign is a parity: two lawful values, related by the
matter--antimatter reflection, so the \emph{content} of the choice
is one bit fixed by the geometry's orientation conventions --- not
a tuning but a labeling. And the magnitude never was a $CP$
quantity: it is the ledger's one throat integer, read at
half-depth, $\etaB \sim e^{-K/2}$ --- \emph{the baryon-to-photon
ratio is the square root of the lock-scale suppression}. The same
$K = 35$ that parks the seesaw partners at $3.85$~TeV and caps the
strong-$CP$ leakage of \S\ref{ssec:theta-zero} counts the
survivors of the annihilation: one integer, three exports --- a
mass, a bound, and the number of us. ``Deterministically
programmed'' is therefore the precise description, not a
rhetorical one: given the constitution, the existence, the sign,
and the size of the matter excess were each forced, by three
different structures, none adjustable --- and the whole claim is
graded, displayed, and mortal in the laboratories of the next
subsection. Nature did not fine-tune the mirrors so that matter
would survive. On this reading there was never a tuner to hire ---
and the number that measures our own existence is not an accident
the universe committed but a theorem it executed.

\subsection{The grades, and the exposure}
\label{ssec:lepto-grade}
What is derived, what is conditional, and what would kill it ---
the register this review applies to every chapter, applied to the
one whose subject is us. \emph{Derived, on the inherited ledger:}
the lock mass and splittings, the uniqueness of the Majorana door,
the seesaw-matched Yukawa, the ledger-free washout threshold, the
strong-washout regime with its erasure of initial conditions, the
prompt crossing and zero dilution, and the sphaleron/entropy
bookkeeping (standard). \emph{Derived-conditional:} the flavored
asymmetry on the inherited phases, carried on the parent's named
partial-resonance hypothesis, whose candidate mechanism (the
flavor-measurement regulator) is identified but not yet closed at
transport level. \emph{The named input:} the on-resonance
condition itself --- the parent cohort's sharpest open item,
displayed rather than hidden. \emph{Postulated:} the exponent
$K/2$ of the throughput compression --- the chain lands on it; the
derivation is an open work order. The assembled result carries
grade D$^{*}$ with an honest factor-$\sim3$ band.

The exposure is unusually rich for a baryogenesis proposal, and
the parent paper counts that richness as the mechanism's chief
virtue: the account of why we exist is due in the data within the
decade, and mortal four separate ways. DUNE and Hyper-Kamiokande
measure $\dPMNS$ to $\pm15^{\circ}$: a value inconsistent with
$196^{\circ}$ falsifies the anchor-plus-deviation letter (the
current normal-ordering fit, $177^{\circ}\pm20$, sits one sigma
away). The next-generation neutrinoless double-beta program
probes the Majorana door: the parities $(0,\pi)$ give $\mbb =
1.72$~meV, and an \emph{observation} at the tens-of-meV level now
within reach would falsify the parity pattern outright. The
appearance asymmetry checks the sign ($\JPMNS < 0$; T2K already
disfavors the opposite). The $3.85$~TeV lock cluster sits within
HL-LHC reach with a pre-declared kill criterion in the cohort's
ledger. And behind all four stands the door itself: one observed
proton decay, or one $n$--$\bar n$ oscillation, breaks the
$\Delta B \in 3\bZ$ arithmetic on which the whole birth story
runs. No other baryogenesis proposal on the market carries a
falsifier list with dates; most carry none at all.

\section{The cosmological sheet: twenty-one entries, and four
that are not predictions}
\label{sec:cosmos-sheet}

The concordance model of cosmology has six parameters. That is the
number usually quoted, and it understates the problem, because the six
are the ones a fit adjusts: the inventory a cosmology must actually
specify --- what the universe is made of, how fast it expands, when the
first atoms formed, what launched it and what it is drifting toward ---
runs to about twenty. None of them is explained by the model that fits
them.

This chapter reports what the same geometry claims about that inventory
\citep{ArisakaA10Cosmos}. The claim is of the same kind as
\S\ref{sec:sm-sheet}'s and it is graded the same way. But it needs one
piece of bookkeeping stated \emph{before} the table rather than after
it, because without that piece the table is misleading. Figure~\ref{fig:cosmic-descent}, the cohort's descent drawn to scale (\citealp{ArisakaA10Cosmos}), sets that inventory in its place: the same fractions and epochs read as one open-system fall from the GUT edge to the de~Sitter future.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a10_cosmic_descent.pdf}
\caption[The cosmic \textsf{GRIP} descent: every key epoch on the radial
axis (reproduced from \citealp{ArisakaA10Cosmos}).]{\textbf{The cosmic
\textsf{GRIP} descent, drawn to scale on} \(r=\ln(E_{\rm GUT}/\mu)\). One
open-system descent from the GUT edge (\(r=0\)) to the de~Sitter future,
read simultaneously in four quantities --- energy \(\mu\), time \(t\), and
the scale factor \(a\), each against the radial depth \(r\). The key
epochs of the cohort's master chronology (\citealp{ArisakaA10Cosmos}) are
drawn to scale and colored, with the late universe magnified beneath the
break so that recombination, structure, and the present epoch remain
legible. Highlighted rows carry the framework's derived markers: the
plateau tilt and tensor ratio (\(n_s=0.9620\), \(r=0.0043\)), the lock
\(M_{\rm lock}=E_{\rm Pl}/(2e^{35})=3.85\)~TeV, the corridor switch
\(z_{\rm sw}=e^{7}\) (integer-ledger marker \(J=7\)), and today's
\(\pi\)-locked inventory \((\Omega_m,\Omega_\Lambda,\Omega_b)=(1/\pi,\,
1-1/\pi,\,1/2\pi^2)\). Every value is a standard cosmological reading of
that chronology; nothing on the axis is fitted.}
\label{fig:cosmic-descent}
\end{figure}

\subsection{Four entries are anchored, and they are not predictions}
\label{ssec:anchored}
The owning paper's inventory has twenty-one entries. Three of them have
no measurement to compare with; one has only an upper bound; four are
\emph{anchored}. An anchored entry is one whose value the framework
takes from observation and then uses downstream. Its deviation from
measurement is zero --- necessarily, by construction, and not because
anything was predicted correctly.

The four are the scalar amplitude $A_s$, the clustering amplitude
$\sigma_8$, the $S_8$ parameter on the microwave-background branch, and
the effective number of neutrino species. They appear in the owning
paper's own deviation plot as open markers at exactly zero, and its
caption says so: \emph{anchored entries sit at $\mathrm{Dev} = 0$ by
construction}. This review repeats the marking rather than trusting the
reader to remember it, because four rows at zero deviation in a table
of twenty-one is precisely the visual that would be misread.

With those four removed, the honest arithmetic is:
\begin{equation}
\underbrace{21}_{\text{entries}}
\;-\;\underbrace{4}_{\text{anchored}}
\;-\;\underbrace{3}_{\text{no measurement}}
\;-\;\underbrace{1}_{\text{bound only}}
\;=\;\underbrace{13}_{\text{genuine comparisons}} ,
\label{eq:cosmos-count}
\end{equation}
to which the owning paper adds four further acoustic quantities that
are, in its own words, \emph{images} of the same fractions ---
correlated with them, not additional freedom --- bringing the number of
deviations it actually scores to seventeen.

\paragraph{And one further observation, which is arithmetic rather than
opinion.} The owning paper grades two of the twenty-one entries fully
\emph{Derived}. They are the tensor-to-scalar ratio and the radiation
deviation. The first has no measured central value, only a bound; the
second is one of the four anchored rows. So \emph{not one} of the
entries this framework grades fully derived is compared against a
measured number. Its own deviation table is more conservative still,
grading the tensor ratio conditional --- on which reading the single
fully derived entry is an anchored one. Every one of the sheet's
strongest agreements is a \emph{conditional} entry.

That is not a refutation of anything. It is the shape of the claim, and
a reader is entitled to have it stated before the numbers rather than
reconstructed from them afterwards.

\subsection{The sheet}
\label{ssec:cosmos-table}
Table~\ref{tab:cosmos-sheet} is the inventory. The anchored rows carry
the word in place of a prediction; the measured column is
\citet{Planck2018} throughout, as reported by the owning paper.

\begin{table}[!t]
\centering
\caption[The cosmological sheet, with grades and anchors.]{\textbf{The cosmological sheet, with grades and anchors.}
Closed forms as the owning paper \citep{ArisakaA10Cosmos} states them;
predictions recomputed here from those forms and the ledger of
\S\ref{ssec:twelve}. \emph{Anchored} marks an entry taken from
observation, whose zero deviation is a construction and not a result.
Grades as in Table~\ref{tab:sm-sheet}. Four of the twenty-one entries
are anchored, three carry no measurement, and one carries only a bound,
so thirteen rows here are genuine comparisons; the owning paper scores
four further acoustic quantities that are correlated images of the same
fractions. The two boldface entries are the sheet's extremes: the
atomic decoupling redshift, agreeing at one part in a thousand, and the
local distance-ladder rate, which the owning paper does not claim ---
see \S\ref{ssec:cosmos-first}.}
\label{tab:cosmos-sheet}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}P{0.14\textwidth}P{0.24\textwidth}P{0.14\textwidth}P{0.16\textwidth}P{0.13\textwidth}P{0.06\textwidth}@{}}
\toprule
\textbf{quantity} & \textbf{closed form} & \textbf{predicted} &
\textbf{measured} & \textbf{deviation} & \textbf{grade} \\
\midrule
\multicolumn{6}{@{}l}{\itshape The $\pi$-locked inventory (five)} \\
$\Omega_m$ & $1/\pi$ & $0.318310$ & $0.315(7)$ & $+1.1\e{-2}$ & D$^*$ \\
$\Omega_\Lambda$ & $1-1/\pi$ & $0.681690$ & $0.685(7)$ & $-4.8\e{-3}$ & D$^*$ \\
$\Omega_b$ & $1/(2\pi^2)$ & $0.050661$ & $0.0493(6)$ & $+2.8\e{-2}$ & D$^*$ \\
$\Omega_{\rm dm}$ & $1/\pi - 1/(2\pi^2)$ & $0.267649$ & $0.2657(70)$ & $+7.3\e{-3}$ & D$^*$ \\
$w$ & $-1$ exactly & $-1$ & $-1.028(32)$ & $+2.7\e{-2}$ & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Baryogenesis and the light elements (two)} \\
$\eta_B$ & $\varepsilon\kappa = e^{-K/2}$ & $7.34\e{-10}$ & $6.12(4)\e{-10}$ & $+1.99\e{-1}$ & D$^*$ \\
${}^7$Li$/$H & SBBN$/\pi$ & $1.59\e{-10}$ & $1.6(3)\e{-10}$ & $-5.3\e{-3}$ & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Expansion and recombination (four)} \\
$H_0^{\rm CMB}$ & inventory inversion & $66.6(2)$ & $67.4(5)$ & $-1.2\e{-2}$ & D$^*$ \\
$H_0^{\rm late}$ & \,$+$ Local-Hole lift & $68.5(6)$ & $73.50(81)$ & $\mathbf{-6.8\e{-2}}$ & D$^*$ \\
$z_{\rm dec}$ & atomic chain & $1091.1$ & $1089.92(25)$ & $\mathbf{+1.1\e{-3}}$ & D$^*$ \\
$\ell_D$ & diffusion integral & $1350(3)$ & $1345(15)$ & $+3.7\e{-3}$ & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape The launch (four); $N^\star = 52.6$} \\
$n_s$ & $1 - 2/N^\star$ & $0.961977$ & $0.9649(42)$ & $-3.0\e{-3}$ & D$^*$ \\
$r$ & $12/N^{\star 2}$ & $0.004337$ & $<0.036$ (95\%) & --- & D$^*$ \\
$A_s$ & \emph{anchored} & \emph{anchored} & $2.10(4)\e{-9}$ & $0$ & A \\
$V_{\rm inf}^{1/4}$ & plateau below $E_{\rm GUT}$ & $8.3\e{15}$\,GeV & not measured & --- & D$^*$ \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape Clustering and radiation (four)} \\
$\sigma_8$ & \emph{anchored} & \emph{anchored} & $0.811(6)$ & $0$ & D$^*$ \\
$S_8^{\rm CMB}$ & \emph{anchored} & \emph{anchored} & $0.832(13)$ & $0$ & D$^*$ \\
$S_8^{\rm late}$ & corridor response & $0.78(1)$ & $0.759(23)$ & $+2.8\e{-2}$ & D$^*$ \\
$\Delta N_{\rm eff}$ & \emph{anchored} & \emph{anchored} & $0.04(18)$ & $0$ & D \\
\addlinespace
\multicolumn{6}{@{}l}{\itshape The dark-matter line (two)} \\
$m_{\rm K}$ & compact-angle relic & $8.1(5)\,\mu$eV & not measured & --- & D$^*$ \\
$f_{\rm K}$ & abundance closure & $7.1(5)\e{11}$\,GeV & not measured & --- & D$^*$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{What a reader will check first}
\label{ssec:cosmos-first}
Four entries carry most of the weight, and again they are of four
different kinds.

\paragraph{The matter fraction is $1/\pi$.} This is the claim the whole
inventory turns on, and it is worth seeing what comes with it for free.
Given $\Omega_m = 1/\pi$ and $\Omega_b = 1/(2\pi^2)$, the flatness of
the universe is not a further claim but an identity,
$\Omega_m + \Omega_\Lambda = 1$ exactly, and the baryon share of matter
is
\begin{equation}
\frac{\Omega_b}{\Omega_m} \;=\; \frac{1}{2\pi} \;=\; 0.1591549 ,
\label{eq:baryon-share}
\end{equation}
against a measured $0.15709(160)$. That last statement is the strongest
one in the chapter, for a reason developed in
\S\ref{ssec:epoch}: it is a \emph{ratio}, and the ratio carries no
epoch input at all.

\paragraph{The first atoms, and two redshifts that must not be
confused.} The program's sharpest zero-parameter reproduction is here,
and it is easy to state wrongly. There are two quantities.

The first is a \emph{corridor switch} at
\begin{equation}
1 + z_{\rm sw} \;=\; e^{J} \;=\; e^{7} \;=\; 1096.633 ,
\label{eq:zsw}
\end{equation}
an exact topological statement carrying no continuous parameter --- the
same integer $J = 7$ that gave $\alpha^{-1}(M_Z) = 2^J$ in
\S\ref{ssec:immediate}. The second is the \emph{atomic} decoupling
redshift, the surface where free electrons vanish into hydrogen. It is a
different quantity, and the framework computes it from its own outputs:
the hydrogen binding energy from the particle sector, a recombination
factor of $53.0$ evaluated on $\Omega_b = 1/(2\pi^2)$, and the present
temperature. The result,
\begin{equation}
\ln(1+z_{\rm dec}) \;=\; \ln\frac{B}{kT_0} - \ln\frac{B}{kT_\star}
\;=\; 10.967 - 3.971 \;=\; 6.996 ,
\label{eq:lnzdec}
\end{equation}
gives $z_{\rm dec} = 1091.1$ against a measured $1089.92(25)$: an
agreement of $0.11\%$ with nothing fitted.

Only the second quantity is compared with the sky. The first is
annotated and not scored. Their separation is itself a computed number,
$\delta = J - \ln(1+z_{\rm dec}) = 0.0041$ e-folds, which is $0.41\%$
in $1+z$ --- a
consistency, not an assumption, and nothing in the owning paper depends
on its being zero. A reader who takes away ``$e^7$ predicts the
microwave sky to a fraction of a percent'' has taken away the wrong
statement; the correct one is that a geometric odometer and an atomic
thermometer, computed independently, agree to four parts in a thousand,
and why they should is not yet a theorem.

\paragraph{The expansion rate: two routes, and one concession.} The
background rate is reached twice. Once by inversion: the microwave sky
measures two densities that do not depend on $h$, and dividing them by
the predicted fractions forces
$H_0^{(0)} = 66.6 \pm 0.2$ --- an inversion rather than a fit, since the
fractions were not adjusted. Once by geometry, with no astronomical
input at all, through a vacuum-energy coefficient that is itself a
ledger monomial,
\begin{equation}
\zeta_\Lambda \;=\; \frac{\pi}{L K} \;=\; \frac{\pi}{175}
\;=\; 0.0179520
\quad\Longrightarrow\quad
H_0^{\rm geom} \;=\; 66.8 \pm 0.2 .
\label{eq:zeta}
\end{equation}
The two agree to three parts in a thousand, and the whole rate collapses to
a single closed form in the integers and the measured constants,
\begin{equation}
H_0 \;=\; e^{-4K}\sqrt{\frac{2\pi^{5}c^{5}}
{3\,L\,K\,(\pi-1)\,\hbar\,G_N}}
\;=\; 66.7\ \text{km\,s}^{-1}\text{Mpc}^{-1},
\label{eq:h0-analytic}
\end{equation}
which is the kind of statement this Part exists to report --- and which
was recomputed here from CODATA values before it was typeset.

Then the concession. The local distance ladder measures $73.50(81)$.
The framework derives the \emph{sign} of the excess and about a third of
its size, from a mild local under-density; the remaining seven percent
it reads as a distance-ladder calibration systematic, and its own words
are that this is \emph{not a number GG derives}. That is the $-6.8\%$
row, the worst in either sheet, and it is printed at full size. The
owning paper adds the sentence that makes the position falsifiable
rather than merely modest: a converged gravitational-wave siren
measurement reading the background near $73$ would refute the inventory
outright.

\paragraph{The quietest failure is not the largest one.} The acoustic
angle $100\,\theta_\star$ deviates by less than half a percent --- a
number that looks like a success beside the others. But $\theta_\star$
is measured to three parts in ten thousand, so that half-percent is a
discrepancy of sixteen standard deviations on the measurement alone, and
still four after a theory tolerance of a tenth of a percent is allowed.
The owning paper prints this, together with a $\chi^2$ comparison
against the microwave band powers that goes the same way, and states
plainly that it is not claimed the exact chart out-fits the concordance
model. This review records the same. A framework with no dials cannot
improve a fit; what it can do is be close without them, and be honest
about the difference.

\subsection{The price of the three absolute fractions: the epoch}
\label{ssec:epoch}
The matter fraction is not a constant of nature. It falls monotonically
from one in the matter era to zero in the de Sitter future, so the
equality $\Omega_m = 1/\pi$ holds \emph{at one instant only}, and which
instant we occupy is fixed by the present temperature $T_0$ --- which
the framework takes from observation.

The owning paper is explicit about this, and the sentence deserves
quoting rather than summarizing: it does not claim that $0.318$ stands
on the footing of $\alpha^{-1} = 137$; it claims that the present
fractions take a forced, coupled $\pi$-form, read at the one epoch
calibrated to the sky.

The arithmetic is small and worth having. At the measured central
values the exact crossing lies slightly in the past --- at a scale
factor $0.9949$, about $75$ million years ago. The measurement error on
$\Omega_m$ blurs that timing by roughly $160$ million years, so the
offset is half a standard deviation and the data are consistent with the
crossing being now. Used once as a declared input, the epoch collapses
three of the four inventory deviations to $(0.00, +1.69, -0.31)\%$ while
the fourth becomes the input itself and scores nothing.

\emph{What this costs, stated plainly.} Three of the twenty-one entries
are absolute present-day fractions, and they inherit one observational
calibration. What survives untouched is the \emph{form} --- the coupled
$\pi$-structure --- and every ratio built from it, of which
Eq.~\eqref{eq:baryon-share} is the sharpest, since it is claimed to hold
at every redshift and can be tested at three of them. A reader deciding
how much weight to put on ``$\Omega_m = 1/\pi$'' should put it on the
form and on the ratios, not on the third decimal place of the absolute
value.

\emph{A disagreement inside the owning papers, recorded rather than
resolved.} One passage argues that ``now'' is counted in integers from
the recombination switch and is therefore not a free choice of cosmic
time. Four other passages state that the epoch is the single declared
calibration input, ``named, conceded, and left open''. This review
follows the second, which is that paper's own honesty split, and notes
that the first would be a considerably stronger claim than anything else
in this chapter.

\subsection{What is genuinely forward-looking}
\label{ssec:cosmos-forward}
Three commitments here are not postdictions of anything, and they are
two-sided --- a measurement on either side of them ends the claim.

The tilt and the tensor ratio come from one number, the e-fold count
$N^\star = 52.6$:
\begin{equation}
n_s \;=\; 1 - \frac{2}{N^\star} \;=\; 0.9620 ,
\qquad
r \;=\; \frac{12}{N^{\star 2}} \;=\; 0.0043 .
\label{eq:ns-r}
\end{equation}
The tilt already agrees at $0.3\%$. The tensor ratio sits a factor of
eight below the present bound, so it is a prediction awaiting an
instrument rather than an agreement, and the owning paper states that a
measurement inconsistent with it \emph{in either direction} falsifies
the derived launch.

The dark-energy equation of state is claimed exactly $-1$, and the
accompanying corollary is unusually unhedged: any confirmed deviation,
in either direction and at any late-time redshift, refutes the reading
outright --- the framework cannot absorb $w \neq -1$.

The dark-matter candidate is a single line. Its mass is
$8.1 \pm 0.5\,\mu$eV and its laboratory frequency $1.95$ GHz, with a
searched band of $1.59$--$2.13$ GHz. Those two intervals are not the
same interval, and the owning paper presents them in two places as
though they were: the mass error is the continuous geometric
uncertainty, while the frequency band is the image of a \emph{discrete}
window, the integer cap loading $\Xi_{\rm cap} \in [35,47]$ around its
benchmark $\Xi_{\rm cap} = K + N + L = 43$. The distinction matters to a
reader who wants to know what a null result would exclude, so this
review separates them. The band also carries a predicted internal
structure --- a splitting of one part in $K$, about $56$ MHz --- which is
the sort of detail that makes a search either decisive or wasted.

All three are two-sided, and all three are dated by the owning
paper against instruments that exist or are funded.
\S\ref{sec:falsifiers} collects them with the dates attached.

\subsection{Grade box}
\label{ssec:cosmos-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Verified here.} Every scored entry of Table~\ref{tab:cosmos-sheet}
was recomputed from its closed form and reproduces the owning paper's
quoted deviation; so did the flatness identity, the baryon share of
Eq.~\eqref{eq:baryon-share}, the two redshifts and their offset, both
routes to the expansion rate including the closed form of
Eq.~\eqref{eq:h0-analytic} from CODATA constants, the epoch arithmetic,
and the counts of Eq.~\eqref{eq:cosmos-count}. Arithmetic, again, not
derivations.

\emph{Reported, at the owning paper's grade.} The closed forms; the
conjectures they are conditional on, of which the closed-packet
integration lemma carries the whole $\pi$-locked inventory; and the
two-stage corridor.

\emph{Printed against the program's interest.} The four anchored rows
and what anchoring means; that no fully derived entry is compared with a
measurement; the $-6.8\%$ local-ladder row and the concession attached
to it; the acoustic angle's four-sigma pull behind a half-percent
deviation; the epoch as a declared input; and three internal
inconsistencies in the owning papers --- the derived-versus-declared
epoch, the two grades carried for the tensor ratio, and the conflation
of the Kaxion mass error with the search band.

\emph{Not claimed anywhere in this chapter.} That twenty-one
cosmological quantities have been derived. Thirteen are compared;
none of the fully derived ones is among them.
\end{quote}


\section{What is time? The chronology of the ledger}
\label{sec:what-is-time}
The cosmological sheet just read the universe's inventory at a
single late moment. This chapter reads the axis those entries are
strung along --- and it exists because the parent treatment
(\PaperPFive{}, Part~IX, in its expanded form) has turned the
oldest question in physics into something this review can report
as it reports everything else: defined, located, quantified, and
falsifiable. The essence, in the review's register.

\subsection{The definition, and where its parts live}
\label{ssec:time-definition}
The parent paper defines time by four endowments, each a theorem-
shaped clause: time is the unique direction that is
\emph{ordered} (the light cone sorts before from after),
\emph{repeating} (the arena's Killing flow ticks --- the reason
frequency and energy exist), \emph{one-way} (the dictionary's
projection is non-invertible --- records accumulate), and
\emph{sheltered} (the spectrum's single side opens the half-plane
where $it$, KMS, and every Euclidean computation lawfully live).
What this review adds from its own capstone
(\S\ref{ssec:symmetry-reading}) is the anatomy: the four clauses
live on the geometry's parts, one each --- order on the cone,
\textbf{the tick on the circle}, \textbf{the arrow on the
interval}, the shore on the spectrum. The middle two are the
load-bearing news, and they are this paper's two fibers wearing
time's clothes: \emph{the tick and the arrow live on different
fibers}. The circle repeats and cannot point (its windings are
two-sided; it knows charge, not direction); the interval points
and cannot repeat (nothing on a fiber without closed loops has a
period). Every physical clock is both at once --- a rotor plus a
ratchet; a caesium phase counted by an escapement --- and that
composite character is why humanity could axiomatize reversible
mechanics (all tick) and irreversible thermodynamics (all arrow)
as two separate, complete, contradictory sciences for three
centuries. They are not rival theories of one time. They are the
physics of two fibers, marginalized onto one boundary axis.

\subsection{The chronology: one line, one kink}
\label{ssec:time-chronology-w1}
The quantitative content --- and the reason this chapter sits
directly after the cosmological sheet --- is that boundary time
and the descent's radial coordinate are two parametrizations of
one motion, with a computable map between them. In the radiation
era, standard kinematics plus $r = \ln(E_{\rm GUT}/T)$ give
\begin{equation}
\frac{d\ln t}{dr} \;=\; 2
\qquad\text{(radiation; exactly)}, \qquad
\frac{d\ln t}{dr} \;=\; \tfrac32
\qquad\text{(matter)},
\label{eq:time-slope-w1}
\end{equation}
so \emph{boundary time is the exponential shadow of the linear
descent}: each unit the read head of \S\ref{ssec:symmetry-reading}
advances, the universe's clock multiplies by $e^{2} \simeq 7.4$.
Figure~\ref{fig:time-line-w1} draws the whole of cosmic history in
those coordinates --- eight epochs on one straight line of slope
two, kinking once to three-halves at matter--radiation equality
--- and the endpoint is an audit that closes:
\begin{equation}
\ln\frac{t_0}{t_{\rm Pl}} \;=\; 140.24
\;=\; \underbrace{36.11}_{\text{launch}}
\;+\; \underbrace{91.60}_{\text{radiation}}
\;+\; \underbrace{12.5}_{\text{matter}+\Lambda}.
\label{eq:time-ledger-w1}
\end{equation}
The age of the universe, in the bulk's own units, is an audited
sum over the descent --- standard kinematics, the derived
reheating temperature, nothing else. Two placements ride along at
canon grade: the lock crossing sits at $r = K = 35.00$ exactly
(half-Planck convention), and the corridor switch sits at
$\ln(1 + z_{\rm sw}) = 7 = J$ --- exactly $J$ e-folds of boundary
scale below today. And one observation is carried at the flagged
grade the parent paper assigns it, no higher: $140.24 = 4K \times
1.0017$ --- \emph{the age of the universe is $e^{4K}$ Planck
times to two parts in a thousand} --- a compression whose
$O(1)$ pieces have no derivation yet, filed beside the e-fold
budget campaign that would supply one.

\begin{figure}[tbp]
\centering
\resizebox{0.757\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth]
\draw[->] (0.4,0) -- (11.2,0) node[right] {$r=\ln(E_{\rm GUT}/T)$};
\draw[->] (0.4,0) -- (0.4,6.1) node[above] {$\ln(t/t_{\rm Pl})$};
\foreach \r in {20,30,40,50,60,70}{
  \draw ({0.4+(\r-15)*0.18},0) -- ({0.4+(\r-15)*0.18},-0.07)
    node[below] {\r};}
\foreach \y in {40,60,80,100,120,140}{
  \draw (0.4,{(\y-25)*0.048}) -- (0.33,{(\y-25)*0.048})
    node[left] {\y};}
\draw[line width=1.10pt, blue!70!black]
  (1.143,0.533) -- (3.976,2.061) -- (4.548,2.364)
  -- (5.796,3.072) -- (6.705,3.565) -- (9.232,4.930);
\draw[line width=1.10pt, red!70!black]
  (9.232,4.930) -- (9.437,5.027) -- (10.697,5.532);
\fill (1.143,0.533) circle (1.6pt)
  node[right=2pt, align=left] {launch $(19.1,\,36.1)$};
\fill (3.976,2.061) circle (1.6pt)
  node[above left] {lock $(r=K)$};
\fill (4.548,2.364) circle (1.4pt) node[below right] {EW};
\fill (5.796,3.072) circle (1.4pt) node[below right] {QCD};
\fill (6.705,3.565) circle (1.4pt) node[below right] {BBN};
\fill (9.232,4.930) circle (1.6pt)
  node[below right] {equality};
\fill (9.437,5.027) circle (1.4pt);
\node[anchor=east] at (9.15,5.19) {recombination};
\fill (10.697,5.532) circle (1.8pt)
  node[below right=1pt and -2pt, align=left]
  {today\\ $(72.3,\,140.24)$};
\node[blue!70!black, rotate=28] at (5.4,3.35)
  {slope $= 2$ (radiation)};
\node[red!70!black, rotate=22] at (10.05,5.02)
  {slope $\tfrac32$};
\draw[->, violet!70!black] (8.25,5.75) -- (9.40,5.10);
\node[violet!70!black, anchor=east, align=center]
  at (8.25,5.82) {$\ln(1{+}z_{\rm sw}) = 7 = J$\\
  ($J$ e-folds below today)};
\end{tikzpicture}}
\caption[The whole history of time, on one line.]{The whole
history of time, on one line: boundary time (Planck units,
logarithmic) against the radial coordinate (half-Planck
convention), for eight epochs from the launch to today. The
radiation era is a straight line of slope exactly $2$
\eqref{eq:time-slope-w1}; the slope kinks to $\tfrac32$ at
equality; the lock crossing sits at $r = K = 35$; the corridor
switch sits $J = 7$ e-folds below today; the endpoint's audited
decomposition is Eq.~\eqref{eq:time-ledger-w1}. Boundary time is
the exponential shadow of the linear descent.~(after
\PaperPFive{})}
\label{fig:time-line-w1}
\end{figure}

\subsection{The clock ladder, and the grades}
\label{ssec:time-grades-w1}
The repetition endowment is quantitative too, because $E =
\hbar\omega$ is the circle's currency statement: energy is
winding rate, and every clock is a winding counter. The ladder
runs from the electron's Compton rate ($1.24\times10^{20}$~Hz ---
every mass is a clock) through caesium ($9.19$~GHz) and the
optical lattices ($\sim10^{15}$~Hz) to the one rung this cohort
contributed before it was sought --- the Kaxion's predicted
$1.95$~GHz line --- and ends at the sky's $H_0^{-1}$: a dynamic
range of $e^{140}$, which is the same number as the audit above,
because clock hierarchies are depth readings of one descent. The
grades travel as always: the slope laws and the ledger
\eqref{eq:time-ledger-w1} are arithmetic on standard kinematics
plus the derived $T_{\rm rh}$ (at its parent grade); the
endowments carry the parent theorems' grades; the $4K$
compression is flagged, not claimed; and the derivation home for
all of it --- the four endowments walked, the KMS quartet worked,
the shore married to the scale rate, and the significance argued
against every previous definition of time --- is \PaperPFive{}
Part~IX, to which this chapter is the ledger's index entry.

\section{What ``no free parameters'' is worth: a comparison}
\label{sec:comparison}

A program that removes the dials invites one question above all
others: compared with what? This chapter answers it, and the answer is
less flattering to the program than the comparison its own papers
draw --- which is why the comparison is redrawn here.

\subsection{The opponent's case, at its strongest}
\label{ssec:steelman}
The contrast usually offered is \emph{zero against $10^{500}$}. It
should not be offered, and this review will not repeat it, for two
reasons.

The first is a category error. The figure $10^{500}$ counts
\emph{vacua} --- discrete solutions of a compactification problem
\citep{Douglas2003,Susskind2003} --- while zero counts \emph{continuous
parameters}. These are quantities of different kinds, and setting them
side by side in one column is the move a well-prepared opponent
dismantles first. It is also not a stable figure: published estimates
for different classes run from $10^{120}$ to $10^{1500}$ to, for a
single elliptic fourfold in F-theory, $10^{272{,}000}$
\citep{TaylorWang2015}. A number that moves by two hundred and seventy
thousand orders of magnitude with the class considered is a feature of
the class, not a constant of the theory.

The second reason is more damaging and must be stated plainly.
\emph{String theory makes the same claim about continuous parameters
that this program does.} It has no adjustable dimensionless coupling
either: what would be a coupling is the expectation value of a field, so
the freedom was never in a dial. On the axis of continuous parameters,
the two programs agree, and the naive comparison is not merely unfair
--- it is a comparison between a program and itself.

Something further is owed before the question is put, because a
contrast is only worth drawing between positions stated at full
strength, and the landscape's case is usually reported by its critics
rather than by its advocates. Stated as its advocates state it, it runs
roughly as follows. A theory of quantum gravity was not constructed in
order to have many vacua; the many vacua were \emph{found}, by people
who would have preferred one, and who published the finding because
that is what one does with an unwelcome result. The same framework has
produced results that survive independently of whether the landscape is
the right reading of it --- a duality between a gravitational theory and
a gauge theory without gravity, and an exact statistical-mechanical
count of the microstates of an extremal black hole, both of which are
now standard tools. And anthropic reasoning within it has one success
of a kind this review is bound to respect: \citet{Weinberg1987}
predicted the order of magnitude of the cosmological constant from
selection considerations \emph{a decade before it was measured}, and
was closer than any first-principles calculation of the day. A
program that asks to be judged by dated forward commitments cannot
dismiss the one dated forward commitment its opponent got right.

The disagreement is therefore not about competence or courage. It is
about what an explanation is. On the landscape's reading, explaining a
constant means exhibiting a mechanism that permits it and a selection
effect that makes our observing it unsurprising. On this review's
reading, that is a weaker thing than exhibiting a mechanism that
\emph{compels} it, and the difference shows up not in elegance but in
what can be lost: a framework of the first kind absorbs a surprising
measurement by moving within its own space of vacua, and a framework of
the second kind cannot. Which is the right standard is a real question,
and it is not settled by either side asserting it.

So the honest question is not who has fewer dials. It is: \emph{once the
dials are gone, where does the freedom go?}

\subsection{Where the freedom goes}
\label{ssec:where-freedom}
Four answers are on offer, and they are genuinely different postures
rather than four positions on one scale.

\paragraph{Leave it as measured input.} The Standard Model and the
concordance model of cosmology take this route. Twenty-eight numbers and
six numbers respectively are measured and written in, and the theories
are silent about why they take the values they do. This is not a defect
of those theories; it is their declared scope.

\paragraph{Add more of it.} The minimal supersymmetric extension is the
clearest case, and its arithmetic is a decomposition rather than an
estimate:
\begin{equation}
\underbrace{124}_{\text{MSSM}}
\;=\;
\underbrace{18}_{\substack{\text{Standard Model}\\\text{incl. }\bar\theta}}
\;+\;
\underbrace{1}_{\text{Higgs sector}}
\;+\;
\underbrace{105}_{\text{genuinely new}} ,
\label{eq:mssm124}
\end{equation}
the hundred and five comprising the gaugino and higgsino sector, the
sfermion masses, and thirty-six mixing angles and forty $CP$-violating
phases among them \citep{PDG2024}. A program adopted partly because it
would explain the Higgs mass ends by multiplying the number of things
requiring explanation by a factor of four. \S\ref{ssec:failures} takes up what the searches
have since done to its preferred region; this is the accounting
alongside it.

\paragraph{Distribute it across a discrete set.} This is the landscape's
answer, and it is a serious one. The freedom is not eliminated but
relocated: from continuous couplings into a choice among vacua, each
giving a different low-energy theory
\citep{BoussoPolchinski2000}. What makes that relocation possible ---
and what makes the resulting freedom genuinely discrete rather than
continuous --- is that the moduli are stabilized
\citep{KKLT2003}; before stabilization they are fields, and their
expectation values are exactly the continuous freedom the program
sets out to remove. Its difficulty is equally
serious and its own literature states it: the framework supplies no
internal mechanism for selecting among them. The swampland program
\citep{Vafa2005} is the most developed attempt to constrain the
proliferation from the low-energy side.

\paragraph{Relocate it into a small integer set, and claim that set is
forced.} This is the program under review. The freedom goes into
$(M,N,L;J,K)$, five integers satisfying the two closure relations of
Eq.~\eqref{eq:closure} --- and the claim is that the choice is not a
choice.

\subsection{The contrast that survives}
\label{ssec:real-contrast}
Stated this way the comparison sharpens rather than dissolves, and
Table~\ref{tab:comparison} is its whole content.

\begin{table}[!t]
\centering
\caption[Four postures toward the same problem.]{\textbf{Four postures toward the same problem.} The first
column is the only one on which the naive comparison is usually drawn,
and it is the least interesting: two of the five entries claim the same
thing there. The argument is in the last two columns. Counts as in
\S\ref{ssec:twelve}, \S\ref{ssec:the-sheet} and Eq.~\eqref{eq:mssm124};
vacuum estimates from \citet{Douglas2003} and \citet{TaylorWang2015}.
\emph{Vacua are not parameters}, and the two are kept in separate
columns here for that reason.}
\label{tab:comparison}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}P{0.17\textwidth}P{0.12\textwidth}P{0.28\textwidth}P{0.34\textwidth}@{}}
\toprule
\textbf{program} & \textbf{continuous parameters} &
\textbf{where the remaining freedom sits} &
\textbf{is the discrete choice claimed forced?} \\
\midrule
Standard Model $+$ general relativity & 28 & nowhere: the parameters
\emph{are} the freedom & not applicable \\
\addlinespace
$\Lambda$CDM & 6 & nowhere; six fitted numbers & not applicable \\
\addlinespace
MSSM & $+105$ & nowhere; the freedom grows & not applicable \\
\addlinespace
String compactifications & none claimed & a discretuum of vacua,
estimated at $10^{120}$ to $10^{272{,}000}$ depending on the class &
\textbf{no} --- and its own literature says the framework supplies no
selection mechanism \\
\addlinespace
This program & none claimed & five integers with two closure
relations & \textbf{claimed, not proved} --- the completion class is,
in the owning paper's words, asserted \\
\bottomrule
\end{tabular}
\end{table}

The contrast that survives is not \emph{zero versus many}. It is a
contrast in two parts.

\emph{First, the size of the discrete set.} One program's residual
freedom is a set of five integers constrained by two equations; the
other's is a discretuum whose size is not known to within hundreds of
thousands of orders of magnitude. That difference is not rhetorical. A
set that small can be written down, and a theory that has written it
down can be caught: every number in Parts~\ref{part:ledger}'s two
sheets is a place where five integers can fail.

\emph{Second, and against the program:} neither has proved its
discrete choice forced. The landscape does not claim to have. This
program does claim it, and has not. Its own statement, quoted in
\S\ref{ssec:not-zero}, is \emph{zero free continuous parameters, plus
one frozen discrete completion class whose necessity is asserted, not
proved.} Until that necessity is a theorem, the difference between the
two programs is a difference in the \emph{size} of an unselected
discrete set, not in whether one has been selected.

That is a narrower claim than the one the cohort's own front matter
makes, and it is the one this review is prepared to defend.

\subsection{What the comparison does not buy}
\label{ssec:not-bought}
Three limits belong here rather than in a footnote, and all three are
stated by the owning papers themselves.

\paragraph{The inputs, not the corrections.} The program derives the
\emph{inputs} of the Standard Model, not the radiative corrections that
act on them. The loop periods of precision quantum electrodynamics and
the non-perturbative hadronic correlators remain, in the owning paper's
words, irreducible four-dimensional data that the geometry does not
compute \citep{ArisakaA9SM}. Deriving a Lagrangian's parameters is not
deriving its predictions; the rest of quantum field theory is still
required, and is still doing most of the work between a parameter and a
measurement.

\paragraph{The trials factor.} A small alphabet of integers and two
mathematical constants can express a great many numbers, and the honest
question is how many targets the alphabet had to hit. That is a single
statistic --- computed globally, for the whole codebook against the
whole declared alphabet at once, not parameter by parameter --- and it
has not been computed. The owning paper names it as the decisive test,
commits to reporting it, and states that if the global $p$-value is
large the program fails by its own standard. Its concession is exact
and deserves quoting:

\begin{quote}\itshape
Until it is computed and reported\ldots\ the codebook's closures are an
honest ansatz, not a proof, and we say so.
\end{quote}

\noindent This review will not improve on that sentence, and notes only
that it is the correct place for a skeptical reader to begin.

It can, however, do one thing more, and consistency requires it. Every
physical claim in Part~\ref{part:ledger} is carried with a kill
criterion, a named instrument and a window, on the principle that a
commitment without a date is not a commitment. The same principle
applied here yields a methodological entry of the same shape, and it
belongs in the open where the physical ones are:

\begin{quote}\small
\textbf{Commitment (methodological).} \emph{What:} the global trials
factor of the codebook --- one statistic, computed for the whole
closure set against the whole declared alphabet at once, with the
alphabet and the target list both fixed and published \emph{before} the
computation is run. \emph{Kills:} a large global $p$-value. \emph{By
whom:} the owning papers, not this review. \emph{Window:} before
journal submission of the codebook, and in any case within the decade
in which the physical falsifiers of Table~\ref{tab:falsifiers} come
due. \emph{Status if unmet:} the closures remain an honest ansatz, and
this paragraph remains the correct citation for a reader who wants to
say so.
\end{quote}

\noindent Nothing about the physics changes if that computation comes
out badly except everything, which is the point of writing it down.

\paragraph{Uniqueness.} The strongest of the standing claims --- that
this is \emph{the} admissible geometry rather than merely \emph{a}
self-consistent one --- remains a conjecture, the spectrum-level
uniqueness theorem being open. \S\ref{ssec:exposure} returns to it as
the premise on which most weight rests.

\subsection{Grade box}
\label{ssec:comparison-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Verified here.} The decomposition of Eq.~\eqref{eq:mssm124}; the
Standard-Model and $\Lambda$CDM counts; the spread of published vacuum
estimates; and that the four entries of the first column of
Table~\ref{tab:comparison} count the same kind of object, which is the
condition for the comparison to mean anything.

\emph{Reported, at the owning papers' grade.} That the integer ledger is
forced.

\emph{Printed against the program's interest.} That the comparison its
own papers draw is a category error; that string theory makes the same
zero-continuous-parameter claim; that the program's discrete choice is
asserted rather than proved, which is the same gap it charges the
landscape with, differing in size and not in kind; that the inputs are
derived and the corrections are not; and that the trials factor is
uncomputed.

\emph{Not claimed anywhere in this chapter.} That this program has
been shown superior to any other. What has been shown is that it is
exposed in a way the others are not --- which is the subject of
\S\ref{sec:falsifiers}.
\end{quote}


\section{The dated falsifiers}
\label{sec:falsifiers}

Everything in Part~\ref{part:ledger} so far has been a claim about
numbers already measured. This chapter is about the numbers that are
not, and it is the shortest chapter in the paper because its content is
a table.

\subsection{Why the dates are the point}
\label{ssec:why-dates}
A theory with no adjustable parameter cannot survive a failed
prediction by adjusting itself. That is the cost of the posture of
\S\ref{sec:zero-parameters}, and the owning papers accept it in terms
that leave no room afterwards: a single failed prediction ends the
framework, because there is nothing left to retune.

Claims of falsifiability are cheap, and most programs make them. Two
things make this one checkable rather than rhetorical. The first is that
the cohort maintains its predictions in a single canonical ledger with a
consistency checker attached, and the checker is run against every paper
--- including this one --- as part of the production gate, so a stale
prediction surviving in one paper after being revised in another is a
detectable error rather than an accident nobody notices. The second is
that the entries carry \emph{dates and instruments}. A kill criterion
with no date is a promise about an unspecified future; a kill criterion
naming an instrument and a decade is a bet.

\subsection{The ledger}
\label{ssec:falsifier-table}
Table~\ref{tab:falsifiers} lists the commitments this review regards as
load-bearing. \emph{Two-sided} means what it says: a measurement
inconsistent with the value in \emph{either} direction falsifies. Most
of these are two-sided, which is the property that distinguishes a
prediction from a bound.

\begin{table}[!t]
\centering
\caption[The dated commitments.]{\textbf{The dated commitments.} Drawn from the cohort's
canonical falsifier ledger, which is a maintained file with a
consistency checker, not a list assembled for this paper. Every row
carries all three of a kill criterion, a named instrument and a window.
The neutrino mass sum is placed first because it is the one most likely
to fire first, and because it is a \emph{floor}: the prediction is the
normal-hierarchy minimum recomputed in \S\ref{ssec:the-sheet}, so a
bound pushing below it has nowhere to go. The proton row is of a
different kind from the rest --- not a value with a tolerance but a
prohibition, which one event ends.}
\label{tab:falsifiers}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}P{0.15\textwidth}P{0.24\textwidth}P{0.24\textwidth}P{0.15\textwidth}P{0.115\textwidth}@{}}
\toprule
\textbf{quantity} & \textbf{committed value} & \textbf{what kills it} &
\textbf{instrument} & \textbf{window} \\
\midrule
$\sum m_\nu$ & $58.5$ meV & a robust bound below $\sim58$ meV &
DESI $+$ CMB & \textbf{near-term} \\
\addlinespace
$z_{\rm dec}$ & the atomic chain, \S\ref{ssec:cosmos-first} & measured
value off that chain by $>0.5\%$ & 21-cm, CMB spectral distortions &
now \\
\addlinespace
$\Omega_m$ & $1/\pi$ & a joint posterior outside $[0.315, 0.322]$ &
CMB-S4, DESI, Roman & late 2020s \\
\addlinespace
$M_{\rm lock}$ & $3.85$ TeV & no resonance in the predicted channel &
HL-LHC & 2026--2032 \\
\addlinespace
$m_{\rm K}$ & $8.1\,\mu$eV; $1.59$--$2.13$ GHz, split $1/K$ & the window
scanned to depth with no line & haloscopes & 2026--2035 \\
\addlinespace
$w$ & $-1$ exactly, two-sided & any confirmed departure, either
direction & DESI, Euclid, Rubin & $\sim$2030 \\
\addlinespace
$H_0$ & background $\approx 67$ & a converged siren catalog reading
$\approx73$ & GW standard sirens & this decade \\
\addlinespace
$r$ & $12/N^{\star 2} = 0.0043$, two-sided & detection well above it,
\emph{or} a null below $\sim10^{-3}$ & CMB-S4, LiteBIRD & 2030--2033 \\
\addlinespace
$n_s$ & $1-2/N^\star = 0.9620$ & a tilt below $\sim0.955$ & CMB &
now--2030 \\
\addlinespace
$\delta_{\rm PMNS}$ & $\approx 196^\circ$ & a central value far from it
at DUNE precision & DUNE, JUNO, Hyper-K & 2030--2035 \\
\addlinespace
$\tau_p$ & absolutely stable & \textbf{one} decay event, any channel &
Hyper-K, DUNE & 2030s \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Which of these is the sharpest}
\label{ssec:sharpest}
Three deserve singling out, for three different reasons.

\emph{The neutrino mass sum, because it is nearly due.} The prediction
is a floor of $58.5$ meV, and the bounds are tightening from above.
This is not a case where a null result is uninformative: the committed
value has no room below it.

\emph{The proton, because it is a prohibition.} Every other row is a
number with a tolerance. This one is not: baryon number is carried by a
massive gauge factor, the selection rule forbids the decay, and a single
event in any channel ends the program with nothing to renegotiate.
Grand unification's characteristic prediction is that the proton decays;
this program predicts the exact opposite, which is as clean a
discriminator as the subject offers.

\emph{The Kaxion line, because it is narrow and the search is running.}
A mass, a frequency window, a coupling on a known line, and an internal
splitting of one part in $K$: four independent handles on the same
object, in a band instruments are scanning now.

\subsection{Grade box}
\label{ssec:falsifier-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Verified here.} That every row of Table~\ref{tab:falsifiers}
carries a kill criterion, an instrument and a window; that the canonical
ledger exists as a file with a checker attached and that the checker
runs in this paper's gate; and the neutrino-mass arithmetic, which is
the same computation as \S\ref{ssec:the-sheet}'s.

\emph{Reported, at the owning papers' grade.} The committed values
themselves, each at the grade its sheet gives it --- and it is worth
noting that most of these rows are \emph{derived-conditional}, not
theorems.

\emph{Printed against the program's interest.} That a list of
falsifiers is not evidence of anything: a theory can be both falsifiable
and false, and being exposed is not the same as being right.

\emph{Not claimed anywhere in this chapter.} That any of these tests has
been passed. None has yet been run at the precision that would decide
it. The owning paper's own summary is the one to end on:

\begin{quote}\itshape
Internal consistency earns the framework a fair hearing; correctness is
decided by the dated falsifiers, and uniqueness by a proof not yet in
hand.
\end{quote}
\end{quote}



\subsection{Part~IV in one paragraph, and what comes next}
\label{ssec:part-iv-close}
\enlargethispage{4\baselineskip}%
Part~IV was the bill: five integers in, and forty-nine numbers set beside
them with a closed form, a measurement, a deviation and a grade apiece.
The grades do not flatter --- six derived, fourteen resting on one named
lemma each, eight asserted, the soft center the charged-fermion masses ---
and between the sheets came the chapters where the ledger does what a
table cannot: the mirrors, $\bar\theta$, the matter that survived, what a
clock counts. Whether any of it hangs together is Part~V's question. It
reads the seven rungs as one argument with a direction, puts the two
halves of the paper into contact for the only time, and then applies to
this program the standard it has just applied to parity, to $CP$, and to
naturalness.

%% ===================================================================
%%  PART V --- DISCUSSIONS
%% ===================================================================
\GGPart[Discussions]{Discussions}{The seven rungs read as one argument;
the ladder cashed against the ledger, and what the family number is
worth beside it; and the honest record of where symmetry-first
reasoning has succeeded and where it has failed}
\label{part:discussions}

\section{The seven rungs as one argument}
\label{sec:one-argument}
Part~II climbed seven steps, one chapter at a time, and each chapter was
written to stand on its own. Read in sequence, however, they are not seven
illustrations of a single principle. They are one argument, and the
argument has a direction. This chapter states it.


\subsection{The arc table, annotated and earned}
\label{ssec:arc-earned}
Table~\ref{tab:arc-promise} was offered in \S\ref{ssec:promise} as a
promise. Here it is again, with the columns Part~II has earned.

\begin{table}[H]
\centering
\caption[The arc, earned.]{\textbf{The arc, earned.} The fourth column
is the one Part~II adds: it records not what the rung says but what it
would cost to break it. Read downward, the cost rises from ``a different
dynamics'' to ``a discontinuity,'' and the fifth column fills: what
begins as a statement about motion ends as a statement about a whole
number.}
\label{tab:arc-earned}\small
\begin{tabular}{@{}P{0.18\textwidth}P{0.155\textwidth}P{0.23\textwidth}P{0.18\textwidth}P{0.145\textwidth}@{}}
\toprule
\textbf{rung} & \textbf{what conserves} & \textbf{why} & \textbf{what
violating it would take} & \textbf{how much is an integer} \\
\midrule
S1 Noether I & a current & a continuous global symmetry of the action &
a different action, or the field equations failing & full --- it
constrains motion \\
S1$'$ Noether II & nothing new & a local symmetry gives identities among
the field equations & nothing: it holds off shell & none \\
S2 gauge & a redundancy & gauge-equivalent configurations are the same
state & nothing: there is no second state to move to & none ---
bookkeeping \\
S3 Bianchi & flux & $F = \dd A$, hence $\dd F = 0$ & abandoning
the statement that the field is a curvature & none --- a geometric
identity \\
S4 Wilson & a holonomy & the loop does not bound, so $F$ does not
determine $A$ & filling in the hole in the space & the \emph{frame} ---
one phase per cycle \\
S5 Ward & the current, inside every correlator & a change of integration
variable cannot change an integral & a measure that is not invariant ---
and then only by an integer & the \emph{failure} \\
S6 \GS{} & the \emph{net} ledger & one failure is arranged to pay
another & an imbalance in the total, not in any entry & the \emph{trade}
\\
S7 \HS{} & an integer class & integers have nowhere continuous to move &
a discontinuous jump of the whole configuration & the \emph{whole
content} \\
\bottomrule
\end{tabular}
\end{table}

The fourth column is the one to read. Noether's first theorem gives a
conservation law that a change of dynamics can remove: write a different
action and the current is different, or gone. That is the ordinary sense
in which a conservation law is a law --- it is contingent on how the
world happens to be arranged, and it constrains how things move. Every
rung above it is progressively harder to remove, and progressively less
about motion.


\subsection{The axis restated: how much of the statement is an integer}
\label{ssec:axis}
Chapter~\ref{sec:two-ideas} asked a question and then left it standing:
why is anything an integer? The ladder answers it, and the answer is
best seen by reading the last column of Table~\ref{tab:arc-earned}
downward on its own.

\begin{quote}\itshape
The sequence is not a list of statements that grow harder to break. It
is a sequence in which the \emph{granularity of failure} coarsens: from
a quantity that can drift by any amount, through statements that cannot
fail at all, to statements that can fail only by a whole number.
\end{quote}

Set out row by row, the progression is this. At S1 there is no integer
anywhere: a conserved current can be made to drift by any amount one
likes, by writing a different action. At S1$'$, S2 and S3 there is still
no integer, and there is no failure either --- these are algebra,
convention and geometry, and nothing in them can go wrong. At S4 an
integer appears for the first time, and it appears as a \emph{frame}
rather than as a failure: the number of independent holonomies a space
carries is a whole number fixed by the shape of the space, and on an
orbifold even the admissible values are discrete. At S5 an integer
appears as a \emph{failure}: an identity that says nothing about motion
turns out to be capable of breaking, and the amount by which it breaks
is the integral of a characteristic class. At S6 the integer becomes a
\emph{trade} --- two failures of that kind, arranged to cancel exactly.
And at S7 there is nothing left but the integer.

Two things follow, and they are the reason this reading was worth the
trouble of adopting.

\emph{First, unbreakable is not the best thing a conservation law can
be.} An identity that cannot fail tells one nothing when something goes
wrong, because nothing can go wrong. An identity that can fail only by a
whole number tells one exactly how much, and a quantity known exactly is
a quantity one can cancel against, carry forward, and audit. That is
what makes Chapter~\ref{sec:gs}'s trade possible at all: one cannot
arrange for two failures to cancel unless one knows each of them
exactly.

\emph{Second, this is why the ladder ends where it does.} The top rung
is not the most rigid because nothing can touch it. It is the most rigid
because everything it says is already a whole number, so there is
nothing continuous left for a dynamics to move. Chapters~\ref{sec:ggep}
onwards ask what an argument of that kind can do when the system in
question is not closed, and the answer they propose is the subject of
Part~\ref{part:turn}.

\subsection{Symmetry's retreat and rigidity's advance}
\label{ssec:retreat}
Two things happen along the sequence, and they happen together.

\emph{Symmetry retreats.} At S1 the symmetry is the whole explanation. At
S1$'$ the symmetry is local, and the yield is an identity rather than a
charge --- the same 1918 paper that gave physics its most celebrated
symmetry argument also showed where the argument stops. At S2 the object
that everyone calls a symmetry turns out to be a redundancy of
description: a local gauge transformation does not carry one physical
state to a different one, and Elitzur's theorem records the consequence
that there is nothing there to break spontaneously
\citep{Elitzur1975}. At S3 no symmetry appears in the derivation at all;
$\dd\dd = 0$ is a fact about the exterior derivative. At S6 a symmetry is
deliberately allowed to fail. And at S5 the argument that forces an
integer --- a phase must close when transported around a loop --- has no
symmetry in it anywhere.

\emph{Rigidity advances.} In the same order, the conserved quantity
becomes harder to disturb. A current can be spoiled by changing the
Lagrangian. An identity cannot be spoiled at all without giving up the
definitions. A ledger can be rebalanced but not evaded. An integer
cannot be moved by any continuous deformation whatsoever.

These two trends are not a coincidence, and the relation between them is
the paper's thesis in one line: \emph{a conservation statement is
exactly as rigid as it is free of dynamics.} A quantity conserved because
of a symmetry of the dynamics inherits the contingency of that dynamics.
A quantity conserved because of what an object \emph{is} inherits
nothing, and can be disturbed by nothing. The price of rigidity is that
the statement stops telling you how anything moves.

It is worth being precise about what this does and does not say. It does
not say that Noether's theorem is superseded; it is used constantly and
correctly, and most of the conservation laws a working physicist applies
in a day are of that first kind. It says that the conservation statements
which survive at the level where the theory is most rigid are not of
that kind ---
and that when physicists reach for the word ``symmetry'' to describe
them, as in ``gauge symmetry,'' the word has quietly changed meaning
along the way.


\subsection{The exhibit: confinement and the immortal proton}
\label{ssec:exhibit}
If the thesis of the last section wanted a single exhibit, the sheet
supplies one, and it is the pair of facts this review spent
\S\ref{sec:triangle} assembling: no experiment has ever isolated a
quark, and no experiment has ever watched a proton die. In the
Standard Model these two facts have entirely different standings, and
neither standing is a symmetry explanation. Confinement is dynamics
--- true on the lattice to high confidence, organized by a lore
(dual superconductors, center vortices, $N$-ality) in which the
$\mathbb{Z}_3$ center decides who is confined, when confinement
holds, and where the disorder lives, three roles the account
inherits from $\mathrm{SU}(3)$ representation theory and does not
further explain. Baryon-number conservation is an \emph{accident} ---
nothing imposes it, every extension of the theory violates it, and
the textbook's own fine print (the $\mathbb{Z}_6$ quotient of
\S\ref{ssec:abelian-ledger}) sits inert beside it. The century's two
sturdiest facts, one owned by lore and one by luck.

The geometric reading of \S\ref{sec:triangle} converts both into
consequences of a single holonomy, and the conversion is precisely
the ladder's move --- \emph{from a statement about dynamics to a
statement about what an object is}. The singlet rule stops being an
output of the disorder and becomes the closure condition of a circle
\eqref{eq:singlet-chain-w1}: a grammar, administered fragment by
fragment. Baryon number stops being a habit and becomes a winding
\eqref{eq:centre-identity-w1}: a sector label no operator can reach.
The two lattice-verified roles of the center, and the fractional
electric charges that have decorated the quark model since 1964, and
the quotient footnote, all become one triangle seen from different
sides --- with the electromagnetic shadow $Q + B \in \mathbb{Z}$
\eqref{eq:em-shadow-w1} welding the sides together. Where grand
unification put its unifier on the boundary, made the $X,Y$ real
four-dimensional particles, and thereby made proton decay mandatory
\citep{GeorgiGlashow1974} --- fifty years of silence pushing the
scale ever upward --- the bulk reading removes the $X,Y$ from the
boundary and writes the conservation law that the same silence
measures. And where the Standard Model's accident merely \emph{prices}
baryon violation by an unknown scale, the winding \emph{forbids} it
at every dimension: any decay signal, at any rate, distinguishes the
two readings forever.

The honest ledger travels with the exhibit, as everywhere in this
review. The lock that underwrites both conversions is the parent
paper's theorem at its own conditional grade, not a settled fact; the
dynamical half of confinement --- the tension's value, the linear
rise, the mass gap --- remains exactly as open here as in the
standard account, and the parent program says so in its own print;
and the exhibit hands over its kill switches --- one free fractional
charge, one proton decay, one $n$--$\bar n$ oscillation --- which are
among the most continuously tested null results in physics. But the
shape of the claim is the shape this Part has been tracking through
every rung: the deepest conservation statements are not symmetries of
a dynamics but rigidities of a kind. The proton, on this reading, is
the flagship: it is not forbidden to decay. There is nothing for it
to decay \emph{into} that carries its winding, and no operator that
can carry the winding away.

And the pair of facts is really a triple. Alongside ``no experiment
has ever isolated a quark'' and ``no experiment has ever watched a
proton die'' stands a third: the universe kept six parts in ten
billion of itself, and we are that remainder. Chapter
\ref{sec:leptogenesis} traced the third fact to the same circle as
the first two --- the pentagon's chirality for the existence of the
bias, a Majorana parity for its sign, the throat depth $e^{-K/2}$
for its size --- so the ledger's account of \emph{what exists} and
its account of \emph{how much of it exists} are one account, read
at three depths. A geometry that forbids the proton's death, on
this reading, is the same geometry that arranged its birth; the
rigidity thesis of \S\ref{ssec:retreat} covers not only why matter
persists but why there was matter to persist at all.

\subsection{Symmetry breaking as symmetry reading: the two fibers}
\label{ssec:symmetry-reading}
The exhibit closed on a claim large enough to deserve its own
subsection, and this review closes its argument by paying it: beneath
every chapter of this paper sits one question, and the ledger's answer
to it is where the framework's argument comes out. The question: \emph{how is symmetry
broken --- how does a universe create something from nothing ---
deterministically and inevitably, when the starting point is
symmetric?} It is one of the oldest questions in physics, and it has
a name: Curie's principle, stated in 1894 \citep{Curie1894} ---
\emph{the symmetries of
the causes are found in the effects}. Taken seriously, it seems to
forbid our universe. If the beginning was symmetric ---
matter--antimatter symmetric, mirror-symmetric, one unified force ---
then where did the asymmetry come from?

The field has three standard escapes, and each pays a cost this Part
has already itemized. \emph{Spontaneous breaking} --- the Mexican
hat: the symmetric state is unstable and the system falls into one of
a \emph{continuous degenerate family} of asymmetric vacua. The cost
is that the choice is random: which vacuum, which direction, which
sign, the theory is silent; different regions choose differently and
the disagreements persist as defects. Curie is evaded by a coin
flip, and nothing about the \emph{outcome} is explained.
\emph{Explicit breaking}: the asymmetry was in the laws all along ---
a CKM phase, a $\theta$-angle, written in by hand. The cost is that
the starting point was never symmetric, and the question is not
answered but renamed as an input. \emph{The anthropic move}: every
choice happens somewhere, and we observe the one compatible with
observers. The cost is that it is not an explanation --- it is the
decision to stop asking (\S\ref{ssec:anthropic}). The ledger's
account, assembled across this paper, constitutes a fourth escape,
and it can now be stated in one sentence whose every clause is a
graded claim of the cohort: \emph{the breaking is spontaneous in its
dynamics --- no asymmetric term is added by hand --- yet
deterministic in its outcome --- no coin is ever flipped --- because
the vacuum manifold is not a continuous hat but a quantized dial,
and the descent that turns the dial is a one-way machine.}

\paragraph{The anatomy: one fiber that remembers, one that forgets.}
The mechanism lives in the two non-spatial fibers of
Chapter~\ref{sec:oggep}'s projection --- the scale interval $I_r$ and
the phase circle $\Stheta$ --- and the division of labor between
them is not a metaphor but a piece of exact mathematics:
\emph{holonomy requires closed loops, and an interval has none.} Set
the two side by side, as Table~\ref{tab:two-fibres} does:

\begin{table}[H]
\centering
\caption[The two non-spatial fibers: what each one can do.]{\textbf{One
fiber that forgets, one that remembers.} The scale interval $I_r$ and
the phase circle $\Stheta$, set against each other row by row. The
division of labor is not a metaphor but a consequence of one fact in
the top two rows: holonomy needs closed loops, and an interval has
none. Everything below follows --- the arrow, the schedule and the
budget on the left; the letters, the windings and the conserved charges
on the right. The last row is where the two meet: Sakharov's third
condition belongs to $I_r$ and his second to $\Stheta$.}
\label{tab:two-fibres}\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{@{}lp{0.36\textwidth}p{0.36\textwidth}@{}}
\toprule
 & \textbf{the scale fiber $I_r$ (5th)} & \textbf{the phase circle
$\Stheta$ (6th)} \\
\midrule
topology & non-compact interval & compact circle \\
closed loops & none --- no loop ever closes & all --- the circle
\emph{is} a closed loop \\
holonomy & none possible & everything: Wilson phases, windings, the
alphabet \\
structure & semigroup --- the flow runs one way, no inverses & group
--- every phase invertible \\
character & irreversible, dissipative, open: the \GDOS{} generator
(\S\ref{ssec:gdos}) & unitary, coherent, memory: the
Aharonov--Bohm/Berry content \\
carries & the arrow (time's and the RG's, unified); the schedule
(gates at definite $r$); the budget (the $e^{-K}$-type depths) & the
letters (pentagon, parities, triangle) and the charges (the
winding $B$) \\
can do & \textbf{forget}: decohere, gate, never return &
\textbf{remember}: quantize, conserve, protect forever \\
Sakharov condition & (3) departure from equilibrium & (2) $CP$
violation; the structure of (1) \\
\bottomrule
\end{tabular}
\end{table}

\noindent A fiber with no closed loops has no gauge-invariant
memory, no quantized charges, no protected phases: transport along
it is pure semigroup, which is how irreversibility enters a
fundamental theory honestly. A fiber that is nothing but closed
loops carries nothing but holonomy: quantized, rephasing-invariant,
topologically protected. \emph{The fifth dimension is where the
universe forgets; the sixth is where it remembers.} And symmetry
breaking that is both \emph{permanent} (which needs forgetting ---
no way back) and \emph{structured} (which needs remembering ---
definite letters, not noise) requires exactly this pair. Two
circles would give a reversible, forever-oscillating world in which
nothing is ever decided; two intervals would give a featureless
slide with nothing to decide \emph{into}. The bulk carries
precisely one of each, and that is the anatomical reason the
breaking in this framework has both an arrow and an alphabet.

\paragraph{Why the outcome is deterministic: three mechanisms,
stacked.} The determinism is not one mechanism but three, and each
closes a different door through which randomness enters the
standard accounts.

\emph{First: the vacuum manifold is a dial, not a hat.} The
gauge-sector breaking of this paper is Wilson-line breaking: the
order parameter is a holonomy on the compact circle. Classically
the family is flat and exactly degenerate; the quantum effective
potential of Eq.~\eqref{eq:hosotani-potential} lifts the degeneracy
--- the dial parks itself (\S\ref{ssec:hosotani}). Consistency
quantizes the admissible settings, so the hat's continuous brim is
replaced by discrete click-stops; and the spectrum's demands select
exactly one --- the three-demand scan of \S\ref{ssec:lock-winding},
with its unique winner. The ``choice'' of broken vacuum is an
arithmetic scan with one survivor: there is no direction to point
randomly in, because a quantized dial has no directions, only
settings. Elitzur's theorem is respected rather than finessed
(\S\ref{ssec:elitzur}): what breaks is never the local redundancy
--- nothing can break that --- but the genuine global structure the
Wilson line carries, the center of \S\ref{ssec:centre}, and it
breaks to a \emph{discrete, scanned, unique} value. That is the
deterministic residue of everything the textbook phrase ``gauge
symmetry breaking'' ever legitimately meant.

\emph{Second: the landscape has a unique floor.} Where the breaking
involves genuine boundary order parameters, the engine of
Chapter~\ref{sec:grip} takes over: the \GFF{} is the landscape;
\GRIP{} is the descent --- \DGI{} at the top, \GSSB{} in the
middle, \GA{} at the bottom. The determinism here is
Morse-theoretic: the landscape's minima are filtered by the
\QiFour{} pillars, \GRIPin{} relaxes within a floor to the unique
ground state --- the \GGA{} --- and \GRIPout{} opens the next
floor. At its stated grades, the cascade has no ties to break by
coin flip: where the hat has a degenerate brim, the \GFF{} has a
gradient and a single deepest basin.

\emph{Third: the descent cannot re-choose.} Determinism of outcome
needs more than uniqueness of selection; it needs the selection to
\emph{stick}. That is the scale fiber's gift: transport along $r$
is a semigroup --- no inverses, no return. Each \GSSB{} event is
gated at a definite $r$; once a gate is crossed, the previous floor
is not disfavored but unreachable --- the prompt crossing and zero
dilution that Chapter~\ref{sec:leptogenesis} used are this
statement at the lock. In the Mexican-hat world, different patches
choose differently and the disagreements persist as domain walls;
in this descent there is nothing patch-local to choose --- the
holonomy is a single bulk setting, not a field's local whim --- and
nothing reversible to regret. The breaking is recorded once, in a
medium that cannot be rewritten.

\paragraph{Curie dissolved: the reading of a groove.} Stack the
three mechanisms and Curie's principle is not violated but
dissolved, because the effect's asymmetry was never absent from the
cause. It was \emph{latent in the arena's discrete data} --- the
odd block integer $L = 5$ that makes the alphabet chiral
(\S\ref{sec:discrete}), the $\mathbb{Z}_3$ center that makes baryon
number a winding (\S\ref{sec:triangle}), the depth $K = 35$ that
prices the lock, the leakage, and the matter
(\S\ref{sec:leptogenesis}). Here one must be precise, because the
phrase ``totally symmetric starting point'' carries two senses and
the argument lives in their separation. The start is symmetric in
every \emph{dynamical} sense: no asymmetric term in the action, no
vacuum chosen, the holonomy unparked, the mirrors unbroken. It is
not --- and cannot be --- symmetric in its \emph{arithmetic},
because a compact quantized fiber necessarily carries discrete
structure, and discrete structure takes sides: a pentagon has no
mirror symmetry to lose, because it never had one --- five is odd.
What the descent does is not \emph{create} asymmetry but
\emph{expose} it, floor by floor, at scheduled addresses.
\emph{Symmetry breaking in this framework is symmetry reading. The
sixth dimension is the tape; the fifth dimension is the read head;
the constitution cut the groove; and the descent plays the record
--- once, forward, and the same way in every possible run.} That is
the precise content of ``how symmetry is broken to create something
from nothing, deterministically and inevitably'': the something was
written; the nothing was the time before the needle reached it.

\paragraph{Sakharov's conditions as anatomy.} The birth of matter,
Chapter~\ref{sec:leptogenesis}'s subject, is the sharpest test of
this reading, and it passes in a form worth displaying. Sakharov's
three conditions \citep{Sakharov1967} are, in every other
framework this review knows, three separately arranged ingredients.
Here they are the anatomy of the bulk --- one condition per
geometric structure, and Table~\ref{tab:sakharov-anatomy} displays the
correspondence:

\begin{table}[H]
\centering
\caption[Sakharov's three conditions as the anatomy of the bulk.]{\textbf{Sakharov's
conditions, read as anatomy.} The three ingredients that every other
framework this review knows arranges separately are here one per
geometric structure --- two carried by the phase circle $\Stheta$ and
one by the scale interval $I_r$, the same two fibers as
Table~\ref{tab:two-fibres}. Leptogenesis is the one event at which all
three act at the same address, $r = K = 35$.}
\label{tab:sakharov-anatomy}\small
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{@{}lp{0.44\textwidth}l@{}}
\toprule
Sakharov (1967) & the ledger's anatomy & fiber \\
\midrule
(1) $B$ violation & the sphaleron door, $\Delta B \in 3\bZ$, $B{-}L$
exact; the winding's protection before and after & $\Stheta$
(topology) $+$ weak sectors \\
(2) $C$, $CP$ violation & the chiral pentagon (existence); the
Majorana parity (sign) & $\Stheta$ (the alphabet) \\
(3) nonequilibrium & the gated descent; the lock crossing at
$r = K$; prompt, zero dilution & $I_r$ (the semigroup) \\
\bottomrule
\end{tabular}
\end{table}

\noindent Sakharov listed what any explanation of our existence
must contain; this bulk simply \emph{is} that list. Leptogenesis is
the one event where all three clauses of the anatomy act at the
same address --- $r = K = 35$ --- which is why, in this framework,
the birth of matter is not merely possible but scheduled: it could
not stall (the crossing is prompt), it could not choose otherwise
(the letters are quantized and the scan has one winner), and it
cannot be undone (the semigroup, the welded door, zero dilution).

\paragraph{The residual bit, honestly.} One two-valued freedom
genuinely survives the arithmetic: the mirror pair --- the twist
and its reflection, the parity read as $\mp$, matter against
antimatter. The honest statement, at the parent papers' grades: the
\emph{content} of that bit is deterministic --- the universe could
not have been unbiased, could not have been biased by a different
amount, and could not carry different biases in different places,
because the setting is one bulk holonomy, not a patchwork of local
choices. The \emph{labeling} of which sign is called matter is a
convention, fixed jointly with the orientation conventions of the
mirror chapter. Whether the bit itself admits a further derivation
--- the descent's own arrow selecting between the two orientations
of the circle --- is an open question this review flags as exactly
that: open, attractive, and unclaimed.

\paragraph{What is new here, stated as structural contrast.} The
register of this paper forbids polemic, so the uniqueness claim is
made the only honest way: by listing what each neighboring
tradition \emph{is}, and where this account differs, so a referee
can check the negative. \emph{Against textbook spontaneous
breaking}: Nambu--Goldstone and Higgs dynamics
\citep{Nambu1960,Goldstone1961} explain the
instability of the symmetric state, and explain it correctly; they
do not, and by construction cannot, explain the outcome --- the
degenerate continuum makes the broken state's identity random and
region-dependent. Here the continuum is replaced by a quantized
dial and the outcome is arithmetic. \emph{Against explicit
breaking}: the Standard Model's asymmetric parameters are inputs;
here the corresponding quantities are letters of a forced alphabet
--- there is no dial to write a value into. \emph{Against the
landscape--anthropic tradition}: there, lawful vacua fill continua
and our own is selected environmentally; here the ``landscape'' is
a Morse function with a unique floor per stage, and nothing is
selected because nothing else was available. \emph{Against
engineered baryogenesis}: the standard mechanisms assemble
Sakharov's conditions from separately chosen parts --- a new scale
here, a new coupling there, phases free; here the three conditions
are the anatomy of the arena, one per structure, and the yield is
priced by the ledger's one depth integer with zero fitted
parameters. Each contrast is checkable, and none of it claims the
neighbors are wrong about what they compute --- only that none of
them contains a mechanism by which a symmetric beginning is
compelled, rather than permitted, to become this world. That
compulsion --- \emph{something from nothing, deterministically and
inevitably, with the determinism graded and the inevitability
falsifiable} --- is, in this review's judgment, the deepest
conclusion the framework supports, and it is the conclusion toward
which every rung of the ladder, every chapter of the turn, and
every entry of the ledger has been climbing.

\paragraph{The fences, kept.} Three, stated plainly. The grades are
inherited, not improved: the cascade's uniqueness stands at its
parents' stated grades, the lock scan at the proton companion's,
the alphabet at the mirror companion's lattice-rigidity clause, the
birth's magnitude at the sheets' D$^{*}$ --- derived subject to one
named lemma --- with its named input and postulated
exponent --- ``deterministic'' is a statement about the dial count,
which is zero, not a promotion of any grade. The two senses of
``symmetric start'' must travel together: symmetric dynamics,
asymmetric arithmetic --- conflate them and the claim sounds either
trivial or false; separate them and it is neither. And the causal
direction is the constitutional one: the descent along the scale
fiber is the bulk fact, and cosmic expansion --- cooling, redshift,
the thermal history --- is its boundary shadow; the two are one
motion read twice. Determinism of this kind is only physics if it
can die, and it hangs from the same four ropes as the birth
(\S\ref{ssec:lepto-grade}): the account of why there is something
rather than nothing --- checkable, in laboratories, this decade.

\subsection{Time, read instead of assumed}
\label{ssec:time-read}
The capstone of the last subsection has a corollary large enough
to state separately, because it closes a circle this review opened
in its first chapter. Every rung of the ladder, every chapter of
the turn, and every entry of the ledger was narrated \emph{in}
time --- broken \emph{then}, locked \emph{then}, born \emph{then}
--- while time itself sat in the background, assumed the way every
physics book assumes it. Chapter~\ref{sec:what-is-time} and its
parent treatment end the assumption: time is the reading itself.
The tick is the circle's repetition; the arrow is the interval's
semigroup; the ordered axis they stamp is the cone's; and ``the
descent plays the record once, forward'' --- the sentence this
review used for symmetry breaking --- \emph{is} the definition of
time, read from the other side. Symmetry breaking is symmetry
reading; time is what the reading feels like from inside.

In the review's judgment three things about this are significant
beyond the cohort. First, the composite anatomy: no previous
account of time --- Newton's flow, Leibniz's relations, Einstein's
order, Boltzmann's statistics, quantum mechanics' external
parameter --- distinguishes the tick from the arrow as separable
structures; each tradition succeeded exactly where its one
endowment ruled and puzzled exactly where the others intruded, and
the three-century standoff between reversible mechanics and
irreversible thermodynamics dissolves into the statement that they
were never about the same fiber. Second, the number: every
previous definition of time said what time is; the parent
treatment's is the first that says \emph{how much of it there is,
and why} --- $e^{140}$ Planck ticks, closed as an audit
\eqref{eq:time-ledger-w1} --- and a definition that can be wrong
about a number is a scientific object in a way no prior definition
of time has ever been. Third, the unification this Part exists to
record: the same two fibers that made symmetry breaking
deterministic (\S\ref{ssec:symmetry-reading}), made the proton
immortal, and priced the matter excess also constitute time's tick
and time's arrow. One anatomy; the conservation laws, the broken
mirrors, the number of survivors, and the direction called time,
read off it together. The review knows of no other framework in
which ``what is a conservation law?'', ``why is symmetry
broken?'', ``why are we here?'', and ``what is time?'' receive
answers that are not merely compatible but are the \emph{same
answer} examined from four sides --- and it is the review's
closing judgment that this convergence, more than any single
number, is what the ledger is worth.

\subsection{Rigidity of kind, freedom of value}
\label{ssec:kind-value}
One rung does not fit the pattern of the last two sections, and the
misfit is worth printing rather than smoothing over, because it is the
single most useful structural observation this Part can make.

Every other rung on the ladder has no dynamical content at all. The
Wilson line of Chapter~\ref{sec:wilson} is not like that. It splits:

\begin{itemize}[leftmargin=1.4em,itemsep=0.3em]
\item its \emph{kind} is fixed by topology and cannot be negotiated --- a
  phase valued on a circle, one for each independent loop, with the
  count of them a whole number determined by the shape of the space and
  by nothing else;
\item its \emph{value} is fixed by dynamics --- a one-loop competition
  among whatever is circulating, whose minimum moves when the field
  content moves.
\end{itemize}

So the two things this Part has been treating as one axis are two axes,
and here they come apart. Rigidity of kind does not imply poverty of
content. A quantity can live in a frame that no dynamics can deform and
still have a value that dynamics alone decides.

That is not an embarrassment for the ladder's thesis; it is a
sharpening of it. What the sequence tracks is the retreat of dynamics
from the \emph{frame}, not its retreat from the world. The frame becomes
progressively more rigid, and progressively more discrete, while the
world inside the frame goes on being as dynamical as it ever was. A
conservation law at the top of the ladder does not say that nothing
happens. It says that whatever happens, happens inside a structure whose
shape was not up for negotiation.

\emph{Why it matters for what follows.} Part~\ref{part:turn} proposes
exactly that arrangement, at a larger scale: an exact ledger whose
integers are fixed by the geometry, and inside it a descent that is
thoroughly dynamical and whose endpoint is a matter of where the
landscape happens to run downhill. A reader inclined to treat that
combination as having it both ways should note that it is not novel to
the proposal. It is the structure of a Wilson line, and it has been in
physics since 1983.

\subsection{Why counting statements survive into a broken-symmetry world}
\label{ssec:counting-survives}
The last observation of Part~II is the one Part~III will use, so it is
worth isolating.

Consider what happens to each kind of conserved quantity when a symmetry
is broken by a small amount --- which is, empirically, the condition of
the actual world. A current conserved by a symmetry that is broken at
order $\epsilon$ is conserved only up to order $\epsilon$: the
conservation degrades smoothly, in proportion to the breaking, and this
is exactly what is observed for isospin, for flavor symmetries, and for
$CP$ itself. A ledger identity behaves differently but still
continuously: the entries move, and the total is what is protected.

An integer does neither. There is no such thing as being conserved up to
order $\epsilon$ when the quantity in question takes values in
$\mathbb{Z}$. Either the integer is what it was or the configuration has
jumped. Small breakings, of which the world evidently has many, cannot
touch it.

This is why counting statements are the observables that survive into a
universe with almost none of its symmetries left intact. Electric charge
comes in exact multiples of one unit, and the equality of the electron
and proton charges has been tested and found consistent with zero
difference at the level of $10^{-21}$ of that unit
\citep{MarinelliMorpurgo1984}; there are exactly three colors; there
are exactly three families of matter. Nothing in the present universe is
symmetric enough to protect those numbers by a Noether argument. They
are protected, if the top of the ladder is the right reading, because
they are integers.

Part~III takes this one step further and asks what happens to the whole
picture when the system under discussion is not closed --- when what is
being described is a boundary with a bulk on the other side of it. The
short answer, and the pivot of this paper, is that the ladder's seven rungs all describe closed worlds, that a closed world can tell you which
integers are \emph{allowed}, and that only an open one can tell you which
integer is \emph{taken}.


\section{The ladder cashed: the same integers, doing unrelated work}
\label{sec:cashed}

\S\ref{ssec:counting-survives} ended Part~II with a claim and no test.
The claim was that an integer is the one kind of conserved statement a
broken world cannot erode, and the evidence offered was that the
quantities still known exactly --- the unit of charge, the number of
colors, the number of families --- are without exception counts.

Part~\ref{part:ledger} supplies the test, and this chapter runs it. It
is the one place in the paper where the argument of Part~II meets a
number, and it is placed here, rather than in Part~II, because it cannot
be made before the sheets exist.

\subsection{The question the ledger makes askable}
\label{ssec:askable}
A skeptic reading Part~II could grant every step and still say: very
well, integers are robust, but that is a fact about arithmetic, not
about the world. Nothing in Part~II required that any particular integer
do any particular job.

Part~\ref{part:ledger} changes what can be asked, because it puts four
working integers on the table --- $N = 3$, $L = 5$, $J = 7$, $K = 35$
--- and then uses them. So the question becomes sharp: \emph{are the
integers at the top of the ladder the same integers as the ones in the
ledger, and how far apart are the jobs they are asked to do?}

\emph{One coincidence, disposed of before it distracts.} The ladder of
Part~\ref{part:ladder} has seven rungs and the ledger contains
$J = 7$. These are not the same seven and this paper does not connect
them. The ladder has seven steps because seven results were found worth
climbing, and a different author would have drawn six or nine; $J$ is
$7$ because $35/5 = 7$. Nothing anywhere in this review uses the
agreement, and it is mentioned here only so that no reader has to spend
a paragraph wondering whether it was being used quietly. A program
whose subject is small integers has to be more careful about accidental
ones than a program without them, not less.

\subsection{Twenty jobs, four integers}
\label{ssec:same-integers}
Table~\ref{tab:same-integers} is the answer, and it is the whole of this
chapter's positive content.

\emph{One absence, and it is deliberate.} The cohort's most quoted
result --- that there are three families --- is not a row of that table.
The integer in the ledger is the color index $N_c$, fixed by the chain
of \S\ref{ssec:twelve} in which no color count appears until it is
concluded; the family count $N_g$ is not a member of the ledger at all,
and is reached by the pincer of \S\ref{ssec:three-gen}, whose premises
have nothing in common with that chain. A table about \emph{re-use}
would misreport the relation between them by listing the family count as
one more job done by the same integer, because that is not what it is.
Two derivations sharing no premise and arriving at the same small number
is a stronger fact than one integer being spent twice, and it is
recorded where it is earned. It is simply not this table's kind of
fact.

\begin{table}[!t]
\centering
\caption[Four integers, twenty jobs.]{\textbf{Four integers, twenty jobs.} Every entry is a
closed form drawn from Tables~\ref{tab:sm-sheet} and
\ref{tab:cosmos-sheet} or from the identities of \S\ref{ssec:twelve},
and each was recomputed in this review before being listed. What the
table is for is not the accuracy of any row --- that is the sheets'
business --- but the \emph{re-use}: the same four integers are asked to
do all twenty, and none is re-chosen between one job and the
next. Twenty entries, eighteen distinct quantities: $\sin^2\theta_W$
and $\zeta_\Lambda$ each appear twice because each draws on two of the
four integers at once, which is a tighter constraint than a row
consuming one and not a double count. The family count is deliberately
not among them; see the text.}
\label{tab:same-integers}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}P{0.08\textwidth}P{0.35\textwidth}P{0.44\textwidth}@{}}
\toprule
& \textbf{where it appears} & \textbf{what kind of quantity that is} \\
\midrule
\multirow{5}{*}{$J = 7$}
 & $\alpha^{-1}(M_Z) = 2^{J}$ & an electromagnetic coupling at the $Z$ pole \\
 & $\alpha^{-1}(0) = 2^{J} + N^{2}$ & the same coupling in the infrared \\
 & $e = \sqrt{4\pi/2^{J}}$ & the charge on the electron \\
 & $1 + z_{\rm sw} = e^{J}$ & the corridor switch at recombination \\
 & $K = LJ$ & the ledger's own closure identity \\
\addlinespace
\multirow{5}{*}{$K = 35$}
 & $M_{\rm lock} = \tfrac12 M_{\rm Pl}e^{-K}$ & a scale a collider might reach \\
 & $\varepsilon\kappa = e^{-K/2}$ & the excess of matter over antimatter \\
 & $\Delta\nu/\nu = 1/K$ & a splitting a microwave cavity might resolve \\
 & $\zeta_\Lambda = \pi/(LK)$ & the energy of the vacuum \\
 & $\Xi_{\rm cap} = K + N + L$ & the dark-matter cap loading \\
\addlinespace
\multirow{5}{*}{$N = 3$}
 & the number of colors & the strong interaction \\
 & $\sin^2\theta_W = L^2/4N^3$ & the weak mixing angle \\
 & $v = M_{\rm lock}(2/L)^{N}$ & the electroweak scale \\
 & $\lambda_H = 2^{-N}$ & the Higgs self-coupling \\
 & $m_3 = 2N\,m_\star$ & the heaviest neutrino \\
\addlinespace
\multirow{5}{*}{$L = 5$}
 & the minimal chiral package & what matter there is \\
 & $\sin^2\theta_W = L^2/4N^3$ & the weak mixing angle \\
 & $\delta_{\rm CKM} = 2\pi/L$ & $CP$ violation in the quark sector \\
 & $\zeta_\Lambda = \pi/(LK)$ & the energy of the vacuum \\
 & $m_\mu = (v/\sqrt2)\,L/2^{J+L+1}$ & the mass of the muon \\
\bottomrule
\end{tabular}
\end{table}

Two pairings in that table deserve to be read against each other.

\emph{$J = 7$ appears in $\alpha^{-1}(M_Z) = 2^{J}$ and in
$1+z_{\rm sw} = e^{J}$.} The first is a coupling measured at a collider
at ninety-one giga-electronvolts. The second is the corridor switch of
\S\ref{ssec:cosmos-first}, a topological marker in the recombination era
at about a quarter of an electronvolt. The care taken there applies
here: the switch is \emph{not} the redshift at which the first atoms
formed --- that is the atomic quantity $z_{\rm dec}$, which the same
section derives by a different route and which sits $0.4\%$ away in
e-folds. What is claimed is only this: one integer appears in a
laboratory coupling and in a cosmological marker, eleven orders of
magnitude apart in energy and established a century apart, once through
a power of two and once through a power of $e$.

\emph{$K = 35$ appears in $M_{\rm lock}$ and in $\Delta\nu/\nu = 1/K$.}
One fixes a mass a collider might produce; the other fixes the internal
splitting of a dark-matter line at $8.1\,\mu$eV, about eighteen orders
of magnitude below it, which a microwave cavity might resolve.

\emph{What this is and is not evidence for.} It is not evidence that any
of the closed forms is right --- \S\ref{ssec:not-bought} was explicit
that the trials factor bounding exactly this kind of impression has not
been computed. What it is, is the structural feature that a fitted
formula cannot imitate. A fit is chosen per observable. These are not:
one small set is fixed, and then spent. Whether it was spent correctly
is what the dated falsifiers of \S\ref{sec:falsifiers} will decide.

\subsection{Breaking as the source of structure}
\label{ssec:breaking-source}
The ladder has a second reading, which this review has approached three
times and now states directly. It is \emph{this paper's} reading; it is
not attributed to the cohort's papers, none of which puts it in these
words.

The conventional register treats broken symmetry as damage. A symmetry
is the beautiful thing; breaking is what the world does to it; and the
job of theory is to find the symmetry underneath the mess. Part~II's
honest record shows where that register has succeeded and where it has
not.

The reading offered here reverses the direction. A perfectly symmetric
configuration has no structure in it to speak of --- nothing
distinguishes one point, one direction, one state from another, and
there is correspondingly nothing to be an observer, or an atom, or a
galaxy. Structure is what a symmetry has to lose in order for anything
to be somewhere. On this reading breaking is not the damage done to a
design; breaking \emph{is} the productive step --- the operation by which a
world with things in it is selected out of a world with none.

What the ladder adds is the mechanism by which such a world can still
have exact statements in it. If every conserved quantity degraded in
proportion to the breaking, a broken world would be a world of
approximations only. The ladder's last three rungs say otherwise: some
statements fail only in whole numbers, and those survive intact.
\emph{So breaking can be the source of structure without being the
destroyer of exactness}, and the two facts --- that the world is broken,
and that it still contains numbers known to twenty-one decimal places
--- stop being in tension. That reconciliation is what this review takes
the ladder to be for.

\medskip
The sharpest available statement of what symmetry-plus-breaking buys is
a ladder of three horizon problems, and it is owned by the cosmology
paper \citep{ArisakaA10Cosmos}. The first --- the classic one --- asked
why causally disconnected regions share the same \emph{conditions}, one
temperature to a part in $10^{5}$; inflation answered it in 1981. The second asks why they hold the same \emph{things} ---
why an electron on the far side of the sky is identical to one here,
though no signal ever compared them; the sameness of the geometry
beneath the things answers it. The third asks why they made the same
\emph{choices}: every disconnected region chose matter over antimatter
with one sign, one yield, one mechanism, and nothing available to
coordinate the choice --- and had the choosing carried any stochastic
element, the sky would show annihilation seams between matter and
antimatter domains, and it shows none. The ladder is graded by how much
of the present subject each rung requires: sameness of conditions needs
an early accelerated phase alone; sameness of things needs identity of
the underlying geometry; sameness of choices needs the \emph{breaking
itself} to be deterministic --- \GSSB{} selecting a unique attractor
rather than sampling a degenerate family. And the choice in question is
the one this review has been calling the productive step: baryogenesis, the
breaking that selects a world with things in it over a world with none.
It threads an exact needle --- asymmetric in precisely the one conserved
number that is global and unwatched by any field, exactly neutral in
every number a gauge field polices, which is why the books close to
those twenty-one decimal places. The universe breaks the symmetry it
may break and keeps the one it must keep: same conditions, same things,
same choices, each rung bought by a deeper layer of the same structure.

\subsection{Against the anthropic move}
\label{ssec:anthropic}
There is a competing account of why the constants take the values they
do, and it must be met rather than ignored, because it is the account
most physicists would reach for first.

The anthropic answer is that the constants could have been otherwise;
that somewhere they are otherwise; and that we find ourselves among the
values compatible with observers, because nobody is available to be
puzzled by the others.

Four objections are worth separating, and the first three are already in
the cohort's literature \citep{ArisakaA11Origin}.

\emph{First, a selection effect is not an explanation.} That
observers can only observe observer-compatible worlds is true and
tells us nothing about why any world has the values it has. The owning
paper's formulation is blunter than this review would have been: it is
like saying that the reason you are alive is that dead people cannot ask
questions.

\emph{Second, if nothing was chosen, there is nothing to select over.}
The anthropic question presupposes a range of values that might have
been taken. A program claiming the values are forced denies the
premise rather than answering the question.

\emph{Third --- and this is the methodological point, not a taste
claim --- the two accounts differ in what happens when they fail.} An
anthropic explanation that runs into trouble can be restated one level
deeper: a wider ensemble, a different measure, a subtler selection. A
framework that has fixed five integers and printed the consequences
cannot do that. Its own paper puts it exactly: the refusal to retreat is
the structural feature that lets it be tested.

\emph{Fourth, and least often noticed, there is a positive datum.} The
argument above is negative --- nothing was available to tune. But the
program's derived launch places the inflationary stretch \emph{before}
any matter existed, which means the sameness of the electron from one
side of the sky to the other was inherited from a single causal patch
rather than arranged region by region. A tuned universe would have to
tune each region separately. An inherited one could not differ between
regions even in principle. That is an argument about \emph{ordering},
and it is available to this program and not to a selection account.

\emph{Where the review parts company with its source.} None of that
establishes the framework. And the owning paper does not hold the line
perfectly: it closes one of its central Parts by saying that the
geometry requires us because, among all admissible geometries, this is
the one admitting an articulator who can say so. The clause after the
comma is selection over an ensemble --- the move the same paper rejects
four times elsewhere. A reader who finds it will conclude that the
anthropic reasoning was smuggled back in at the end. This review reports
the sentence rather than repeating it.

\subsection{The family number, in proportion}
\label{ssec:n3-proportion}
One result of the program is quoted more than any other, and this
review has been careful with it for that reason.

That there are exactly three families of matter is the most memorable
single number the framework produces. It is memorable because it is a
small whole number attached to a famous open question, and because the
argument for it --- a lower bound from the existence of $CP$ violation
and an upper bound from a budget --- can be stated in two sentences.
\S\ref{ssec:three-gen} states it, with all three of its fences printed.

It is worth saying plainly what it is not. In proportion it is
\emph{one row of forty-nine}: one entry in a Standard-Model sheet of
twenty-eight and a cosmological sheet of twenty-one. It is graded
derived-conditional, not derived. And its lower leg consumes an
empirical input --- the observed fact that $CP$ violation exists --- so
even the pincer is not a statement from pure geometry.

Set beside the weak mixing angle, which is the one entry in either sheet
that is a theorem end to end, or beside the twenty-one-entry
cosmological inventory, the family number is neither the most secure
result nor the largest. A reader who takes it as the program's
headline has taken the most quotable claim for the strongest one. The
sheets do not support that ordering, and neither does this review.

\subsection{Grade box}
\label{ssec:cashed-grade}
\begin{quote}\small
\textbf{Grade box: what this chapter has and has not established.}

\emph{Verified here.} Every entry of Table~\ref{tab:same-integers},
recomputed from its closed form; the two separations quoted, in orders
of magnitude; and the proportion of \S\ref{ssec:n3-proportion} --- one
row of forty-nine.

\emph{Argued here, as this review's own reading.} That breaking is the
source of structure rather than damage to it. This is not attributed to
the cohort's papers, and \S\ref{ssec:breaking-source} says so.

\emph{Reported, at the owning paper's grade.} The four objections to the
anthropic move, three of which are that paper's and one of which ---
the ordering argument --- this review regards as its strongest and least
used.

\emph{Printed against the program's interest.} That the re-use of
Table~\ref{tab:same-integers} is not evidence that any closed form is
correct, and that the statistic which would bound the impression it
creates has not been computed; and that the owning paper reintroduces a
selection argument in its own closing passage.

\emph{Not claimed anywhere in this chapter.} That the integers are
forced. That remains what it was in \S\ref{ssec:not-zero}: asserted.
\end{quote}


\section{Symmetry, breaking, and the honest record}
\label{sec:honest-record}
A review that only celebrates symmetry will lose its best readers, and it
will deserve to. The record of symmetry-first reasoning is not uniformly
good. It contains some of the most productive ideas in the history of
science and at least three cautionary episodes, two of them settled
against the symmetry and one still running. This chapter prints all
three. It comes before the conclusions on purpose: the standard applied
here to the tradition is the standard a reader is entitled to apply to
the present program, and it is set out first so that nothing later can
quietly relax it.


\subsection{A perfectly symmetric universe is empty}
\label{ssec:empty}
Begin with the observation that makes the rest of the chapter necessary.

A universe in exact possession of all the symmetries one might wish for
it would contain nothing worth describing. If matter and antimatter
entered on exactly equal terms, they would have annihilated into
radiation, and there would be no atoms; the standard analysis of what is
required to avoid that outcome lists violation of baryon number, of $C$
and $CP$, and departure from thermal equilibrium \citep{Sakharov1967} --- three broken
conditions, not three symmetries. If space were exactly homogeneous,
there would be no galaxies, because there would be no denser place for
matter to fall toward; structure grows from the departures from
homogeneity, not from the homogeneity. If the microscopic laws were
exactly symmetric under reversal of time, there would be no
distinguishable past. And if reflection were an exact symmetry of
chemistry, the two mirror forms of every handed molecule would be
equally represented, and the biochemistry described in
\S\ref{ssec:chirality} could not work the way it does.

The point is not rhetorical. It is a constraint on what any theory built
on symmetry may claim. A principle that asserts only symmetry predicts a
world with no contents. Whatever the correct statement of the role of
symmetry in physics is, it must have a second half.


\subsection{Where symmetry-first reasoning failed: parity, CP, naturalness}
\label{ssec:failures}
Table~\ref{tab:failures} lists them together with what each cost and how
long the cost went unnoticed. The three episodes below are chosen because
in each of them a symmetry
was believed on grounds that were partly aesthetic, and in each the
belief was tested.

\begin{table}[H]
\centering
\caption[Three episodes in which a symmetry was believed and
tested.]{Three episodes in which a symmetry believed partly on
aesthetic grounds was put to an experiment. The first two are settled
against the symmetry. The third is not a refutation of supersymmetry;
it is a refutation of the argument that predicted where the
superpartners would be.}
\label{tab:failures}\small
\begin{tabular}{@{}P{0.16\textwidth}P{0.26\textwidth}P{0.22\textwidth}P{0.26\textwidth}@{}}
\toprule
\textbf{symmetry} & \textbf{why it was believed} & \textbf{how it was
tested} & \textbf{outcome} \\
\midrule
Parity & Never questioned; reflection invariance seemed a property of
space itself & Angular distribution of electrons from oriented
$^{60}$Co & Violated, and maximally, in the weak interaction
\citep{LeeYang1956,Wu1957} \\
$CP$ & Adopted as the natural repair once parity had failed & Search for
$K_L \to \pi^+\pi^-$, a decay $CP$ forbids & Violated at the
$10^{-3}$ level \citep{Christenson1964} \\
Naturalness & A symmetry (supersymmetry) invoked to protect a scale from
quantum corrections & Direct searches at the Large Hadron Collider &
No superpartners; the region the argument preferred is now excluded
\citep{ATLASsusy2026} \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Parity, 1956--57.} Reflection invariance was so plainly a
property of space that nobody had troubled to test it in the weak
interaction. \citet{LeeYang1956} made the striking observation that the
evidence simply did not exist, and proposed experiments. Within months,
C.-S. Wu and collaborators had cooled a sample of $^{60}$Co in a
magnetic field and counted the electrons emitted along and against the
nuclear spin; the counts differed \citep{Wu1957}. A companion
experiment on muon decay, reported in the same issue and on the
following page, found the same thing \citep{Garwin1957}. What makes the
episode
instructive is not that a symmetry failed but \emph{how} it failed. The
violation was not a small correction to a good approximation. It was
maximal: the weak interaction couples to one handedness of the fermion
field and not at all to the other. The most confident symmetry in the
subject turned out to be not slightly wrong but entirely absent, and it
had gone untested precisely because it was too obviously true to check.

\paragraph{$CP$, 1964.} The repair was immediate and elegant: if
reflection alone is not a symmetry, perhaps reflection combined with
exchange of particles and antiparticles is. For eight years that held.
Then \citet{Christenson1964} observed the long-lived neutral kaon
decaying into two pions --- a channel that $CP$ conservation forbids ---
at a rate of order $10^{-3}$ of its total. The second symmetry was
broken too, and this time weakly rather than maximally, which is in its
own way harder to accommodate: a symmetry that is exact is
comprehensible, and a symmetry that is absent is comprehensible, but a
symmetry violated by a small and apparently arbitrary amount asks for an
explanation that half a century has not fully supplied.

There is a debt to record here. The derivation of the family number in
this program rests, in one of its three legs, on the fact that $CP$
violation is real: a mixing matrix that admits a physical
$CP$-odd phase requires at least three families
\citep[the Kobayashi--Maskawa observation,][]{KobayashiMaskawa1973}. The
input consumed is a single sign --- $CP$ violation exists --- and not a
fitted number, but it is an input, and it is an input supplied by a
\emph{broken} symmetry. Had the 1964 experiment come out the other way,
the lower bound of \S\ref{ssec:three-gen} would not exist. That is worth
stating plainly in a chapter about symmetry's failures: one of them is
load-bearing here.

\paragraph{Naturalness, and the present.} The third case is the live
one. For roughly forty years the dominant expectation in particle
physics was that the electroweak scale must be protected from large
quantum corrections by a symmetry --- the criterion was given its
standard formulation by \citet{tHooft1980}, and that the symmetry would be
supersymmetry, with superpartners near a TeV. The argument was
quantitative and honest about its own exposure: the lighter the top
squark, the smaller the residual cancellation required. Searches at the
Large Hadron Collider have found nothing. In the simplified models the
collaborations report against, gluino masses below roughly $2.2$~TeV and
top-squark masses below roughly $1.2$~TeV are now excluded
\citep{ATLASsusy2026}.

The honest reading is narrower than the popular one. Supersymmetry as
such is not excluded, and cannot be by searches of this kind; what has
been excluded is the region that the naturalness argument singled out.
Because the residual cancellation scales as the square of the
superpartner mass, moving the top squark from a few hundred GeV to above
a TeV costs about an order of magnitude in the very quantity the
argument was trying to keep small. The prediction has not been falsified
in the way parity was falsified. It has been rendered uninteresting on
its own terms --- which, for a principle whose entire appeal was that it
made a scale unsurprising, is close enough to the same thing.


\subsection{Two criteria from one paper, and how they aged}
\label{ssec:two-criteria}
The naturalness episode of \S\ref{ssec:failures} has a companion that is
rarely told beside it, and the two together make this chapter's point
more sharply than either alone.

The lectures in which the naturalness criterion received its standard
formulation \citep{tHooft1980} contain a second criterion, by the same
author, in the same year. It is the \emph{anomaly matching} condition:
if a global symmetry has an anomaly, the coefficient of that anomaly
computed from the elementary fields of a theory must equal the
coefficient computed from whatever massless composite states the theory
produces at low energy --- no matter how violently the dynamics
rearranges itself in between.

Both were proposed as constraints on what a theory may look like. Both
were influential. They have not aged alike.

\emph{Naturalness} rests on a symmetry argument: a small parameter is
acceptable if setting it to zero enlarges a symmetry, so that the
smallness is protected. Its most quantitative application pointed at new
states near a TeV, and \S\ref{ssec:failures} records the outcome.

\emph{Anomaly matching} rests on an integer. The coefficient is a sum of
cubes of charges over a discrete spectrum, exact at one loop; the
renormalization-group flow that connects the two ends is continuous; and
a continuous function taking values in the integers cannot move. The
argument is four decades old and has not needed amending. Its most
celebrated application is the one \S\ref{ssec:fatal} computes: the width
of the neutral pion, from which the number of colors can be read off.

\begin{quote}\itshape
One paper, one author, one year. The criterion built on a symmetry that
might or might not be there has been excluded in the region it
preferred. The criterion built on an integer is still exact.
\end{quote}

This is the pattern of \S\ref{ssec:pattern} in its cleanest available
form, and it is worth being careful about what it does and does not
show. It does not show that 't~Hooft was careless; both criteria were
reasonable when proposed, and one of them is among the most useful tools
in the subject. It shows something narrower and more useful: that of two
constraints written down together by the same person, the one whose
content was a whole number outlived the one whose content was an
aesthetic judgment about what a Lagrangian should look like. That is
the ladder's thesis, tested on a case where the confounding variables ---
author, year, ambition, community --- are held fixed.

\emph{One caution, and it belongs here.} Anomaly matching applies to the
anomalies of \emph{global} symmetries, which are physical. Gauge
anomalies are not matched; they must cancel outright, for the reason
\S\ref{ssec:fatal} gives. The two must not be run together.

\subsection{The pattern: symmetry read off the data succeeds; symmetry imposed for beauty does not}
\label{ssec:pattern}
Set the three failures beside the successes of Part~II and a pattern
appears that is sharper, and more useful, than ``symmetry reasoning is
unreliable.''

Every symmetry principle in Part~II that survived was read \emph{off}
something already in evidence. Noether was answering a specific technical
puzzle about energy in general relativity, not proposing an aesthetic
program. The gauge principle was extracted from a phase that was
already known to be unobservable. The Bianchi identity was not adopted;
it is what a curvature is. The Green--Schwarz mechanism was invented
under duress, to cancel an anomaly that had been computed and would
otherwise have killed the theory. Even Dirac's argument, the most
apparently metaphysical of them, is a consistency requirement on an
object whose existence was being entertained, not a preference.

Every failure in this chapter went the other way. Parity was assumed
because reflection is a property of the space one draws diagrams in.
$CP$ was assumed because it was the nearest available repair. Naturalness
was, in the end, the demand that a number not be surprising.

So the operative distinction is not symmetry against no symmetry. It is
this: \emph{is the invariance a description of something measured, or a
requirement imposed because the alternative would be ugly?} The first
kind arrives with the measurement that could have gone the other way,
and therefore with its own falsifier attached. The second kind arrives
with nothing but the taste of the person proposing it, and taste has a
poor record against data.

There is a second-order lesson, and Part~II has already supplied it.
The results on the ladder that have lasted best are the ones that are
not symmetry statements at all. An identity cannot be violated because there
is nothing in it to violate; an integer cannot drift because there is
nowhere continuous for it to drift to. Rigidity of that kind is not a
matter of taste, and it is not exposed to the failure mode this chapter
has been describing. That is the strongest argument this paper can make
for taking the top of the ladder seriously, and it is an argument about
logic rather than beauty.


\subsection{A third route to the same problem, and its own fence}
\label{ssec:third-route}
Honesty about a failed argument requires saying what the alternatives
are, and one of them has appeared in this paper already. It is recorded
here with its grade, and the grade is the point of the subsection.

The problem naturalness was addressing is that a scalar mass receives
corrections that are not small, so that a light scalar looks like an
accident. Supersymmetry answers it with a symmetry: pair every particle
with a partner, and the corrections cancel. That answer is
\S\ref{ssec:failures}'s subject and its preferred region is excluded.

Chapter~\ref{sec:wilson} exhibits a different kind of answer. If the
scalar is a Wilson-line phase, no local counterterm can generate a
potential for it, because it is not the value of any local
gauge-invariant operator; the potential it does acquire is finite,
non-local, and calculable. The protection comes from topology and
locality rather than from a symmetry, and unlike a symmetry it has
nothing to break. Be exact about
what kind of protection this is, because the circle protects two
different things two different ways. A winding number cannot move at
all: nothing continuous acts on an integer, and that is the protection
the seventh rung supplies. A Wilson-line phase is a \emph{continuous}
modulus --- it moves, and the quantities built from it run like the
addresses they are. What non-locality buys there is not fixity but
\emph{finiteness}: a number the theory owes a calculation for, rather
than a number that absorbs a divergence.

Read alongside \S\ref{ssec:two-criteria} the parallel is close enough to
be worth naming: one route to the problem is a symmetry argument and one
is an integer-and-topology argument, and this paper's whole thesis is
about which of those tends to survive.

\begin{quote}\small
\textbf{The fence, and it is not optional.} That the one-loop potential
for a Wilson-line phase is finite is a \emph{theorem}, and it is proved
in Appendix~\ref{app:wilson}. That this resolves the hierarchy problem
is \emph{open}. No realization with the observed spectrum is
established; the searches that excluded the natural region for
superpartners are pushing the scale of these models up as well; and this
paper takes no position on whether the route succeeds. It is recorded
because a chapter that prints the failure of one answer and suppresses
the existence of others would be making a rhetorical point rather than
an honest one.
\end{quote}

\emph{And the same standard, applied here.} A reader will notice that
the alternative just described is congenial to this paper's thesis, and
should treat that as a reason for suspicion rather than comfort. The
thesis predicts that integer-and-topology arguments outlast symmetry
arguments; an example that fits the thesis and has not yet been tested
is not evidence for it. It becomes evidence only if the route is
realized and survives, and it becomes evidence against if the route is
realized and fails. Until then it is a candidate, and \S\ref{ssec:exposure}
is where this paper says what would count against its own case.

\subsection{What this program therefore exposes of itself}
\label{ssec:exposure}
A chapter that applies this standard to others must apply it here.

\emph{The ladder is a construction, and this is the place to say so.}
Nobody handed down these seven rungs. A reviewer chose them, and chose
the order, and the order is what carries the argument --- so the
selection is not innocent of the conclusion it reaches. A different
reviewer could build a different ladder: one that put spontaneous
symmetry breaking on it as a rung in its own right, or the
renormalization group, or Wigner's classification, and such a ladder
might well arrive somewhere else. Two things are offered in defense,
and neither is a proof. The first is that each rung here is a result
the discipline already agrees on, taken on its own terms, with no
weight placed on any reading of it that its authors would dispute; what
is chosen is the sequence, not the physics. The second is that the
ordering is not a matter of taste but of a stated criterion --- what it
would cost to violate the statement --- which is checkable rung by rung
and which \S\ref{ssec:arc-earned} tabulates so that a reader can
disagree with a specific entry rather than with an impression. A reader
who rejects the criterion should reject the ladder, and Part~II's
individual chapters survive that rejection intact.

\emph{On the good side of the line.} The ledger integers $(M, N, L; J, K)
= (1, 3, 5; 7, 35)$ are outputs of closure identities, not choices: they
satisfy $K = LJ = M^2 + N^2 + L^2$ with $J = 2N + 1$, and every one of
those relations is arithmetic that a reader can check in a minute
\citep{ArisakaA2GG6}. The four coefficients of the modified identity of
\S\ref{ssec:bi6} are fixed by a trace ratio, $(L - N)/L = 2/5$, and not
fitted to anything \citep{ArisakaA3BI6}. The family number is compared
against a measured mass spectrum rather than asserted
\citep{ArisakaP2N3}. Each of these is a symmetry-and-counting statement
read off a structure, in the sense of \S\ref{ssec:pattern}, and each
carries a way of being wrong.

\emph{On the risky side of the line.} One premise of this program is
exactly of the kind this chapter has been warning about. The axiom of
maximum symmetry and minimum freedom is a minimality principle: it
selects, among structures that satisfy every other constraint, the one
that carries no unforced content. It does real work. The trace-ratio
identity $(L - N)/L = 2/5$ is satisfied by the whole family $(L, N) =
(5k, 3k)$ for every positive integer $k$; the selection of $k = 1$, and
hence of $\SUfive{}$ rather than $\SU(10)$, is made by that minimality
axiom and not by the ratio \citep{ArisakaA3BI6}. A reader who declines
the axiom is entitled to the whole family. The proposition that
observation is a projection from a bulk to a boundary is likewise an
axiom, not a derived statement, and the open-system reading of Part~III
stands on it.

Naming these is the point. A premise imposed for economy is not
automatically wrong --- economy is not the same kind of appeal as beauty,
and it has a better record --- but it is the premise a hostile reader
should attack first, and this paper would rather point at it than have it
found. Every claim imported into Part~III accordingly carries a grade,
and the grades are collected in Appendix~\ref{app:dictionary} so that a
reader can see at a glance which parts of the structure would survive the
loss of which premise.



\subsection{Part~V in one paragraph, and what comes next}
\label{ssec:part-v-close}
\looseness=-1
Part~V did three things. It read the ladder as one argument and found the
trend: climbing it, a conservation law becomes harder to violate and says
less about how anything moves --- not two trends but one seen from either
side. It cashed the ladder against the ledger --- four integers, twenty
jobs, none re-chosen between one job and the next --- while saying plainly
what that is and is not evidence for, since the trials factor that would
bound this kind of impression has not been computed. And it printed the record of symmetry-first reasoning honestly,
all three cautionary episodes included, arriving at a distinction sharper
than ``symmetry reasoning is unreliable'': an invariance read \emph{off}
something measured arrives with its own falsifier attached, while an
invariance imposed because the alternative would be ugly arrives with
nothing but the taste of the person proposing it. Then it turned that
standard on this program and named the minimality axiom as the premise to
attack first.

%% v15.1: MY \looseness=-1 REMOVED, AND THE ENLARGEMENT WITH IT.
%% The looseness closed a two-line orphan and printed the folio
%% through the words `and refuse' in this paragraph's last line --
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%% GG-Report-369 as a fix that had worked.  Removing the looseness
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%% \enlargethispage{2\baselineskip} is enough to reach the folio on
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%% page below the text block, and it is not safe on Letter.  Both
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%% is a thin page rather than a collision.
Part~VI says what is left standing. It separates the two kinds of claim
this paper makes --- a review of established physics, owing nothing to
anything proposed here, and a proposal carried at declared grades --- so
that a reader may keep the first and refuse the second; and it closes in
my own voice.

%% ===================================================================
%%  PART VI --- CONCLUSIONS
%% ===================================================================
\GGPart[Conclusions]{Conclusions}{What has been established and under
which fences --- followed by the single passage in which I speak in
my own voice}
\label{part:conclusions}

\section{What has been established, and under which fences}
\label{sec:established}
Two different kinds of claim have been made in this paper, and they carry
different weights. This chapter separates them.


\subsection{The ladder as a review result}
\label{ssec:ladder-result}
Nothing in Part~II is new. Noether's theorems are from 1918, the gauge
principle and its correct interpretation are settled, the Bianchi
identity is older than general relativity, the \GS{} mechanism is from
1984, and the \HS{} construction is from 2005. Every result is standard
and every derivation is in the literature.

What Part~II contributes is a reading. The seven results, taken in
historical order, form a sequence with two monotone trends: the
conservation statements become harder to violate and they say less about
motion, and these two trends are the same trend seen from opposite sides.
The reading has a consequence that is not merely aesthetic. If it is
right, then the search for deeper conservation laws should be expected to
turn up fewer symmetry arguments and more consistency requirements, and
the observables at the bottom of the theory should be expected to be
integers rather than continuous parameters. Both of those are checkable
statements about where physics is going, and both are, so far, what has
happened.

This much a reader may take as a review result, at the ordinary standard
of a review: it rests on established mathematics and asks only that the
sequence be read in order.

\emph{Physics intuition.} There is a way of putting the reading that
requires no formalism at all. Ask, of any conservation law, what it would
take to break it. For Noether's currents the answer is: write a different
theory. For the Bianchi identity: stop calling the field a curvature. For
the \GS{} ledger: fail to balance a total. For an integer class: perform
a discontinuous jump. Those four answers are in order of increasing
absurdity, and that ordering --- which needs no mathematics to perceive
--- is the whole of what Part~II established.


\subsection{The \GGTheory{} turn as a proposal, at declared grades}
\label{ssec:turn-grades}
Part~III is not a review. It is a proposal, and its claims are of several
different kinds. Table~\ref{tab:grades} records which is which, with the
member paper that owns each. The grade vocabulary is the one set out in
the Preface: \emph{axiom} is assumed, \emph{theorem} is proved from
what precedes it, \emph{derived-conditional} is proved subject to a named
clause, and \emph{open} is unsettled and labeled as such. Two rows carry
a qualifier rather than one of the four words: \GDOS{} is graded
\emph{theorem-level}, developed from the axioms to that standard rather
than proved as a single statement, and \GRIP{} is not one claim at all
but a framework whose components are graded separately.

\begin{table}[H]
\centering
\caption[The grades of the claims imported into Part~III.]{The claims
Part~III uses, with their grades and the member papers that own them.
Nothing in this paper re-derives any of them; the full ledger, with the
premises each grade depends on, is Appendix~\ref{app:dictionary}. The
last row is set apart because it is the only one this paper reports
without using: \S\ref{ssec:qi4-fence} states what depends on it, which
is one marked clause of Chapter~\ref{sec:grip} and nothing else.}
\label{tab:grades}\small
\begin{tabular}{@{}P{0.30\textwidth}P{0.30\textwidth}P{0.30\textwidth}@{}}
\toprule
\textbf{claim used in Part~III} & \textbf{grade} & \textbf{home} \\
\midrule
\GGEP{}: geometry and gauge structure are one object & axiom & the
constitution \citep{ArisakaA1Constitution} \\
\OGGEP{}: observation is a projection from bulk to boundary & axiom &
the constitution \citep{ArisakaA1Constitution} \\
The \OGNfive{} ledger $(1,3,5;7,35)$ & theorem, from closure identities &
the bulk paper \citep{ArisakaA2GG6} \\
\BIsix{} closure, the coefficient vector, the trace-ratio lock & theorem
& the anomaly paper \citep{ArisakaA3BI6} \\
Selection of $k = 1$ in the family $(5k, 3k)$ & derived-conditional, on
the minimality axiom & the anomaly paper \citep{ArisakaA3BI6} \\
\GDOS{}: the open-system boundary law & theorem-level development from
the axioms & the boundary paper \citep{ArisakaA5GDOS} \\
\GRIP{} as a relaxation cascade & framework, with its components graded
separately & the cascade paper \citep{ArisakaA6Gi4} \\
The Born rule & theorem at the relaxation level, conditional on four
named premises, one analytic residue open & the quantum-mechanics paper
\citep{ArisakaA8QM} \\
$N_g = 3$ & derived-conditional: an empirical input, a coupling clause, and one topological residue & the family-number paper
\citep{ArisakaP2N3} \\
The $\kappa$-bridge & theorem on the benchmark background; two residues open off it & the family-number paper's Appendix~H \citep{ArisakaP2N3} \\
\addlinespace
The \QiFour{} pillars, and the selection of a ground attractor among the
attractors & open throughout; stated in \S\ref{sec:qi4} and checked
nowhere in this paper & the extension paper \citep{ArisakaA7Qi4} \\
\bottomrule
\end{tabular}
\end{table}

\emph{Physics intuition: what a grade is for.} A grade is not a hedge.
It is a statement about which other claims would go down with this one,
and the fourth column of Table~\ref{tab:H-ledger}, in
Appendix~\ref{app:dictionary}, makes that explicit. A
reader who declines the projection axiom loses Part~III and keeps
Part~II. A reader who declines the minimality axiom keeps almost
everything and loses the privilege of one integer over its multiples.
Neither reader has to argue that anything is false, and neither is
obliged to accept the rest as a package. That modularity is the practical
purpose of grading, and it is why the ledger is printed rather than
summarized.

Two entries deserve emphasis because they are the ones most easily
misread. The Born rule is a \emph{result} of the architecture rather than
an input to it, but it is a result at a declared grade with a named
analytic residue outstanding, and the direction of dependence runs from
the axioms through the boundary law to it --- not the other way. Nothing
in this paper founds quantum mechanics, and \S\ref{ssec:born} says so
explicitly. And the family number is conditional in three separate ways,
each printed in \S\ref{ssec:three-gen}: one of them is the empirical fact
of $CP$ violation, which is to say a broken symmetry, which is the theme
of this paper arriving in the accounts.


\subsection{What kind of claim each Part makes}
\label{ssec:kinds-of-claim}

Three different kinds of claim have been made in this paper, and a great
deal of confusion about programs of this sort comes from not
distinguishing them. It is worth doing so explicitly, because the
distinction determines what would count as an objection.

\paragraph{A review result.} Part~II claims that seven standard results,
read in historical order, form a sequence with two monotone trends. That
is a claim about the literature, and it can be wrong in exactly one way:
by misreading what those results say. It asks the reader to grant no
premises, and it survives the failure of everything in Part~III. A critic
who wishes to attack it must argue that the sequence does not have the
shape claimed --- for instance, that Noether's second theorem is not
really an off-shell identity, or that gauge invariance is not really a
redundancy. Those are arguments about established mathematics, and they
can be settled.

\paragraph{A proposal at declared grades.} Part~III claims that a
particular architecture stands at the end of that sequence. This is a
different kind of claim, and it is conditional in a way the first is not:
it rests on axioms, and Table~\ref{tab:H-ledger} in
Appendix~\ref{app:dictionary} states which results
fail if which premise is withdrawn. A critic here does not need to show
that anything is false. It is enough to decline a premise --- and
\S\ref{ssec:exposure} identifies the premise most worth declining. This
is a weaker claim than the first in the sense that it asks for more, and
a stronger one in the sense that it says more.

\paragraph{A wager.} \S\ref{sec:concluding} is neither of these. A wager
is not a proposition with a truth value; it is a decision about what to
spend a career on, made under uncertainty, and its correctness is not the
sort of thing that can be established in advance. What can be asked of a
wager is that it be \emph{payable} --- that it commit its holder to
outcomes that could go against them. A program that arranges to be
unfalsifiable has not made a wager; it has merely expressed a preference.

\paragraph{Why the distinction matters.} Most disputes about programs of this kind are really disputes about which
of the three is on offer, conducted by parties who have not noticed that
they disagree about that.
A reader who takes Part~III's proposal for a review result will find it
outrageously underdetermined. A reader who takes
\S\ref{sec:concluding}'s wager for a proposal will find it unsupported,
and will be right to, because a wager is not supported --- it is placed.
This paper's practice of printing a grade beside every claim exists to
make the sorting mechanical rather than a matter of interpretation.

\paragraph{Agreement is not fit, and the difference is the whole
question.} One methodological point deserves emphasis because it is
where this program's evidence, such as it is, actually lives. A
successful \emph{fit} consumes a degree of freedom: a parameter is
adjusted, a curve passes through the points, and the achievement is
bounded by how many parameters were available. A successful
\emph{agreement} consumes nothing. When two computations with no shared
working parts are made to produce the same object and do, no dial was
turned, and the outcome was available to come out otherwise.
\S\ref{ssec:kappa} is an instance --- a topological pushforward from six
dimensions and the Berry bundle of ordinary four-dimensional quantum
mechanics, computed independently on a benchmark background, returning
the same line bundle. How
much weight that carries is a matter for judgment. That it is a
different kind of evidence from a fit is not.

\subsection{The argument in everyday terms}
\label{ssec:everyday}

For a reader who wants the shape of the argument and not its machinery,
the whole of this paper can be carried in four pictures. None of them is
a proof of anything; each is a way of holding a technical claim in mind
long enough to decide whether to go and check it.

\paragraph{The till and the ledger.} The oldest conservation laws say
that nothing leaves the till. Count the money in the morning, count it at
night, and the numbers agree. The \GS{} mechanism of
Chapter~\ref{sec:gs} replaces this with a bookkeeper's rule that is
weaker locally and stronger globally: money may move between accounts,
individual accounts may go into deficit, and what must hold is that the
\emph{total} closes at the end of the day. That is not a relaxation of
discipline. It is a stricter discipline applied to a larger object, and
it is the sense in which this paper is called \emph{The Books Must
Balance}.

\paragraph{Dials and counts.} Some quantities in physics are dials: they
can be nudged a little, and a small disturbance moves them a little.
Masses, couplings and mixing angles are dials. Other quantities are
counts: the number of colors, the number of families, the number of
units of charge. A count cannot be nudged --- there is no such thing as
moving from three to three-and-a-tiny-bit --- so a disturbance either
leaves it alone or breaks something. The observation of
\S\ref{ssec:counting-survives} is that we live in a world where almost
everything has been nudged, and that the quantities still known exactly
are, without exception, the counts. That is not a coincidence to be
admired; it is a clue about which kind of statement survives a world like
this one, and hence about which kind of statement the deepest laws should
be expected to be.

\paragraph{The budget.} Why are there three families rather than four?
Chapter~\ref{sec:grip}'s answer is not that a fourth is forbidden. It is
that a fourth is unaffordable. Reaching each successive family costs
something from an account fixed in advance, the costs rise faster than
the account does, and the sum runs out after three. The distinction
between \emph{forbidden} and \emph{unaffordable} is the distinction
between a rule and a budget, and \S\ref{ssec:exclusion} argues that
physics has had a great deal to say of the first kind and almost nothing
of the second --- because it has always written its laws for closed
systems, and only an open system has an account to draw on.

\paragraph{The shadow.} The projection of Chapter~\ref{sec:oggep} is the
hardest of the four to hold, and \S\ref{ssec:bulk} gives the picture: a
shadow on a wall is not a smaller room next door. Asking where the extra
dimensions are is like asking which corner of the wall the object is
hiding in. The useful questions about a shadow are what its outline
records and what the projection discards, and the program's claim is
that both questions have exact answers here.

\paragraph{What to take if you take one thing.} That conservation laws
became more rigid as they became less dynamical; that what survives at
the end of that process is an integer; and that an integer is the only
kind of quantity a broken world cannot erode. Everything else in this
paper is either the demonstration of that or a proposal about what it
implies.

\subsection{The one-sentence reading}
\label{ssec:one-sentence}
If a reader takes one sentence from this paper, it should be this one.

\begin{quote}
Conservation laws did not become unbreakable as they became less
dynamical; they became \emph{discretely} breakable, until at the top of
the ladder what is conserved is an integer and the reason is that
integers have nowhere continuous to move --- and the proposal of this
program is that such integers are what a four-dimensional boundary
reads off an exact six-dimensional ledger, with the breaking of symmetry
as the mechanism of the reading rather than as damage to it.
\end{quote}

The change of one word in that sentence --- \emph{discretely}, in place
of a claim that the statements cannot fail --- is what the two rungs
added in the middle of the ladder buy.
Section~\ref{ssec:axis} sets it out: an identity that cannot fail says
nothing when something goes wrong, and an identity that fails only by a
whole number says exactly how much. It is the second kind that a ledger
can be built on.

The first clause is a review of established physics. The second is a
proposal at declared grades, with its premises named in
Table~\ref{tab:grades} and its exposures named in
\S\ref{ssec:exposure}. A reader may accept the first and reject the
second without inconsistency, and the paper has been arranged so that
this is easy to do.



\emph{And what stands behind all of it.} The argument is finished at this
point; what follows the personal remark is apparatus, and it is there to be
used rather than read straight through. Part~\ref{part:appendices} is
fifteen appendices at the level of a first-year graduate course, each one
opening from its standard textbook, climbing to exactly what the body
needed and no further, and closing with problems and full solutions --- so
that a reader who rejects every proposal in Part~\ref{part:turn} still
leaves with a usable course. Part~\ref{part:qa} answers the hundred and two
questions an attentive reader actually arrives with, at the level of the
body. Part~\ref{part:references} lists every work cited, each entry saying
what that work established and where these pages first lean on it. A claim
in this paper that a reader wants to attack can be traced through those
three Parts to its owner.

\section{Concluding remark: designed, or forced?}
\label{sec:concluding}
\begin{quote}\small\itshape
What follows is a personal remark. It is the only passage in this paper
written in the first person and the only place where a view is expressed
that physics cannot adjudicate. Nothing anywhere else in the paper leans
on it, and a reader who disagrees with every word of this section may
still hold the rest to its stated grades.
\end{quote}

\bigskip

\paragraph{What is not a matter of opinion.}
Let me begin with the part that is not mine.

The laws of our universe are of a very particular type. They are highly
constrained. They exhibit invariances that nobody put there. And,
repeatedly and across unrelated domains, those invariances have turned
out to encode \emph{counting} statements rather than arbitrary
parameters: the charge of the electron is one unit and not $1.0000003$;
there are three colors; there are three families; the anomaly
coefficients of the Standard Model cancel exactly, as an identity between
integers. The world could have been arranged so that its deepest
constants were merely numbers. It was not. That is an empirical fact
about the universe I have spent a career measuring, and it is the fact
this entire program is built on.

It is worth being precise about how strange this is. Nothing in the
structure of quantum field theory requires it. A perfectly consistent
theory can be written down in which the proton's charge differs from the
electron's in the twenty-second decimal place, in which the number of
colors is not an integer in any meaningful sense because there is no
gauge group to count, in which every parameter is a real number with no
preferred value. Such theories are not pathological. They are simply not
what we find. The world we have is more rigid than it needs to be, and
the rigidity shows up as integers.

\paragraph{The part that is mine.}
I find it very difficult to regard that as an accident.

When I look at the ledger of Chapter~\ref{sec:gs} --- four coefficients
locked by a single ratio, closing on a set of integers that then
reappear, unbidden, in the family count and in the mass spectrum
\citep{ArisakaA9SM} --- I do not experience it as a coincidence that
happens to have a compact description. I experience it as the mark of a
necessity --- and for most of my working life I would have called that
necessity a design. I have thought this since long before I had a
formalism to attach it to, and the formalism has not weakened it. What
the formalism has done is take the word away, and I will come to that.

Part~\ref{part:ledger} is what changed while this paper was being
written, and it changed what I can say here. The conviction used to rest
on the \emph{shape} of the laws --- that they are rigid, that they close,
that they count. It now rests on something a reader can check line by
line. Table~\ref{tab:same-integers} is the passage I would put in front
of a skeptic: one integer, $J = 7$, fixing a coupling measured at a
collider at ninety-one giga-electronvolts and, through
$1 + z_{\rm sw} = e^{J}$, a topological marker in the recombination
era --- eleven orders of magnitude and a century apart. That marker is
the corridor switch and not the epoch of atom formation; the two are
distinguished where they are derived (\S\ref{ssec:cosmos-first}), and
the distinction is not one I want blurred in my own favor. I am aware
that \S\ref{ssec:same-integers} also prints the reason that even this
is not yet evidence. Both things are true, and I would rather hold them
together than choose.

I am aware that this is a report of an experience and not an argument,
and I want to be careful not to dress it as one. What I can say in its
defense is only this: the experience is the same one that has, in my
working life, most often preceded a correct guess. It is not a proof. It
is the thing that made me look.

\paragraph{The alternative, stated at full force.}
There is a well-developed answer to everything I have just said, and I
would rather state it properly than leave it as a shadow.

The answer runs: of course the constants look tuned, because we are
looking from inside a region where they permit observers. If the
underlying theory has an enormous landscape of possible vacua, and if
they are realized somewhere, then no explanation of any particular value
is required beyond the observation that we could not have found ourselves
where the values forbid us. The apparent design is a selection effect,
and the sense of astonishment is what a selection effect feels like from
the inside.

I do not think this is a foolish position, and I want to concede what it
gets right. It is right that selection effects are real and that physics
has been fooled by them before. It is right that our sample is one. And
it is right that a feeling of astonishment is very weak evidence about
anything.

Where I part company is on what the argument explains. A selection
effect can account for a value lying in a habitable range. It is much
less good at accounting for a value being an \emph{integer}, or for four
coefficients being locked to each other by a ratio, or for the same
integer arriving from two directions that share no machinery. Those are
not statements that some parameter fell in a window. They are statements
about the \emph{kind} of structure the laws have, and habitability seems
to me to have very little to say about kind. A universe with three
families and a universe with three-point-one families are not both
options one of which was selected; the second is not a universe at all.

I hold that as an objection, not a refutation. If the landscape picture
is right, my objection is a failure of imagination on my part, and I
would want to know it.

\paragraph{Wigner's question.}
Behind all of this sits the question \citet{Wigner1960} asked, and I do
not think this paper can avoid it. Why should mathematics work?

Two answers are usually given. The first is that mathematics is a
language we invented for describing patterns, so finding that it
describes patterns is no more surprising than finding that a hammer hits
things. The second is that the mathematical structures are in some sense
what the world is made of, and our finding them is discovery rather than
invention.

I have never been able to hold the first answer for long. It accounts
comfortably for the mathematics we build to fit what we already know. It
accounts very badly for the recurring experience of the opposite order:
a structure worked out for internal reasons, decades before anyone needed
it, turning out to be the thing physics required. The identity in
Chapter~\ref{sec:bianchi} was written down in the 1880s in pure
differential geometry, mislaid twice, and then turned out to force the
shape of Einstein's field equations. Nobody was fitting anything. The
quadratic refinement of Chapter~\ref{sec:hs} was constructed by
topologists with no interest in six-dimensional physics, and it is the
same function as the toll ladder of a relaxation cascade.

I do not know how to make that pattern comfortable on the first answer,
and I have stopped trying.

\paragraph{The refinement, which is the part I would ask a skeptic to
take seriously.}
For most of my life I would have said the necessity is symmetry. I no
longer
think that is right, and Chapter~\ref{sec:honest-record} is why.

A universe in exact possession of all its symmetries is empty. There is
no matter in it, because matter and antimatter canceled; there is no
structure in it, because structure grows from inhomogeneity; there is no
chemistry of the kind that builds a cell, because the molecules of life
are handed and a symmetric world has no preferred hand. Perfect symmetry
is not a description of a perfect world. It is a description of nothing
at all.

So whatever is forced here, it is not symmetry. It is symmetry \emph{plus
exactly enough breaking to permit existence} --- and that is a far more
delicate and more interesting condition than the thing I used
to admire. Too much symmetry and the universe is sterile. Too little and
there is no ledger to balance, no integer to protect, nothing that stays
put long enough to be a law. What we appear to have is a world driven
into the narrow strip between those two failures, in which the breaking is
severe enough to produce a matter asymmetry, a hand, an arrow of time,
and a periodic table, and yet controlled enough that the books still
close. \GRIP{} is my attempt to write down that strip as a mechanism: a
cascade whose content is breaking, and whose conservation laws survive
not as symmetries of the state it ends in but as identities of the whole
descent.

I want to claim something modest for this refinement and nothing more.
It is a better version of my own conviction than the one I held before,
it is more specific, and --- this is the part that matters --- it is
closer to being checkable. ``The world is designed'' commits me to
nothing: it names no step, forbids no measurement, and cannot be taken
away from me by anything a telescope could find. ``What is forced is
symmetry plus exactly the breaking that permits existence'' commits me
to a mechanism, and a mechanism can fail. That is why I have stopped
using the older word --- not because the conviction weakened, but
because it finally became specific enough to lose. The cohort states
the same conclusion in four words: not contingent, not lucky, not
designed; forced \citep{ArisakaA11Origin}. I would rather be judged by
that sentence than by the one I began with, and for exactly the reason
that it is the one that can be taken away from me.

\paragraph{Beauty, and what it has cost.}
Chapter~\ref{sec:honest-record} lists three occasions on which physicists
were confident about a symmetry for reasons that were partly aesthetic,
and were wrong. I take that history personally rather than as a caution
about other people.

The uncomfortable fact is that I would have been wrong about parity. I am
fairly sure of it. Reflection invariance is exactly the sort of thing I
find beautiful, and in 1955 I would have seen no reason to test it. The
lesson I draw is not that beauty is a bad guide --- it has been an
extraordinary guide --- but that it is a guide with a characteristic
failure mode, and that the failure mode is precisely the case where the
principle is imposed rather than read off. That is why
\S\ref{ssec:exposure} names the minimality axiom of this program by
name, and says plainly that it is on the wrong side of the line drawn in
Chapter~\ref{sec:honest-record}. I would rather print that than have a
referee find it.

\paragraph{What would change my mind.}
A conviction that cannot be moved is not worth holding, so let me say
what would move this one.

A fourth light family would do it, immediately and completely. So would a
demonstration that the coefficients of Chapter~\ref{sec:gs} are not
locked --- that the ratio can be shifted without cost, and that what
looks like a determination is a convention. So would the collapse of the
agreement reported in \S\ref{ssec:kappa}: if the two independent routes
turn out to share a hidden input, then there was only ever one
computation and the agreement was an artifact of my own bookkeeping. And,
more slowly but just as surely, so would a decade in which the
program's declared falsifiers were tested and the numbers came back on
the wrong side.

I would like to record that I expect to find out. The claims in the
member papers are not safe claims.

\paragraph{Where I leave it.}
Physics cannot tell you whether the books balance because someone is
keeping them. I do not think it ever will, and I have tried in this paper
to keep that question in a single section where a reader can decline it
without losing anything else.

What physics can tell you --- what I believe this paper shows it is now
in a position to tell you --- is that the books do balance, that the
balancing is exact, that the entries are integers, and that the breaking
of symmetry is not the flaw in the accounts but the reason there is
anything to count.

Whether that is a fact about the world or a fact about someone's
intention is not a question I can settle, and I will not pretend
otherwise. But I notice that I have spent a career being surprised in one
direction and not the other, and at some point a run of surprises in one
direction stops being a run.

\medskip
The keystone paper of the cohort ends on a sentence I will not improve
by paraphrasing, and it is the belief this whole program was built to
test \citep{ArisakaA11Origin}:

\begin{quote}\itshape
For fifty years I have believed that nothing about us is accidental ---
that nature restricts unnecessary freedom, making inevitable decisions.
That belief now has its deepest anchor. The universe had no choices left
to make: not in its dials, not in its integers, not even in the number it
is written in. And what it built with that necessity --- Cosmos, Life,
and Mind --- is therefore not a lucky throw but the unique reading of the
one geometry, at the grades this paper has recorded. That is exactly as
far as the theorems reach; it is also far enough to have been worth a
life.
\end{quote}

\medskip
And one thing more, which belongs to this paper rather than to that one,
because it is about the subject this paper has: for fifty years I was not
looking for the symmetry of nature. I was looking for the origin of the
precise symmetry breaking that allowed us to be here. A world that kept
all its symmetries would have nothing in it to find. What had to be
explained was never the symmetry --- it was the exactness of the
breaking: that it went just so far and no further, and left a world with
atoms in it, and with us.

\medskip
\hfill --- K.~A.

%% ===================================================================
%%  ACKNOWLEDGMENTS
%%  (cohort convention (b): after the Conclusions, before the Technical
%%  Appendices -- see the placement note in ucla-arisaka-gg-11.sty)
%% ===================================================================
\GGAcknowledgments

\medskip
One acknowledgment belongs to this paper in particular. A review owes a
debt an original paper does not: the century it walks through was
written by other people, and the value of these pages lies mostly in
having read them in one order. Where a rung of the ladder is carried at
a grade below \emph{derived}, that grade was set by the parent paper and
is not softened here; where a step is open --- the minimality premise
above all --- it is named where a reader can find it rather than left to
be discovered. The reader who wants to attack this paper should attack
those joints.

And a debt of a plainer kind. Almost nothing in Part~II is mine.
Noether saw in 1918 what the two theorems separate and what follows from
the difference; Bianchi's identity was written before anyone knew what it
would be needed for; Wilson gave the loop, Ward and Takahashi the
identity that survives quantization, Green and Schwarz the cancellation
that costs an integer, and Hopkins and Singer the refinement at the top
of the climb. The experimenters are owed as much as the theorists: Wu's
team, who took the trouble to cool a cobalt source and found that the
world is not mirror-symmetric, and Christenson, Cronin, Fitch and Turlay,
who found the smaller failure that made the larger question askable. Two
of the three cautionary episodes in \S\ref{sec:honest-record} are
cautionary only because somebody measured. If these pages are of use, it
is because those people did the work and left it in a state where it
could be read in one order.


%% ===================================================================
%%  PART VII --- TECHNICAL APPENDICES
%% ===================================================================
\GGAppendixPart[Technical appendices]{Technical Appendices}{Fifteen
appendices at the level of a first-year graduate course --- each opening
from the standard textbooks, climbing to exactly what the body needed,
and closing with problems and full solutions}
\label{part:appendices}

\paragraph{Prerequisites and recommended textbooks.} This Part assumes
advanced undergraduate mechanics, electrodynamics and quantum mechanics,
a first course in special relativity, and some acquaintance with
Lagrangian field theory. It does \emph{not} assume differential forms,
fiber bundles, anomalies, open-system dynamics, or algebraic topology of
any kind; each is built here from its standard textbook starting point.

\paragraph{How the fifteen are arranged.} The first seven follow the
ladder of Part~II in its own order: forms
(Appendix~\ref{app:forms}), Noether's two theorems
(Appendix~\ref{app:noether}), connections and holonomy
(Appendix~\ref{app:connections}), Wilson lines
(Appendix~\ref{app:wilson}), Ward identities
(Appendix~\ref{app:ward}), anomalies
(Appendix~\ref{app:anomalies}), and differential cohomology
(Appendix~\ref{app:diffcohom}). Appendix~\ref{app:bianchi} supplies
the third rung's identity, which the ladder leans on and which no
earlier appendix derived. Appendix~\ref{app:opensys} is the technical
support for the whole of Part~III: density matrices, completely
positive maps, the Lindblad generator, and the semigroup structure on
which the review's central structural claim rests. The last five
support Part~IV one chapter at a time --- the parameter count
(Appendix~\ref{app:params}), the discrete symmetries
(Appendix~\ref{app:discrete}), confinement and strong $CP$
(Appendix~\ref{app:confine}), baryogenesis
(Appendix~\ref{app:lepto}), and the chronology and the arrow
(Appendix~\ref{app:time}) --- and the dictionary closes the Part.

\paragraph{How to use them.} Each of the first fourteen is
self-contained and runs in the same four movements. \emph{Step~1} builds the standard material
from its textbook starting point, at length and with the derivations
written out, so that a reader who rejects everything this review
proposes still has a usable course. \emph{Step~2} says what the
standard treatment assumes and what it leaves open, fairly and without
caricature. \emph{Step~3} says what the body does with it. Then come
problems with full worked solutions --- eighty-two in all across the
fourteen --- because a derivation one has not used is a derivation one
has not learned.

\begin{itemize}[leftmargin=1.4em,itemsep=0.2em]
\item \emph{Primary.} \citet{PeskinSchroeder1995} for field theory and
  anomalies; \citet{Nakahara2003} for forms, bundles, connections and
  characteristic classes; \citet{MisnerThorneWheeler1973} for the
  gravitational side of Appendix~\ref{app:connections}.
\item \emph{A gentler first pass.} \citet{Baez1994} for forms and gauge
  fields with the geometry kept visible; \citet{Zee2010} for field theory
  told physically.
\item \emph{In depth.} \citet{BottTu1982} for de Rham theory and the
  cohomological machinery of Appendix~\ref{app:diffcohom};
  \citet{ChengLi1984} for the gauge-theory structure and group theory
  behind Chapter~\ref{sec:gauge}; \citet{Weinberg1996} for anomalies at
  full rigor.
\item \emph{Specialist, for Appendix~\ref{app:diffcohom} only.}
  \citet{CheegerSimons1985} and \citet{HopkinsSinger2005}, which are the
  primary literature rather than textbooks and are read last.
\end{itemize}

\noindent
Each appendix follows the same three-step plan: \textbf{Step 1}, the
construction as a graduate student meets it; \textbf{Step 2}, what that
construction assumes and cannot explain; \textbf{Step 3}, what the
argument of this paper does with it. Where a result is imported from a
cohort paper rather than derived, the appendix says so at the point of
use and names the owner.

\section{Differential forms from scratch}
\label{app:forms}
\GGteachbox{Why vector calculus runs out above three dimensions; forms as
the things one integrates; the wedge product and the exterior derivative;
$\dd\dd = 0$ proved in components and understood structurally; Stokes'
theorem as the single statement that replaces four; and Maxwell's
equations in two lines. The pay-off is Chapter~\ref{sec:bianchi}: after
this appendix, $\dd F = \dd\dd A = 0$ is a triviality rather than a
slogan.}

\GGreadalong{\citet{Nakahara2003}, Ch.~5, for the systematic treatment;
\citet{Baez1994}, Part~II, for the same material with the physics kept in
view; \citet{MisnerThorneWheeler1973}, Ch.~4, for the classical
presentation in a gravitational setting.}

\subsection{Step 1 --- where vector calculus runs out}
\label{ssec:A-step1}

An undergraduate learns three derivative operators --- $\nabla$,
$\nabla\times$, $\nabla\cdot$ --- and two identities,
\begin{equation}
\nabla \times (\nabla f) = 0, \qquad
\nabla \cdot (\nabla \times \mathbf{V}) = 0 ,
\label{eq:A-vector-identities}
\end{equation}
which are presented as separate facts to be memorized. They are the same
fact, and this appendix is largely the story of why.

The reason vector calculus works at all in three dimensions is a
numerical accident. An antisymmetric object with $k$ indices in $n$
dimensions has $\binom{n}{k}$ independent components, and in $n = 3$
these are
\begin{equation}
\binom{3}{0},\ \binom{3}{1},\ \binom{3}{2},\ \binom{3}{3}
 \;=\; 1,\ 3,\ 3,\ 1 .
\label{eq:A-binom3}
\end{equation}
Objects with one index and objects with two antisymmetric indices both
have three components, so both can be written as ``vectors'' and one may
define a cross product taking two of the first kind to something with
three components. In four dimensions the row is $1, 4, 6, 4, 1$: a
two-index antisymmetric object has six components, not four, and the
cross product has nowhere to go. This is not a technicality about
notation. It is why the electromagnetic field, which is genuinely a
two-index antisymmetric object, splits artificially into $\mathbf{E}$ and
$\mathbf{B}$ in a way that depends on the observer.

\emph{Physics intuition: what a form is for.} A $k$-form is the kind of
thing you integrate over a $k$-dimensional region. A $1$-form is what you
integrate along a curve --- $\mathbf{F}\cdot\dd\boldsymbol{\ell}$ is a
$1$-form, and \emph{work} is its integral. A $2$-form is what you
integrate over a surface --- $\mathbf{B}\cdot\dd\mathbf{A}$ is a
$2$-form, and \emph{flux} is its integral. A $3$-form is what you
integrate over a volume; \emph{charge} is its integral. The formalism is
not an abstraction imposed on physics; it is the observation that
physics has been sorting its quantities by what they get integrated over
since Faraday.

\subsection{Step 1 (continued) --- wedge and exterior derivative}
\label{ssec:A-wedge}

\paragraph{Statement.} A $k$-form on an $n$-dimensional manifold is a
completely antisymmetric tensor field with $k$ lower indices, written
\begin{equation}
\omega \;=\; \frac{1}{k!}\,\omega_{\mu_1 \cdots \mu_k}\,
 \dd x^{\mu_1} \wedge \cdots \wedge \dd x^{\mu_k} ,
\label{eq:A-kform}
\end{equation}
where the wedge $\wedge$ is an associative product that is antisymmetric
on $1$-forms, $\dd x^\mu \wedge \dd x^\nu = -\,\dd x^\nu \wedge \dd x^\mu$.
The sign rule for general degrees is
\begin{equation}
\alpha \wedge \beta = (-1)^{pq}\,\beta \wedge \alpha ,
\qquad \alpha \in \Omega^p, \ \beta \in \Omega^q .
\label{eq:A-wedge-sign}
\end{equation}

The exterior derivative is the unique operator $\dd : \Omega^k \to
\Omega^{k+1}$ that acts on functions as the ordinary gradient, obeys the
graded Leibniz rule
\begin{equation}
\dd(\alpha \wedge \beta)
 = \dd\alpha \wedge \beta + (-1)^{p}\, \alpha \wedge \dd\beta ,
\label{eq:A-leibniz}
\end{equation}
and satisfies $\dd\dd = 0$. In components it is the antisymmetrized
partial derivative,
\begin{equation}
(\dd\omega)_{\mu_0 \mu_1 \cdots \mu_k}
 = (k+1)\,\partial_{[\mu_0} \omega_{\mu_1 \cdots \mu_k]} .
\label{eq:A-d-components}
\end{equation}

\paragraph{Worked example: grad, curl and div are one operator.} In
$n = 3$, apply Eq.~\eqref{eq:A-d-components} three times. On a $0$-form
$f$ it gives $(\dd f)_i = \partial_i f$, the gradient. On a $1$-form
$A = A_i \dd x^i$ it gives $(\dd A)_{ij} = \partial_i A_j - \partial_j
A_i$, whose three independent components are the curl. On a $2$-form it
gives the divergence. Three operators, one definition, and the reason
they looked like three is Eq.~\eqref{eq:A-binom3}.

\subsection{Step 1 (concluded) --- \texorpdfstring{$\dd\dd = 0$}{dd = 0}, and why it is the whole story}
\label{ssec:A-ddzero}

\paragraph{Statement.} $\dd\dd\omega = 0$ for every form $\omega$.

\paragraph{Proof.} Apply Eq.~\eqref{eq:A-d-components} twice. The result
carries two derivative indices antisymmetrized against each other,
\begin{equation}
(\dd\dd\omega)_{\mu\nu\rho\cdots}
 \;\propto\; \partial_{[\mu}\partial_{\nu}\,\omega_{\rho\cdots]} ,
\label{eq:A-ddzero}
\end{equation}
and $\partial_\mu \partial_\nu$ is symmetric under exchange of $\mu$ and
$\nu$ while the bracket antisymmetrizes. A symmetric object contracted
with an antisymmetric one vanishes. \qed

That is the entire proof, and it is worth pausing on how little went into
it: no metric, no connection, no signature, no dimension. Both identities
in Eq.~\eqref{eq:A-vector-identities} are instances --- the first is
$\dd\dd f = 0$ on a $0$-form, the second is $\dd\dd A = 0$ on a
$1$-form --- and they were never two facts.

\emph{Physics intuition.} $\dd\dd = 0$ is the statement that a boundary
has no boundary. The boundary of a disc is a circle; the circle has no
endpoints. The boundary of a ball is a sphere; the sphere has no edge.
Eq.~\eqref{eq:A-leibniz} and Stokes' theorem below make this precise, and
once one has seen it the identity stops being an algebraic accident and
becomes a fact about shapes.

\subsection{Step 1 (closing) --- Stokes' theorem, and Maxwell in two lines}
\label{ssec:A-stokes}

\paragraph{Statement.} For any $(k-1)$-form $\omega$ on a compact
oriented $k$-dimensional region $M$ with boundary $\partial M$,
\begin{equation}
\int_M \dd\omega \;=\; \oint_{\partial M} \omega .
\label{eq:A-stokes}
\end{equation}

The fundamental theorem of calculus, Green's theorem, the Kelvin--Stokes
theorem and the divergence theorem are the cases $k = 1, 2, 2, 3$ of
Eq.~\eqref{eq:A-stokes}. Four theorems, one statement, and the pairing of
Eq.~\eqref{eq:A-stokes} with $\dd\dd = 0$ is the duality noted above:
$\dd$ is to forms what $\partial$ is to regions.

\paragraph{Worked example: Maxwell.} Assemble the potential into a
$1$-form $A = A_\mu \dd x^\mu$ and set $F = \dd A$. Then all of
electromagnetism is
\begin{equation}
\dd F = 0 , \qquad \dd \star F = \star J ,
\label{eq:A-maxwell}
\end{equation}
where $\star$ is the Hodge dual, the operation that exchanges $k$-forms
for $(n-k)$-forms using the metric. Count: $F$ has $\binom{4}{2} = 6$
components, the electric and magnetic fields; $\dd F$ has
$\binom{4}{3} = 4$, which are $\nabla\cdot\mathbf{B} = 0$ and the three
components of Faraday's law; $\dd\star F$ has $4$, which are Gauss's law
and the three components of Amp\`ere--Maxwell.

The asymmetry between the two equations in Eq.~\eqref{eq:A-maxwell} is
the subject of Chapter~\ref{sec:bianchi}, and it is now visible in the
notation: the first requires no metric and no $\star$, and follows from
$F = \dd A$ alone; the second needs the metric, needs a source, and comes
from varying an action.


\subsection{Step 1 (further) --- the Hodge star, and where the metric enters}
\label{ssec:A-hodge}

Everything to this point has used no metric. The exterior derivative, the
wedge, $\dd\dd = 0$ and Stokes' theorem are all statements about a smooth
manifold and nothing more. That is worth dwelling on, because exactly one
of the two halves of Eq.~\eqref{eq:A-maxwell} breaks the pattern.

\paragraph{Statement.} On an oriented $n$-manifold with a metric there is
a unique linear map $\star : \Omega^k \to \Omega^{n-k}$ fixed by
\begin{equation}
\alpha \wedge \star\beta
 = \langle \alpha, \beta\rangle \, \mathrm{vol} ,
\qquad \alpha, \beta \in \Omega^k ,
\label{eq:A-hodge-def}
\end{equation}
with $\langle\cdot,\cdot\rangle$ the induced inner product on $k$-forms
and $\mathrm{vol}$ the volume form. In components,
\begin{equation}
(\star\omega)_{\mu_{k+1}\cdots\mu_n}
 = \frac{\sqrt{|g|}}{k!}\,
   \epsilon_{\mu_{k+1}\cdots\mu_n}{}^{\nu_1\cdots\nu_k}\,
   \omega_{\nu_1 \cdots \nu_k} .
\label{eq:A-hodge-comp}
\end{equation}
The metric appears twice --- in $\sqrt{|g|}$ and in raising the indices
of $\epsilon$ --- and there is no way to define $\star$ without it.

\paragraph{Why this settles the law/identity split.} Return to
Eq.~\eqref{eq:A-maxwell}. The first equation, $\dd F = 0$, contains no
$\star$: it is metric-free, dimension-blind, and follows from $F = \dd A$.
The second, $\dd\star F = \star J$, contains two: it knows the geometry,
it knows the signature, and it changes form in every dimension. That is
the formal statement of what \S\ref{ssec:not-a-law} argues in words ---
one is a fact about what a curvature is, the other a fact about how this
particular world is arranged.

A useful consequence for Chapter~\ref{sec:gs}: in $2m$ dimensions
$\star$ maps $\Omega^m$ to itself, so a middle-degree form can be
self-dual. In four dimensions that is the instanton condition
$F = \star F$; in six it is the self-duality available to the three-form
$H_3$, which is why a six-dimensional tensor sector behaves the way
Eq.~\eqref{eq:bi6} needs it to.

\emph{Physics intuition.} The star is the operation that turns ``the
thing you integrate over a surface'' into ``the thing you integrate over
its complement''. Flux through a surface and current through the
complementary directions are the same information seen twice, and it
takes a metric to know which directions are complementary. Strip the
metric away and half of electromagnetism survives untouched; that half is
the third rung of this paper's ladder.
\subsection{Step 2 --- what the formalism assumes}
\label{ssec:A-step2}

Two assumptions are buried in the above and are worth surfacing, because
Part~II turns on both.

\emph{That $F$ is exact, not merely closed.} A form with $\dd\omega = 0$
is \emph{closed}; one with $\omega = \dd\eta$ is \emph{exact}. Every
exact form is closed, by $\dd\dd = 0$. The converse holds only locally
(the Poincar\'e lemma) and can fail globally: on a space with a hole
there exist closed forms that are not exact, and the space of such forms
modulo exact ones is the de Rham cohomology $H^k_{\mathrm{dR}}(M)$.

That failure is not a defect. It is the entire subject of
Appendix~\ref{app:diffcohom}, and physically it is the Aharonov--Bohm
effect (\S\ref{ssec:third-datum}) and Dirac's monopole
(\S\ref{ssec:dirac}). When \S\ref{ssec:no-monopoles} says that magnetic
charge is the obstruction to writing $F = \dd A$ globally, the technical
content is precisely that $F$ is closed but not exact.

\emph{That the manifold is orientable and the integrals converge.}
Standard, and assumed throughout without further comment.

\subsection{Step 3 --- what the body does with it}
\label{ssec:A-step3}

Chapter~\ref{sec:bianchi} needs exactly three things from this appendix,
and they are now available at no further cost.

\begin{enumerate}[leftmargin=1.5em,itemsep=0.25em]
\item $\dd F = 0$ follows from $F = \dd A$ with no dynamics, no
  Lagrangian and no symmetry --- Eq.~\eqref{eq:A-ddzero}.
\item The identity has $\binom{4}{3} = 4$ independent components, which
  are two of Maxwell's four equations.
\item Its failure mode is not violation but non-exactness: the identity
  survives, in patches, and what changes is the global bookkeeping.
\end{enumerate}

Chapter~\ref{sec:gs} then needs one thing more, which is that
Eq.~\eqref{eq:A-leibniz} permits an object whose exterior derivative is
prescribed to be non-zero without contradiction, provided that object is
not itself the exterior derivative of anything.

\subsection{Problems and worked solutions}
\label{ssec:A-problems}

\medskip\noindent\textbf{A.1. Show that the two identities of
Eq.~\eqref{eq:A-vector-identities} are the same statement.}

\noindent\emph{Solution.} \emph{Step 1.} Write the first as $\dd\dd f =
0$ for a $0$-form $f$: $(\dd f)_i = \partial_i f$, and
$(\dd\dd f)_{ij} = \partial_i \partial_j f - \partial_j \partial_i f = 0$,
which in three-dimensional notation reads $\nabla \times \nabla f = 0$.
\emph{Step 2.} Write the second as $\dd\dd A = 0$ for a $1$-form: the
components of $\dd A$ are the curl, and the components of $\dd(\dd A)$
are the divergence of the curl. \emph{Step 3.} Both are
Eq.~\eqref{eq:A-ddzero} with $k = 0$ and $k = 1$. \emph{Physics
intuition:} the same fact appears twice in three dimensions only because
$\binom{3}{1} = \binom{3}{2}$ lets one write both a $1$-form and a
$2$-form as a ``vector''.

\medskip\noindent\textbf{A.2. In $n$ dimensions, what is the highest
degree of a non-zero form, and how many independent components does it
have?}

\noindent\emph{Solution.} Degree $n$, with $\binom{n}{n} = 1$ component.
A top form is a single function times $\dd x^1 \wedge \cdots \wedge
\dd x^n$, which is why the volume element is a top form and why
integration over the whole manifold makes sense only for forms of degree
$n$. \emph{Physics intuition:} there are only so many independent
directions to antisymmetrize over; asking for $n+1$ of them in $n$
dimensions forces a repeat, and a repeat kills an antisymmetric object.

\medskip\noindent\textbf{A.3. Verify Eq.~\eqref{eq:A-wedge-sign} for
$p = q = 1$, and deduce that a $1$-form wedged with itself vanishes.}

\noindent\emph{Solution.} \emph{Step 1.} For $1$-forms $\alpha, \beta$
the rule $\dd x^\mu \wedge \dd x^\nu = -\dd x^\nu \wedge \dd x^\mu$ gives
$\alpha \wedge \beta = -\beta \wedge \alpha$, which is
Eq.~\eqref{eq:A-wedge-sign} at $(-1)^{1\cdot 1}$. \emph{Step 2.} Set
$\beta = \alpha$: then $\alpha \wedge \alpha = -\alpha \wedge \alpha$, so
$\alpha \wedge \alpha = 0$. \emph{Step 3.} Note the contrast with even
degree: a $2$-form need \emph{not} square to zero, and $F \wedge F$ is
the object that carries the instanton number.
\emph{Physics intuition:} this is why the topological terms in gauge
theory involve $F \wedge F$ and never $A \wedge A$ on its own.

\medskip\noindent\textbf{A.4. Show that $\dd$ commutes with pullback,
and explain why this makes $\dd F = 0$ observer-independent.}

\noindent\emph{Solution.} \emph{Step 1.} For a map $\phi$ and a form
$\omega$, $\phi^{*}(\dd\omega) = \dd(\phi^{*}\omega)$; this follows
directly from the chain rule applied to
Eq.~\eqref{eq:A-d-components}. \emph{Step 2.} A change of coordinates or
of observer is a pullback. \emph{Step 3.} If $\dd F = 0$ in one frame it
is zero in every frame, with no transformation law to check.
\emph{Physics intuition:} the statement contains no metric, so there is
nothing in it for a boost to act on --- which is exactly the sense in
which Chapter~\ref{sec:bianchi} calls it an identity rather than a law.

\medskip\noindent\textbf{A.5. On the punctured plane
$\mathbb{R}^2 \setminus \{0\}$, show that
$\omega = (x\,\dd y - y\,\dd x)/(x^2+y^2)$ is closed but not exact, and
compute $\oint\omega$ on a circle about the origin.}

\noindent\emph{Solution.} \emph{Step 1.} A direct computation gives
$\dd\omega = 0$ away from the origin. \emph{Step 2.} In polar coordinates
$\omega = \dd\theta$, and $\theta$ is not a single-valued function on the
punctured plane, so $\omega$ is not $\dd$ of anything globally defined.
\emph{Step 3.} $\oint_{|z| = r}\omega = 2\pi$ for every $r$, independent
of the radius --- a topological invariant, not a geometric one.
\emph{Physics intuition:} this is the Aharonov--Bohm effect stripped of
all physics. The $2\pi$ is a winding number, and \S\ref{ssec:dirac}'s
integer is the same integer.

\medskip\noindent\textbf{A.6. Count the independent components of $F$,
$\dd F$ and $\dd \star F$ in six dimensions, and say which of Maxwell's
statements survives the change of dimension unchanged.}

\noindent\emph{Solution.} \emph{Step 1.} $\binom{6}{2} = 15$,
$\binom{6}{3} = 20$, $\binom{6}{5} = 6$. \emph{Step 2.} $\dd F = 0$
survives verbatim: its derivation used neither the metric nor the
dimension. \emph{Step 3.} $\dd\star F = \star J$ changes character
entirely, since $\star$ depends on the dimension.
\emph{Physics intuition:} the half of Maxwell that is an identity is
dimension-blind, and the half that is a law is not --- which is why
Chapter~\ref{sec:gs} can port the identity to six dimensions without
importing the dynamics.

\GGtakeaway{\emph{Standard result:} the exterior calculus, with
$\dd\dd = 0$ and Stokes' theorem, replacing the three operators and four
integral theorems of vector calculus. \emph{What it assumed:} an
orientable manifold, and --- crucially --- that the field strength is
\emph{exact} rather than merely closed. \emph{The reading this paper
takes:} $\dd F = 0$ is an identity, not a law, because
Eq.~\eqref{eq:A-ddzero} uses nothing but the symmetry of mixed partial
derivatives; and the failure of exactness, not of closedness, is what
magnetic charge means. \emph{Continue in:}
Appendix~\ref{app:connections} for what $A$ is geometrically, and
Appendix~\ref{app:diffcohom} for the closed-but-not-exact story told
properly.}


\section{Noether's two theorems, derived in full}
\label{app:noether}
\GGteachbox{The variational calculus at the level needed; Noether's first
theorem derived with the boundary terms kept rather than discarded; the
three classical currents computed explicitly; the second theorem derived
in full; and the two theorems run side by side on one field theory, so
that the difference between a charge and an identity is a computation
rather than an assertion. The pay-off is Chapter~\ref{sec:noether}, and
in particular \S\ref{ssec:noether-ii}.}

\GGreadalong{\citet{PeskinSchroeder1995}, \S2.2, for the first theorem in
its usual field-theory form; \citet{Weinberg1996}, Ch.~7, for the
constrained-system machinery behind the second;
\citet{KosmannSchwarzbach2011} for the history and for a modern
translation of the 1918 paper itself.}

\subsection{Step 1 --- variational calculus, in the form the theorems need}
\label{ssec:B-step1}

Let the fields be $\phi^a(x)$, with Lagrangian density
$\mathcal{L}(\phi^a, \partial_\mu \phi^a)$ and action
$S = \int \dd^4x\, \mathcal{L}$. Under an arbitrary variation
$\delta\phi^a$,
\begin{equation}
\delta S = \int \dd^4 x
 \left[
   \underbrace{\left(
     \frac{\partial \mathcal{L}}{\partial \phi^a}
     - \partial_\mu \frac{\partial \mathcal{L}}
       {\partial (\partial_\mu \phi^a)}\right)}_{\textstyle E_a}
   \delta\phi^a
   \;+\; \partial_\mu
   \left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^a)}
     \delta\phi^a\right)
 \right] .
\label{eq:B-variation}
\end{equation}
The first bracket is the Euler--Lagrange expression $E_a$; the second is
a total derivative. The usual next move is to declare the variation to
vanish on the boundary and discard the second term. \emph{Do not discard
it.} Both of Noether's theorems live in the interplay between the two
terms, and the standard derivation obscures the second theorem precisely
by throwing away the piece that carries it.

\subsection{Step 1 (continued) --- the first theorem, boundary terms kept}
\label{ssec:B-first}

\paragraph{Statement.} Let the action be invariant under a continuous
transformation $\delta_\epsilon \phi^a = \epsilon\, \Delta^a(\phi,
\partial\phi)$ with $\epsilon$ a \emph{constant}, in the sense that
$\delta_\epsilon \mathcal{L} = \epsilon\, \partial_\mu K^\mu$ for some
$K^\mu$. Then
\begin{equation}
j^\mu \;=\;
 \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^a)}\,\Delta^a
 \;-\; K^\mu
\label{eq:B-current}
\end{equation}
satisfies $\partial_\mu j^\mu = -\,E_a \Delta^a$, and therefore
$\partial_\mu j^\mu = 0$ whenever the field equations $E_a = 0$ hold.

\paragraph{Proof.} Substitute $\delta\phi^a = \epsilon \Delta^a$ into
Eq.~\eqref{eq:B-variation}. By hypothesis the left-hand side is
$\epsilon \int \dd^4x\, \partial_\mu K^\mu$. Equating integrands and
dividing by $\epsilon$,
\begin{equation}
\partial_\mu K^\mu
 = E_a \Delta^a
 + \partial_\mu\!\left(
    \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^a)}\Delta^a
   \right) ,
\end{equation}
which rearranges to $\partial_\mu j^\mu = -E_a \Delta^a$ with $j^\mu$ as
in Eq.~\eqref{eq:B-current}. \qed

Two features of this deserve comment. First, the conservation is
\emph{on shell} and the derivation shows exactly why: the term that
spoils it is proportional to the field equations. Second, $j^\mu$ is not
unique --- adding $\partial_\nu \Theta^{\mu\nu}$ with
$\Theta^{\mu\nu} = -\Theta^{\nu\mu}$ changes nothing, since the extra
divergence vanishes identically. Improvement terms of that kind are used
constantly in practice, and they are harmless for the same reason
$\dd\dd = 0$ (Appendix~\ref{app:forms}).

\subsection{Step 1 (continued) --- the three classical currents, computed}
\label{ssec:B-three}

\paragraph{Time translation.} Take $\delta \phi^a = \epsilon\,
\partial_0 \phi^a$. Then $\delta\mathcal{L} = \epsilon\, \partial_0
\mathcal{L}$, so $K^\mu = \delta^\mu_{\ 0}\mathcal{L}$, and
Eq.~\eqref{eq:B-current} gives
\begin{equation}
j^0 \;=\;
 \frac{\partial \mathcal{L}}{\partial \dot\phi^a}\,\dot\phi^a
 - \mathcal{L}
 \;=\; \mathcal{H} ,
\label{eq:B-energy}
\end{equation}
the Hamiltonian density. \S\ref{ssec:noether-i} does the mechanical case
of this on the harmonic oscillator, where it turns $T - V$ into $T + V$.

\paragraph{Space translation.} Take $\delta\phi^a = \epsilon^i
\partial_i \phi^a$; the same computation gives the momentum density
$T^{0}{}_{i}$, and the general object is the canonical
energy--momentum tensor
\begin{equation}
T^{\mu}{}_{\nu} =
 \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^a)}
 \partial_\nu \phi^a - \delta^\mu{}_\nu \mathcal{L} ,
\qquad \partial_\mu T^{\mu}{}_{\nu} = 0 .
\label{eq:B-emt}
\end{equation}

\paragraph{Rotation, and the phase.} Lorentz transformations give the
angular-momentum current, which contains an orbital piece built from
Eq.~\eqref{eq:B-emt} and a spin piece from the transformation of the
field indices. The internal phase rotation $\delta\phi = \ii\epsilon\phi$
on a complex scalar gives Eq.~\eqref{eq:scalar-current} directly from
Eq.~\eqref{eq:B-current}, with $K^\mu = 0$ because $\mathcal{L}$ is
strictly invariant rather than invariant up to a divergence.

\emph{Physics intuition.} The four cases differ only in what
$\Delta^a$ is, and Eq.~\eqref{eq:B-current} is a machine: feed it the
transformation, turn the handle, receive the current. What makes the
theorem remarkable is not any individual output but that the handle
exists at all.

\subsection{Step 1 (concluded) --- the second theorem, in full}
\label{ssec:B-second}

Now let the parameter be a function, $\epsilon \to \epsilon(x)$, and
suppose the action is invariant for \emph{every} such function. Write the
transformation in the general form the theorem requires,
\begin{equation}
\delta \phi^a = R^a{}_{(0)}\,\epsilon + R^{a\mu}{}_{(1)}\,\partial_\mu \epsilon ,
\label{eq:B-local-transf}
\end{equation}
where $R_{(0)}$ and $R_{(1)}$ are local functions of the fields. Because
the variation is a symmetry for arbitrary $\epsilon$, the on-shell part
of Eq.~\eqref{eq:B-variation} must vanish for arbitrary $\epsilon$:
\begin{equation}
0 = \int \dd^4x \; E_a
 \left( R^a{}_{(0)}\epsilon + R^{a\mu}{}_{(1)} \partial_\mu \epsilon\right)
 = \int \dd^4x \; \epsilon
 \left( E_a R^a{}_{(0)} - \partial_\mu\!\left(E_a R^{a\mu}{}_{(1)}\right)\right) ,
\label{eq:B-second-derivation}
\end{equation}
integrating by parts in the second step and dropping the surface term
because $\epsilon$ may be taken of compact support. Since $\epsilon(x)$
is arbitrary, the bracket must vanish identically:
\begin{equation}
E_a R^a{}_{(0)} - \partial_\mu\!\left(E_a R^{a\mu}{}_{(1)}\right)
 \;\equiv\; 0 .
\label{eq:B-noether-ii}
\end{equation}

\paragraph{Statement.} Equation~\eqref{eq:B-noether-ii} is Noether's
second theorem. It is an identity among the Euler--Lagrange expressions
and their derivatives, holding for \emph{every} field configuration
whether or not it solves the field equations. It contains no new
conserved charge.

\paragraph{Worked example: electromagnetism.} For
$\mathcal{L} = -\tfrac14 F_{\mu\nu}F^{\mu\nu}$ with
$\delta A_\mu = \partial_\mu \epsilon$, the transformation has
$R_{(0)} = 0$ and $R^{\mu}{}_{(1)\nu} = \delta^\mu{}_\nu$. The
Euler--Lagrange expression is $E^\nu = \partial_\mu F^{\mu\nu}$, and
Eq.~\eqref{eq:B-noether-ii} reduces to
\begin{equation}
\partial_\nu \partial_\mu F^{\mu\nu} \equiv 0 ,
\end{equation}
which is Eq.~\eqref{eq:noether-ii}. The identity is now derived rather
than merely observed, and its status as an identity is visible in the
derivation: it came from the requirement that the variation vanish for
arbitrary $\epsilon(x)$, and nothing was put on shell.

\subsection{Step 2 --- what the two theorems assume, and what follows}
\label{ssec:B-step2}

The first theorem assumes a symmetry with finitely many constant
parameters and delivers one current per parameter. The second assumes a
symmetry with arbitrary functions and delivers one identity per function.
The distinction is not a matter of degree.

Three consequences are worth naming for the body.

\emph{Underdetermination.} Equation~\eqref{eq:B-noether-ii} says that the
field equations are not independent. For electromagnetism, one
combination of the four equations $E^\nu = 0$ is a constraint rather than
an evolution equation, which is why $A_\mu$ has four components and two
propagating degrees of freedom.

\emph{The energy problem in general relativity.} With general covariance
the arbitrary functions are the four $\xi^\mu(x)$ of a coordinate change,
and Eq.~\eqref{eq:B-noether-ii} becomes the contracted Bianchi identity
$\nabla_\mu G^{\mu\nu} \equiv 0$ of Eq.~\eqref{eq:div-G}. The
gravitational ``energy conservation law'' is that identity, which is why
it holds off shell and why no local, covariant gravitational
energy--momentum tensor exists. This is the problem Hilbert and Klein
posed and Noether solved, as \S\ref{ssec:goettingen} records.

\emph{No charge from a redundancy.} Because the second theorem yields no
current, there is no gauge charge to measure --- consistent with
Elitzur's theorem (\S\ref{ssec:elitzur}) and with the reading of gauge
invariance as redundancy.


\subsection{Step 2 (further) --- improvement, and where the missing modes went}
\label{ssec:B-improvement}

Two loose ends from \S\ref{ssec:B-first} and \S\ref{ssec:B-second}
deserve closing, because between them they explain what happened to the
degrees of freedom that Eq.~\eqref{eq:B-noether-ii} removed.

\paragraph{Improvement, and the Belinfante tensor.} The canonical
energy--momentum tensor of Eq.~\eqref{eq:B-emt} is in general neither
symmetric nor gauge invariant --- for electromagnetism it contains a term
$F^{\mu\lambda}\partial_\nu A_\lambda$, which is not. Both defects are
repaired by an improvement,
\begin{equation}
T^{\mu\nu}_{\text{Bel}}
 = T^{\mu\nu}_{\text{can}} + \partial_\lambda \Theta^{\lambda\mu\nu},
\qquad \Theta^{\lambda\mu\nu} = -\Theta^{\mu\lambda\nu} ,
\label{eq:B-belinfante}
\end{equation}
whose divergence vanishes identically by the argument of
problem~B.2. The improved tensor is symmetric, gauge invariant, and for
electromagnetism reduces to the familiar
$T^{\mu\nu} = F^{\mu\lambda}F^{\nu}{}_{\lambda}
 - \tfrac14 \eta^{\mu\nu}F^2$.

The moral is not that the canonical tensor is wrong. It is that
Noether's theorem determines a current only up to an identically
conserved piece, and that physics --- here, coupling to gravity, which
demands a symmetric source --- picks the representative. The freedom is
exactly the freedom of Chapter~\ref{sec:gauge}, one level down.

\paragraph{The constraint, and the counting.} Equation~%
\eqref{eq:B-noether-ii} said that one combination of the field equations
is not independent. In Hamiltonian language that appears as a
\emph{constraint}. For electromagnetism the momentum conjugate to $A_0$
is
\begin{equation}
\pi^0 = \frac{\partial \mathcal{L}}{\partial \dot A_0} = 0 ,
\label{eq:B-primary}
\end{equation}
a \emph{primary} constraint: it holds identically, with no reference to
the equations of motion. Requiring it to be preserved in time generates a
\emph{secondary} constraint, which is Gauss's law
$\partial_i \pi^i = 0$. The two constraints are first class, and each
removes one field and one momentum from the physical phase space.

Now the count, which is problem~B.6 done properly. The phase space has
$4$ fields and $4$ momenta, so $8$ functions per point. Two first-class
constraints remove $2 \times 2 = 4$. What survives is $4$ phase-space
functions, that is $2$ configuration degrees of freedom --- the two
polarizations of light.

\emph{Physics intuition.} The missing modes were never there. A local
symmetry does not delete degrees of freedom that a theory had; it
reveals that the description had more variables than the theory had
content, and Eq.~\eqref{eq:B-primary} is where the surplus announces
itself. This is Chapter~\ref{sec:gauge}'s redundancy reading in its
Hamiltonian form, and it is why \S\ref{ssec:elitzur} can say there is
nothing there to break.
\subsection{Step 3 --- the two theorems side by side}
\label{ssec:B-step3}
Chapter~\ref{sec:noether} needs the two theorems in one view, and
Table~\ref{tab:B-two-theorems} is that view: the same theory, the same
Lagrangian, and two entirely different outputs according to whether the
symmetry parameter is a constant or a function. The last two rows are
the ones the rest of the ladder leans on.

\begin{table}[H]
\centering
\caption[Noether's two theorems compared.]{The two theorems, on the same
theory. The last two rows are the ones Chapter~\ref{sec:noether} builds
on, and the empty cell is the historical starting point of the whole
subject. Two columns, one theorem each, compared row by row: the parameter each takes, the output each gives, whether it holds on shell or off, what it constrains, and then the electromagnetic and gravitational cases worked in both columns. Setting them side by side is the point, because the two are routinely quoted as one theorem and they do not say the same thing.}
\label{tab:B-two-theorems}\small
\begin{tabular}{@{}P{0.22\textwidth}P{0.33\textwidth}P{0.33\textwidth}@{}}
\toprule
 & \textbf{first theorem} & \textbf{second theorem} \\
\midrule
parameter & constants $\epsilon^A$, finitely many &
functions $\epsilon^A(x)$, infinitely many \\
output & a current per parameter,
Eq.~\eqref{eq:B-current} & an identity per function,
Eq.~\eqref{eq:B-noether-ii} \\
holds & on shell & off shell \\
what it constrains & the motion & the equations \\
electromagnetic case & $\partial_\mu j^\mu = 0$, charge conservation &
$\partial_\mu\partial_\nu F^{\mu\nu} \equiv 0$ \\
gravitational case & --- (no covariant local current) &
$\nabla_\mu G^{\mu\nu} \equiv 0$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Problems and worked solutions}
\label{ssec:B-problems}

\medskip\noindent\textbf{B.1. Derive Eq.~\eqref{eq:B-energy} for
$\mathcal{L} = \tfrac12 (\partial\phi)^2 - V(\phi)$ and confirm that the
conserved density is $T + V$ rather than $T - V$.}

\noindent\emph{Solution.} \emph{Step 1.}
$\partial\mathcal{L}/\partial\dot\phi = \dot\phi$. \emph{Step 2.}
Eq.~\eqref{eq:B-energy} gives $j^0 = \dot\phi^2 - [\tfrac12\dot\phi^2 -
\tfrac12(\nabla\phi)^2 - V] = \tfrac12\dot\phi^2 + \tfrac12(\nabla\phi)^2
+ V$. \emph{Step 3.} Every sign in the potential has flipped relative to
$\mathcal{L}$. \emph{Physics intuition:} the Legendre transform in
Eq.~\eqref{eq:B-energy} is exactly the operation that turns a Lagrangian
into a Hamiltonian, and Noether's theorem has rediscovered it without
being told to.

\medskip\noindent\textbf{B.2. Show that adding
$\partial_\nu \Theta^{\mu\nu}$ with $\Theta$ antisymmetric leaves both
$\partial_\mu j^\mu$ and the total charge unchanged.}

\noindent\emph{Solution.} \emph{Step 1.} The divergence is unchanged
because $\partial_\mu\partial_\nu \Theta^{\mu\nu} = 0$ by symmetry
against antisymmetry --- the same argument as
Eq.~\eqref{eq:A-ddzero}. \emph{Step 2.} The charge changes by
$\int \dd^3x\, \partial_i \Theta^{0i}$, a total spatial divergence.
\emph{Step 3.} That integral vanishes for fields falling off at
infinity. \emph{Physics intuition:} the improvement freedom is the
current-level version of gauge freedom, and it is why one may symmetrize
the energy--momentum tensor without changing any physics.

\medskip\noindent\textbf{B.3. A complex scalar has a global $U(1)$ and,
coupled to electromagnetism, a local one. Compute what each yields, and
state in one sentence why they differ.}

\noindent\emph{Solution.} \emph{Step 1.} Global: $\Delta = \ii\phi$,
$K^\mu = 0$, so Eq.~\eqref{eq:B-current} gives
Eq.~\eqref{eq:scalar-current}, a current whose integral is the charge.
\emph{Step 2.} Local: the transformation acquires the
$\partial_\mu\epsilon$ piece of Eq.~\eqref{eq:B-local-transf} acting on
$A_\mu$, and the second theorem applies, giving
Eq.~\eqref{eq:B-noether-ii}. \emph{Step 3.} The difference is that an
arbitrary function may be taken of compact support, which is what forces
the bracket in Eq.~\eqref{eq:B-second-derivation} to vanish pointwise.
\emph{Physics intuition:} a constant parameter can only tell you
something about the total; a free function tells you something at every
point, and something true at every point about arbitrary configurations
is an identity.

\medskip\noindent\textbf{B.4. Show that general covariance plus the
second theorem gives $\nabla_\mu G^{\mu\nu} \equiv 0$, and explain why
this means gravitational energy is not localizable.}

\noindent\emph{Solution.} \emph{Step 1.} Under
$\delta g_{\mu\nu} = \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu$ the
Einstein--Hilbert action is invariant for arbitrary $\xi^\mu(x)$.
\emph{Step 2.} The Euler--Lagrange expression is $G^{\mu\nu}$, and
Eq.~\eqref{eq:B-noether-ii} with $R_{(1)}$ read off the transformation
gives $\nabla_\mu G^{\mu\nu} \equiv 0$. \emph{Step 3.} Because this is an
identity rather than a conservation law, there is no independent current
whose density could be called the local energy of the field.
\emph{Physics intuition:} one can always choose coordinates in which the
gravitational field vanishes at a point, so any local energy density
could be transformed away --- and the second theorem is the formal
statement of that fact.

\medskip\noindent\textbf{B.5. Why does the standard textbook derivation
of Noether's theorem, which discards the surface term, make the second
theorem invisible?}

\noindent\emph{Solution.} \emph{Step 1.} The standard derivation sets
$\delta\phi = 0$ on the boundary and drops the second term of
Eq.~\eqref{eq:B-variation}. \emph{Step 2.} But the second theorem is
obtained by \emph{keeping} arbitrary $\epsilon(x)$, including
configurations of compact support, and reading off the coefficient of
$\epsilon$ --- Eq.~\eqref{eq:B-second-derivation}. \emph{Step 3.} A
derivation that has already restricted the variations cannot see the
consequence of allowing all of them. \emph{Physics intuition:} the
discarded term is not a technicality; it is where the information about
local symmetry lives, and \S\ref{ssec:textbook-stop} argues that
discarding it is what makes the rest of the ladder look like a series of
exceptions.

\medskip\noindent\textbf{B.6. Count the independent field equations for
$A_\mu$ in four dimensions, and account for the discrepancy with the
number of propagating modes.}

\noindent\emph{Solution.} \emph{Step 1.} Four equations $E^\nu = 0$, one
per component. \emph{Step 2.} Eq.~\eqref{eq:B-noether-ii} removes one:
the divergence of the equations vanishes identically, so only three are
independent. \emph{Step 3.} Of the four components of $A_\mu$, one is
pure gauge and one is fixed by a constraint rather than evolved, leaving
$4 - 1 - 1 = 2$ propagating polarizations. \emph{Physics intuition:} the
identity and the redundancy remove one degree of freedom each, and they
are two faces of the same local symmetry --- which is exactly
\S\ref{ssec:gauge-intuition}'s claim that redundancy is what reconciles
locality with unitarity.

\GGtakeaway{\emph{Standard result:} Noether's first theorem, one current
per constant parameter, conserved on shell. \emph{What it assumed:}
that the symmetry parameters are constants, and --- in the usual
derivation --- that the surface term may be discarded. \emph{What the
second theorem adds:} for symmetries with arbitrary functions, no current
at all, but an identity among the field equations that holds off shell
(Eq.~\eqref{eq:B-noether-ii}); electromagnetism and general relativity
are the two standard instances, and the second is the problem Noether was
actually asked to solve. \emph{Continue in:}
Appendix~\ref{app:connections} for the geometry behind the local case,
and Chapter~\ref{sec:noether} for what the distinction does to the rest
of the ladder.}


\section{Connections, curvature, and holonomy}
\label{app:connections}
\GGteachbox{Parallel transport, worked on the sphere until the answer is
obvious; the covariant derivative as the bookkeeping that makes
comparison at a distance possible; curvature as the failure of transport
to commute, derived as a commutator; holonomy, and the theorem that
curvature does \emph{not} determine it; the non-abelian case and the
Wilson loop; Aharonov--Bohm as the experiment that makes the third datum
physical; and the contracted second Bianchi identity, derived, which
\S\ref{ssec:einstein} quotes. The pay-off is
Chapters~\ref{sec:gauge}, \ref{sec:bianchi} and \S\ref{ssec:third-datum}.}

\GGreadalong{\citet{Nakahara2003}, Ch.~7 and~10, for connections on
fiber bundles; \citet{MisnerThorneWheeler1973}, Ch.~8 and~11, for the
gravitational case with the geometry drawn;
\citet{WuYang1975} for the statement that the holonomies \emph{are} the
gauge field.}

\subsection{Step 1 --- parallel transport, on a sphere}
\label{ssec:C-step1}

Take a vector tangent to the Earth at the north pole, pointing along the
Greenwich meridian. Carry it south to the equator, keeping it as
parallel to itself as the surface permits --- never rotating it about the
local normal. Turn and carry it a quarter of the way round the equator.
Then carry it back north to the pole.

It has rotated by ninety degrees.

Nothing was done to it. At every step it was transported as parallel as
the surface allowed, and it came back turned. The rotation is not a
property of the vector, of the path length, or of the traveler; it is a
property of the \emph{surface}, and it is the cleanest possible
demonstration that on a curved space there is no observer-independent way
to compare vectors at different points.

\paragraph{Statement.} On the unit sphere, parallel transport around a
closed loop rotates a tangent vector by an angle equal to the
\emph{solid angle} enclosed by the loop:
\begin{equation}
\Delta\varphi \;=\; \Omega_{\text{enclosed}}
 \;=\; \oint \left(1 - \cos\theta\right)\dd\phi .
\label{eq:C-solid-angle}
\end{equation}
For the polar cap of half-angle $\theta$ the enclosed solid angle is
$2\pi(1 - \cos\theta)$, and for the octant walked above it is
$4\pi/8 = \pi/2$ --- the ninety degrees, obtained without any computation
beyond dividing a sphere into eight equal pieces.

\emph{Physics intuition.} Eq.~\eqref{eq:C-solid-angle} is worth
memorizing because it is the whole subject in miniature: a
\emph{local} quantity (the curvature, here constant and equal to one)
integrated over a region produces a \emph{global} observable (the
rotation) attached to the region's boundary. Every result in this
appendix, and most of Chapter~\ref{sec:hs}, is a version of that
sentence.

\subsection{Step 1 (continued) --- the covariant derivative}
\label{ssec:C-covderiv}

To differentiate a vector field one must subtract vectors at neighboring
points, and the transport rule above says that subtraction is not
defined until a rule for comparison is supplied. That rule is the
\emph{connection}. Writing it in components,
\begin{equation}
\nabla_\mu V^\rho
 = \partial_\mu V^\rho + \Gamma^{\rho}{}_{\mu\sigma} V^\sigma ,
\label{eq:C-covderiv}
\end{equation}
where the second term is the correction that undoes the change of basis.
For a metric-compatible torsion-free connection the coefficients are
determined by the metric,
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
 = \tfrac12 g^{\rho\sigma}
   \left(\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu}
        - \partial_\sigma g_{\mu\nu}\right) .
\label{eq:C-christoffel}
\end{equation}

The gauge case of Chapter~\ref{sec:gauge} is
Eq.~\eqref{eq:C-covderiv} with the internal index in place of the
spacetime one and $-\ii q A_\mu$ in place of $\Gamma$: the covariant
derivative $D_\mu = \partial_\mu - \ii q A_\mu$ of
Eq.~\eqref{eq:covderiv} is a connection, and $A_\mu$ is its
coefficient. The two constructions are not analogous; they are the same
construction on different bundles, which is the observation
\GGEP{} (\S\ref{ssec:ggep-stated}) elevates to an axiom.

\subsection{Step 1 (continued) --- curvature as a commutator}
\label{ssec:C-curvature}

\paragraph{Statement.} The failure of covariant derivatives to commute is
tensorial, and defines the curvature:
\begin{equation}
[\nabla_\mu, \nabla_\nu]\,V^\rho = R^{\rho}{}_{\sigma\mu\nu}\,V^\sigma ,
\qquad
R^{\rho}{}_{\sigma\mu\nu}
 = \partial_\mu \Gamma^{\rho}{}_{\nu\sigma}
 - \partial_\nu \Gamma^{\rho}{}_{\mu\sigma}
 + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma}
 - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma} .
\label{eq:C-riemann}
\end{equation}
In the gauge case the same computation gives
\begin{equation}
[D_\mu, D_\nu] = -\ii q F_{\mu\nu} ,
\label{eq:C-Fcommutator}
\end{equation}
which is Eq.~\eqref{eq:F}, and in the non-abelian case
Eq.~\eqref{eq:H-nonabelian} with its commutator term.

The physical reading is the sphere again: transporting around an
infinitesimal closed loop in the $\mu\nu$ plane returns the vector
rotated by the curvature times the enclosed area. Curvature \emph{is}
infinitesimal holonomy.

\paragraph{The Bianchi identities, derived.} Antisymmetrizing
Eq.~\eqref{eq:C-riemann} over its last three indices and using the
Jacobi identity for the commutator gives the first (algebraic) identity
$R^{\rho}{}_{[\sigma\mu\nu]} = 0$; applying $\nabla$ and antisymmetrizing
gives the second,
\begin{equation}
\nabla_{[\lambda} R_{\mu\nu]\rho\sigma} = 0 ,
\label{eq:C-bianchi2}
\end{equation}
which is Eq.~\eqref{eq:bianchi-2}. Contracting twice with the metric and
using the symmetries of the Riemann tensor produces
\begin{equation}
\nabla_\mu \left(R^{\mu\nu} - \tfrac12 g^{\mu\nu} R\right)
 \;=\; \nabla_\mu G^{\mu\nu} \;\equiv\; 0 ,
\label{eq:C-divG}
\end{equation}
Eq.~\eqref{eq:div-G}. The derivation used the metric only to raise
indices; the identity itself descends from the Jacobi identity for
covariant derivatives, which is why \S\ref{ssec:not-a-law} classifies it
with $\dd\dd = 0$ rather than with a field equation.

\subsection{Step 1 (concluded) --- holonomy, and what curvature omits}
\label{ssec:C-holonomy}

For a closed loop $\gamma$ the total effect of transport is the
\emph{holonomy}. In the abelian case it is a phase,
\begin{equation}
\mathrm{Hol}(\gamma) = \exp\!\left(\ii q \oint_\gamma A\right) ,
\label{eq:C-holonomy}
\end{equation}
and in the non-abelian case a group element given by the path-ordered
exponential, whose trace is the Wilson loop:
\begin{equation}
W(\gamma) = \mathrm{Tr}\,
 \mathcal{P}\exp\!\left(\ii g \oint_\gamma A_\mu \dd x^\mu\right) .
\label{eq:C-wilson}
\end{equation}

\paragraph{Statement (the point of the appendix).} Curvature does not
determine holonomy. If $\gamma$ bounds a surface $\Sigma$ then Stokes'
theorem gives $\oint_\gamma A = \int_\Sigma F$ and the holonomy is fixed
by the curvature. If $\gamma$ does not bound --- because the space has a
hole, or a compact direction --- then no such relation exists, and two
configurations with identical curvature everywhere can have different
holonomies.

That is a mathematical statement with an experimental consequence, and
\S\ref{ssec:C-ab} is the experiment.

\subsection{Step 2 --- Aharonov--Bohm, and what the formalism assumes}
\label{ssec:C-ab}

Confine a magnetic field entirely inside a long solenoid and send an
electron beam around both sides. Outside the solenoid $F = 0$ on every
path the electron takes. The relative phase of the two partial beams is
nonetheless
\begin{equation}
\Delta\varphi = \frac{q}{\hbar}\oint_\gamma A\cdot \dd\boldsymbol{\ell}
 = \frac{q\,\Phi}{\hbar} ,
\label{eq:C-ab}
\end{equation}
periodic in the enclosed flux with period $h/q$
\citep{AharonovBohm1959}. The effect is observed, at the predicted size.

Three things follow, and all three are used in Part~II.

\emph{The potential is not merely a calculational device.} A description
in terms of $F$ alone is incomplete, since $F$ vanishes where the physics
happens.

\emph{But the potential is not observable either.} Only
Eq.~\eqref{eq:C-holonomy} is, and it is invariant under
$A \to A + \dd\Lambda$ provided $\Lambda$ is single-valued.

\emph{So the honest object is the holonomy.} \citet{WuYang1975} state
this as a structural principle: the complete gauge-invariant description
of a gauge field is the collection of its non-integrable phase factors.
Neither $A$ (too much) nor $F$ (too little) is that description.

\paragraph{What is assumed.} That the bundle is smooth and the loop
piecewise smooth; that the transport is unitary, which is what makes
Eq.~\eqref{eq:C-holonomy} a phase rather than a general complex number;
and, for the gravitational case, that the connection is metric-compatible
and torsion-free --- a choice, not a theorem, and one that
Eq.~\eqref{eq:C-christoffel} encodes.


\subsection{Step 2 (further) --- bundles, transition functions, and clutching}
\label{ssec:C-bundles}

The two-cap argument of \S\ref{ssec:dirac} used, without naming it, the
machinery that makes ``a gauge field'' a global object rather than a
local one. It is worth naming, because Chapter~\ref{sec:hs} needs it.

\paragraph{Statement.} A principal $G$-bundle over $X$ is specified by an
open cover $\{U_\alpha\}$ together with transition functions
$g_{\alpha\beta} : U_\alpha \cap U_\beta \to G$ satisfying the cocycle
condition
\begin{equation}
g_{\alpha\beta}\, g_{\beta\gamma}\, g_{\gamma\alpha} = e
\quad \text{on } U_\alpha \cap U_\beta \cap U_\gamma .
\label{eq:C-cocycle}
\end{equation}
A connection is a collection of local one-forms $A_\alpha$ agreeing on
overlaps up to a gauge transformation by $g_{\alpha\beta}$; the curvature
$F$ is globally defined even though $A$ is not.

\paragraph{Clutching, and where the integer lives.} For $X = S^2$ take
two caps overlapping in a band about the equator. The bundle is then
determined by a single transition function on the band, and since the
band retracts to a circle, by a map $S^1 \to G$. For $G = U(1)$ such maps
are classified up to deformation by their winding number,
\begin{equation}
\pi_1(U(1)) = \mathbb{Z} ,
\label{eq:C-pi1}
\end{equation}
and \emph{that} integer is Dirac's $n$. The construction --- build a
bundle by gluing two trivial pieces with a map on the overlap --- is
called clutching, and Eq.~\eqref{eq:C-pi1} is why the monopole charge is
an integer and not a real number.

Note the shape of the argument, because it recurs at the top of the
ladder. Nothing was quantized by a dynamical mechanism. A quantity was
identified with a homotopy class, and homotopy classes of maps into
$U(1)$ happen to be integers. \S\ref{ssec:top-of-ladder}'s claim that the
top rung has no dynamics in it is, in this instance, the observation that
Eq.~\eqref{eq:C-pi1} is a fact about circles.

\paragraph{The non-abelian case, briefly.} For $G = SU(2)$ over $S^4$
the relevant group is $\pi_3(SU(2)) = \mathbb{Z}$, and the integer is the
instanton number $\frac{1}{8\pi^2}\int \mathrm{Tr}\,F\wedge F$. The same
pattern: a topological classification, an integer, and an action that
cannot be lowered continuously below the corresponding bound. That bound
is the Bogomolnyi bound \citep{Bogomolny1976}, and the tolls of \S\ref{ssec:solvency} are
instances of it.

\emph{Physics intuition.} A bundle is what you get when a field cannot be
described by one formula everywhere and you are honest about it. The
transition functions are the bookkeeping of the honesty, and
Eq.~\eqref{eq:C-cocycle} is the requirement that the bookkeeping be
consistent on triple overlaps --- the same kind of consistency
requirement that, three chapters later, produces an integer with no
symmetry in sight.
\subsection{Step 3 --- what the body does with it}
\label{ssec:C-step3}

Chapter~\ref{sec:gauge} needs Eq.~\eqref{eq:C-covderiv} and
Eq.~\eqref{eq:C-Fcommutator}: the connection is forced by locality, and
the curvature is what is observable. Chapter~\ref{sec:bianchi} needs
Eq.~\eqref{eq:C-divG}: the identity that forces the shape of Einstein's
equations. Chapter~\ref{sec:hs} needs
\S\ref{ssec:C-holonomy}: that holonomy is a third datum, independent of
curvature wherever loops fail to bound. And Chapter~\ref{sec:oggep}
needs the observation that a holonomy is evaluated on a \emph{loop} ---
that is, on the boundary of a region rather than inside it --- which is
why Figure~\ref{fig:seam} puts it in the third row.

\subsection{Problems and worked solutions}
\label{ssec:C-problems}

\medskip\noindent\textbf{C.1. Verify Eq.~\eqref{eq:C-solid-angle} for the
octant, and for a loop of constant latitude $\theta$.}

\noindent\emph{Solution.} \emph{Step 1.} The octant is one eighth of the
sphere, so $\Omega = 4\pi/8 = \pi/2$: a quarter turn, matching the
walk described in \S\ref{ssec:C-step1}. \emph{Step 2.} For constant
latitude, $\oint(1-\cos\theta)\dd\phi = 2\pi(1-\cos\theta)$.
\emph{Step 3.} As $\theta \to 0$ this goes to zero like $\pi\theta^2$,
the enclosed area --- curvature times area, as
\S\ref{ssec:C-curvature} says. \emph{Physics intuition:} a small loop
sees a flat world; the holonomy is second order in the loop size, which
is why local physics never notices curvature.

\medskip\noindent\textbf{C.2. Show that $F_{\mu\nu}$ defined by
Eq.~\eqref{eq:C-Fcommutator} is gauge covariant in the non-abelian case
but not gauge invariant, and say what is invariant instead.}

\noindent\emph{Solution.} \emph{Step 1.} Under $A_\mu \to U A_\mu U^{-1}
+ \tfrac{\ii}{g}U\partial_\mu U^{-1}$ one finds
$F_{\mu\nu} \to U F_{\mu\nu} U^{-1}$. \emph{Step 2.} That is covariance,
not invariance: $F$ changes unless $U$ commutes with it.
\emph{Step 3.} Traces such as $\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$, and
the Wilson loop Eq.~\eqref{eq:C-wilson}, are invariant.
\emph{Physics intuition:} in the abelian case the group is
one-dimensional and covariance collapses to invariance, which is why
electromagnetism lets one treat $F$ itself as the observable and
Yang--Mills does not.

\medskip\noindent\textbf{C.3. Derive Eq.~\eqref{eq:C-divG} from
Eq.~\eqref{eq:C-bianchi2}.}

\noindent\emph{Solution.} \emph{Step 1.} Write
Eq.~\eqref{eq:C-bianchi2} out as three cyclic terms.
\emph{Step 2.} Contract with $g^{\lambda\rho}g^{\mu\sigma}$; the
symmetries of the Riemann tensor turn two of the three terms into Ricci
tensors and the third into the Ricci scalar, giving
$2\nabla_\mu R^{\mu\nu} - \nabla^\nu R = 0$. \emph{Step 3.} Rearranging,
$\nabla_\mu(R^{\mu\nu} - \tfrac12 g^{\mu\nu}R) = 0$.
\emph{Physics intuition:} the factor of $\tfrac12$ in the Einstein
tensor is not a choice --- it is whatever the contraction produced, and
it is the reason $G^{\mu\nu}$ rather than $R^{\mu\nu}$ appears on the
left of the field equations.

\medskip\noindent\textbf{C.4. Two configurations outside a solenoid have
the same $F$ everywhere but fluxes differing by $h/q$. Are they
physically distinguishable?}

\noindent\emph{Solution.} \emph{Step 1.} By Eq.~\eqref{eq:C-ab} the
phases differ by $2\pi$. \emph{Step 2.} A phase of $2\pi$ is no phase at
all, so the interference patterns coincide. \emph{Step 3.} They are
\emph{not} distinguishable --- which is the content of the flux quantum.
\emph{Physics intuition:} the observable is not the flux but the flux
modulo $h/q$, and the integer counting how many quanta have been passed
is exactly the integer of \S\ref{ssec:dirac}.

\medskip\noindent\textbf{C.5. On a space where every loop bounds, show
that holonomy carries no information beyond curvature. What is the
topological name for such a space?}

\noindent\emph{Solution.} \emph{Step 1.} If $\gamma = \partial\Sigma$
then Stokes gives $\oint_\gamma A = \int_\Sigma F$. \emph{Step 2.} So the
holonomy of every loop is determined by $F$. \emph{Step 3.} A space in
which every loop bounds is simply connected --- first homotopy group
trivial, hence first de Rham cohomology trivial.
\emph{Physics intuition:} the third datum is a statement about the
\emph{shape} of the space, not about the field, which is why
Chapter~\ref{sec:oggep} can argue that a theory formulated on a closed
simply connected world would never have needed it.

\medskip\noindent\textbf{C.6. Why does \S\ref{ssec:ggep-buys} say that
\GGEP{} collapses two rungs into one? Answer using
Eqs.~\eqref{eq:C-riemann} and \eqref{eq:C-Fcommutator}.}

\noindent\emph{Solution.} \emph{Step 1.} Both curvatures are commutators
of the same covariant derivative, differing only in which bundle the
index lives on. \emph{Step 2.} Both therefore satisfy a Bianchi identity,
by the same Jacobi argument. \emph{Step 3.} If the two connections are
components of one connection, there is one curvature and one identity.
\emph{Physics intuition:} Part~II told the story twice because textbooks
teach the two bundles separately; the axiom asserts that the repetition
was the artifact.

\GGtakeaway{\emph{Standard result:} connections, curvature as the
commutator of covariant derivatives, holonomy as the transport around a
loop, and the two Bianchi identities. \emph{What it assumed:}
metric compatibility and vanishing torsion in the gravitational case;
smoothness and unitarity throughout. \emph{The reading this paper takes:}
curvature determines holonomy only where loops bound, so the holonomy is
an independent third datum --- and it is the datum evaluated on a
boundary rather than in a region, which is why it survives into
Part~III as the thing an open system cannot discard. \emph{Continue in:}
Appendix~\ref{app:diffcohom}, which assembles all three data into one
object, and \S\ref{ssec:third-datum}.}


\section{Wilson lines and the one-loop phase potential}
\label{app:wilson}
\GGteachbox{A constant gauge potential along a compact direction whose
field strength vanishes identically and which nonetheless cannot be
gauged away; the Kaluza--Klein tower whose masses move with it, which is
what makes it physical; the mode sum resummed by hand, exhibiting the
phase-dependence as exponentially suppressed in the ultraviolet and the
potential as finite with no regulator; the Coleman--Weinberg potential
that results, and the sign that decides where it minimizes; and the
center symmetry, which is the Elitzur-compatible sense in which a Wilson
line breaks anything at all. The pay-off is
Chapter~\ref{sec:wilson} and \S\ref{ssec:third-datum}.}

\GGreadalong{\citet{PeskinSchroeder1995}, Ch.~11, for the effective
action and the one-loop effective potential; \citet{Nakahara2003},
Ch.~10, for holonomy over a base that is not simply connected;
\citet{ChengLi1984}, Ch.~4, for the group theory of \SU(N) used in
\S\ref{ssec:D-centre}; then the primary literature ---
\citet{Manton1979} and \citet{Hosotani1983} for the mechanism, and
\citet{Hosotani1989} for the systematic treatment. The last two are
papers, not textbooks, and are read after the first three.}

\subsection{Step 1 --- the flat connection, and why it is not pure gauge}
\label{ssec:D-flat}

Take a five-dimensional spacetime $M_4 \times S^1$ with coordinates
$x^M = (x^\mu, y)$, $\mu = 0,1,2,3$, flat metric, and the fifth
coordinate periodic,
\begin{equation}
y \;\sim\; y + L .
\label{eq:D-circle}
\end{equation}
On it put a gauge field that is as boring as a gauge field can be
without being zero: every component vanishes except the one along the
circle, and that one is constant,
\begin{equation}
A_\mu = 0 \quad (\mu = 0,1,2,3), \qquad
A_y = \frac{\theta}{gL} = \text{constant} ,
\label{eq:D-Ay}
\end{equation}
with $g$ the gauge coupling and $\theta$ a real number, introduced here
as a parameter and shown below to be an observable. For the abelian case
$A_y$ is a number; for $\SU(N)$ take
$A_y = \theta\, T/(gL)$ with $T$ one fixed generator, and every
statement below goes through with $\theta$ replaced by the eigenvalues of
$\theta T$.

\paragraph{The field strength.} Compute every component of
\begin{equation}
F_{MN} = \partial_M A_N - \partial_N A_M + \ii g\left[A_M, A_N\right]
\label{eq:D-Fdef}
\end{equation}
in turn. For $M,N$ both among $0,1,2,3$: all three terms contain $A_\mu$,
which vanishes. For $M = \mu$, $N = y$: the first term is
$\partial_\mu A_y = 0$ because $A_y$ does not depend on $x^\mu$; the
second is $\partial_y A_\mu = 0$ because $A_\mu = 0$; the third is
$[A_\mu, A_y] = 0$ for the same reason. For $M = N$ the antisymmetry of
Eq.~\eqref{eq:D-Fdef} gives zero. In the non-abelian case the one
remaining possibility, $[A_y, A_y]$, vanishes because a matrix commutes
with itself. Hence
\begin{equation}
F_{MN} = 0 \qquad \text{for every } M,N \text{, at every point.}
\label{eq:D-Fzero}
\end{equation}
The configuration is \emph{flat}: it carries no field strength anywhere.

\paragraph{The Wilson line.} The holonomy around the circle, in the
notation of Eq.~\eqref{eq:C-holonomy}, is
\begin{equation}
W \;=\; \exp\!\left(\ii g \oint_{S^1} A_y \,\dd y\right)
  \;=\; \exp\!\left(\ii g \cdot \frac{\theta}{gL}\cdot L\right)
  \;=\; e^{\ii\theta} .
\label{eq:D-wilsonline}
\end{equation}
The coupling and the circumference both cancel, which is the first sign
that $\theta$ and not $A_y$ is the quantity of interest.

\paragraph{The gauge transformation that would remove it.} Use the
convention of Problem~C.2, under which a gauge transformation with group
element $\Omega$ acts as
\begin{equation}
A_M \;\longrightarrow\; \Omega A_M \Omega^{-1}
 + \frac{\ii}{g}\,\Omega\,\partial_M \Omega^{-1} .
\label{eq:D-gaugeact}
\end{equation}
Take
\begin{equation}
\Omega(y) = \exp\!\left(-\,\frac{\ii\,\theta\, y}{L}\right) .
\label{eq:D-gaugefn}
\end{equation}
Then $\Omega^{-1} = e^{\ii\theta y/L}$,
$\partial_y \Omega^{-1} = (\ii\theta/L)\,e^{\ii\theta y/L}$, so
$\Omega\,\partial_y\Omega^{-1} = \ii\theta/L$ and the second term of
Eq.~\eqref{eq:D-gaugeact} equals $-\theta/(gL)$. It cancels
Eq.~\eqref{eq:D-Ay} exactly:
\begin{equation}
A_y \;\longrightarrow\; \frac{\theta}{gL} - \frac{\theta}{gL} \;=\; 0 .
\label{eq:D-removed}
\end{equation}
So the configuration looks like pure gauge, and by every local test it
is: on any interval of the circle short of the whole, $\Omega$ is a
perfectly good gauge function and $A_y$ is a perfectly good pure gauge.

\paragraph{Statement (the point of the subsection).} A gauge
transformation must be single-valued on the space it acts on, because
the fields it acts on are. Eq.~\eqref{eq:D-gaugefn} obeys
$\Omega(y+L) = e^{-\ii\theta}\,\Omega(y)$, so
\begin{equation}
\Omega(y+L) = \Omega(y)
\qquad\Longleftrightarrow\qquad
e^{-\ii\theta} = 1
\qquad\Longleftrightarrow\qquad
\theta \in 2\pi\mathbb{Z} .
\label{eq:D-single}
\end{equation}
For $\theta$ not a multiple of $2\pi$ the only function that removes
$A_y$ is not a gauge transformation. The field strength vanishes
everywhere, Eq.~\eqref{eq:D-Fzero}, and the configuration is
\emph{not} pure gauge. What is left over is Eq.~\eqref{eq:D-wilsonline},
and it is gauge invariant.

\paragraph{Why the residue is invariant.} Let $\Omega = e^{\ii\alpha(y)}$
be any \emph{single-valued} abelian gauge transformation. Then
Eq.~\eqref{eq:D-gaugeact} reads $A_y \to A_y + (1/g)\,\partial_y\alpha$,
and Eq.~\eqref{eq:D-wilsonline} picks up the factor
$\exp(\ii\oint \dd\alpha)$. But a single-valued $\alpha$ returns to
itself around the circle, so
\begin{equation}
\oint_{S^1} \dd\alpha \;=\; \alpha(L) - \alpha(0) \;=\; 0 ,
\label{eq:D-loopdlambda}
\end{equation}
and $W \to W$. Every single-valued gauge transformation leaves
Eq.~\eqref{eq:D-wilsonline} alone. Multi-valued ones shift $\theta$ by
multiples of $2\pi$, which by Eq.~\eqref{eq:D-single} is the statement
that $\theta$ is defined modulo $2\pi$ and not more finely.

This is Appendix~\ref{app:connections} in its sharpest form. Curvature
determines holonomy only where loops bound (\S\ref{ssec:C-holonomy}); the
circle of Eq.~\eqref{eq:D-circle} does not bound; so the holonomy is free
of the curvature, and here it is the \emph{only} thing there is.

\emph{Physics intuition.} A gauge transformation is a change of
bookkeeping, and bookkeeping has to close. The function
Eq.~\eqref{eq:D-gaugefn} is a change of bookkeeping that fails to close
by $e^{-\ii\theta}$ after one lap, and the failure to close is not a
defect of the function but a property of the space. Every result in this
appendix is a consequence of that one sentence.

\subsection{Step 1 (continued) --- the Kaluza--Klein tower}
\label{ssec:D-tower}

A phase that no local measurement can see is worth little. This
subsection shows that $\theta$ moves a spectrum, which is as physical as
a quantity gets.

Put a complex scalar $\Phi(x,y)$ of charge $q$ on the same space,
periodic around the circle, and expand it in the modes that periodicity
allows:
\begin{equation}
\Phi(x,y) \;=\; \frac{1}{\sqrt{L}}
 \sum_{k\in\mathbb{Z}} \phi_k(x)\, e^{2\pi \ii k y / L} .
\label{eq:D-modes}
\end{equation}
The covariant derivative along the circle is
$D_y = \partial_y + \ii q g A_y$, so with Eq.~\eqref{eq:D-Ay}
\begin{equation}
D_y \Phi
 \;=\; \frac{\ii}{L\sqrt{L}} \sum_{k\in\mathbb{Z}}
 \left(2\pi k + q\theta\right) \phi_k(x)\, e^{2\pi\ii k y/L} .
\label{eq:D-Dy}
\end{equation}
Integrating the five-dimensional kinetic term
$|D_M\Phi|^2$ over $y$ makes the modes orthonormal and leaves, for each
$k$, a four-dimensional kinetic term plus a mass term with
\begin{equation}
m_k \;=\; \frac{\left|\,2\pi k + q\theta\,\right|}{L} ,
\qquad k \in \mathbb{Z} .
\label{eq:D-mass}
\end{equation}
Three properties of Eq.~\eqref{eq:D-mass} carry the whole argument.

\paragraph{(a) The spectrum moves with $\theta$, so $\theta$ is
physical.} Differentiating, $\partial m_k/\partial\theta = \pm q/L \ne 0$
for $q \ne 0$: every level shifts, and the shifts are not a common
translation, since the levels approach and cross as $\theta$ runs. A
quantity that changes a mass spectrum is measurable in principle by
measuring the spectrum. No appeal to the potential $A_y$ is needed, and
none is legitimate --- $A_y$ is not gauge invariant, whereas
Eq.~\eqref{eq:D-mass} is.

\paragraph{(b) Physics is $2\pi$ periodic in $\theta$.} Send
$\theta \to \theta + 2\pi$ in Eq.~\eqref{eq:D-mass}:
\begin{equation}
2\pi k + q(\theta + 2\pi)
 \;=\; 2\pi\,(k + q) + q\theta .
\label{eq:D-shift}
\end{equation}
For integer $q$ the map $k \mapsto k+q$ is a bijection of $\mathbb{Z}$,
so the \emph{set} of masses is unchanged; only the labels move. For unit
charge the relabeling is $k \to k+1$, level by level. The spectrum after
the shift is the spectrum before it, so nothing measurable distinguishes
$\theta$ from $\theta + 2\pi$. This is Eq.~\eqref{eq:D-single} again,
arrived at from the matter sector instead of the gauge sector, and it is
why $\theta$ lives on a circle rather than a line. (If some field carried
a charge that was not an integer multiple of the unit charge, the period
would be correspondingly larger; the statement as given assumes integer
charges.)

\paragraph{(c) The physical range is half the circle.} Send
$\theta \to -\theta$. Then $2\pi k - q\theta = -(2\pi(-k) + q\theta)$,
and taking the absolute value in Eq.~\eqref{eq:D-mass} together with the
relabeling $k \to -k$ returns the original set. The spectrum is even,
so $\theta$ and $-\theta$ are indistinguishable and the fundamental
domain is
\begin{equation}
0 \;\le\; \theta \;\le\; \pi .
\label{eq:D-domain}
\end{equation}
The two endpoints, $W = +1$ and $W = -1$ in Eq.~\eqref{eq:D-wilsonline},
are the two points fixed by the reflection, and \S\ref{ssec:D-potential}
finds the minimum at one or the other.

\paragraph{Aharonov--Bohm, said again.} Eq.~\eqref{eq:D-mass} is the
Aharonov--Bohm effect of \S\ref{ssec:C-ab} with the interferometer
replaced by the compact dimension. There, a charge $q$ encircling a
solenoid acquired the phase $q\Phi/\hbar$ from a region in which
$F = 0$, and the interference pattern was periodic in the enclosed flux
with period $h/q$. Here, a charge $q$ propagating once around the circle
acquires the phase $q\theta$ from a configuration in which $F = 0$
everywhere, and the spectrum is periodic in $\theta$ with period
$2\pi/q$ in the same sense --- Eq.~\eqref{eq:D-shift} is the flux
quantum written in dimensionless variables. The difference is not one of
principle but of geography: the solenoid makes the loop fail to bound by
removing a region of space, and the compactification makes it fail to
bound by identifying two points. \citet{WuYang1975} is the statement that
in both cases the honest object is the phase factor.

\emph{Physics intuition.} The tower is a ruler and $\theta$ slides it.
The ruler's markings are $2\pi/L$ apart, and sliding it by a whole
marking is not sliding it at all --- which is why the observable is an
angle and why an angle is exactly what a Wilson line returns.

\subsection{Step 1 (concluded) --- the resummation, by hand}
\label{ssec:D-resum}

Any one-loop quantity built from Eq.~\eqref{eq:D-mass} is a sum over
$k \in \mathbb{Z}$, and every such sum diverges. The next subsection
needs to know \emph{where} the divergence sits, and the answer is the
content of one identity, derived here in full:
\begin{equation}
\sum_{k\in\mathbb{Z}} \frac{1}{(k+b)^2 + x^2}
 \;=\; \frac{\pi \,\sinh 2\pi x}
 {x\left(\cosh 2\pi x - \cos 2\pi b\right)} ,
\qquad x > 0 ,
\label{eq:D-resum}
\end{equation}
in which $b$ will be $q\theta/2\pi$ and $x$ a dimensionless mass.

\paragraph{Step (i): factorize and split.} Over the complex numbers
$(k+b)^2 + x^2 = (k+b+\ii x)(k+b-\ii x)$, so
\begin{equation}
\frac{1}{(k+b)^2+x^2}
 \;=\; \frac{1}{2\ii x}
 \left[\frac{1}{k+b-\ii x} - \frac{1}{k+b+\ii x}\right] .
\label{eq:D-partial}
\end{equation}
Neither term on the right is separately summable over $\mathbb{Z}$; the
combination is, and the symmetric (principal-value) summation used below
is the one under which each term is defined.

\paragraph{Step (ii): the cotangent identity.} The standard
partial-fraction expansion of the cotangent, valid for $z$ not an
integer and summed symmetrically, is
\begin{equation}
\sum_{k\in\mathbb{Z}} \frac{1}{k+z}
 \;=\; \lim_{K\to\infty} \sum_{k=-K}^{K} \frac{1}{k+z}
 \;=\; \pi \cot \pi z .
\label{eq:D-cot}
\end{equation}
It is the same identity that, restricted to $z = \ii x$, gives the
familiar $\sum_k 1/(k^2+x^2) = (\pi/x)\coth \pi x$; the derivation below
recovers that case as a check.

\paragraph{Step (iii): apply it twice.} Putting
Eq.~\eqref{eq:D-cot} into Eq.~\eqref{eq:D-partial} with
$z = b \mp \ii x$,
\begin{equation}
\sum_{k\in\mathbb{Z}} \frac{1}{(k+b)^2+x^2}
 \;=\; \frac{\pi}{2\ii x}
 \Big[\cot\pi(b - \ii x) - \cot\pi(b + \ii x)\Big] .
\label{eq:D-twocot}
\end{equation}

\paragraph{Step (iv): the difference of cotangents.} Write
$A = \pi b$ and $B = \pi x$ and put the difference over a common
denominator:
\begin{equation}
\cot(A-\ii B) - \cot(A + \ii B)
 = \frac{\cos(A-\ii B)\sin(A+\ii B) - \cos(A+\ii B)\sin(A-\ii B)}
        {\sin(A-\ii B)\,\sin(A+\ii B)} .
\label{eq:D-cotdiff}
\end{equation}
The numerator is the sine of the difference of the two arguments,
$\sin\big[(A+\ii B) - (A - \ii B)\big] = \sin 2\ii B = \ii \sinh 2B$.
The denominator is handled by
$\sin P \sin Q = \tfrac12\left[\cos(P-Q) - \cos(P+Q)\right]$ with
$P = A - \ii B$ and $Q = A + \ii B$, which gives
$\tfrac12\left[\cosh 2B - \cos 2A\right]$. Hence
\begin{equation}
\cot(A-\ii B) - \cot(A+\ii B)
 \;=\; \frac{2\ii \sinh 2B}{\cosh 2B - \cos 2A} .
\label{eq:D-cotdiff2}
\end{equation}

\paragraph{Step (v): collect.} Substituting
Eq.~\eqref{eq:D-cotdiff2} into Eq.~\eqref{eq:D-twocot}, the two factors
of $\ii$ cancel and Eq.~\eqref{eq:D-resum} follows. Two checks. Setting
$b = 0$ and using $\sinh 2u = 2\sinh u \cosh u$ and
$\cosh 2u - 1 = 2\sinh^2 u$ returns $(\pi/x)\coth \pi x$, the identity
quoted in Step (ii). Letting $x \to 0$ at $b = 0$, the right-hand side
behaves as $1/x^2$, which is the divergent $k = 0$ term of the left-hand
side and nothing else.

\paragraph{The exponential form.} Eq.~\eqref{eq:D-resum} is closed but
its message is hidden. Set $u = e^{-2\pi x}$, so that
$\sinh 2\pi x = (u^{-1}-u)/2$ and
$\cosh 2\pi x - \cos 2\pi b = (1 - 2u\cos 2\pi b + u^2)/(2u)$, and
therefore
\begin{equation}
\frac{\sinh 2\pi x}{\cosh 2\pi x - \cos 2\pi b}
 \;=\; \frac{1 - u^2}{1 - 2u\cos 2\pi b + u^2} .
\label{eq:D-uform}
\end{equation}
Now expand. With $z = e^{2\pi\ii b}$ the denominator factorizes as
$(1-uz)(1-u z^{-1})$, and summing the two geometric series
\begin{equation}
\begin{aligned}
1 + \frac{uz}{1-uz} + \frac{u z^{-1}}{1 - u z^{-1}}
 &\;=\; \frac{(1-uz)(1-uz^{-1}) + uz(1-uz^{-1}) + uz^{-1}(1-uz)}
             {(1-uz)(1-uz^{-1})} \\[2pt]
 &\;=\; \frac{1-u^2}{1 - 2u\cos 2\pi b + u^2}
\end{aligned}
\label{eq:D-geometric}
\end{equation}
--- the numerator collapses because the three cross terms leave
$1 - u^2$ exactly. The left-hand side of Eq.~\eqref{eq:D-geometric} is
$1 + \sum_{w\ge 1} u^w (z^w + z^{-w})$, so combining with
Eq.~\eqref{eq:D-resum},
\begin{equation}
\sum_{k\in\mathbb{Z}} \frac{1}{(k+b)^2+x^2}
 \;=\; \frac{\pi}{x}\left[\,1
 \;+\; 2\sum_{w=1}^{\infty} e^{-2\pi w x}\cos(2\pi w b)\right] .
\label{eq:D-resum-exp}
\end{equation}
This is the Poisson-resummed form of the same identity, and it is the
one that matters.

\paragraph{Statement (the point of the subsection).} In
Eq.~\eqref{eq:D-resum-exp} the entire large-$x$ growth sits in the term
$\pi/x$, which does not contain $b$. Every $b$-dependent term carries a
factor $e^{-2\pi w x}$ with $w \ge 1$. Equivalently, in the closed form
Eq.~\eqref{eq:D-resum} the $b$-dependence enters only through
$\cos 2\pi b$ in the denominator, where it is added to $\cosh 2\pi x$
and is therefore negligible against it once $x$ is large. So:

\begin{itemize}[leftmargin=1.4em,itemsep=0.2em]
\item the ultraviolet divergence of the mode sum is
  \emph{$b$-independent}, hence $\theta$-independent, hence identical for
  every value of the Wilson phase;
\item it therefore cancels exactly from the difference
  $V(\theta) - V(0)$, and from every $\theta$-derivative of an effective
  potential;
\item the $\theta$-dependent part of that potential is finite with no
  regulator, no counterterm and no renormalization condition.
\end{itemize}

That is why the potential computed in \S\ref{ssec:D-potential} is
finite, in a five-dimensional theory that is not renormalizable, without
apology. The finiteness is a theorem about the sum, not an estimate.

\paragraph{The Ward-identity reading.} There is a structural reason
behind the arithmetic, and it is worth stating in the language of
Chapter~\ref{sec:gauge}. A counterterm in a local field theory is the
spacetime integral of a polynomial in the fields and their derivatives
\emph{at a point}. The quantity $\theta$ is not the value of any such
polynomial: by Eqs.~\eqref{eq:D-Fzero} and \eqref{eq:D-wilsonline} it is
the holonomy of a loop that does not bound, invisible to every local
gauge-invariant operator, since all of them are built from $F_{MN}$ and
its covariant derivatives and all of those vanish. Gauge invariance ---
the Ward identity --- restricts the counterterms to gauge-invariant local
operators; no such operator can depend on $\theta$; so no counterterm can
generate or absorb a potential for $\theta$. The divergences and the
$\theta$-dependence live in disjoint places because locality and
topology put them there.

Two honest qualifications. First, the statement concerns the
$\theta$-\emph{dependent} part only: the $\theta$-independent vacuum
energy remains divergent and is untouched by anything here. Second, the
argument as given is a one-loop statement made from a mode sum; its
all-orders form is the standard non-renormalization argument for a
non-local order parameter, and is not re-derived in this paper.

\emph{Physics intuition.} Ultraviolet physics is short-distance physics,
and short distances cannot tell that the circle closes. The
$\theta$-dependence is a statement about going all the way around, so it
is suppressed by exactly the cost of a mode traveling the whole
circumference, $e^{-2\pi w x}$ for $w$ laps. Divergences come from small
loops; $\theta$ is visible only to large ones.

\subsection{Step 2 --- the Coleman--Weinberg potential, and where it minimizes}
\label{ssec:D-potential}

Nothing so far has made $\theta$ dynamical. It was a parameter labeling
a family of flat configurations, all with the same classical action,
since Eq.~\eqref{eq:D-Fzero} makes the Yang--Mills action vanish for
every $\theta$. Classically the family is exactly degenerate. Quantum
mechanically it is not, because the tower of Eq.~\eqref{eq:D-mass} has a
zero-point energy and that energy depends on $\theta$.

\paragraph{The one-loop vacuum energy.} For one real bosonic degree of
freedom of charge $q$, the one-loop effective potential is the sum over
the tower of the logarithm of the Euclidean inverse propagator,
\begin{equation}
V(\theta) \;=\; \frac{1}{2}\sum_{k\in\mathbb{Z}}
 \int\!\frac{\dd^4 p}{(2\pi)^4}\,
 \ln\!\left(p^2 + m_k^2\right) ,
\label{eq:D-vac}
\end{equation}
with $m_k$ from Eq.~\eqref{eq:D-mass}. A fermionic degree of freedom
contributes the same expression with the opposite overall sign, which is
the only place statistics enters and the only thing that will matter.

\paragraph{Proper time.} Use
$\ln X = -\int_0^\infty (\dd t/t)\, e^{-tX}$, up to an
$X$-independent constant that carries no $\theta$, together with the
Gaussian integral
$\int \dd^4p\,(2\pi)^{-4} e^{-t p^2} = 1/(16\pi^2 t^2)$. Then
\begin{equation}
V(\theta) \;=\; -\,\frac{1}{32\pi^2}
 \int_0^\infty \dd t\; t^{-3}
 \sum_{k\in\mathbb{Z}} e^{-t\,m_k^2} .
\label{eq:D-schwinger}
\end{equation}

\paragraph{Poisson resummation.} The mode sum is a Gaussian in $k$, and
Poisson resummation trades the sum over levels for a sum over
\emph{windings}:
\begin{equation}
\sum_{k\in\mathbb{Z}} e^{-t(2\pi k + q\theta)^2/L^2}
 \;=\; \frac{L}{2\pi}\sqrt{\frac{\pi}{t}}
 \sum_{w\in\mathbb{Z}} e^{\ii w q\theta}\, e^{-w^2 L^2/(4t)} .
\label{eq:D-poisson}
\end{equation}
The $w = 0$ term carries no $\theta$ and produces the divergence at
$t \to 0$; it is precisely the $\pi/x$ of
Eq.~\eqref{eq:D-resum-exp} and it is dropped. Each $w \ne 0$ term is
finite:
\begin{equation}
\int_0^\infty \dd t\; t^{-7/2}\, e^{-w^2L^2/(4t)}
 \;=\; \Gamma\!\left(\tfrac52\right)
 \left(\frac{4}{w^2L^2}\right)^{5/2}
 \;=\; \frac{3\sqrt{\pi}}{4}\cdot\frac{32}{|w|^5 L^5} ,
\label{eq:D-gamma}
\end{equation}
where the substitution $t \to c/t$ with $c = w^2L^2/4$ turns the integral
into $\Gamma(5/2)\,c^{-5/2}$ and $\Gamma(5/2) = 3\sqrt{\pi}/4$.
Assembling Eqs.~\eqref{eq:D-schwinger}--\eqref{eq:D-gamma} and pairing
$w$ with $-w$ to make the sum real,
\begin{equation}
V(\theta) \;=\; \mp\,\frac{3}{4\pi^2 L^4}
 \sum_{w=1}^{\infty} \frac{\cos\!\left(w q \theta\right)}{w^5}
 \;+\; \left(\theta\text{-independent}\right) ,
\label{eq:D-potential}
\end{equation}
the upper sign for each bosonic degree of freedom and the lower for each
fermionic one obeying the same periodic boundary condition.

\paragraph{What is convention-dependent, and what is not.} The
coefficient $3/(4\pi^2L^4)$ and the sign assignment in
Eq.~\eqref{eq:D-potential} depend on choices that a review must not
disguise: the normalization of Eq.~\eqref{eq:D-modes}, the counting of
physical polarizations (a five-dimensional gauge field contributes three,
a Dirac fermion four, and so on), the representation of the charged
field, which fixes the set of effective charges $q$ appearing in the
tower, and the boundary conditions, since fields antiperiodic around the
circle have $2\pi k \to 2\pi(k+\tfrac12)$ in Eq.~\eqref{eq:D-mass} and
hence $\theta \to \theta + \pi$ in Eq.~\eqref{eq:D-potential}. What is
\emph{not} convention-dependent is the shape: the weight $w^{-5}$, the
periodicity, the evenness, and the correlation between statistics and the
location of the minimum. In $d$ dimensions with one compact direction the
exponent is $d$; five dimensions gives $w^{-5}$.

\paragraph{The shape function.} Define
\begin{equation}
f(\theta) \;\equiv\; \sum_{w=1}^{\infty} \frac{\cos w\theta}{w^5} ,
\label{eq:D-fdef}
\end{equation}
so that Eq.~\eqref{eq:D-potential} at unit charge is $\mp c\, f(\theta)$
with $c = 3/(4\pi^2L^4) > 0$. Four properties, each elementary.

\begin{itemize}[leftmargin=1.4em,itemsep=0.2em]
\item \emph{Absolute convergence.} $|\cos w\theta| \le 1$, so the series
  is dominated term by term by $\sum_w w^{-5} = \zeta(5)$, and
  \begin{equation}
  |f(\theta)| \;\le\; \zeta(5) \;=\; 1.0369277551 .
  \label{eq:D-zeta5}
  \end{equation}
  Convergence is absolute and uniform; $f$ is continuous, and
  differentiable term by term since $\sum_w w^{-4}$ converges too.
\item \emph{Periodicity and parity.} Each term is $2\pi$ periodic and
  even in $\theta$, so $f$ is. This is Eqs.~\eqref{eq:D-shift} and
  \eqref{eq:D-domain} reappearing in the potential, as they must.
\item \emph{The maximum.} At $\theta = 0$ every term equals $w^{-5}$ and
  $f(0) = \zeta(5) = 1.0369277551$, which by
  Eq.~\eqref{eq:D-zeta5} is the largest value $f$ can take.
\item \emph{The minimum.} At $\theta = \pi$, $\cos w\pi = (-1)^w$ and
  \begin{equation}
  f(\pi) \;=\; \sum_{w=1}^{\infty}\frac{(-1)^w}{w^5}
   \;=\; -\left(1 - 2^{-4}\right)\zeta(5)
   \;=\; -\,\tfrac{15}{16}\,\zeta(5) \;=\; -\,0.9721197704 .
  \label{eq:D-fpi}
  \end{equation}
  That this is the global minimum follows from monotonicity: term-by-term
  differentiation gives
  $f'(\theta) = -\sum_{w\ge1}\sin(w\theta)/w^4$, and the sine series
  $\sum_{w\ge1}\sin(w\theta)/w^4$ is strictly positive on $(0,\pi)$ ---
  write it as $\sum_w w^{-3}\left[\sin(w\theta)/w\right]$ and apply Abel
  summation, using that the partial sums of $\sum \sin(w\theta)/w$ are
  positive on $(0,\pi)$ and that $w^{-3}$ decreases to zero. So $f$
  decreases strictly from $\theta = 0$ to $\theta = \pi$, and by parity
  and periodicity those are its only stationary points.
\end{itemize}

\paragraph{Where it minimizes.} With the bosonic sign,
$V = -c f(\theta)$ is minimized where $f$ is largest:
\begin{equation}
\theta_{\min} = 0 , \qquad W = e^{\ii\theta_{\min}} = 1 .
\label{eq:D-minboson}
\end{equation}
With the fermionic sign for matter periodic around the circle ---
adjoint fermions being the standard case --- $V = +c f(\theta)$ is
minimized where $f$ is smallest:
\begin{equation}
\theta_{\min} = \pi , \qquad W = e^{\ii\theta_{\min}} = -1 .
\label{eq:D-minfermion}
\end{equation}
The two endpoints of Eq.~\eqref{eq:D-domain} are the two answers, and
which one is selected is decided by a competition of signs among the
degrees of freedom running in the loop. Problem~D.4 does the competition
with a general count.

\paragraph{Statement (the Hosotani mechanism).} The four-dimensional
theory obtained by compactification contains the gauge fields that
commute with the holonomy $W$ and gives a mass to the rest, since by
Eq.~\eqref{eq:D-mass} a field of non-zero effective charge acquires
$m \ge |q\theta|/L$ even in its lowest mode. When $W = \mathbf{1}$ the
commutant is the whole group and the four-dimensional gauge symmetry is
the five-dimensional one. When $W \ne \mathbf{1}$ the commutant is a
proper subgroup and the pattern of massless vectors is correspondingly
smaller. Neither outcome was written into the Lagrangian: the Lagrangian
contains a group, a compactification and a list of matter fields, and the
realized symmetry is the output of Eq.~\eqref{eq:D-potential}. Which
gauge symmetry is realized is therefore a dynamical consequence of the
matter content, not an input. That is the mechanism of
\citet{Manton1979} and \citet{Hosotani1983}, treated systematically in
\citet{Hosotani1989}.

\paragraph{Grades.} Two statements of this subsection are theorems and
one is not, and the distinction is not decorative. The finiteness of the
$\theta$-dependent potential, \S\ref{ssec:D-resum}, is a theorem. The
shape Eq.~\eqref{eq:D-potential} and the location of its minimum, given a
field content, follow from it by computation and are equally secure. The
proposal that this mechanism resolves the hierarchy problem by
identifying the four-dimensional Higgs field with $A_y$ --- gauge-Higgs
unification, so that the Higgs mass is protected by the same argument
that made $V(\theta)$ finite --- is \emph{open}: no realization with the
observed spectrum is established, and this review takes no position on
it. The program and its difficulties are surveyed by
\citet{HatanakaInamiLim1998}.

\emph{Physics intuition.} The classical theory could not decide where
$\theta$ sat, so the vacuum fluctuations decided. Bosons and fermions
push in opposite directions because their loops enter with opposite
signs, and the answer is whichever side has more degrees of freedom. It
is the same logic as Coleman--Weinberg symmetry breaking in four
dimensions \citep{ColemanWeinberg1973}, run on an angle instead of a
field value, and with the
divergences already removed by the topology.

\subsection{Step 2 (further) --- the center, and the Polyakov loop}
\label{ssec:D-centre}

Section~\ref{ssec:elitzur} established that a local gauge symmetry
cannot break, and that the language of ``breaking a gauge symmetry'' is
a picture rather than a fact. A Wilson line is nevertheless described
everywhere as an order parameter for symmetry breaking. Both statements
are correct, and this subsection says why.

\paragraph{The center.} The center of a group is the subgroup commuting
with everything. For $\SU(N)$ it consists of the multiples of the
identity that have unit determinant,
\begin{equation}
Z\!\left(\SU(N)\right)
 = \left\{\, z\,\mathbf{1} \;:\; z^N = 1 \,\right\}
 \;\cong\; \mathbb{Z}_N ,
\label{eq:D-centre}
\end{equation}
the $N$-th roots of unity times the identity: $\det(z\mathbf{1}) = z^N$,
so $z^N = 1$ is the whole condition.

\paragraph{The twisted transformations.} On the compactified space
consider transformations that are periodic only up to a center element,
\begin{equation}
\Omega(y + L) \;=\; z\,\Omega(y) , \qquad z^N = 1 .
\label{eq:D-twist}
\end{equation}
By Eq.~\eqref{eq:D-single} such an $\Omega$ is \emph{not} single-valued
and therefore \emph{not} a gauge transformation. But the gauge field
transforms in the adjoint, Eq.~\eqref{eq:D-gaugeact}, and there $\Omega$
appears only in the combinations $\Omega \cdots \Omega^{-1}$ and
$\Omega\,\partial\Omega^{-1}$, in both of which the constant central
factor $z$ cancels. The transformed configuration is periodic and the
action is unchanged. So Eq.~\eqref{eq:D-twist} generates a genuine
symmetry of the pure gauge theory --- a $\mathbb{Z}_N$ acting on the
whole spacetime at once, with a single parameter $z$ and not a function
of position. It is a \emph{global} symmetry, and it is exactly the kind
of object Eq.~\eqref{eq:D-single} declared not to be gauge.

\paragraph{The Polyakov loop.} Let
\begin{equation}
P(x) \;=\; \mathcal{P}\exp\!\left(\ii g \int_{y}^{y+L}
 A_y(x,y')\,\dd y'\right)
\label{eq:D-polyakov}
\end{equation}
be the untraced holonomy of Eq.~\eqref{eq:C-wilson} around the compact
direction. Under an ordinary gauge transformation it conjugates,
$P \to \Omega(y) P \Omega(y+L)^{-1}$ with $\Omega(y+L) = \Omega(y)$, so
$\tr P$ is gauge invariant. Under Eq.~\eqref{eq:D-twist} the endpoint
carries the extra factor and, $z$ being central,
\begin{equation}
\tr P \;\longrightarrow\; z^{-1}\,\tr P .
\label{eq:D-ptransf}
\end{equation}
$\tr P$ is therefore gauge invariant but \emph{not} center invariant:
precisely the profile of an order parameter.

\paragraph{The fixed point.} If $\tr P$ is invariant under
Eq.~\eqref{eq:D-ptransf} for some $z \ne 1$ then
$(1 - z^{-1})\tr P = 0$, and since $z \ne 1$,
\begin{equation}
\tr P = 0 .
\label{eq:D-fixedpoint}
\end{equation}
Zero is the only value fixed by a non-trivial center element, so
\begin{itemize}[leftmargin=1.4em,itemsep=0.2em]
\item $\langle \tr P\rangle = 0$ is the center-symmetric phase --- for
  the thermal circle, the confined phase, in the sense of
  \citet{Wilson1974};
\item $\langle \tr P\rangle \ne 0$ breaks the center spontaneously, to
  one of $N$ degenerate vacua related by Eq.~\eqref{eq:D-ptransf}.
\end{itemize}

\paragraph{Statement (Elitzur compatibility).} Elitzur's theorem
(\S\ref{ssec:elitzur}) says that a symmetry whose parameter is an
independent function of position cannot break, because the integration
over that freedom kills any non-invariant quantity exactly. The center
symmetry Eq.~\eqref{eq:D-twist} has one parameter $z$ for the entire
system; the integration does not factorize; the theorem does not apply,
and the symmetry may break in the ordinary way. Its order parameter,
Eq.~\eqref{eq:D-polyakov}, is gauge invariant --- as \S\ref{ssec:elitzur}
insisted any honest order parameter must be --- and it is non-local: it
is the holonomy of a loop that wraps a non-contractible cycle and
therefore cannot be shrunk to a point. That is the whole of the
Elitzur-compatible sense in which a Wilson line ``breaks a symmetry.''
What breaks is a global discrete symmetry, and what does the breaking is
a non-local observable. The gauge redundancy is not involved at either
end.

One further consistency check, which is also a limitation. If matter in
the fundamental representation is present, $z$ no longer cancels, the
center is broken explicitly, and $\tr P$ ceases to be an order parameter
at all --- the same conclusion \S\ref{ssec:elitzur} reached from the
lattice gauge-Higgs system, and by the same reasoning: an order parameter
survives only as long as the symmetry it is ordering does.

\emph{Physics intuition.} A gauge transformation that fails to close by a
center element is invisible to everything built from the gauge field
locally, and visible to exactly one thing --- a loop that goes all the
way around. So the theory possesses a symmetry that only a global
observable can detect, and a phase transition that only a global
observable can see. The Wilson phase of \S\ref{ssec:D-flat} and the
Polyakov loop are the same construction asked two different questions.

\subsection{Step 3 --- what the body does with it}
\label{ssec:D-body}

Chapter~\ref{sec:wilson} needs Eqs.~\eqref{eq:D-Fzero} and
\eqref{eq:D-wilsonline}: zero curvature everywhere, one gauge-invariant
phase per non-contractible cycle, and no local statement able to see it.
It needs Eq.~\eqref{eq:D-potential} for the second half of the rung ---
that the phase is not a free label but a field with a computed potential,
so that the realized symmetry is an output.

\S\ref{ssec:G-sequences} supplies the address. The kernel of the
curvature map in Eq.~\eqref{eq:G-seq1} is the group of \emph{flat}
classes, $H^{k-1}(X;\mathbb{R}/\mathbb{Z})$, and the configuration of
Eq.~\eqref{eq:D-Ay} is an element of it. The two ends of that one exact
sequence are the two phenomena this paper keeps side by side.

\emph{The circle.} For $X = S^1$ the Betti numbers are $b_1 = 1$ and
$b_2 = 0$. In degree two, $\Omega^2(S^1) = 0$ --- a one-manifold carries
no two-forms --- so the curvature map of Eq.~\eqref{eq:G-seq1} has zero
target and \emph{every} degree-two differential character is flat. The
group of them is
\begin{equation}
\check{H}^{2}(S^1) \;\cong\; H^{1}(S^1;\mathbb{R}/\mathbb{Z})
 \;\cong\; \mathbb{R}/\mathbb{Z} ,
\label{eq:D-h1s1}
\end{equation}
one circle's worth: exactly the $\theta$ of Eq.~\eqref{eq:D-Ay} modulo
$2\pi$, as Eq.~\eqref{eq:D-single} required. A Wilson line, and no flux.
\citet{CheegerSimons1985} is where the flat classes are given their name.

\emph{The two-sphere.} For $X = S^2$ the Betti numbers are $b_1 = 0$ and
$b_2 = 1$. Then $H^1(S^2;\mathbb{R}/\mathbb{Z}) = 0$, so there are no
flat classes and no Wilson line at all, and what survives is the integer
of Eq.~\eqref{eq:G-dirac}. A flux, and no Wilson line.

Dirac and Hosotani are therefore not two mechanisms but two kernels ---
the two ends of Eq.~\eqref{eq:G-seq1}, read from opposite sides, on
spaces whose first and second Betti numbers are exchanged. Which is
available is settled by the shape of the compact space before any field
is written down, which is the sense in which \S\ref{ssec:top-of-ladder}
calls the top rung free of dynamics. The use of the flat end to break a
symmetry on a compactification that is not simply connected is standard
in the higher-dimensional literature \citep{Witten1985}, and
Chapter~\ref{sec:hs} needs only the structural half of it.

\subsection{Problems and worked solutions}
\label{ssec:D-problems}

\medskip\noindent\textbf{D.1. Compute every component of $F_{MN}$ for the
configuration Eq.~\eqref{eq:D-Ay} and show that all of them vanish.
Where in the computation would a non-constant $A_y$ have shown up?}

\noindent\emph{Solution.} \emph{Step 1.} For $M, N \in \{0,1,2,3\}$ all
three terms of Eq.~\eqref{eq:D-Fdef} contain a factor $A_\mu = 0$, so
$F_{\mu\nu} = 0$. \emph{Step 2.} For the mixed component,
$F_{\mu y} = \partial_\mu A_y - \partial_y A_\mu + \ii g[A_\mu, A_y]$;
the first term vanishes because $A_y$ is independent of $x^\mu$, the
second and third because $A_\mu = 0$. \emph{Step 3.} $F_{yy} = 0$ by
antisymmetry, and in the non-abelian case $[A_y, A_y] = 0$ because a
matrix commutes with itself. Hence Eq.~\eqref{eq:D-Fzero}. A $y$-dependent
$A_y$ would have appeared in $\partial_\mu A_y$ only if it also depended
on $x^\mu$; a $y$-dependence alone still gives $F = 0$, and can in fact
be removed by a single-valued gauge transformation --- only the
\emph{average} of $A_y$ around the circle, which is $\theta/gL$, is
physical. \emph{Physics intuition:} the field strength measures the
failure of transport to commute over an infinitesimal loop, and there are
no infinitesimal loops in this configuration that see anything; the only
loop that sees anything is the one that goes all the way around, and it
is not infinitesimal.

\medskip\noindent\textbf{D.2. Derive the tower Eq.~\eqref{eq:D-mass},
and show by explicit relabeling that the spectrum is invariant under
$\theta \to \theta + 2\pi$. Show also that it is even in $\theta$.}

\noindent\emph{Solution.} \emph{Step 1.} Periodicity of $\Phi$ forces the
expansion Eq.~\eqref{eq:D-modes} in $e^{2\pi\ii k y/L}$ with
$k \in \mathbb{Z}$; acting with $D_y = \partial_y + \ii q g A_y$ and
using $A_y = \theta/(gL)$ gives Eq.~\eqref{eq:D-Dy}, in which each mode
is multiplied by $\ii(2\pi k + q\theta)/L$. \emph{Step 2.} The term
$|D_y\Phi|^2$ in the five-dimensional kinetic term therefore contributes,
after integrating over $y$ with the orthonormal modes, a four-dimensional
mass term with $m_k^2 = (2\pi k + q\theta)^2/L^2$, which is
Eq.~\eqref{eq:D-mass}. \emph{Step 3.} Under $\theta \to \theta + 2\pi$,
Eq.~\eqref{eq:D-shift} gives $2\pi k + q\theta \to 2\pi(k+q) + q\theta$;
since $k \mapsto k+q$ is a bijection of $\mathbb{Z}$ for integer $q$, the
set of masses is unchanged and only the labeling has shifted. Under
$\theta \to -\theta$ the relabeling $k \to -k$ together with the
absolute value in Eq.~\eqref{eq:D-mass} returns the same set, so the
spectrum is even and the physical range is Eq.~\eqref{eq:D-domain}.
\emph{Physics intuition:} the Wilson phase does not create or destroy
states, it slides the whole tower along a ruler whose marks are
$2\pi/L$ apart; sliding by a whole mark permutes the labels and changes
nothing, which is why the observable is an angle.

\medskip\noindent\textbf{D.3. Derive Eq.~\eqref{eq:D-resum} by hand, and
from it obtain Eq.~\eqref{eq:D-resum-exp}. Read off the large-$x$
behavior of the $b$-dependent part.}

\noindent\emph{Solution.} \emph{Step 1.} Factorize
$(k+b)^2 + x^2 = (k+b+\ii x)(k+b-\ii x)$ and split into partial
fractions, Eq.~\eqref{eq:D-partial}. Apply the cotangent identity
Eq.~\eqref{eq:D-cot} to each piece with $z = b \mp \ii x$, which is
legitimate under symmetric summation, giving Eq.~\eqref{eq:D-twocot}.
\emph{Step 2.} Put the difference of cotangents over a common
denominator, Eq.~\eqref{eq:D-cotdiff}; the numerator is
$\sin 2\ii B = \ii \sinh 2B$ and the denominator is
$\tfrac12(\cosh 2B - \cos 2A)$ with $A = \pi b$, $B = \pi x$. Substituting
Eq.~\eqref{eq:D-cotdiff2} into Eq.~\eqref{eq:D-twocot} and canceling the
factors of $\ii$ gives Eq.~\eqref{eq:D-resum}. Check at $b = 0$: the
double-angle identities reduce it to $(\pi/x)\coth\pi x$.
\emph{Step 3.} Set $u = e^{-2\pi x}$ and $z = e^{2\pi\ii b}$, note that
the denominator is $(1-uz)(1-uz^{-1})/(2u)$, and sum the two geometric
series as in Eq.~\eqref{eq:D-geometric} to obtain
Eq.~\eqref{eq:D-resum-exp}. The $b$-dependent terms carry
$e^{-2\pi w x}$, $w \ge 1$; the $b$-independent term is $\pi/x$ and
carries the whole ultraviolet growth. \emph{Physics intuition:} the sum
splits into a piece that knows only the local density of modes and a
piece that knows the circumference, and only the second can depend on the
Wilson phase --- which is why the divergence is $\theta$-blind and the
$\theta$-dependent potential is finite without a regulator.

\medskip\noindent\textbf{D.4. Suppose the loop contains one bosonic
effective degree of freedom of unit charge and $n_f$ fermionic ones with
the same charge and the same periodic boundary condition. Where does
$V(\theta)$ minimize as a function of $n_f$, and what happens at the
crossover?}

\noindent\emph{Solution.} \emph{Step 1.} By
Eq.~\eqref{eq:D-potential} each bosonic degree of freedom contributes
$-c f(\theta)$ and each fermionic one $+c f(\theta)$, with
$c = 3/(4\pi^2L^4) > 0$ and $f$ from Eq.~\eqref{eq:D-fdef}. Summing,
\begin{equation}
V(\theta) \;=\; c\left(n_f - 1\right) f(\theta) .
\label{eq:D-competition}
\end{equation}
\emph{Step 2.} By \S\ref{ssec:D-potential}, $f$ decreases strictly from
$f(0) = \zeta(5) = 1.0369277551$ to
$f(\pi) = -\tfrac{15}{16}\zeta(5) = -0.9721197704$, and has no other
stationary points on $[0,\pi]$. So for $n_f > 1$ the prefactor is
positive and $V$ is minimized where $f$ is smallest, at
$\theta_{\min} = \pi$, hence $W = -1$; for $n_f < 1$, that is $n_f = 0$,
the prefactor is negative and $V$ is minimized where $f$ is largest, at
$\theta_{\min} = 0$, hence $W = +1$. \emph{Step 3.} At $n_f = 1$ the
prefactor vanishes and $V(\theta) \equiv 0$ at one loop: the potential is
flat, every $\theta$ is degenerate, and the one-loop calculation does not
determine the vacuum. That is the crossover, and it is the Bose--Fermi
cancellation seen from the compact direction --- equal numbers of bosonic
and fermionic degrees of freedom with equal masses contribute equal and
opposite zero-point energies at every level of the tower.
\emph{Physics intuition:} the vacuum energy is a vote, bosons and
fermions voting opposite ways with weights given by their degree-of-freedom
counts, and the elected value of $\theta$ is a property of the electorate
rather than of the Lagrangian's gauge group. At a tie, the one-loop
answer is silent and one must go further, which is exactly the situation
supersymmetric compactifications engineer on purpose.

\medskip\noindent\textbf{D.5. Using the Betti numbers, count the Wilson
phases and the flux integers available in degree two on $S^1$, $S^2$ and
$T^2$, and say in each case which of Dirac and Hosotani is realized.}

\noindent\emph{Solution.} \emph{Step 1.} By Eq.~\eqref{eq:G-seq1} the
flat classes are counted by $H^{1}(X;\mathbb{R}/\mathbb{Z})$, which for
these spaces --- all torsion-free in the relevant degree --- is
$(\mathbb{R}/\mathbb{Z})^{b_1}$: one circle of Wilson phase per
independent one-cycle. By Eq.~\eqref{eq:G-seq2} the topological classes
are counted by $H^{2}(X;\mathbb{Z}) \cong \mathbb{Z}^{b_2}$: one flux
integer per independent two-cycle. \emph{Step 2.} For $S^1$,
$b_1 = 1$ and $b_2 = 0$: one Wilson phase, no flux integer, and
Eq.~\eqref{eq:D-h1s1} says every class is flat --- Hosotani only. For
$S^2$, $b_1 = 0$ and $b_2 = 1$: no Wilson phase, one flux integer ---
Dirac only, and Eq.~\eqref{eq:G-dirac} is the whole content. For $T^2$,
$b_1 = 2$ and $b_2 = 1$: two Wilson phases, one around each cycle, and
one flux integer --- both at once. \emph{Step 3.} The three cases
exhaust the possibilities in this degree: a space can have loops that do
not bound, surfaces that do not bound, both, or neither.
\emph{Physics intuition:} the question ``is there a Wilson line here?''
is not a question about the gauge field but about the space, and it is
answered by counting holes before any dynamics is specified --- which is
why Dirac's integer and Hosotani's angle sit in the same exact sequence
rather than in competing chapters.

\GGtakeaway{\emph{Standard result:} a flat connection along a compact
direction is labeled by a gauge-invariant angle, the Wilson phase; the
Kaluza--Klein spectrum depends on it, so it is observable; its one-loop
potential is finite without a regulator and takes the form
Eq.~\eqref{eq:D-potential}; and the center symmetry, of which the
Polyakov loop is the order parameter, is global and may therefore break.
\emph{What it assumed:} one compact direction with the simplest topology,
integer charges, periodic boundary conditions except where stated, and a
one-loop computation. \emph{The reading this paper takes:} the Wilson
phase is the third datum of \S\ref{ssec:third-datum} made dynamical ---
invisible to curvature, invisible to every local counterterm, and yet
carrying a computed potential that decides which gauge symmetry the
four-dimensional theory displays. \emph{Grades:} the finiteness is a
theorem; the identification of the four-dimensional Higgs with $A_y$ as a
resolution of the hierarchy problem is open. \emph{Continue in:}
Appendix~\ref{app:diffcohom}, \S\ref{ssec:G-sequences}, for the exact
sequence in which this phase and Dirac's integer are the two kernels, and
Chapter~\ref{sec:wilson} for the rung itself.}


\section{Ward identities, derived}
\label{app:ward}
\GGteachbox{The master Ward identity derived from the path integral by a
local change of integration variables, with the contact terms kept rather
than dropped; its electron--photon specialization, the Ward--Takahashi
identity, verified at tree level and pushed to $Z_1 = Z_2$ and the
species-independence of the renormalized charge; transversality of the
vacuum polarization and the masslessness of the photon; then Step~2 on
the one assumption the derivation makes --- that the integration measure
is invariant --- which is where an anomaly enters, and the BRST
statement of the same identity, whose nilpotency $s^2 = 0$ is what forces
the descent equations. The pay-off is Chapter~\ref{sec:ward}, and the
hand-off is Appendix~\ref{app:anomalies}.}

\GGreadalong{\citet{PeskinSchroeder1995}, \S7.4 for the Ward--Takahashi
identity and \S16.4 for BRST; \citet{ChengLi1984} for the Ward
identities of quantum electrodynamics and their use in the
renormalization program; \citet{Weinberg1996}, Ch.~17, for the
Slavnov--Taylor identities in the non-abelian case at full generality;
and the two original papers, \citet{Ward1950} and \citet{Takahashi1957},
which are both short and both readable.}

\subsection{Step 1 --- a local change of variables, contact terms kept}
\label{ssec:E-derivation}

Noether's first theorem (Appendix~\ref{app:noether}) gives a classical
statement: $\partial_\mu j^\mu = 0$ on shell. The quantum theory does not
work with field configurations, it works with correlation functions, and
the question is what survives. The answer is stronger than a naive
translation would suggest, and the derivation is three lines of honest
bookkeeping.

\paragraph{Setup.} Let the theory contain fields $\phi_k$ carrying charges
$q_k$ under a global $\mathrm{U}(1)$, and write correlation functions as
path integrals,
\begin{equation}
\big\langle T\, \phi(x_1)\cdots\phi(x_n) \big\rangle
 = \frac{1}{Z}\int \mathcal{D}\phi \;
   \phi(x_1)\cdots\phi(x_n)\, \mathrm{e}^{\ii S[\phi]} ,
\qquad
Z = \int \mathcal{D}\phi\, \mathrm{e}^{\ii S[\phi]} .
\label{eq:E-correlator}
\end{equation}
Write the infinitesimal transformation in the form
\begin{equation}
\delta \phi_k(x) = \epsilon(x)\, q_k\, \phi_k(x) ,
\label{eq:E-transf}
\end{equation}
with $\epsilon(x)$ an \emph{arbitrary} infinitesimal function of compact
support, and let $J^\mu$ be the Noether current built from
Eq.~\eqref{eq:E-transf} by Appendix~\ref{app:noether}'s
Eq.~\eqref{eq:B-current}. The factor of $\ii$ that a phase rotation
carries has been left outside the charge, so that it will appear
explicitly on the right-hand side below; with the other convention,
$\delta\phi_k = \ii\epsilon q_k \phi_k$, the same identity reads with
$+q_k$ and a current larger by $\ii$. Nothing depends on the choice, and
it is worth fixing once.

\paragraph{Step (i): the action.} Appendix~\ref{app:noether} already did
this computation. Substituting $\delta\phi^a = \epsilon\,\Delta^a$ into
Eq.~\eqref{eq:B-variation} and using
$\partial_\mu j^\mu = -E_a \Delta^a$ gives, with no assumption that the
fields solve anything,
\begin{equation}
\delta S = \int \dd^4 y \; \big(\partial_\mu \epsilon(y)\big) J^\mu(y)
 = -\int \dd^4 y \; \epsilon(y)\, \partial_\mu J^\mu(y) ,
\label{eq:E-deltaS}
\end{equation}
the second step an integration by parts, legitimate because $\epsilon$
has compact support. For \emph{constant} $\epsilon$ the right-hand side
vanishes and one recovers the statement that the action is invariant;
the whole content of what follows is that $\epsilon$ was not taken
constant.

\paragraph{Step (ii): the operators.} The product of fields also changes:
\begin{equation}
\delta\big[\phi(x_1)\cdots\phi(x_n)\big]
 = \sum_{k=1}^{n} q_k\, \epsilon(x_k)\; \phi(x_1)\cdots\phi(x_n) .
\label{eq:E-deltaF}
\end{equation}
These are the \emph{contact terms}. They are the entire point, and the
standard derivation of the classical conservation law never sees them
because it never inserts any operators.

\paragraph{Step (iii): the measure, and the one fact used.} Now use the
fact that a \emph{change of the name of an integration variable cannot
change the value of an integral}. Relabeling $\phi \to \phi + \delta\phi$
inside Eq.~\eqref{eq:E-correlator} leaves the number on the left
untouched, so the first-order variation of the integrand must integrate
to zero:
\begin{equation}
0 = \int \mathcal{D}\phi \; \mathrm{e}^{\ii S}
  \Big[
    \delta\big(\phi(x_1)\cdots\phi(x_n)\big)
    + \ii \, \phi(x_1)\cdots\phi(x_n)\, \delta S
  \Big] ,
\label{eq:E-zero}
\end{equation}
\emph{provided} $\mathcal{D}\phi' = \mathcal{D}\phi$. Hold that proviso;
\S\ref{ssec:E-measure} is about nothing else.

\paragraph{Step (iv): collect.} Insert Eq.~\eqref{eq:E-deltaS} and
Eq.~\eqref{eq:E-deltaF} into Eq.~\eqref{eq:E-zero}, divide by $Z$, and
write $\epsilon(x_k) = \int \dd^4 y\, \epsilon(y)\,\delta^4(y - x_k)$ so
that both terms are integrals against the same arbitrary function:
\begin{equation}
0 = \int \dd^4 y \; \epsilon(y)
 \left[
   \sum_k q_k\, \delta^4(y - x_k)\,
     \big\langle T\, \phi(x_1)\cdots\phi(x_n)\big\rangle
   \;-\; \ii\,
     \big\langle T\, \partial_\mu J^\mu(y)\, \phi(x_1)\cdots\phi(x_n)
     \big\rangle
 \right] .
\end{equation}
Because $\epsilon(y)$ is arbitrary, the bracket vanishes pointwise. Since
the path integral produces the time-ordered product with the derivative
already acting on it, this is
\begin{equation}
\boxed{\;
\partial_\mu \big\langle T\, J^\mu(x)\,
  \phi(x_1)\cdots\phi(x_n)\big\rangle
 = -\,\ii \sum_{k=1}^{n} q_k\, \delta^4(x - x_k)\,
   \big\langle T\, \phi(x_1)\cdots\phi(x_n)\big\rangle \; }
\label{eq:E-master}
\end{equation}

\paragraph{Statement.} Equation~\eqref{eq:E-master} is the master Ward
identity. Away from the insertion points it says
$\partial_\mu \langle T J^\mu \cdots\rangle = 0$: the classical
conservation law holds inside correlation functions. At the insertion
points it says something the classical statement could not: the current's
divergence is a sum of delta functions weighted by the charges of
whatever operators are sitting there.

Two features of Eq.~\eqref{eq:E-master} carry the argument of
Chapter~\ref{sec:ward}, and both are visible in the derivation rather
than asserted after it.

\emph{(a) The right-hand side is pure kinematics.} It contains delta
functions and charges. It contains no coupling constant, no mass, no
loop expansion, and no reference to the Lagrangian beyond the fact that
it was invariant. Change the dynamics --- add interactions, resum,
go to strong coupling --- and the right-hand side does not move. This is
what makes a Ward identity a constraint on a quantum field theory rather
than a result within one, and it is why every consequence derived below
is exact to all orders.

\emph{(b) The derivation used one assumption, and it can fail.} Step~(iii)
assumed $\mathcal{D}\phi' = \mathcal{D}\phi$. Everything else was
calculus. The invariance of the classical action was used in
Eq.~\eqref{eq:E-deltaS}; the invariance of the \emph{measure} was used in
Eq.~\eqref{eq:E-zero}, and it is a separate assumption, because
quantizing requires a choice of measure and the Lagrangian does not make
that choice. When it fails, Eq.~\eqref{eq:E-master} acquires an extra
term on the right, and that extra term is an anomaly. The failure is
worked out in \S\ref{ssec:E-measure} and its consequences are the subject
of Appendix~\ref{app:anomalies}.

\subsection{Step 1 (continued) --- Ward--Takahashi, and $Z_1 = Z_2$}
\label{ssec:E-WT}

Specialize Eq.~\eqref{eq:E-master} to quantum electrodynamics with two
insertions: an electron field $\psi(y)$ and its conjugate
$\bar\psi(z)$, of opposite charge. With the electron's charge normalized
to unity,
\begin{equation}
\partial_\mu^{\,x}
 \big\langle T\, J^\mu(x)\, \psi(y)\, \bar\psi(z) \big\rangle
 = -\,\ii \left[ \delta^4(x - y) - \delta^4(x - z) \right]
   \big\langle T\, \psi(y)\, \bar\psi(z)\big\rangle .
\label{eq:E-WTposition}
\end{equation}
This is already the identity; the rest is Fourier transformation.

\paragraph{To momentum space.} Define the full electron propagator by
\begin{equation}
S(p) = \int \dd^4 x\; \mathrm{e}^{\ii p\cdot x}\,
 \big\langle T\, \psi(x)\,\bar\psi(0)\big\rangle ,
\label{eq:E-prop-def}
\end{equation}
and let $\Gamma^\mu(p+q,p)$ be the full amputated vertex, so that ---
with momentum $q$ entering at the current, $p$ entering at the $\bar\psi$
leg and $p+q$ leaving at the $\psi$ leg, and with the overall
momentum-conserving delta function stripped --- the Fourier transform of
the three-point function on the left of Eq.~\eqref{eq:E-WTposition} is
\begin{equation}
\widetilde{G}^\mu(p+q,p) \;=\; S(p+q)\; \Gamma^\mu(p+q,p)\; S(p) .
\label{eq:E-vertex-def}
\end{equation}
Two things now happen on the two sides of
Eq.~\eqref{eq:E-WTposition}. On the left, $\partial_\mu^{\,x}$ acting on
$\mathrm{e}^{\ii q\cdot x}$ brings down a factor $-\ii q_\mu$. On the
right, each delta function removes one integration, setting the current's
argument equal to one of the two field arguments; the three-point
function then collapses to a two-point function, and what survives is one
full propagator carried at the momentum of the leg that was \emph{not}
touched. Hence
\begin{equation}
-\,\ii\, q_\mu\, S(p+q)\,\Gamma^\mu(p+q,p)\, S(p)
 = -\,\ii \big[ S(p) - S(p+q) \big] .
\label{eq:E-WTmomentum}
\end{equation}
Multiply on the left by $S^{-1}(p+q)$ and on the right by $S^{-1}(p)$.
The result is the Ward--Takahashi identity \citep{Takahashi1957}:
\begin{equation}
q_\mu\, \Gamma^\mu(p+q,\, p)
 \;=\; S^{-1}(p+q) \;-\; S^{-1}(p) .
\label{eq:E-WT}
\end{equation}
It is exact: no order in perturbation theory was assumed anywhere, and
the right-hand side of Eq.~\eqref{eq:E-master} contributed only the two
delta functions.

\paragraph{Verification at tree level.} Take
$\Gamma^\mu = \gamma^\mu$ and $S^{-1}(p) = \slashed{p} - m$. The
left-hand side of Eq.~\eqref{eq:E-WT} is
\begin{equation}
q_\mu \gamma^\mu = \slashed{q} .
\end{equation}
The right-hand side is
\begin{equation}
\big(\slashed{p} + \slashed{q} - m\big) - \big(\slashed{p} - m\big)
 = \slashed{q} ,
\label{eq:E-tree}
\end{equation}
because $v \mapsto \slashed{v}$ is linear in $v$. The two sides
agree, and they agree \emph{identically in $m$}: the mass appears in both
inverse propagators with the same coefficient and cancels in the
difference. Not even the Clifford algebra was needed --- linearity
sufficed.

That cancellation is worth a sentence, because it is the whole difference
between the vector case and the axial one. The vector identity relates
the vertex to a \emph{difference} of inverse propagators, and a mass term
is common to both ends of the difference, so the identity cannot see it.
The axial current is not so lucky: for the axial vertex the corresponding
relation involves $S^{-1}(p+q)\gamma_5 + \gamma_5 S^{-1}(p)$, an
\emph{anticommutator}, in which the mass terms add rather than cancel and
leave a residue $2m\gamma_5$. That residue is the classical statement
$\partial_\mu j_5^\mu = 2\ii m\, \bar\psi\gamma_5\psi$: the axial symmetry
is already broken by a mass before any quantum question is asked. What
\S\ref{ssec:E-measure} adds is that it is broken again, and differently,
when the mass is zero.

\paragraph{Ward's own form: the forward limit.} Expand
Eq.~\eqref{eq:E-WT} for small $q$. The right-hand side is
\begin{equation}
S^{-1}(p+q) - S^{-1}(p)
 = q_\nu \frac{\partial S^{-1}(p)}{\partial p_\nu} + \mathcal{O}(q^2) ,
\end{equation}
so that $q_\mu \Gamma^\mu(p,p) = q_\nu\, \partial S^{-1}/\partial p_\nu$
for every $q$, whence
\begin{equation}
\Gamma^\mu(p,\, p) \;=\; \frac{\partial S^{-1}(p)}{\partial p_\mu} .
\label{eq:E-forward}
\end{equation}
This is the differential identity \citet{Ward1950} wrote down first, five
years before the general relation Eq.~\eqref{eq:E-WT} that contains it.
Read physically, it says that emitting a zero-momentum photon from an
electron line is the same operation as differentiating the electron's
inverse propagator: the photon at zero momentum measures the electron's
response to a shift of its own momentum, and nothing else.

\paragraph{Statement ($Z_1 = Z_2$).} Let $Z_1$ be the renormalization
constant of the vertex and $Z_2$ that of the electron field, defined by
\begin{equation}
\Gamma^\mu = Z_1^{-1}\gamma^\mu + (\text{finite}) ,
\qquad
S^{-1}(p) = Z_2^{-1}\big(\slashed{p} - m\big) + (\text{finite}) .
\label{eq:E-Zdefs}
\end{equation}
Then $Z_1 = Z_2$, to all orders.

\paragraph{Proof.} Substitute Eq.~\eqref{eq:E-Zdefs} into
Eq.~\eqref{eq:E-WT}. The left-hand side contributes
$Z_1^{-1}\slashed{q}$ to the divergent part; the right-hand side
contributes $Z_2^{-1}[(\slashed{p}+\slashed{q}) - \slashed{p}]
= Z_2^{-1}\slashed{q}$, the mass terms canceling as in
Eq.~\eqref{eq:E-tree}. The two divergent structures are the same matrix,
$\slashed{q}$, so their coefficients must agree:
$Z_1^{-1} = Z_2^{-1}$. \qed

\paragraph{Consequence: the renormalized charge.} In terms of the bare
charge $e_0$ and the three constants, the renormalized electric charge is
\begin{equation}
e \;=\; \frac{Z_2}{Z_1}\, \sqrt{Z_3}\; e_0
 \;\overset{Z_1 = Z_2}{=}\; \sqrt{Z_3}\; e_0 ,
\label{eq:E-charge}
\end{equation}
where $Z_3$ renormalizes the photon field. Look at what has just
happened. $Z_1$ and $Z_2$ are properties of the \emph{matter} field ---
they depend on its mass, its spin, its other interactions, everything
about the species. They have canceled. What remains, $Z_3$, is a
property of the \emph{photon}, and the photon is one object shared by
every charged species in the world.

Therefore, for any two species $a$ and $b$,
\begin{equation}
\frac{e_a}{e_b}
 = \frac{\sqrt{Z_3}\; e_{0a}}{\sqrt{Z_3}\; e_{0b}}
 = \frac{e_{0a}}{e_{0b}} ,
\label{eq:E-universality}
\end{equation}
at every scale. Charge \emph{ratios} do not run. An electron and a proton
--- objects that share no other property, one elementary and one a
strongly bound composite of three quarks and an indefinite number of
gluons --- carry charges whose ratio is protected from all of that
dynamics by Eq.~\eqref{eq:E-universality}.

\paragraph{Grade, stated plainly.} Equation~\eqref{eq:E-universality} is
\emph{universality}, and it is \emph{not} quantization. It says that
whatever the ratio $e_{0a}/e_{0b}$ is, the dynamics will not change it.
It does not say the ratio is $-1$, and nothing in this appendix says so.
A world in which the proton's charge exceeded the electron's in magnitude
by one part in $10^{22}$ satisfies every equation of this subsection.
Measurement excludes such a world at the level of $10^{-21}$
\citep{MarinelliMorpurgo1984}, and the explanation of that number is not
here. Ward protects the ratio from dynamics; Dirac (\S\ref{ssec:dirac},
Eq.~\eqref{eq:dirac}) protects it from continuity, and it takes both.
The distinction is the hinge of \S\ref{ssec:integers} and is worked
again as problem~E.5.

\subsection{Step 1 (concluded) --- transversality, and the photon mass}
\label{ssec:E-transverse}

Apply Eq.~\eqref{eq:E-master} with no charged operators inserted at all,
to the two-point function of the current. The right-hand side is then
empty and the identity is the flat statement
$\partial_\mu \langle T J^\mu(x) J^\nu(0)\rangle = 0$. In momentum space,
with $\Pi^{\mu\nu}(q)$ the one-particle-irreducible photon self-energy,
\begin{equation}
q_\mu\, \Pi^{\mu\nu}(q) \;=\; 0 .
\label{eq:E-transversality}
\end{equation}

\paragraph{Statement.} Equation~\eqref{eq:E-transversality} forces
\begin{equation}
\Pi^{\mu\nu}(q) \;=\; \big(q^2 \eta^{\mu\nu} - q^\mu q^\nu\big)\,
 \Pi(q^2)
\label{eq:E-transverse-form}
\end{equation}
for a single scalar function $\Pi(q^2)$.

\paragraph{Proof.} $\Pi^{\mu\nu}$ is a Lorentz tensor built from the only
objects available: the metric $\eta^{\mu\nu}$ and the single vector
$q^\mu$. Hence
$\Pi^{\mu\nu} = A(q^2)\,\eta^{\mu\nu} + B(q^2)\, q^\mu q^\nu$ with two
unknown functions. Contract with $q_\mu$:
\begin{equation}
q_\mu \Pi^{\mu\nu} = \big[A(q^2) + q^2 B(q^2)\big]\, q^\nu = 0 .
\end{equation}
Since $q^\nu \ne 0$ for generic $q$, $A = -q^2 B$. Writing
$B \equiv -\Pi(q^2)$ gives Eq.~\eqref{eq:E-transverse-form}. Two form
factors have become one. \qed

\paragraph{Verification, component by component.} The contraction is
worth doing in the open, because the cancellation is not a limit or an
approximation but an algebraic identity holding for each value of the
free index separately. With
$q^2 = \eta^{\alpha\beta} q_\alpha q_\beta$,
\begin{equation}
q_\mu \big(q^2\eta^{\mu\nu} - q^\mu q^\nu\big)
 = q^2 q^\nu - q^2 q^\nu = 0 ,
\qquad \nu = 0, 1, 2, 3 ,
\label{eq:E-contraction}
\end{equation}
and there is no step in between: the two terms are the same object with
opposite signs, for $\nu = 0$, for $\nu = 1$, for $\nu = 2$, and for
$\nu = 3$. Nothing was assumed about $\Pi(q^2)$, so nothing about the
dynamics can spoil it.

\paragraph{Why the photon stays massless.} The full photon propagator is
built by summing the geometric series of self-energy insertions,
\begin{equation}
D^{\mu\nu}(q) = \frac{-\ii\, \eta^{\mu\nu}}
 {q^2\big(1 - \Pi(q^2)\big)} + \big(\text{gauge terms} \propto q^\mu q^\nu\big) ,
\label{eq:E-propagator}
\end{equation}
where the transverse structure of Eq.~\eqref{eq:E-transverse-form} has
factored one power of $q^2$ out of every insertion. The physical mass is
the location of the pole. A mass $m_\gamma$ would require the pole to sit
at $q^2 = m_\gamma^2 \neq 0$, that is, it would require the self-energy to
contain a term $m_\gamma^2\, \eta^{\mu\nu}$ with no accompanying $q^2$.
Equation~\eqref{eq:E-transverse-form} forbids exactly that structure: the
only tensor allowed already carries the factor $q^2$, so the pole cannot
move off the origin unless $\Pi(q^2)$ itself has a pole at $q^2 = 0$ to
cancel it. This holds order by order and to all orders, because
Eq.~\eqref{eq:E-transversality} did not come from a diagram.

\paragraph{The honest exception, and why it is not a violation.} The
proviso in the last sentence is not decoration. If $\Pi(q^2)$ does
develop a $1/q^2$ pole --- if the spectrum contains a massless scalar
coupling to the current --- then $q^2 \Pi(q^2) \to m_\gamma^2$ and the
photon is massive with Eq.~\eqref{eq:E-transversality} intact throughout.
That is the Higgs mechanism (\S\ref{ssec:higgs}), and in two dimensions it
happens without any scalar being put in by hand, the massless pole being
generated by the fermions themselves. In every case the identity holds;
what changes is the spectrum feeding it. A Ward identity constrains the
form of an amplitude, not the content of a theory.

\emph{Physics intuition.} This is rigidity with no symmetry left to
break. Gauge invariance is a redundancy of description
(\S\ref{ssec:elitzur}), so there is no gauge charge to measure and
nothing for a vacuum to fail to be invariant under; and yet the photon's
masslessness is protected to all orders. What does the protecting is an
identity among amplitudes, not an unbroken symmetry, and the pattern ---
protection surviving the retreat of symmetry --- is the one
\S\ref{ssec:retreat} follows up the whole ladder.

\subsection{Step 2 --- what the derivation assumes: the measure}
\label{ssec:E-measure}

Return to Step~(iii) of \S\ref{ssec:E-derivation}. The one assumption was
$\mathcal{D}\phi' = \mathcal{D}\phi$. \citet{Fujikawa1979} observed that
for fermions this can be false, and that when it is false the failure is
computable and finite.

\paragraph{Why the vector measure is safe.} A Grassmann change of
variables $\psi' = M\psi$ carries a Jacobian $(\det M)^{-1}$ --- the
inverse of the bosonic one, which is the only fact about Grassmann
integration needed here. Under a \emph{vector} rotation,
\begin{equation}
\psi \to \mathrm{e}^{\ii\alpha}\psi , \qquad
\bar\psi \to \bar\psi\, \mathrm{e}^{-\ii\alpha} ,
\label{eq:E-vector-rot}
\end{equation}
the two factors are $(\det \mathrm{e}^{\ii\alpha})^{-1}$ and
$(\det \mathrm{e}^{-\ii\alpha})^{-1}$. They are reciprocals of one
another, and their product is exactly $1$, with no regularization
required and no residue to discuss. The vector measure is invariant
because $\psi$ and $\bar\psi$ rotate \emph{oppositely}.

\paragraph{Why the axial measure is not.} Under an \emph{axial} rotation
the conjugate field rotates with the \emph{same} sign, and this is
forced, not chosen. Since $\gamma_5^\dagger = \gamma_5$ and
$\{\gamma_5, \gamma^0\} = 0$,
\begin{equation}
\bar\psi' = \psi'^\dagger \gamma^0
 = \psi^\dagger \mathrm{e}^{-\ii\alpha\gamma_5}\gamma^0
 = \psi^\dagger \gamma^0\, \mathrm{e}^{+\ii\alpha\gamma_5}
 = \bar\psi\, \mathrm{e}^{\ii\alpha\gamma_5}
\qquad \text{for} \quad
\psi \to \mathrm{e}^{\ii\alpha\gamma_5}\psi .
\label{eq:E-axial-rot}
\end{equation}
The anticommutation of $\gamma_5$ with $\gamma^0$ has flipped a sign that
did not flip in Eq.~\eqref{eq:E-vector-rot}. The two Jacobians therefore
\emph{add} instead of canceling, and
\begin{equation}
\mathcal{D}\psi'\,\mathcal{D}\bar\psi'
 = \mathcal{D}\psi\,\mathcal{D}\bar\psi \;
   \exp\!\Big[-2\ii \operatorname{Tr}\big(\alpha \gamma_5\big)\Big] .
\label{eq:E-jacobian}
\end{equation}
Naively $\operatorname{Tr}(\alpha\gamma_5)$ is zero times infinity:
$\gamma_5$ is traceless over spinor indices, and the trace over spacetime
is $\delta^4(0)$. The product is meaningless until regularized, and the
regulator must be chosen. Fujikawa's choice is the only natural one ---
regulate with the eigenvalues $\lambda_n$ of the Dirac operator itself,
\begin{equation}
\operatorname{Tr}\big(\alpha\gamma_5\big)
 = \lim_{M \to \infty} \sum_n \int \dd^4 x\; \alpha(x)\,
   \varphi_n^\dagger(x)\, \gamma_5\, \mathrm{e}^{-\lambda_n^2/M^2}\,
   \varphi_n(x) ,
\label{eq:E-regtrace}
\end{equation}
and the limit is finite, gauge invariant, and non-zero:
\begin{equation}
\lim_{M\to\infty}\sum_n
 \varphi_n^\dagger \gamma_5 \mathrm{e}^{-\lambda_n^2/M^2}\varphi_n
 = \frac{1}{32\pi^2}\,
   \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma} ,
\label{eq:E-fujikawa}
\end{equation}
the gauge coupling having been absorbed into $F_{\mu\nu}$ throughout.
Twice the right-hand side of Eq.~\eqref{eq:E-fujikawa} --- the factor of
two being the one carried by Eq.~\eqref{eq:E-jacobian} --- is the anomaly
of Eq.~\eqref{eq:F-anomaly}, obtained here without a single diagram.

\paragraph{Statement (what an anomaly is).} An anomaly is the failure of
a Ward identity. Precisely: when Eq.~\eqref{eq:E-jacobian} is non-trivial,
the extra term propagates through Step~(iii) of
\S\ref{ssec:E-derivation} and Eq.~\eqref{eq:E-master} acquires an
additional, operator-valued piece on the right,
\begin{equation}
\partial_\mu \big\langle T\, J_5^\mu(x)\, \cdots \big\rangle
 = \big(\text{contact terms}\big)
 + \frac{1}{16\pi^2}
   \big\langle T\, \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}(x)\,
   \cdots\big\rangle .
\label{eq:E-anomalous-ward}
\end{equation}
The added term is not small, not a correction, and not removable by any
choice of counterterm; it is a different equation.

\paragraph{And the failure is an integer.} The quantity that appeared is
not an arbitrary function. Integrated over a closed Euclidean spacetime,
\begin{equation}
\frac{1}{32\pi^2}\int \dd^4 x\;
 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}
 \;=\; n_+ - n_- \;\in\; \mathbb{Z} ,
\label{eq:E-index}
\end{equation}
the difference between the numbers of left- and right-handed zero modes
of the Dirac operator: the Atiyah--Singer index
\citep{AtiyahSinger1963}. The integrand is a
characteristic class, its integral over a closed manifold is a topological
invariant, and topological invariants of this kind are integers. So the
amount by which a Ward identity fails is an integer times a fixed
rational --- fixed, because the $1/32\pi^2$ in
Eq.~\eqref{eq:E-index} is not adjustable, and the coefficient is exact at
one loop by the Adler--Bardeen theorem \citep{AdlerBardeen1969}.

\paragraph{Hand-off.} That is the whole reason anomaly cancellation is a
Diophantine condition on the matter content rather than a tuning problem,
and it is where this appendix stops. Appendix~\ref{app:anomalies} takes it
from here: \S\ref{ssec:F-measure} is the same Jacobian told as one of
three independent derivations of the same coefficient,
\S\ref{ssec:F-descent} organizes the failures by degree, and
\S\ref{ssec:F-bi6} does the six-dimensional arithmetic. The one sentence
to carry across is that the identity which fails there is the identity
derived here --- Table~\ref{tab:E-vector-axial} runs the two cases in
parallel, so that the one line where they part company is visible --- and
that it fails at exactly the step flagged in
\S\ref{ssec:E-derivation}(b).

\begin{table}[H]
\centering
\caption[The vector and axial identities compared.]{The same derivation
run on the two currents. Every entry in the right-hand column follows
from the single sign flip in Eq.~\eqref{eq:E-axial-rot}, and the last two
rows are what Appendix~\ref{app:anomalies} is about. Two columns, the vector current and the axial current, with the same derivation run down both. Reading across any row shows where the two part company; reading down the right-hand column shows that everything in it descends from one sign.}
\label{tab:E-vector-axial}\small
\begin{tabular}{@{}P{0.20\textwidth}P{0.35\textwidth}P{0.35\textwidth}@{}}
\toprule
 & \textbf{vector current} & \textbf{axial current} \\
\midrule
transformation &
$\psi \to \mathrm{e}^{\ii\alpha}\psi$, conjugate rotates oppositely &
$\psi \to \mathrm{e}^{\ii\alpha\gamma_5}\psi$, conjugate rotates with the
same sign \\
classical status &
conserved for any mass &
conserved only for $m = 0$; otherwise $2\ii m\,\bar\psi\gamma_5\psi$ \\
measure &
two Jacobians cancel exactly, Eq.~\eqref{eq:E-vector-rot} &
two Jacobians add, Eq.~\eqref{eq:E-jacobian} \\
quantum status &
identity holds, Eq.~\eqref{eq:E-master} &
identity fails by Eq.~\eqref{eq:E-fujikawa} \\
size of the failure & --- & an integer, Eq.~\eqref{eq:E-index} \\
observable &
photon massless, charge universal &
$\pi^0 \to \gamma\gamma$, problem~E.3 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Step 2 (further) --- BRST, and a Bianchi identity for the anomaly}
\label{ssec:E-brst}

Everything so far was abelian. The non-abelian generalization needs one
piece of machinery, and it is introduced here as a statement plus a
worked illustration rather than a construction; \citet{Weinberg1996},
Ch.~17, builds it properly.

\paragraph{Statement (the BRST differential).} After gauge fixing, a
non-abelian gauge theory retains a single global fermionic symmetry
generated by an operator $s$ that acts on the gauge field, the matter
field, and the ghost field $c^a$ by
\begin{equation}
s A_\mu^a = \partial_\mu c^a + g f^{abc} A_\mu^b c^c ,
\qquad
s \psi = \ii g\, c^a T^a \psi ,
\qquad
s c^a = -\tfrac12\, g\, f^{abc}\, c^b c^c ,
\label{eq:E-brst-rules}
\end{equation}
with $s$ odd, so that it anticommutes with anything Grassmann-odd. Its
defining property is
\begin{equation}
s^2 = 0 .
\label{eq:E-nilpotent}
\end{equation}

\paragraph{Worked illustration of Eq.~\eqref{eq:E-nilpotent}.} Take the
ghost, where the computation is three lines and shows exactly what
$s^2 = 0$ costs. Because $c^b$ is odd,
$s(c^b c^c) = (s c^b) c^c - c^b (s c^c)$, so
\begin{equation}
s^2 c^a
 = -\tfrac12 g f^{abc}\Big[\big(sc^b\big) c^c - c^b \big(sc^c\big)\Big]
 = -\tfrac12\, g^2\, f^{abc} f^{cde}\; c^b c^d c^e ,
\label{eq:E-ssc}
\end{equation}
the two brackets having become equal after relabeling $b \leftrightarrow
c$ and using the antisymmetry of $f^{abc}$. Now the ghosts anticommute,
so the product $c^b c^d c^e$ is totally antisymmetric in $(b,d,e)$ and
projects out the totally antisymmetric part of $f^{abc}f^{cde}$ in those
three indices. That part is
\begin{equation}
f^{abc}f^{cde} + f^{adc}f^{ceb} + f^{aec}f^{cbd} \;=\; 0 ,
\label{eq:E-jacobi}
\end{equation}
which is the Jacobi identity of the Lie algebra. Hence $s^2 c^a = 0$, and
the same identity does the same work for $s^2 A_\mu^a = 0$. Nilpotency is
not an extra assumption: it \emph{is} the Jacobi identity, wearing
different clothes.

\paragraph{Statement (Slavnov--Taylor).} The non-abelian generalization of
Eq.~\eqref{eq:E-master} is that the effective action $\Gamma$ --- the
generating functional of one-particle-irreducible amplitudes --- is BRST
invariant,
\begin{equation}
s\,\Gamma \;=\; 0 .
\label{eq:E-slavnov-taylor}
\end{equation}
This is the Slavnov--Taylor identity \citep{Slavnov1972}, obtained
independently by \citet{Taylor1971}. In the abelian limit the ghosts
decouple and Eq.~\eqref{eq:E-slavnov-taylor} reduces to
Eq.~\eqref{eq:E-master}; away from it, Eq.~\eqref{eq:E-slavnov-taylor} is
what makes a non-abelian gauge theory renormalizable, because it relates
the divergences of different vertices exactly as
Eq.~\eqref{eq:E-WT} related $Z_1$ to $Z_2$.

\paragraph{The consistency condition.} Suppose Eq.~%
\eqref{eq:E-slavnov-taylor} fails, and write the failure as
\begin{equation}
s\,\Gamma \;=\; \mathcal{A} .
\label{eq:E-anomaly-functional}
\end{equation}
Apply $s$ to both sides. The left-hand side is $s^2 \Gamma = 0$ by
Eq.~\eqref{eq:E-nilpotent}, so
\begin{equation}
s\, \mathcal{A} \;=\; 0 .
\label{eq:E-wz-cocycle}
\end{equation}
Equation~\eqref{eq:E-wz-cocycle} is the Wess--Zumino consistency
condition \citep{WessZumino1971} in its BRST form --- the same statement
as Eq.~\eqref{eq:F-wz}, which said that two gauge variations of a single
functional cannot depend on the order in which they are performed. It is
a \emph{cocycle} condition: the anomaly is closed under $s$. And a
candidate anomaly of the form $\mathcal{A} = s\,\mathcal{B}$ with
$\mathcal{B}$ a local functional is \emph{exact} and can be removed by
adding $-\mathcal{B}$ as a counterterm. So the genuine anomalies are
the cohomology classes of $s$: closed modulo exact.

\paragraph{The structural point.} Compare
Eq.~\eqref{eq:E-nilpotent} with $\dd\dd = 0$
(Appendix~\ref{app:forms}, Eq.~\eqref{eq:A-ddzero}). They are the same
algebraic fact --- a differential of square zero --- appearing once on
spacetime forms and once on the space of field configurations. From
$\dd\dd = 0$ the body extracted the Bianchi identity
$\dd F = 0$ (Chapter~\ref{sec:bianchi}); from $s^2 = 0$ one extracts
Eq.~\eqref{eq:E-wz-cocycle}, which is a Bianchi identity for the anomaly.
The anomaly is not free to be any functional whatever: it is constrained
by an identity of exactly the kind the rest of this paper is about.

That constraint is what forces the descent. Combining the two
differentials into $\dd + s$, which squares to zero because each factor
does and they anticommute, and applying the result to the invariant
polynomial $I_{D+2}$, generates precisely the chain
\begin{equation}
I_{D+2} = \dd\, \omega_{D+1} ,
\qquad
s\, \omega_{D+1} = \dd\, \omega_D^1 ,
\qquad
s\, \omega_D^1 = 0 ,
\label{eq:E-descent-brst}
\end{equation}
whose first two members are Eq.~\eqref{eq:F-descent1} and
Eq.~\eqref{eq:F-descent2}, and whose last member is
Eq.~\eqref{eq:E-wz-cocycle}. The descent equations of
\S\ref{ssec:F-descent} are therefore not a construction one may or may
not adopt; they are the solution of a cocycle condition that
Eq.~\eqref{eq:E-nilpotent} makes unavoidable, which is what
\S\ref{ssec:F-wz} asserted and this subsection supplies the algebra for.

\paragraph{Grade.} Equations~\eqref{eq:E-nilpotent},
\eqref{eq:E-slavnov-taylor} and \eqref{eq:E-wz-cocycle} are theorems and
are stated here without proof; the illustration
Eq.~\eqref{eq:E-ssc}--\eqref{eq:E-jacobi} is complete as it stands.

\subsection{Step 3 --- what the body does with it}
\label{ssec:E-body}

Chapter~\ref{sec:ward} needs four things from this appendix, and they are
now available.

\begin{enumerate}[leftmargin=1.5em,itemsep=0.25em]
\item \emph{The identity itself, with its right-hand side.}
  Equation~\eqref{eq:E-master}: pure kinematics, exact to all orders,
  and derived from one assumption that can fail.
\item \emph{Two faces of one Noether-II identity.} The geometry half is
  $\partial_\mu \partial_\nu F^{\nu\mu} \equiv 0$
  (Eq.~\eqref{eq:noether-ii}), which vanishes by antisymmetry alone,
  needs no field equation, and holds for every configuration whatever.
  The matter half is $\partial_\mu J^\mu = 0$, which holds on shell for
  the matter and off shell for the gauge field --- the gauge field's
  equation of motion $\partial_\mu F^{\mu\nu} = J^\nu$ makes the current's
  conservation follow from the geometry half without any use of the
  matter dynamics. One local symmetry, one identity, read twice.
\item \emph{The gravitational parallel.} \S\ref{ssec:einstein} already
  ran the same argument on the second Bianchi identity: contracting it
  gives $\nabla_\mu G^{\mu\nu} \equiv 0$ (Eq.~\eqref{eq:div-G}), and the
  field equation then forces $\nabla_\mu T^{\mu\nu} = 0$ on the source.
  The logical shape is identical --- an off-shell identity on the
  geometry side, transmitted to the matter side by the field equation ---
  and Appendix~\ref{app:noether}'s Table~\ref{tab:B-two-theorems} is
  where the two columns are set against each other.
\item \emph{The open-system reading.} Part~\ref{part:turn} takes the
  right-hand side of Eq.~\eqref{eq:E-master} and adds a term to it: when
  the boundary theory is not the whole story, the current's divergence
  acquires a flux, and the identity becomes a budget rather than a
  prohibition (\S\ref{ssec:pivot}). The mechanism by which a bulk
  supplies exactly the missing flux is anomaly inflow
  \citep{CallanHarvey1985}, and its use in this program is worked out in \citet[Appendix~H]{ArisakaP2N3}. What is imported there is
  Eq.~\eqref{eq:E-master} with a source, and what makes the import legal
  is that the left-hand side never changed.
\end{enumerate}

\subsection{Problems and worked solutions}
\label{ssec:E-problems}

\medskip\noindent\textbf{E.1. Verify the tree-level Ward--Takahashi
identity Eq.~\eqref{eq:E-WT} by explicit gamma-matrix algebra, and say
which property of the Dirac algebra was actually used.}

\noindent\emph{Solution.} \emph{Step 1.} At tree level
$\Gamma^\mu = \gamma^\mu$, so the left-hand side is
$q_\mu\gamma^\mu = \slashed{q}$. \emph{Step 2.} With
$S^{-1}(p) = \slashed{p} - m$, and with
$(p+q)_\mu\gamma^\mu = \slashed{p} + \slashed{q}$ by linearity, the
right-hand side is
$(\slashed{p} + \slashed{q} - m) - (\slashed{p} - m)
 = \slashed{q}$.
\emph{Step 3.} The two agree, and the only property used was
\emph{linearity} of $v \mapsto \slashed{v}$; the anticommutator
$\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}$ never entered, and neither
did the value of $m$, which canceled between the two terms.
\emph{Physics intuition:} the identity constrains a \emph{difference} of
inverse propagators, and a mass sits identically in both ends of the
difference, so the vector identity is blind to it --- whereas the axial
identity involves an anticommutator with $\gamma_5$ in which the same
two mass terms add to $2m\gamma_5$ instead of canceling, which is why
one identity survives quantization and the other does not.

\medskip\noindent\textbf{E.2. Show that Eq.~\eqref{eq:E-transverse-form}
is the unique tensor structure allowed by
Eq.~\eqref{eq:E-transversality}, and contract it to zero for each value
of the free index.}

\noindent\emph{Solution.} \emph{Step 1.} By Lorentz covariance the only
available tensors are $\eta^{\mu\nu}$ and $q^\mu q^\nu$ --- there is no
other vector in the problem --- so
$\Pi^{\mu\nu} = A\eta^{\mu\nu} + B q^\mu q^\nu$ with $A, B$ functions of
$q^2$ alone. \emph{Step 2.} Contracting,
$q_\mu\Pi^{\mu\nu} = (A + q^2 B)q^\nu$, which must vanish for generic
$q$, so $A = -q^2 B$; writing $B = -\Pi(q^2)$ gives
Eq.~\eqref{eq:E-transverse-form}, and two unknown functions have become
one. \emph{Step 3.} Contract the answer:
$q_\mu(q^2\eta^{\mu\nu} - q^\mu q^\nu) = q^2 q^\nu - q^2 q^\nu = 0$,
separately and identically for $\nu = 0, 1, 2, 3$, as
Eq.~\eqref{eq:E-contraction} records. \emph{Physics intuition:}
transversality spends one of the two form factors, and the one that
survives is a single function whose only chance of moving the photon's
pole off $q^2 = 0$ is to have a pole of its own there --- which is
precisely what the Higgs mechanism supplies and what nothing in
perturbative quantum electrodynamics does.

\medskip\noindent\textbf{E.3. The neutral pion decays through the axial
anomaly. Using
$\Gamma = \dfrac{\alpha^2 m_{\pi^0}^3}{64\pi^3 f_\pi^2}
 \left(\dfrac{N_c}{3}\right)^{\!2}$
with $\alpha = 1/137.036$, $m_{\pi^0} = 134.9768$~MeV and
$f_\pi = 92.21$~MeV, compute the width for $N_c = 3$ and for $N_c = 1$,
and compare with the measured $7.802 \pm 0.117$~eV.}

\noindent\emph{Solution.} \emph{Step 1.} Assemble the pieces:
$\alpha^2 = 5.32514\times 10^{-5}$;
$m_{\pi^0}^3 = 2.459107\times 10^{6}~\text{MeV}^3$;
$64\pi^3 = 1984.402$; $f_\pi^2 = 8502.684~\text{MeV}^2$.
\emph{Step 2.} For $N_c = 3$ the bracket is $1$, and
\begin{equation}
\Gamma = \frac{5.32514\times10^{-5} \times 2.459107\times10^{6}}
 {1984.402 \times 8502.684}\ \text{MeV}
 = \frac{130.951}{1.687274\times 10^{7}}\ \text{MeV}
 = 7.761\ \text{eV} ,
\label{eq:E-pi0}
\end{equation}
against the measured $7.802 \pm 0.117$~eV --- agreement to $0.5\%$, which
is $0.35$ standard deviations, with no free parameter anywhere in the
calculation. \emph{Step 3.} For $N_c = 1$ the bracket is $(1/3)^2 = 1/9$
and $\Gamma = 0.862$~eV, low by a factor of exactly nine and wrong by
some sixty standard deviations. \emph{Physics intuition:} the number of
colors is being read off a decay rate, and it can be read off because
the anomaly coefficient is an integer count of fermion species rather
than a coupling that could be adjusted --- the width is quadratic in
that count, so a wrong count is not slightly wrong but wrong by a
factor. This is the observable face of Eq.~\eqref{eq:E-index}, and it is
the reason \S\ref{ssec:F-triangle} can call the anomaly's coefficient a
measurement rather than a convention.

\medskip\noindent\textbf{E.4. Show that the anomaly coefficient of a
global symmetry cannot change under renormalization-group flow
\citep{tHooft1980}, and say in one sentence why an integer is what makes
the argument work.}

\noindent\emph{Solution.} \emph{Step 1.} Weakly gauge the global symmetry
by coupling it to a background gauge field, and add \emph{spectator}
fermions --- fields charged under the global symmetry but inert under the
strong dynamics --- chosen so that the total anomaly coefficient
vanishes. By \S\ref{ssec:fatal} the resulting theory is consistent, and
it is consistent at every scale, since anomaly freedom is not a statement
about a scale. \emph{Step 2.} Flow to the infrared. The strong dynamics
confines and produces massless composite fermions; the spectators, by
construction, do not feel it, so their contribution
$\mathcal{A}_{\text{spec}}$ is literally the same object at both ends of
the flow. \emph{Step 3.} Consistency at the top requires
$\mathcal{A}_{\text{UV}} + \mathcal{A}_{\text{spec}} = 0$ and
consistency at the bottom requires
$\mathcal{A}_{\text{IR}} + \mathcal{A}_{\text{spec}} = 0$; subtracting,
$\mathcal{A}_{\text{IR}} = \mathcal{A}_{\text{UV}}$. The massless
composite spectrum must reproduce a number computed from the elementary
fields. \emph{Physics intuition:} the argument works because
$\mathcal{A}$ is an integer --- a sum of cubes of charges over a discrete
spectrum, exact at one loop by Adler--Bardeen \citep{AdlerBardeen1969}
--- and a renormalization-group flow is continuous, so a continuous
function taking values in the integers cannot move; had the coefficient
been a continuous quantity the matching would say nothing, and the same
observation is what \S\ref{ssec:counting-survives} generalizes.

\medskip\noindent\textbf{E.5. Show that $Z_1 = Z_2$ makes the renormalized
charge species-independent, and then show that this does \emph{not}
imply that the charges of different species are integer multiples of one
another.}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:E-charge}, $e = (Z_2/Z_1)\sqrt{Z_3}\, e_0$, and
$Z_1 = Z_2$ for \emph{every} charged species separately, because the
Ward--Takahashi identity Eq.~\eqref{eq:E-WT} was derived for each species
separately. The species-dependent factors therefore cancel one by one,
and the only surviving renormalization is $Z_3$, which belongs to the
photon. \emph{Step 2.} Hence for any two species
$e_a/e_b = e_{0a}/e_{0b}$ at every scale, Eq.~\eqref{eq:E-universality}:
the ratio of two charges is the same at all energies, however different
the two species are and whatever interactions they have. \emph{Step 3.}
But the bare ratio is an input, not an output. Nothing above forbids
$e_{0a}/e_{0b} = -1 + 10^{-22}$; that world satisfies
Eq.~\eqref{eq:E-master}, Eq.~\eqref{eq:E-WT} and
Eq.~\eqref{eq:E-universality} exactly, and differs from this one only in
a number nobody put there. The observed integrality
(\S\ref{ssec:integers}, and the $10^{-21}$ bound of
\citet{MarinelliMorpurgo1984}) is therefore \emph{not} explained by any
Ward identity. \emph{Physics intuition:} universality is a statement
about \emph{running} and quantization is a statement about \emph{values};
a dynamical argument can protect a ratio from change but only a
topological one can select it, which is why Chapter~\ref{sec:ward} is a
rung below Chapter~\ref{sec:hs} and not a substitute for it.

\GGtakeaway{\emph{Standard result:} the master Ward identity
Eq.~\eqref{eq:E-master}, derived from a local change of integration
variables with the contact terms kept; its consequences
$q_\mu\Gamma^\mu = S^{-1}(p+q) - S^{-1}(p)$, $Z_1 = Z_2$, the
species-independence of the renormalized charge, and the transversality
that keeps the photon massless to all orders. \emph{What it assumed:}
that the path-integral measure is invariant under the transformation ---
one assumption, isolated at Eq.~\eqref{eq:E-zero}, everything else being
calculus. \emph{What follows when the assumption fails:} the identity
fails by Eq.~\eqref{eq:E-fujikawa}, the failure is the integral of a
characteristic class and hence an integer, Eq.~\eqref{eq:E-index}; and
the nilpotency $s^2 = 0$ of the BRST differential imposes on that failure
a Bianchi identity of its own, Eq.~\eqref{eq:E-wz-cocycle}, which is what
forces the descent. \emph{What it does not give:} charge quantization ---
Eq.~\eqref{eq:E-universality} protects a ratio, it does not choose one.
\emph{Continue in:} Appendix~\ref{app:anomalies} for what the failure
costs and how it is organized, and Chapter~\ref{sec:hs} for the argument
that does select the integer.}


\section{Anomalies, told three ways}
\label{app:anomalies}
\GGteachbox{The chiral anomaly derived three independent ways --- by
counting Landau levels, by the triangle diagram, and by the
non-invariance of the path-integral measure --- so that the reader can
see that the coefficient is not an artifact of any one method. Then the
descent equations, at exactly the level Chapter~\ref{sec:gs} needs, and
the \BIsix{} coefficient arithmetic carried out by hand. The pay-off is
\S\S\ref{ssec:anomalies}--\ref{ssec:bi6}.}

\GGreadalong{\citet{PeskinSchroeder1995}, Ch.~19, for the triangle and
for Fujikawa's argument; \citet{Weinberg1996}, Ch.~22, for anomalies at
full rigor including the descent; \citet{NielsenNinomiya1983} for the
spectral-flow picture used first below;
\citet{GreenSchwarz1984} for the mechanism itself.}

\subsection{Step 1, first telling --- counting Landau levels}
\label{ssec:F-landau}

This is the derivation used in the body, done here from the degeneracy
up.

\paragraph{The lowest Landau level.} A particle of charge $e$ in a
uniform field $B\hat{z}$ has transverse motion quantized into Landau
levels. The degeneracy per unit transverse area is the number of flux
quanta per unit area,
\begin{equation}
\frac{\text{states}}{\text{area}}
 = \frac{B}{\Phi_0} = \frac{eB}{2\pi} ,
\label{eq:F-degeneracy}
\end{equation}
in units $\hbar = c = 1$, with $\Phi_0 = 2\pi/e$ the flux quantum of
\S\ref{ssec:third-datum}. For a massless fermion the lowest level is
special: it has no transverse zero-point energy, its dispersion is
$E = \pm p_z$ with the sign fixed by the handedness, and it is therefore
a one-dimensional \emph{chiral} mode.

\paragraph{Spectral flow.} Switch on $E\hat z$ parallel to $B$. Each
mode's momentum obeys $\dd p_z/\dd t = eE$, so the whole one-dimensional
spectrum slides. States cross zero energy at a rate given by the
one-dimensional density of states times the drift velocity,
\begin{equation}
\frac{\text{crossings}}{\text{length}\times\text{time}}
 = \frac{1}{2\pi}\frac{\dd p_z}{\dd t} = \frac{eE}{2\pi} .
\label{eq:F-flow}
\end{equation}

\paragraph{The anomaly.} Multiplying Eq.~\eqref{eq:F-degeneracy} by
Eq.~\eqref{eq:F-flow} gives the rate at which particles of the given
handedness appear:
\begin{equation}
\frac{\dd n}{\dd t}
 = \frac{eB}{2\pi}\cdot\frac{eE}{2\pi}
 = \frac{e^2}{4\pi^2}\,\mathbf{E}\cdot\mathbf{B} ,
\label{eq:F-anomaly}
\end{equation}
which is Eq.~\eqref{eq:anomaly-rate}. In covariant form,
$\partial_\mu j^\mu_5 = (e^2/16\pi^2)\,
\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$.

\emph{Physics intuition.} The step that does the damage is the one that
looks like bookkeeping: the Dirac sea is bottomless, so states can be
drawn from it indefinitely. In a lattice regularization the sea has a
bottom and the anomaly cancels exactly --- which is the Nielsen--Ninomiya
theorem, and the reason a chiral gauge theory is hard to put on a
lattice.

\subsection{Step 1, second telling --- the triangle}
\label{ssec:F-triangle}

The historical derivation \citep{Adler1969,BellJackiw1969} computes the
three-point function of one axial and two vector currents. The result is
that no regularization preserves both the vector and the axial Ward
identities: one may keep either, but not both.

The structure is worth stating even without the computation. The
amplitude is superficially linearly divergent, so a shift of the loop
momentum --- normally legitimate --- produces a finite surface term. The
anomaly is that surface term. It is therefore \emph{exactly} one loop:
there are no higher-order corrections, because the ambiguity that
produces it exists only at the level of the shift. This is the
Adler--Bardeen theorem \citep{AdlerBardeen1969}, and it is why the
coefficient in
Eq.~\eqref{eq:F-anomaly} is an exact statement rather than a leading
approximation.

The physical choice is forced. Gauge invariance of the vector current is
not optional (\S\ref{ssec:fatal}), so the anomaly is assigned to the
axial current, and the observable consequence is the decay
$\pi^0 \to \gamma\gamma$ at the rate the anomaly predicts.

\subsection{Step 1, third telling --- the measure}
\label{ssec:F-measure}

\citet{Fujikawa1979} locates the anomaly in the path-integral measure. A
chiral rotation of the fermion fields is a change of integration
variables, and a change of variables has a Jacobian. For a chiral
rotation the Jacobian is not one; regularized with the Dirac operator's
own eigenvalues, it is precisely $\exp$ of the anomaly.

In words, and this is the version worth carrying: the classical action is
invariant, but the \emph{measure} is not, and the path integral is the
action \emph{and} the measure. A symmetry of the Lagrangian is not
automatically a symmetry of the quantum theory, because quantizing
requires a choice of measure and that choice can break what the
Lagrangian preserved.

\emph{Physics intuition: why three derivations.} The three tellings
share no machinery --- one counts states in a magnetic field, one shifts
a loop momentum, one computes a Jacobian --- and they agree on the
coefficient. That is the strongest kind of evidence available for a
result of this type, and it is the same evidential pattern that
\S\ref{ssec:kappa} points to at the top of the ladder.

\subsection{Step 2 --- the descent equations}
\label{ssec:F-descent}

Chapter~\ref{sec:gs} needs one structural fact about how anomalies are
organized, and it is a statement about degrees.

Start from a gauge-invariant polynomial in the curvature,
$I_{D+2}$, of degree $D+2$ in a $D$-dimensional theory. It is closed,
$\dd I_{D+2} = 0$, and by the Poincar\'e lemma locally exact:
\begin{equation}
I_{D+2} = \dd\,\omega_{D+1} ,
\label{eq:F-descent1}
\end{equation}
with $\omega_{D+1}$ the Chern--Simons form. That form is \emph{not}
gauge invariant; its gauge variation is itself closed and hence locally
exact,
\begin{equation}
\delta_\epsilon\, \omega_{D+1} = \dd\, \omega^1_{D} ,
\label{eq:F-descent2}
\end{equation}
and $\omega^1_D$ --- of degree exactly $D$, integrable over spacetime ---
is the anomaly. The chain of degrees is
\begin{equation}
(D+2) \;\longrightarrow\; (D+1) \;\longrightarrow\; D ,
\end{equation}
so in ten dimensions $12 \to 11 \to 10$, and in six dimensions
$8 \to 7 \to 6$.

\paragraph{Statement (what the \GS{} mechanism needs).} If the
degree-$(D+2)$ polynomial \emph{factorizes},
\begin{equation}
I_{D+2} = X_4 \wedge X_{D-2} ,
\label{eq:F-factorisation}
\end{equation}
then the anomaly can be canceled by a two-form field $B$ whose field
strength satisfies $\dd H_3 = X_4$ and which contributes a counterterm
proportional to $\int B \wedge X_{D-2}$. Factorization is the whole
condition, and it is a strong arithmetic requirement on the gauge group
and matter content --- in ten dimensions it forces, among other things,
the dimension $496$ discussed in \S\ref{ssec:let-it-fail}.


\subsection{Step 2 (further) --- why the descent is forced}
\label{ssec:F-wz}

Equation~\eqref{eq:F-descent1} and Eq.~\eqref{eq:F-descent2} were
presented as a chain one may write down. In fact the chain is forced, and
the reason is a consistency condition that costs nothing to state and
constrains a great deal.

\paragraph{Statement (Wess--Zumino \citep{WessZumino1971}).} Let $G(\epsilon)$ denote the gauge
variation of the effective action under a transformation with parameter
$\epsilon$. Because gauge transformations form a group, their variations
must satisfy the same algebra:
\begin{equation}
\delta_{\epsilon_1} G(\epsilon_2) - \delta_{\epsilon_2} G(\epsilon_1)
 = G\!\left([\epsilon_1, \epsilon_2]\right) .
\label{eq:F-wz}
\end{equation}
Any candidate anomaly must satisfy Eq.~\eqref{eq:F-wz}. This is not a
dynamical requirement --- it is the statement that the variation of a
single functional cannot depend on the order in which two
transformations are performed.

\paragraph{Why this forces the descent.} Equation~\eqref{eq:F-wz} says
that $G$ is a cocycle for the gauge algebra. Solutions of a cocycle
condition modulo trivial ones (those removable by a local counterterm)
form a cohomology, and the descent chain
Eqs.~\eqref{eq:F-descent1}--\eqref{eq:F-descent2} is precisely the
construction that produces the non-trivial solutions from the invariant
polynomial. In other words, the anomaly is not free to be any functional
of the fields; it is essentially unique in each dimension, and the
descent is the algorithm that finds it.

The same construction runs for gravitational anomalies, where the
invariant polynomial is built from the curvature two-form of the tangent
bundle \citep{AlvarezGaumeWitten1984}; the $\tr R^2$ term of
Eq.~\eqref{eq:F-X4} is its trace.

Three consequences are used in Chapter~\ref{sec:gs}.

\begin{enumerate}[leftmargin=1.5em,itemsep=0.25em]
\item \emph{Uniqueness up to counterterms.} There is nothing to tune. One
  cannot arrange a smaller anomaly by adjusting the theory's local
  couplings; only the matter content changes the answer.
\item \emph{Additivity over species.} The anomaly is linear in the
  contributions of the individual fermions, which is why the cancellation
  conditions Eq.~\eqref{eq:sm-anomaly} are \emph{sums} over the spectrum,
  and hence why they are Diophantine.
\item \emph{The counterterm is available.} The \GS{} mechanism works
  because $\int B \wedge X_{D-2}$ is a legitimate local term whose gauge
  variation is exactly of the descent form. The mechanism is not a trick;
  it is the use of the one freedom the cohomology leaves.
\end{enumerate}

\emph{Physics intuition.} Wess--Zumino is a consistency requirement of
the same species as Dirac's single-valuedness (\S\ref{ssec:dirac}) and
the cocycle condition Eq.~\eqref{eq:C-cocycle}: go around a closed path
in the space of descriptions and come back to where you started. That
this is the third appearance of the same demand, in three quite different
settings, is not an accident of exposition --- it is the pattern
\S\ref{ssec:retreat} identifies as what replaces symmetry as one climbs.
\subsection{Step 3 --- the \texorpdfstring{\BIsix{}}{BI-6} arithmetic, by hand}
\label{ssec:F-bi6}

The six-dimensional instance used in Part~III has
\begin{equation}
X_4 = \tr R^2 - \TR F_5^2
 - \gamma\, F_B^2 - \gamma\, F_{Qi}^2 ,
\qquad \gamma = \frac{L-N}{L} ,
\label{eq:F-X4}
\end{equation}
and the reader should now check every number in it.

\begin{enumerate}[leftmargin=1.5em,itemsep=0.25em]
\item \emph{The ledger.} $(M,N,L) = (1,3,5)$ gives
  $M^2+N^2+L^2 = 1+9+25 = 35$ and $J = 2N+1 = 7$, so $LJ = 35$ also. Two
  routes to the same integer.
\item \emph{The ratio.} $\gamma = (5-3)/5 = 2/5$, so both abelian
  coefficients are $-2/5$ and there is no independent second parameter.
\item \emph{Primitivity.} The coefficient vector is
  $(1, -1, -\tfrac25, -\tfrac25)$; the least positive integer clearing
  every denominator is $5$, since $5 \times \tfrac25 = 2$. So the
  integral lift carries the factor $5$, and $\gcd(2,5) = 1$ makes it
  primitive.
\item \emph{What is not fixed.} The ratio is satisfied by
  $(L,N) = (5k, 3k)$ for every positive integer $k$. The selection
  $k = 1$ is made by an axiom, not by Eq.~\eqref{eq:F-X4}
  (\S\ref{ssec:exposure}).
\end{enumerate}

That is the complete arithmetic of the coefficients, and it takes four
lines. Everything else about the six-dimensional identity --- closure,
the global analysis, the refinement --- is the subject of the member
paper \citep{ArisakaA3BI6} and is not re-derived anywhere in this review.

\subsection{Problems and worked solutions}
\label{ssec:F-problems}

\medskip\noindent\textbf{F.1. Redo Eq.~\eqref{eq:F-anomaly} for a fermion
of charge $Q e$. By what factor does the rate change?}

\noindent\emph{Solution.} \emph{Step 1.} Eq.~\eqref{eq:F-degeneracy}
scales as $Q$; Eq.~\eqref{eq:F-flow} scales as $Q$. \emph{Step 2.} The
product scales as $Q^2$. \emph{Step 3.} For $Q = 2$ the rate is four
times larger. \emph{Physics intuition:} the anomaly is quadratic in the
charge for an abelian vector current and \emph{cubic} for the gauge
anomaly of a chiral theory, where all three legs of the triangle carry
the charge --- which is why \S\ref{ssec:fatal}'s condition is
$\sum q_i^3 = 0$ and not $\sum q_i^2 = 0$.

\medskip\noindent\textbf{F.2. Show that a single Weyl fermion of unit
charge cannot be gauged, and that two of opposite charge can but are not
chiral.}

\noindent\emph{Solution.} \emph{Step 1.} $\sum q^3 = 1 \ne 0$, so the
gauge anomaly is non-zero and the theory does not exist.
\emph{Step 2.} With charges $\{+1,-1\}$, $\sum q = 0$ and
$\sum q^3 = 0$. \emph{Step 3.} But a Dirac mass term $q + (-q) = 0$ is
gauge invariant, so the pair can acquire a mass and leave the spectrum.
\emph{Physics intuition:} anomaly freedom is necessary and not
sufficient for interesting chiral matter; \S\ref{ssec:fatal} exhibits the
smallest genuinely chiral solution, and it needs five fields.

\medskip\noindent\textbf{F.3. Verify the four Standard-Model conditions
of Eq.~\eqref{eq:sm-anomaly} using Table~\ref{tab:sm-anomaly}.}

\noindent\emph{Solution.} \emph{Step 1.} $\sum Y_6 = 12 - 24 + 12 - 12 +
12 = 0$. \emph{Step 2.} $\sum Y_6^3 = 48 - 1536 + 192 - 432 + 1728 = 0$.
\emph{Step 3.} Doublets: $3(+2) + 1(-6) = 0$; triplets:
$2(+2) + (-8) + (+4) = 0$. \emph{Physics intuition:} the third column is
the striking one --- the cubes are large and unrelated-looking, and they
cancel exactly, which is very hard to regard as coincidence.

\medskip\noindent\textbf{F.4. In $D = 6$, what are the degrees in the
descent chain, and what degree must the factor $X_{D-2}$ have?}

\noindent\emph{Solution.} \emph{Step 1.} $(D+2, D+1, D) = (8, 7, 6)$.
\emph{Step 2.} Eq.~\eqref{eq:F-factorisation} needs
$4 + (D-2) = D+2$, so $X_{D-2} = X_4$. \emph{Step 3.} In six dimensions
the anomaly polynomial must factorize as $X_4 \wedge X_4$ --- the same
four-form twice. \emph{Physics intuition:} that is why the
six-dimensional case is the one in which a single four-form carries the
whole mechanism, and it is the structural reason
Eq.~\eqref{eq:F-X4} has the shape it does.

\medskip\noindent\textbf{F.5. Why is the anomaly coefficient exact at one
loop?}

\noindent\emph{Solution.} \emph{Step 1.} The anomaly arises from the
surface term generated by shifting the loop momentum in a linearly
divergent integral. \emph{Step 2.} That ambiguity exists only at one
loop; higher loops are convergent enough that no such term arises.
\emph{Step 3.} Hence the Adler--Bardeen theorem: no corrections.
\emph{Physics intuition:} an exact coefficient is what makes anomaly
cancellation a Diophantine condition on integers rather than an
approximate constraint that could be repaired order by order --- and
that exactness is what lets Chapter~\ref{sec:gs} treat it as a ledger.

\medskip\noindent\textbf{F.6. Check the four numbers of
\S\ref{ssec:F-bi6} without a calculator, and state which of them is
\emph{not} determined by the trace ratio.}

\noindent\emph{Solution.} \emph{Step 1.} $5 \times 7 = 35$;
$1 + 9 + 25 = 35$; $(5-3)/5 = 2/5$; $5 \times 2/5 = 2$.
\emph{Step 2.} All four follow from $(M,N,L) = (1,3,5)$. \emph{Step 3.}
The choice of $(1,3,5)$ over $(1,6,10)$ is \emph{not} determined by the
ratio, which admits $(5k,3k)$ for all $k$. \emph{Physics intuition:} the
ratio fixes the relationship between the coefficients; an axiom fixes the
scale, and \S\ref{ssec:exposure} names that axiom as the program's most
exposed premise.

\GGtakeaway{\emph{Standard result:} the chiral anomaly, with the same
coefficient from three unrelated derivations, exact at one loop; and the
descent equations organizing anomalies by degree,
$(D+2) \to (D+1) \to D$. \emph{What it assumed:} a bottomless Dirac sea
in the first telling, a shift of loop momentum in the second, a
regularized measure in the third --- and their agreement is the evidence
that none of the three assumptions is doing the work. \emph{The reading
this paper takes:} an anomalous \emph{gauge} symmetry is not a small
effect but a non-existent theory, which is what makes anomaly
cancellation a hard integer condition and lets \S\ref{ssec:currency}
treat conservation as a ledger. \emph{Continue in:}
\S\ref{ssec:let-it-fail} for the mechanism, and
Appendix~\ref{app:diffcohom} for what happens to the integers at the top
of the ladder.}


\section{Differential cohomology and quadratic refinements}
\label{app:diffcohom}
\GGteachbox{The three data of a gauge field --- an integer class, a
curvature form, a holonomy phase --- assembled into a single object, the
differential cocycle; the two exact sequences that say precisely what
that object is made of; Dirac quantization recovered as the simplest
case; quadratic refinements over the integers, worked at the
characteristic element $a = 1$; Wu classes and the dimension count that
makes six dimensions special; and what \HS{} adds to all of it. This is
the hardest appendix in the paper, and it is the one
Chapter~\ref{sec:hs} stands on.}

\GGreadalong{\citet{BottTu1982}, Ch.~I--II, for de Rham theory and the
exact sequences; \citet{Nakahara2003}, Ch.~11, for characteristic
classes; then the primary literature ---
\citet{CheegerSimons1985} for differential characters,
\citet{Witten1997} for the shift, and \citet{HopkinsSinger2005} for the
functor. The last three are papers, not textbooks, and are read after
the first two.}

\subsection{Step 1 --- three data, one object}
\label{ssec:G-step1}

Appendix~\ref{app:connections} left a gauge field described by three
things that are not independent and not reducible to one another:

\begin{itemize}[leftmargin=1.4em,itemsep=0.2em]
\item a \emph{characteristic class} $c \in H^{k}(X;\mathbb{Z})$, an
  integer-valued cohomology class --- the winding number of
  \S\ref{ssec:dirac}, the Chern class of the bundle;
\item a \emph{curvature} $\omega \in \Omega^{k}(X)$, a closed
  differential form with integral periods;
\item a \emph{holonomy} $h$, the phase attached to each cycle, which
  interpolates between them.
\end{itemize}

A \emph{differential cocycle} is the triple $(c, h, \omega)$ together
with one compatibility condition. In the model due to
\citet{HopkinsSinger2005} the condition is written
\begin{equation}
\delta c = 0, \qquad
\dd \omega = 0, \qquad
\delta h = \omega - c ,
\label{eq:G-cocycle}
\end{equation}
where $\delta$ is the cochain coboundary --- a cochain assigns a number
to each simplex of a triangulation, and $\delta$ takes a cochain to its
value on boundaries, the combinatorial twin of $\dd$ --- and the last
equation says that
the holonomy is the object whose failure to be closed measures exactly
the difference between the smooth data and the integral data. Two
cocycles are equivalent if they differ by a coboundary, and the
equivalence classes form the \emph{differential cohomology group}
$\check{H}^{k}(X)$.

\emph{Physics intuition.} Compare a real number written as an integer
part plus a fractional part. Neither piece alone determines the number;
one needs both, plus the rule relating them. Equation~\eqref{eq:G-cocycle}
is that arrangement one degree up: $c$ is the integer part, $\omega$ the
continuous part, and $h$ the rule. Every quantization statement in this
paper is a statement about the integer part.

\subsection{Step 1 (continued) --- the two exact sequences}
\label{ssec:G-sequences}

What $\check{H}^{k}$ is made of is stated exactly by two short exact
sequences, and they are worth reading slowly because between them they
contain the whole subject.

\paragraph{Sequence one: the curvature map.}
\begin{equation}
0 \;\longrightarrow\;
 H^{k-1}(X;\mathbb{R}/\mathbb{Z})
 \;\longrightarrow\;
 \check{H}^{k}(X)
 \;\xrightarrow{\ \ \text{curv}\ \ }\;
 \Omega^{k}_{\mathbb{Z}}(X)
 \;\longrightarrow\; 0 .
\label{eq:G-seq1}
\end{equation}
Read left to right: the kernel of the curvature map is the group of
\emph{flat} classes --- configurations with zero field strength but
non-trivial holonomy. Those are exactly the Aharonov--Bohm
configurations of \S\ref{ssec:C-ab}. The sequence says that they are
classified by $H^{k-1}(X;\mathbb{R}/\mathbb{Z})$, and in particular that
they exist whenever that group is non-zero, which is a statement about
the topology of $X$ and not about the field.

\paragraph{Sequence two: the characteristic-class map.}
\begin{equation}
0 \;\longrightarrow\;
 \Omega^{k-1}(X)/\Omega^{k-1}_{\mathbb{Z}}(X)
 \;\longrightarrow\;
 \check{H}^{k}(X)
 \;\xrightarrow{\ \ \text{char}\ \ }\;
 H^{k}(X;\mathbb{Z})
 \;\longrightarrow\; 0 .
\label{eq:G-seq2}
\end{equation}
Here the kernel is the group of \emph{topologically trivial}
configurations --- those with vanishing integer class, which can be
deformed continuously to nothing. The sequence says that these are
parametrized by forms modulo forms with integral periods, i.e. by
continuous data.

\paragraph{Statement (the point).} The two sequences pull the object
apart in two complementary directions. The first isolates what has no
curvature; the second isolates what has no integer. What is left in both
cases is continuous and deformable. \emph{The only part of a differential
cocycle that cannot be continuously deformed is the class in
$H^{k}(X;\mathbb{Z})$}, and that is the whole content of
\S\ref{ssec:cocycle}'s claim that the top of the ladder separates the
movable from the immovable.

\subsection{Step 1 (continued) --- Dirac as the simplest case}
\label{ssec:G-dirac}

Take $X = S^2$ surrounding a monopole, $k = 2$. Then
$H^2(S^2;\mathbb{Z}) = \mathbb{Z}$ and
$H^1(S^2;\mathbb{R}/\mathbb{Z}) = 0$. Equation~\eqref{eq:G-seq1} therefore
says there are no flat non-trivial configurations --- consistent with
$S^2$ being simply connected --- and Eq.~\eqref{eq:G-seq2} says the
configurations are labeled by an integer, with continuous data on top.

That integer is Dirac's $n$. The quantization condition
\begin{equation}
\frac{1}{2\pi}\oint_{S^2} F \;=\; n \in \mathbb{Z}
\label{eq:G-dirac}
\end{equation}
is the statement that the curvature's periods are integral, which is the
condition $\omega \in \Omega^{k}_{\mathbb{Z}}$ appearing on the right of
Eq.~\eqref{eq:G-seq1}. Nothing has been added to \S\ref{ssec:dirac}; the
machinery has simply named the group that the argument was computing in.

\subsection{Step 2 --- quadratic refinements, worked}
\label{ssec:G-refinement}

A pairing on a lattice $\Lambda$ is a bilinear map
$\langle\cdot,\cdot\rangle$. A \emph{quadratic refinement} of it is a
function $q : \Lambda \to \mathbb{Z}$ whose failure to be additive
reproduces the pairing:
\begin{equation}
q(x+y) - q(x) - q(y) + q(0) = \langle x, y \rangle .
\label{eq:G-refinement}
\end{equation}

\paragraph{Statement.} On $\Lambda = \mathbb{Z}$ with the unit pairing
$\langle x, y\rangle = xy$, the integer-valued solutions of
Eq.~\eqref{eq:G-refinement} with $q(0) = 0$ are exactly
\begin{equation}
q_a(x) = \frac{x^2 - a x}{2}, \qquad a \text{ odd} ,
\label{eq:G-qa}
\end{equation}
and $a = 1$ is the smallest positive choice, giving
$q(x) = x(x-1)/2$.

\paragraph{Proof.} Substituting Eq.~\eqref{eq:G-qa} into
Eq.~\eqref{eq:G-refinement} gives $xy$ for any $a$, so the functional
equation fixes the quadratic part and leaves the linear coefficient free.
Integrality is what fixes $a$: $x^2 - ax = x(x-a)$ is even for every
integer $x$ if and only if $x$ and $x-a$ have the same parity for every
$x$, which requires $a$ to be odd. For $a$ even the function fails at
every odd $x$ --- at $a = 2$ and $x = 1$ it returns $-\tfrac12$. \qed

The integer $a$ is the \emph{characteristic element}, and it is the
arithmetic shadow of the half-shift of \S\ref{ssec:witten-shift}. The
elementary signature is the increment: $x(x-1)$ steps by $2x$, always
even, while $x^2$ steps by $2x+1$, always odd.

\subsection{Step 2 (continued) --- Wu classes and the six-dimensional count}
\label{ssec:G-wu}

Which $a$ is \emph{available} is not a matter of taste; it is fixed by
the topology, through the Wu classes.

\paragraph{Statement (Wu's formula).} On a closed $n$-manifold $M$ the
Wu classes $\nu_j \in H^j(M;\mathbb{Z}_2)$ are defined by
\begin{equation}
\mathrm{Sq}^{j}(x) = \nu_j \smile x
\quad\text{for all } x \in H^{n-j}(M;\mathbb{Z}_2) ,
\label{eq:G-wu-def}
\end{equation}
where $\mathrm{Sq}^j$ is the Steenrod square --- a natural operation
raising the degree of a mod-2 class by $j$, agreeing with the cup square
when $j$ is the degree of the class and vanishing when $j$ exceeds it,
which are the only two properties used below ---
\citep{MilnorStasheff1974}. Because $\mathrm{Sq}^j$
annihilates classes of degree less than $j$, the definition forces
\begin{equation}
\nu_j = 0 \quad\text{whenever}\quad 2j > n .
\label{eq:G-wu-vanish}
\end{equation}

On $n = 6$ this gives $\nu_4 = \nu_5 = \nu_6 = 0$. In particular
$\nu_4 = 0$: the degree-four Wu class vanishes identically, so a
degree-four Wu structure exists canonically and the characteristic
element it supplies is the trivial one. On $n = 8$ the count fails, since
$2\cdot 4 = 8$ is not greater than $8$; there $\nu_4$ need not vanish and
the structure must be supplied as extra data.

\emph{Physics intuition.} Dimension is entering through
\emph{availability} rather than through counting. In most of physics the
dimension controls how many components something has; here it controls
whether a structure exists for free or has to be chosen. Six is on the
favorable side of that line for degree-four data, and that is the whole
of \S\ref{ssec:wu}'s claim.

\subsection{Step 3 --- what \texorpdfstring{\HS{}}{Hopkins--Singer} adds}
\label{ssec:G-hs}

Everything above is about a single manifold. \citet{HopkinsSinger2005}
work with \emph{families}: a fiber bundle $E \to S$ whose fibers are the
manifolds of interest, and a construction that integrates differential
cohomology along the fibers.

\paragraph{Statement (the typing).} The pushforward takes a differential
cocycle of degree $2k$ on the total space to one of degree $4k - d$ on
the base, where $d$ is the fiber dimension, and the construction requires
a Wu structure of degree $2k$. Their Theorem~2.17 is stated for
$d = 4k - i$ with $i \le 2$. For a fiber dimension of six,
\begin{equation}
4k - i = 6, \quad i \le 2
 \;\;\Longrightarrow\;\; (k, i) = (2, 2) ,
\label{eq:G-typing}
\end{equation}
uniquely. So the refinement sits on $\check{H}^4$ --- Wu degree four,
which \S\ref{ssec:G-wu} has just shown exists canonically in six
dimensions --- and the output lands in $\check{H}^2$ of the base, which
is a \emph{line bundle}.

Not a number, not a higher gerbe: a line bundle. That is the coincidence
\S\ref{ssec:kappa} reports, and Eq.~\eqref{eq:G-typing} is why six is the
dimension in which it can happen.

\paragraph{The shift, restated.} With fermions present the flux is not
integral but shifted \citep{Witten1997}:
\begin{equation}
\left[\frac{G}{2\pi}\right] - \frac{\lambda}{2}
 \in H^4(X;\mathbb{Z}), \qquad \lambda = \tfrac12 p_1 ,
\label{eq:G-shift}
\end{equation}
so the displacement of the lattice is $\lambda/2 = p_1/4$ --- worth
stating carefully, since it is routinely misquoted as $p_1/2$. A shifted
lattice is exactly what Eq.~\eqref{eq:G-qa} with $a \ne 0$ describes, and
the two statements are the topological and arithmetic faces of one fact.


\subsection{Step 3 (further) --- differential characters, and the pushforward}
\label{ssec:G-characters}

Two constructions were named in Chapter~\ref{sec:hs} and used without
being written down. Both are short.

\paragraph{Cheeger--Simons differential characters.} Let $Z_{k-1}(X)$ be
the group of smooth $(k-1)$-cycles. A differential character of degree
$k$ is a homomorphism
\begin{equation}
\chi : Z_{k-1}(X) \longrightarrow \mathbb{R}/\mathbb{Z}
\label{eq:G-character}
\end{equation}
for which there exists a differential form $\omega \in \Omega^{k}(X)$
with
\begin{equation}
\chi(\partial \Sigma) = \int_{\Sigma} \omega
 \quad \bmod \mathbb{Z}
\qquad \text{for every } k\text{-chain } \Sigma .
\label{eq:G-character-curv}
\end{equation}
The form $\omega$ is the character's curvature, and it is unique.

Read Eq.~\eqref{eq:G-character} physically and it is the holonomy of
\S\ref{ssec:C-holonomy}: a rule assigning a phase to every closed loop.
Read Eq.~\eqref{eq:G-character-curv} and it is Stokes' theorem, holding
modulo integers rather than exactly --- which is precisely the freedom
that lets the phase be well defined while the potential is not. The
group of such characters is naturally isomorphic to the differential
cohomology $\check{H}^{k}(X)$ of \S\ref{ssec:G-step1}, so
\citet{CheegerSimons1985} and the cocycle model describe the same object.
The generalization of this picture to arbitrary generalized cohomology
theories, and the argument that it is the right home for charge
quantization in physics, is due to \citet{Freed2002}.

\emph{Physics intuition.} The whole of differential cohomology is the
formalization of one sentence: \emph{the holonomy is the physics, the
curvature is its derivative, and the integer is what the two cannot
agree about.} Equation~\eqref{eq:G-character-curv} is that sentence,
written as a congruence.

\paragraph{The pushforward, as integration over the fiber.} For an
ordinary differential form on a bundle $E \to S$ with fiber dimension
$d$, integration over the fiber lowers the degree by $d$ and commutes
with $\dd$ up to boundary terms. The \HS{} construction is the
differential-cohomology refinement of that operation:
\begin{equation}
\int_{E/S} \;:\; \check{H}^{\,k}(E)
 \;\longrightarrow\; \check{H}^{\,k-d}(S) ,
\label{eq:G-pushforward}
\end{equation}
defined when the fibration carries the appropriate orientation data --- a
Wu structure of degree $k$, which is where \S\ref{ssec:G-wu}'s dimension
count enters.

Everything Chapter~\ref{sec:hs} claims follows from putting numbers into
Eq.~\eqref{eq:G-pushforward}. With $d = 6$ and $k = 4$ the output is
$\check{H}^{-2}$, which is empty; with $d = 6$ and the typing of
Eq.~\eqref{eq:G-typing}, $k = 2\cdot 2 = 4$ on the total space and the
answer lands in $\check{H}^{2}(S)$ --- the group of line bundles with
connection on the base. The physical statement of \S\ref{ssec:kappa} is
that this line bundle, computed from six-dimensional data, coincides with
one the four-dimensional theory constructs by an entirely different
route.

\paragraph{What the quadratic refinement is for, in this setting.} The
pushforward of Eq.~\eqref{eq:G-pushforward} is not automatically well
defined: the integration over the fiber of a \emph{quadratic} object
requires a consistent choice of sign, and that choice is exactly the
characteristic element $a$ of Eq.~\eqref{eq:G-qa}. So the arithmetic of
\S\ref{ssec:G-refinement} and the topology of \S\ref{ssec:G-wu} are not
two independent inputs. The Wu class decides which $a$ is available; $a$
makes the pushforward well defined; and the pushforward is what produces
the line bundle. Three steps, one chain, and it terminates in an object a
boundary theory can use.
\subsection{Problems and worked solutions}
\label{ssec:G-problems}

\medskip\noindent\textbf{G.1. Using Eq.~\eqref{eq:G-seq1}, decide whether
a flat but topologically non-trivial gauge field can exist on
$S^2$, on $T^2$, and on the punctured plane.}

\noindent\emph{Solution.} \emph{Step 1.} Flat non-trivial classes are
counted by $H^{k-1}(X;\mathbb{R}/\mathbb{Z})$; take $k = 2$, so we need
$H^1$. \emph{Step 2.} $H^1(S^2) = 0$: none. $H^1(T^2) = \mathbb{Z}^2$:
yes, a two-parameter family --- the Wilson lines around the two cycles.
$H^1$ of the punctured plane is $\mathbb{Z}$: yes, one --- the
Aharonov--Bohm phase. \emph{Step 3.} The physics tracks the topology
exactly. \emph{Physics intuition:} the flat sector is where holonomy
lives without curvature, and problem~A.5 computed its generator by hand.

\medskip\noindent\textbf{G.2. Show that Eq.~\eqref{eq:G-refinement} with
$q(0) = 0$ forces $q(-x) = q(x) + a x$, and check at $a = 1$.}

\noindent\emph{Solution.} \emph{Step 1.} Put $y = -x$ in
Eq.~\eqref{eq:G-refinement}: $q(0) - q(x) - q(-x) = -x^2$.
\emph{Step 2.} With $q(0) = 0$, $q(-x) = x^2 - q(x)$. \emph{Step 3.}
Substituting Eq.~\eqref{eq:G-qa}, $x^2 - (x^2 - ax)/2 = (x^2+ax)/2 =
q(x) + ax$; at $a = 1$ this is $q(-x) = q(x) + x$, checked directly at
$x = 3$: $q(-3) = 6 = 3 + 3$. \emph{Physics intuition:} the refinement is
not an even function, and the amount by which it fails to be even is the
characteristic element --- the shift, showing up as an asymmetry.

\medskip\noindent\textbf{G.3. Verify that $a = 2$ fails integrality and
identify the smallest $x$ at which it does.}

\noindent\emph{Solution.} \emph{Step 1.} $q_2(x) = x(x-2)/2$.
\emph{Step 2.} At $x = 1$, $q_2(1) = -\tfrac12$. \emph{Step 3.} It fails
at every odd $x$, and $x = 1$ is the smallest positive one.
\emph{Physics intuition:} the parity of $a$ is the whole condition, and
Eq.~\eqref{eq:G-wu-vanish} is what decides which parities the topology
makes available.

\medskip\noindent\textbf{G.4. Solve Eq.~\eqref{eq:G-typing} for a fiber
dimension of eight, and say what goes wrong.}

\noindent\emph{Solution.} \emph{Step 1.} $4k - i = 8$ with $i \le 2$
gives $(k,i) = (2,0)$. \emph{Step 2.} The output then lands in
$\check{H}^0$ of the base --- a number, not a bundle. \emph{Step 3.} And
by Eq.~\eqref{eq:G-wu-vanish}, $\nu_4$ need not vanish in eight
dimensions, so the Wu structure is no longer canonical.
\emph{Physics intuition:} in eight dimensions the construction gives an
invariant of the whole family rather than an object living over it, and
the family-moduli reading of \S\ref{ssec:kappa} has nothing to attach
itself to.

\medskip\noindent\textbf{G.5. Show that $\lambda/2 = p_1/4$ and explain
why $p_1/2$ is the wrong statement.}

\noindent\emph{Solution.} \emph{Step 1.} By definition
$\lambda = \tfrac12 p_1$. \emph{Step 2.} Eq.~\eqref{eq:G-shift} shifts by
$\lambda/2$. \emph{Step 3.} So the shift is
$\tfrac12 \cdot \tfrac12 p_1 = p_1/4$. \emph{Physics intuition:} the
error arises from confusing the spin characteristic class $\lambda$ with
the Pontryagin class $p_1$ itself; the harness carries this as an
explicit check precisely because the slip is common.

\medskip\noindent\textbf{G.6. The function $q(x) = x(x-1)/2$ appears
three times in this program. Name the three, and say what would follow
if the three were \emph{not} the same function.}

\noindent\emph{Solution.} \emph{Step 1.} As the $a = 1$
Wu--Chern--Simons refinement of the six-dimensional tensor sector
(\S\ref{ssec:bi6}); as the toll ladder of the relaxation cascade
(\S\ref{ssec:solvency}); and as the \HS{} integer model
(\S\ref{ssec:qx}). \emph{Step 2.} If they differed, the cascade's budget
would not be denominated in the same currency as the bulk's
quantization, and the agreement reported in \S\ref{ssec:kappa} would have
no reason to hold. \emph{Step 3.} They are the same function, and that
identity is what \S\ref{ssec:kappa} converts into a check.
\emph{Physics intuition:} coincidences of this kind are either evidence
or a sign that the same assumption has been used twice --- and
distinguishing the two is precisely what the bridge's independent route is for.

\GGtakeaway{\emph{Standard result:} differential cohomology as the theory
of the triple $(c, h, \omega)$, with two exact sequences isolating the
flat and the topologically trivial sectors; quadratic refinements over
the integers with an odd characteristic element; Wu's formula and the
vanishing $\nu_j = 0$ for $2j > n$. \emph{What it assumed:} closed
oriented manifolds, and --- for the family construction --- the typing
constraint of Eq.~\eqref{eq:G-typing}. \emph{The reading this paper
takes:} the only part of a differential cocycle that cannot be
continuously deformed is its integer class, which is the precise form of
the claim that the top of the ladder is rigid; and six dimensions is
distinguished not by a count of components but by the canonical existence
of the degree-four Wu structure. \emph{Continue in:} \S\ref{ssec:kappa}
for the one place where the whole apparatus is put to work, and
Appendix~\ref{app:dictionary} for the grade at which that result is held.}


\section{The Bianchi identity, derived and placed}
\label{app:bianchi}
\GGteachbox{The exterior derivative and $\dd\dd = 0$ recalled and proved;
$F = \dd A$ and the abelian identity $\dd F = 0$ in one line, then in
components, then checked against the two homogeneous Maxwell equations;
the non-abelian curvature $F = \dd A + A \wedge A$ and its covariant
identity $DF = 0$, derived twice --- once with forms and once from the
Jacobi identity; the two gravitational identities, with the double
contraction that produces $\nabla_\mu G^{\mu\nu} \equiv 0$ written out
index by index. Throughout, the line between an identity and a field
equation is drawn and kept. The pay-off is Chapter~\ref{sec:bianchi},
and the modified identity of Chapter~\ref{sec:gs}.}

\GGreadalong{\citet{MisnerThorneWheeler1973}, Ch.~15, for the
gravitational identities with the geometry drawn; \citet{Nakahara2003},
Ch.~5 and~10, for the exterior calculus and the bundle statement;
\citet{Baez1994} for the cleanest short account of why $F = \dd A$
already contains the answer; and \citet{Voss1880} for the paper that had
the result first.}

\subsection{Step 1 --- the exterior derivative, and why $\dd\dd = 0$}
\label{ssec:H-step1}

Everything in this appendix follows from one fact about one operator, so
the fact is worth proving rather than quoted. Appendix~\ref{app:forms}
builds the exterior calculus; what is needed here is a page of it.

A $k$-form is a totally antisymmetric tensor field written with a basis
of wedge products,
\begin{equation}
\omega \;=\; \frac{1}{k!}\,
 \omega_{\mu_1 \cdots \mu_k}\,
 \dd x^{\mu_1} \wedge \cdots \wedge \dd x^{\mu_k} ,
\label{eq:H-kform}
\end{equation}
the $1/k!$ compensating for the fact that the sum runs over all index
values rather than over ordered ones. The wedge is associative and
graded-commutative: $\dd x^\mu \wedge \dd x^\nu = - \dd x^\nu \wedge
\dd x^\mu$, and in particular $\dd x^\mu \wedge \dd x^\mu = 0$. The
exterior derivative raises the degree by one,
\begin{equation}
\dd\omega \;=\; \frac{1}{k!}\,
 \partial_\nu \omega_{\mu_1 \cdots \mu_k}\,
 \dd x^\nu \wedge \dd x^{\mu_1} \wedge \cdots \wedge \dd x^{\mu_k} .
\label{eq:H-extderiv}
\end{equation}

\paragraph{Statement.} $\dd\dd\omega = 0$ for every form $\omega$.

\paragraph{Proof.} Do the $0$-form case first, where nothing can hide.
For a function $f$, Eq.~\eqref{eq:H-extderiv} gives
$\dd f = \partial_\mu f\, \dd x^\mu$, and applying it again,
\begin{equation}
\dd \dd f \;=\; \partial_\nu \partial_\mu f \;
 \dd x^\nu \wedge \dd x^\mu \;=\; 0 ,
\label{eq:H-ddzero}
\end{equation}
because the coefficient is symmetric under $\mu \leftrightarrow \nu$ and
the basis element is antisymmetric, and the contraction of a symmetric
object with an antisymmetric one vanishes term by term. The general case
is the same sentence with more indices: applying
Eq.~\eqref{eq:H-extderiv} twice puts two derivative indices in front of
the same antisymmetric string, and $\partial_\rho \partial_\nu$
is symmetric while $\dd x^\rho \wedge \dd x^\nu$ is not. \qed

That is the entire engine. Everything below is
Eq.~\eqref{eq:H-ddzero} wearing different clothes: the symmetry of mixed
partial derivatives, set against the antisymmetry of a wedge.

\paragraph{The converse, which will matter in Step 2.} A form with
$\dd\omega = 0$ is called \emph{closed}; one of the form $\omega = \dd
\eta$ is called \emph{exact}. Equation~\eqref{eq:H-ddzero} says exact
implies closed. The converse is the Poincar\'e lemma: on a region that
can be contracted continuously to a point, closed implies exact. Note
what the lemma requires --- a statement about the \emph{shape} of the
region, not about the form. On a region with a hole the converse can
fail, and the failure is measured by de Rham cohomology
(Appendix~\ref{app:diffcohom}). Keep that asymmetry in view: the
identity of this appendix is free, and its converse is not.

\subsection{Step 1 (continued) --- $F = \dd A$, and the identity in one line}
\label{ssec:H-abelian}

Write the abelian gauge potential as a $1$-form $A = A_\mu \dd x^\mu$
and define the field strength as its exterior derivative:
\begin{equation}
F \;=\; \dd A \;=\; \partial_\mu A_\nu\, \dd x^\mu \wedge \dd x^\nu
 \;=\; \tfrac12 F_{\mu\nu}\, \dd x^\mu \wedge \dd x^\nu ,
\qquad
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu .
\label{eq:H-FdA}
\end{equation}
The factor of $\tfrac12$ in the middle expression is the $1/k!$ of
Eq.~\eqref{eq:H-kform} for $k = 2$, and the antisymmetrization on the
right is forced by the wedge, not imposed by hand.

Now apply $\dd$:
\begin{equation}
\dd F \;=\; \dd\dd A \;=\; 0 .
\label{eq:H-dF}
\end{equation}

That is the derivation. It took one line, it used
Eq.~\eqref{eq:H-ddzero} and nothing else, and it is worth pausing on how
little went into it. No action was varied. No symmetry was named. No
field equation was used. No assumption was made about the dimension of
spacetime, its signature, the presence or absence of sources, or the
dynamics of $A$. The single input is the first equality of
Eq.~\eqref{eq:H-FdA}: that $F$ is the derivative of something.

In components, Eq.~\eqref{eq:H-dF} reads
\begin{equation}
\partial_{[\lambda} F_{\mu\nu]} \;=\; 0 ,
\qquad \text{equivalently} \qquad
\partial_\lambda F_{\mu\nu}
 + \partial_\mu F_{\nu\lambda}
 + \partial_\nu F_{\lambda\mu} \;=\; 0 ,
\label{eq:H-cyclic}
\end{equation}
the cyclic sum being the antisymmetrization written out, since
$F_{\mu\nu}$ is already antisymmetric in its own pair.

\paragraph{How many equations is this?} A $3$-form in $D$ dimensions has
$\binom{D}{3}$ independent components, so Eq.~\eqref{eq:H-cyclic} is
that many statements: $4$ in four dimensions, $10$ in five, $20$ in six,
$120$ in ten, $165$ in eleven. In four dimensions the count matches the
familiar pair of homogeneous Maxwell equations exactly --- one scalar
plus one three-vector --- which is the check of the next section.

There is a second way to write Eq.~\eqref{eq:H-cyclic} that is often
more convenient. Define the dual $\tilde F^{\mu\nu} = \tfrac12
\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$. Then the cyclic identity is
equivalent to
\begin{equation}
\partial_\mu \tilde F^{\mu\nu} \;=\; 0 ,
\label{eq:H-dual}
\end{equation}
which looks like a conservation law and is not one: no action produced
it, and no symmetry stands behind it. Chapter~\ref{sec:bianchi} is about
precisely that difference, and Eq.~\eqref{eq:H-dual} is the trap the
chapter is disarming --- an equation with the syntax of a conservation
law and none of its provenance.

\subsection{Step 1 (continued) --- what the four components say}
\label{ssec:H-maxwell}

Fix conventions: $F_{0i} = E_i$ and $F_{ij} = -\epsilon_{ijk}B_k$, with
Latin indices spatial.

\paragraph{The purely spatial triple.} Take $(\lambda,\mu,\nu) =
(1,2,3)$ in Eq.~\eqref{eq:H-cyclic}:
\begin{equation}
\partial_1 F_{23} + \partial_2 F_{31} + \partial_3 F_{12}
 \;=\; -\left(\partial_1 B_1 + \partial_2 B_2 + \partial_3 B_3\right)
 \;=\; -\,\nabla\!\cdot\!\mathbf{B} \;=\; 0 .
\label{eq:H-divB}
\end{equation}
No magnetic charge. One equation, from the one triple with no time
index in it.

\paragraph{The three triples containing time.} Take $(\lambda,\mu,\nu) =
(0,i,j)$:
\begin{equation}
\partial_0 F_{ij} + \partial_i F_{j0} + \partial_j F_{0i}
 \;=\; -\epsilon_{ijk}\,\partial_t B_k
   - \partial_i E_j + \partial_j E_i \;=\; 0 .
\label{eq:H-faraday-raw}
\end{equation}
Use $\epsilon_{ijk}(\nabla \times \mathbf{E})_k = \partial_i E_j -
\partial_j E_i$, which follows from
$\epsilon_{ijk}\epsilon_{kab} = \delta_{ia}\delta_{jb} -
\delta_{ib}\delta_{ja}$. Equation~\eqref{eq:H-faraday-raw} becomes
\begin{equation}
\epsilon_{ijk}\left[
 \partial_t B_k + (\nabla \times \mathbf{E})_k \right] \;=\; 0 ,
\qquad\text{that is}\qquad
\nabla \times \mathbf{E} + \partial_t \mathbf{B} \;=\; 0 .
\label{eq:H-faraday}
\end{equation}
Faraday's law. Three equations, from the three triples containing the
time direction. Four altogether, matching the count $\binom{4}{3} = 4$.

\paragraph{And the other four Maxwell equations.} They are $\dd \star F =
\star J$, and they are of a completely different kind. They come from
varying an action; they involve the sources; they can be modified by
adding terms to the Lagrangian; they are, in the vocabulary of
Chapter~\ref{sec:bianchi}, \emph{laws}. Equations
\eqref{eq:H-divB} and \eqref{eq:H-faraday} came from a definition and
cannot be modified at all without changing what $F$ means.

\emph{Physics intuition.} A student meeting Maxwell's equations for the
first time is given four boxes of equal size, and nothing in the
presentation suggests that two of them are of a different species. They
are. Half of Maxwell's equations are content about the world, tested by
measurement and adjustable in principle; the other half are the receipt
for having written the field as the curl of a potential. That split is
not a curiosity of electromagnetism. It runs through everything above
this rung.

\subsection{Step 1 (continued) --- the non-abelian case, derived twice}
\label{ssec:H-nonab}

For a non-abelian group the potential is Lie-algebra valued, $A = A_\mu^a
T^a \dd x^\mu$, and the curvature acquires a quadratic term:
\begin{equation}
F \;=\; \dd A + A \wedge A ,
\qquad
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu] ,
\label{eq:H-nacurv}
\end{equation}
where $A \wedge A$ means wedge on the form indices and matrix
multiplication on the algebra indices, so that $A \wedge A =
\tfrac12 [A_\mu, A_\nu]\,\dd x^\mu \wedge \dd x^\nu$ --- non-zero
precisely because the matrices do not commute. The naive statement
$\dd F = 0$ is now false; what replaces it is a covariant identity.

For Lie-algebra-valued forms of degrees $p$ and $q$ define the graded
bracket $[\alpha,\beta] = \alpha\wedge\beta - (-1)^{pq}\beta\wedge\alpha$,
and the covariant exterior derivative $D\beta = \dd\beta + [A,\beta]$.

\paragraph{Statement.} $DF = \dd F + [A,F] = 0$.

\paragraph{First derivation: the forms.} Since $A$ is a $1$-form and $F$
a $2$-form, $pq = 2$ and $[A,F] = A \wedge F - F \wedge A$. Also, for
a $1$-form, $\dd(A \wedge A) = \dd A \wedge A - A \wedge \dd A$. Then
\begin{align}
\dd F &= \dd\dd A + \dd(A\wedge A)
       = \dd A \wedge A - A \wedge \dd A ,
\label{eq:H-nastep1}\\
A \wedge F &= A \wedge \dd A + A \wedge A \wedge A ,
\label{eq:H-nastep2}\\
F \wedge A &= \dd A \wedge A + A \wedge A \wedge A ,
\label{eq:H-nastep3}
\end{align}
and adding Eq.~\eqref{eq:H-nastep1} to
Eq.~\eqref{eq:H-nastep2} minus Eq.~\eqref{eq:H-nastep3}, every term
cancels against a partner:
\begin{equation}
DF \;=\; \dd F + A\wedge F - F \wedge A \;=\; 0 .
\label{eq:H-DF}
\end{equation}
Note where the cubic terms went: $A\wedge A \wedge A$ appears once with
each sign and cancels without needing to be evaluated. \qed

\paragraph{Second derivation: the Jacobi identity.} Write the covariant
derivative as $D_\mu = \partial_\mu + A_\mu$, so that
$[D_\mu, D_\nu] = F_{\mu\nu}$ by Eq.~\eqref{eq:H-nacurv}. The Jacobi
identity for any three operators,
\begin{equation}
\big[D_\lambda,[D_\mu,D_\nu]\big]
 + \big[D_\mu,[D_\nu,D_\lambda]\big]
 + \big[D_\nu,[D_\lambda,D_\mu]\big] \;=\; 0 ,
\label{eq:H-jacobi}
\end{equation}
holds identically --- expand the nine double commutators and watch them
cancel in pairs. Substituting $[D_\mu,D_\nu] = F_{\mu\nu}$ turns
Eq.~\eqref{eq:H-jacobi} into
\begin{equation}
D_\lambda F_{\mu\nu} + D_\mu F_{\nu\lambda} + D_\nu F_{\lambda\mu}
 \;=\; 0 ,
\label{eq:H-nacyclic}
\end{equation}
which is Eq.~\eqref{eq:H-DF} in components and reduces to
Eq.~\eqref{eq:H-cyclic} when the algebra is abelian.

The second derivation is the more instructive one, because it says what
the identity \emph{is}: the associativity of the operators whose
commutator defines the curvature. Nothing about gauge theory was used,
and that is why the same argument works for gravity in the next section
without modification.

\subsection{Step 1 (concluded) --- the gravitational identities}
\label{ssec:H-grav}

Gravity has two identities of this family, and they are not the same
statement.

\paragraph{The first (algebraic) identity.} For a torsion-free
connection,
\begin{equation}
R^{\rho}{}_{[\sigma\mu\nu]} \;=\; 0 ,
\qquad\text{that is}\qquad
R^{\rho}{}_{\sigma\mu\nu} + R^{\rho}{}_{\mu\nu\sigma}
 + R^{\rho}{}_{\nu\sigma\mu} \;=\; 0 .
\label{eq:H-bianchi1}
\end{equation}
It involves no derivatives of the curvature; it is a symmetry of the
Riemann tensor, and it is what reduces the naive component count to the
familiar $D^2(D^2-1)/12$ --- twenty in four dimensions, split ten and
ten between Ricci and Weyl.

\paragraph{The second (differential) identity.}
\begin{equation}
\nabla_{[\lambda} R_{\mu\nu]\rho\sigma} \;=\; 0 ,
\qquad\text{that is}\qquad
\nabla_\lambda R_{\mu\nu\rho\sigma}
 + \nabla_\mu R_{\nu\lambda\rho\sigma}
 + \nabla_\nu R_{\lambda\mu\rho\sigma} \;=\; 0 .
\label{eq:H-bianchi2}
\end{equation}
Two derivations are available and both are short. The first is
Eq.~\eqref{eq:H-jacobi} again, with $D \to \nabla$: the curvature is the
commutator of covariant derivatives, and the Jacobi identity for that
commutator is Eq.~\eqref{eq:H-bianchi2}.

The second is worth doing because it makes the identity visibly a
statement about mixed partials. At any point $p$ choose Riemann normal
coordinates, in which $\Gamma^{\rho}{}_{\mu\nu}(p) = 0$ and $g_{\mu\nu} =
\eta_{\mu\nu} + O(x^2)$. At $p$ the covariant derivative reduces to the
partial derivative, and the curvature reduces to
\begin{equation}
R_{\rho\sigma\mu\nu}(p) = \tfrac12\left(
 \partial_\mu \partial_\sigma g_{\rho\nu}
+\partial_\nu \partial_\rho g_{\sigma\mu}
-\partial_\nu \partial_\sigma g_{\rho\mu}
-\partial_\mu \partial_\rho g_{\sigma\nu}\right) .
\label{eq:H-normal}
\end{equation}
Differentiate once more and antisymmetrize over the three indices
$[\lambda\mu\nu]$. Every term of Eq.~\eqref{eq:H-normal} carries either
$\partial_\mu$ or $\partial_\nu$; adding $\partial_\lambda$ in front
produces a pair of derivatives symmetric in two of the three indices
being antisymmetrized, and dies. Since Eq.~\eqref{eq:H-bianchi2} is a
tensor equation and holds at an arbitrary $p$ in one coordinate system,
it holds everywhere in all of them. \qed

\paragraph{Counting the second identity.} Naively
Eq.~\eqref{eq:H-bianchi2} is $\binom{D}{3}\binom{D}{2}$ equations ---
$24$ in four dimensions. They are not independent: antisymmetrizing over
four of the five indices gives a combination that vanishes identically
once Eq.~\eqref{eq:H-bianchi1} is used, which removes
$D\binom{D}{4}$ of them. What is left is
\begin{equation}
\binom{D}{3}\binom{D}{2} - D\binom{D}{4}
 \;=\; \frac{D^2 (D^2-1)(D-2)}{24} ,
\label{eq:H-count2}
\end{equation}
equal to $20$ for $D = 4$, $75$ for $D=5$, $210$ for $D=6$ and $3300$
for $D=10$. (The two sides of Eq.~\eqref{eq:H-count2} agree for every
$D$; problem H.1 asks for the check.)

\paragraph{The contraction, done index by index.} Take the convention
$R_{\mu\nu} = g^{\alpha\beta}R_{\beta\mu\alpha\nu}$ for the Ricci
tensor. Contract Eq.~\eqref{eq:H-bianchi2} with $g^{\nu\rho}$. The three
terms give, in order,
\begin{equation}
-\nabla_\lambda R_{\mu\sigma}
 \;+\; \nabla_\mu R_{\lambda\sigma}
 \;+\; \nabla^{\rho} R_{\lambda\mu\rho\sigma} \;=\; 0 ,
\label{eq:H-once}
\end{equation}
the minus sign in the first term coming from
$g^{\nu\rho}R_{\mu\nu\rho\sigma} = -g^{\nu\rho}R_{\nu\mu\rho\sigma} =
-R_{\mu\sigma}$. Now contract Eq.~\eqref{eq:H-once} with
$g^{\mu\sigma}$. The first term gives $-\nabla_\lambda R$; the second
gives $\nabla^\mu R_{\lambda\mu}$; the third gives
$\nabla^\rho R_{\rho\lambda}$, using the pair symmetry
$R_{\lambda\mu\rho\sigma} = R_{\rho\sigma\lambda\mu}$ and the fact that
contracting the second and fourth slots of the Riemann tensor also
produces the Ricci tensor. Hence
\begin{equation}
2\,\nabla^\mu R_{\mu\lambda} \;=\; \nabla_\lambda R ,
\qquad\text{that is}\qquad
\nabla_\mu \left(R^{\mu\nu} - \tfrac12 g^{\mu\nu}R\right)
 \;=\; \nabla_\mu G^{\mu\nu} \;\equiv\; 0 .
\label{eq:H-divG}
\end{equation}
This is the contracted identity that Chapter~\ref{sec:bianchi} uses and
that Appendix~\ref{app:connections} quotes. The identity sign is not
decoration: Eq.~\eqref{eq:H-divG} holds for \emph{every} metric, solution
or not, smooth or ugly, in vacuum or not.

\paragraph{Which are identities, and which are field equations.} Say it
flatly, because the whole rung turns on it.
Equations~\eqref{eq:H-dF}, \eqref{eq:H-DF}, \eqref{eq:H-bianchi1},
\eqref{eq:H-bianchi2} and \eqref{eq:H-divG} are identities. They follow
from the definitions of the objects in them, they hold off shell, and no
measurement can conflict with them. Equations $\dd\star F = \star J$ and
$G^{\mu\nu} = \kappa T^{\mu\nu}$ are field equations. They follow from
an action, they hold on shell, they involve coupling constants that had
to be measured, and they could have been otherwise. Table~\ref{tab:H-split}
sets the four cases side by side.

\begin{table}[H]
\centering
\caption[Identities and field equations, four sectors.]{The same split,
four times. The first three rows are standard; the fourth is
Chapter~\ref{sec:gs}'s move, included here so that the reader can see
exactly which column it changes. Nothing in the third column was ever
violated --- the fourth row simply stops being an instance of it, because
$H_3$ stops being the derivative of anything.}
\label{tab:H-split}\small
\begin{tabular}{@{}P{0.17\textwidth}P{0.21\textwidth}P{0.22\textwidth}P{0.28\textwidth}@{}}
\toprule
\textbf{sector} & \textbf{curvature} & \textbf{identity (off shell)} &
\textbf{field equation (on shell)} \\
\midrule
abelian gauge & $F = \dd A$ & $\dd F = 0$, Eq.~\eqref{eq:H-dF} &
$\dd \star F = \star J$, from an action \\
non-abelian gauge & $F = \dd A + A\wedge A$ &
$DF = 0$, Eq.~\eqref{eq:H-DF} &
$D \star F = \star J$, from an action \\
gravitation & $R$ from $\Gamma$ &
$\nabla_{[\lambda}R_{\mu\nu]\rho\sigma} = 0$ and
$\nabla_\mu G^{\mu\nu} = 0$ &
$G^{\mu\nu} = \kappa T^{\mu\nu}$, from an action \\
tensor sector, \GS{} &
$H_3 = \dd B + \omega_3$ &
none: the premise is declined &
$\dd H_3 = X_4$, imposed by consistency \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Step 2 --- what the identity assumes, and what it does not}
\label{ssec:H-step2}

\paragraph{What is assumed: a potential.} The single premise of
Eq.~\eqref{eq:H-dF} is that $F$ is the exterior derivative of a
$1$-form. That is a stronger assumption than it looks, and it is
\emph{local}. On a contractible patch the Poincar\'e lemma of
\S\ref{ssec:H-step1} makes the premise free: $\dd F = 0$ there
guarantees an $A$ with $F = \dd A$, so on a patch the premise and the
identity are equivalent and nothing has been assumed. Globally the
premise is a real restriction. It says the bundle is trivial --- that one
$A$ works everywhere. When it does not, one covers the space in patches,
writes $F = \dd A_{(\alpha)}$ on each, and glues with transition
functions. The identity survives untouched in every patch, and what
changes is a global datum: an integer.

That is the whole content of the monopole discussion of
Chapter~\ref{sec:bianchi}. The identity is not the sort of thing a
monopole can violate. A monopole is the statement that no single $A$
exists on the sphere surrounding it, and the integer measuring the
obstruction is the first Chern number (Appendix~\ref{app:diffcohom}).

\paragraph{What is not assumed: everything else.} The list is worth
having explicitly, since the point of the rung is the shortness of the
list of inputs.

\emph{No symmetry.} Nothing was invariant under anything. Contrast
Appendix~\ref{app:noether}, where a current exists because a
transformation leaves an action alone; here there is no transformation
and no current in that sense.

\emph{No Lagrangian.} No action was written, so no action can be
modified to change the result.

\emph{No dynamics, and no field equation.} Equation~\eqref{eq:H-dF}
holds for an arbitrary $A$, including one that solves nothing.

\emph{No dimension, signature, or sources.} $\binom{D}{3}$ changes with
$D$; the identity does not.

\emph{No quantum mechanics.} It is a statement about smooth functions.

Put these together and the status of the rung is fixed. In
Appendix~\ref{app:noether}'s language, a Noether current is conserved
\emph{on shell} because the offending term is proportional to the field
equations. Equation~\eqref{eq:H-cyclic} has no offending term to be
proportional to anything. It is conservation with the dynamics removed.

\paragraph{The historical point.} The identity is universally called
Bianchi's, from the 1902 paper in which he proved it by direct
calculation \citep{Bianchi1902}. He was not first. A contracted form
appears in \citet{Voss1880}, twenty-two years earlier, in a paper on the
transformation of quadratic differential forms with no physical
motivation at all; Ricci obtained the identities independently around
1889 and then lost track of them. Chapter~\ref{sec:bianchi} tells the
story; what the appendix adds is the observation that a result of this
kind is exactly the sort that gets mislaid. It predicts nothing, it
resolves no puzzle anyone had, and it looks like a manipulation. Its
importance became visible only when somebody needed to know which tensors
have identically vanishing divergence --- which is to say, in 1915.

\subsection{Step 3 --- the third rung, and the identity deliberately broken}
\label{ssec:H-step3}

\paragraph{Why the review puts this rung third.} Chapter~\ref{sec:noether}
gives conservation from symmetry; Chapter~\ref{sec:gauge} gives a local
symmetry that turns out to be a redundancy and yields no charge at all.
Chapter~\ref{sec:bianchi} is where a reader first meets a
conservation-like statement with \emph{nothing} of the Noether apparatus
behind it. Equation~\eqref{eq:H-dual} has the form of a conservation law
and none of its ancestry: no invariance, no action, no on-shell
qualifier. It is the first rung at which protection is bought by
exactness rather than by symmetry, and it is therefore the first evidence
for the review's thesis that the ladder does not end where the textbooks
stop it (Chapter~\ref{sec:two-ideas}).

The rung also fixes the register for what comes above it. An identity
constrains what a theorist may write down and predicts nothing about the
world. Equation~\eqref{eq:H-divG} is the standard demonstration: it does
not derive general relativity, it does not fix Newton's constant, and it
permits the cosmological term. What it does is exclude nearly every
alternative, because a geometrical side with non-vanishing divergence
would impose conditions on matter beyond
$\nabla_\mu T^{\mu\nu} = 0$, and those conditions fail for sources one is
entitled to consider. Among tensors built from the metric and its first
two derivatives and linear in the second derivatives, the
divergence-free combinations in four dimensions are $G^{\mu\nu}$ and
$g^{\mu\nu}$, and that is very nearly the whole freedom.

\paragraph{Breaking it on purpose, and what that costs.}
Chapter~\ref{sec:gs} is the rung at which this apparently unbreakable
structure is broken deliberately. In ten dimensions the antisymmetric
tensor field has field strength $H_3$, classically $\dd B$, hence
classically obeying $\dd H_3 = 0$ by Eq.~\eqref{eq:H-ddzero}. The
\GS{} mechanism takes instead $\dd H_3 = X_4$ with $X_4$ a four-form
built from curvatures, and arranges the resulting gauge variation to
cancel the fermion loop's anomaly (Appendix~\ref{app:anomalies}).

Read with this appendix in hand, nothing was violated. What happened is
that the premise of \S\ref{ssec:H-step2} was declined:
\begin{equation}
H_3 \;=\; \dd B + \omega_3 ,
\qquad
\dd \omega_3 = X_4
\;\;\Longrightarrow\;\;
\dd H_3 = X_4 ,
\label{eq:H-modified}
\end{equation}
with $\omega_3$ a Chern--Simons form. Once $H_3$ is not the exterior
derivative of anything, Eq.~\eqref{eq:H-ddzero} has nothing to act on
and $\dd H_3$ is under no obligation to vanish. The identity is intact;
its premise no longer applies to this object.

The cost is exact and should be stated as a cost. $H_3$ is no longer
gauge invariant: under a gauge transformation $\omega_3$ shifts by an
exact form, so $B$ must shift to compensate, and every term in the action
built from $H_3$ acquires a variation. One buys anomaly cancellation and
one pays with the gauge invariance of a field one had taken to be
elementary.

\paragraph{The program's instance.} The cohort's version of
Eq.~\eqref{eq:H-modified} lives in six rather than ten dimensions
\citep{ArisakaA3BI6}, with
\begin{equation}
\dd H_3 \;=\; X_4
 \;=\; \tr R^2 - \TR F_5^2
   - \tfrac{2}{5} F_B^2 - \tfrac{2}{5} F_{Qi}^2 ,
\label{eq:H-bi6}
\end{equation}
whose coefficient vector is $(1, -1, -\tfrac25, -\tfrac25)$. The two
abelian coefficients are one number, not two: they are a ratio of traces
that for the ledger integers $(M,N,L) = (1,3,5)$ reads $(L-N)/L =
2/5$, and the same integers satisfy $J = 2N+1 = 7$ and
$K = LJ = M^2 + N^2 + L^2 = 35$, checkable in a line as $5 \times 7 = 35$
and $1 + 9 + 25 = 35$. Chapter~\ref{sec:zero-parameters} is where that
kind of arithmetic is audited; the point here is structural rather than
numerical. A four-form in six dimensions has $\binom{6}{4} = 15$
components, and Eq.~\eqref{eq:H-bi6} is the statement that all of them
are fixed by curvatures with locked coefficients. What was an identity
at this rung becomes, four rungs up, a ledger with entries.

\emph{Physics intuition.} There is a temptation to read
Eq.~\eqref{eq:H-modified} as physics cheating: an identity was
inconvenient, so it was repealed. That reading is wrong in a way worth
fixing, because it recurs at every rung above. Identities are not
repealed; premises are. The move is the same one made when a conserved
current is improved by $\partial_\nu \Theta^{\mu\nu}$, or when a
Hamiltonian is shifted by a constant: the object one thought was
fundamental turns out to be defined only up to something, and the physics
lives in the equivalence class. What Green and Schwarz added is that the
\emph{identity} can be the thing one is defined up to.

\subsection{Problems and worked solutions}
\label{ssec:H-problems}

\medskip\noindent\textbf{H.1. Verify that the two sides of
Eq.~\eqref{eq:H-count2} agree, and evaluate both counts for $D = 4$ and
$D = 6$. Then state how many equations $\dd F = 0$ is in the same two
dimensions.}

\noindent\emph{Solution.} \emph{Step 1.} For $D = 4$:
$\binom{4}{3}\binom{4}{2} = 4 \times 6 = 24$ and
$4\binom{4}{4} = 4$, so the left side is $20$; the right side is
$16 \times 15 \times 2 / 24 = 20$. \emph{Step 2.} For $D = 6$:
$\binom{6}{3}\binom{6}{2} = 20 \times 15 = 300$ and
$6\binom{6}{4} = 6 \times 15 = 90$, so the left side is $210$; the right
side is $36 \times 35 \times 4 / 24 = 210$. \emph{Step 3.} The gauge
identity is a $3$-form equation, $\binom{4}{3} = 4$ components in four
dimensions and $\binom{6}{3} = 20$ in six. \emph{Physics intuition:} the
gravitational identity grows much faster with dimension than the gauge
one because it carries a spectator index pair; in four dimensions the
coincidence that both the Riemann tensor and its differential identity
have twenty components is exactly that --- a coincidence.

\medskip\noindent\textbf{H.2. Derive $\nabla\!\cdot\!\mathbf{B} = 0$ and
Faraday's law from Eq.~\eqref{eq:H-cyclic}, keeping every sign, and then
say precisely why the other two Maxwell equations cannot be obtained the
same way.}

\noindent\emph{Solution.} \emph{Step 1.} With $F_{ij} =
-\epsilon_{ijk}B_k$ the spatial triple gives
Eq.~\eqref{eq:H-divB} directly, the three terms assembling
$-\nabla\!\cdot\!\mathbf{B}$. \emph{Step 2.} With $F_{0i} = E_i$, so
$F_{i0} = -E_i$, the triples $(0,i,j)$ give
Eq.~\eqref{eq:H-faraday-raw}; applying the $\epsilon\epsilon$ contraction
turns the two electric terms into $-\epsilon_{ijk}(\nabla\times
\mathbf{E})_k$ and yields Eq.~\eqref{eq:H-faraday}. \emph{Step 3.} The
remaining Maxwell equations involve $\star F$, which requires a metric,
and $J$, which requires a matter sector; both enter through an action.
No amount of differentiating a definition produces a coupling to a
source. \emph{Physics intuition:} $\dd F = 0$ tells you about the field
alone; $\dd \star F = \star J$ tells you how the field answers to
matter. The first is grammar, the second is content.

\medskip\noindent\textbf{H.3. Derive $DF = 0$ from the Jacobi identity,
and check that it reduces to Eq.~\eqref{eq:H-cyclic} when the group is
$U(1)$. Then explain why the derivation makes no reference to gauge
theory.}

\noindent\emph{Solution.} \emph{Step 1.} With
$D_\mu = \partial_\mu + A_\mu$ one has $[D_\mu,D_\nu] = F_{\mu\nu}$ by
Eq.~\eqref{eq:H-nacurv}; the cross terms
$\partial_\mu A_\nu - \partial_\nu A_\mu$ come from the derivative acting
on the potential and the $[A_\mu,A_\nu]$ from the matrices.
\emph{Step 2.} Equation~\eqref{eq:H-jacobi} is an identity for any three
operators; substituting the inner commutator gives
Eq.~\eqref{eq:H-nacyclic}. \emph{Step 3.} For $U(1)$ all commutators of
$A$'s vanish, $D_\mu F_{\nu\lambda} \to \partial_\mu F_{\nu\lambda}$, and
Eq.~\eqref{eq:H-nacyclic} becomes Eq.~\eqref{eq:H-cyclic}.
\emph{Physics intuition:} the input is associativity --- that
$(D_\lambda D_\mu) D_\nu$ and $D_\lambda (D_\mu D_\nu)$ are the same
operator. Any curvature defined as a commutator satisfies a Bianchi
identity, which is why the gravitational and gauge cases are one theorem
and not two.

\medskip\noindent\textbf{H.4. Carry out the double contraction leading to
Eq.~\eqref{eq:H-divG}, and then explain why $\nabla_\mu T^{\mu\nu} = 0$
in general relativity is a consequence rather than an extra assumption.}

\noindent\emph{Solution.} \emph{Step 1.} Contracting
Eq.~\eqref{eq:H-bianchi2} with $g^{\nu\rho}$ gives
Eq.~\eqref{eq:H-once}; the sign of the first term comes from swapping the
first index pair of the Riemann tensor before recognizing the Ricci
tensor. \emph{Step 2.} Contracting again with $g^{\mu\sigma}$ gives
$-\nabla_\lambda R + 2\nabla^\mu R_{\mu\lambda} = 0$, which is
Eq.~\eqref{eq:H-divG}. \emph{Step 3.} Take the divergence of
$G^{\mu\nu} = \kappa T^{\mu\nu}$. The left side vanishes identically by
Eq.~\eqref{eq:H-divG}, so the right side must vanish too --- the matter
conservation law is forced by the field equation together with an
identity about geometry. \emph{Physics intuition:} this is why
Einstein's equations could not have had a different geometrical side.
Any candidate with non-vanishing divergence would impose a condition on
matter over and above conservation, and generic sources would refuse it.

\medskip\noindent\textbf{H.5. A magnetic monopole of charge $g$ sits at
the origin. Show that $\dd F = 0$ still holds everywhere it is defined,
and locate exactly where the global statement fails.}

\noindent\emph{Solution.} \emph{Step 1.} Away from the origin the field
is smooth and $\dd F = 0$ holds pointwise, since it is an identity for
any $F$ that is locally the curvature of a potential; on the northern
cap of a sphere around the origin such an $A_{(N)}$ exists, and on the
southern cap an $A_{(S)}$ does. \emph{Step 2.} The two disagree on the
overlap by a gauge transformation. Stokes' theorem applied to each cap
separately gives $\oint_{S^2} F = \oint_{\text{equator}}(A_{(N)} -
A_{(S)})$, which is the winding of the transition function --- an
integer times $2\pi$, hence the quantization $qg = 2\pi n$ in units with
$\hbar = 1$ \citep{Dirac1931}. \emph{Step 3.} The failure is not in
$\dd F = 0$, which never breaks; it is in the assumption that one $A$
covers the sphere. \emph{Physics intuition:} the identity is local and
the monopole is global, so they cannot collide. The integer is not a
violation of a conservation law but a label for the way the local
descriptions are stitched, which is precisely the reading
Chapter~\ref{sec:hs} generalizes.

\medskip\noindent\textbf{H.6. Given $H_3 = \dd B + \omega_3$ with
$\dd\omega_3 = X_4$, verify Eq.~\eqref{eq:H-modified} and work out what
must happen to $B$ under a gauge transformation. Then check the ledger
arithmetic behind the coefficients in Eq.~\eqref{eq:H-bi6}.}

\noindent\emph{Solution.} \emph{Step 1.} $\dd H_3 = \dd\dd B +
\dd\omega_3 = 0 + X_4$ by Eq.~\eqref{eq:H-ddzero}, so the modified
identity is not an extra postulate but a computation once
$\omega_3$ is included. \emph{Step 2.} Under a gauge transformation the
Chern--Simons form shifts by an exact piece,
$\omega_3 \to \omega_3 + \dd \Lambda_2$; for $H_3$ to be the object the
theory intends, $B$ must shift by $-\Lambda_2$, so $B$ is not gauge
invariant and neither is any term built from it alone. \emph{Step 3.}
The coefficients: $(L - N)/L = (5-3)/5 = 2/5$ for
$(M,N,L) = (1,3,5)$; and the closure relations $J = 2N + 1 = 7$,
$K = LJ = 5 \times 7 = 35 = 1 + 9 + 25 = M^2 + N^2 + L^2$ hold, so the
vector $(1,-1,-\tfrac25,-\tfrac25)$ contains one free-looking number and
that number is a ratio of integers already spent elsewhere.
\emph{Physics intuition:} the whole trade is visible in Step 1. One
sentence of algebra converts an unbreakable identity into a controlled
failure, and the invoice arrives in Step 2 as the lost gauge invariance
of $B$. Nothing was obtained for nothing, which is the honest way to read
every mechanism at this level of the ladder.

\GGtakeaway{\emph{Standard result:} $F = \dd A$ forces $\dd F = 0$, and
$\dd\dd = 0$ is the whole derivation; in components this is exactly
$\nabla\!\cdot\!\mathbf{B} = 0$ and Faraday's law, $\binom{D}{3}$
equations in $D$ dimensions. The non-abelian version $DF = 0$ and the
gravitational $\nabla_{[\lambda}R_{\mu\nu]\rho\sigma} = 0$ are the same
theorem, both following from the Jacobi identity for the operators whose
commutator is the curvature; contracting the latter twice gives
$\nabla_\mu G^{\mu\nu} \equiv 0$. \emph{What it assumed:} only that the
field strength is locally the derivative of a potential --- no symmetry,
no Lagrangian, no dynamics, no field equation, no dimension. \emph{What
it leaves open:} the global question. Whether one potential covers the
space is not settled by the identity, and the obstruction is an integer.
\emph{Continue in:} Chapter~\ref{sec:bianchi} for the placement of the
rung, Chapter~\ref{sec:gs} and Appendix~\ref{app:anomalies} for the
identity broken on purpose, and Appendix~\ref{app:diffcohom} for the
integer that the local statement cannot see.}

\section{Open systems --- density matrices, channels, and the semigroup}
\label{app:opensys}
\GGteachbox{Why a state has to be a matrix rather than a vector, argued
from ignorance and from entanglement separately; the partial trace,
defined and then worked on two qubits until the loss is visible;
complete positivity, with the transpose counterexample computed to the
last eigenvalue; the Kraus representation and the Choi construction that
proves it; the Born--Markov derivation of the Lindblad equation with
every approximation named as it is made; and the dephasing qubit solved
in closed form. The pay-off is Part~III --- Chapters~\ref{sec:ggep},
\ref{sec:oggep}, \ref{sec:grip} and \ref{sec:qi4} --- whose central claim
is a statement about a map that has no inverse.}

\GGreadalong{\citet{Lindblad1976} for the general theorem and
\citet{GoriniKossakowskiSudarshan1976} for the finite-dimensional
version, read together since each supplies what the other assumes;
\citet{Gleason1957} for the sharp statement of what can be proved about
probabilities without further input; and \citet{ArisakaA8QM} for the
program's own use of both.}

\subsection{Step 1 --- why a state has to be a matrix}
\label{ssec:I-step1}

A vector in a Hilbert space is not a general state, and there are two
independent reasons. They are usually run together, and separating them
is the first step of this appendix.

\paragraph{Reason one: ignorance.} A source emits $|\psi_1\rangle$ with
probability $p_1$ and $|\psi_2\rangle$ with probability $p_2$. No vector
describes this. Superposing them is wrong: $\sqrt{p_1}|\psi_1\rangle +
\sqrt{p_2}|\psi_2\rangle$ has interference terms that a classical mixture
does not. What does describe it is
\begin{equation}
\rho \;=\; \sum_i p_i \,|\psi_i\rangle\!\langle \psi_i| ,
\qquad p_i \ge 0, \quad \sum_i p_i = 1 ,
\label{eq:I-rho}
\end{equation}
because expectation values of every observable come out right:
\begin{equation}
\langle A \rangle \;=\; \sum_i p_i \langle \psi_i| A |\psi_i\rangle
 \;=\; \tr(\rho A) .
\label{eq:I-expect}
\end{equation}

\paragraph{Reason two: subsystems.} This one is deeper, because it
survives even when nothing is unknown. Take the two-qubit state
\begin{equation}
|\Phi^+\rangle \;=\; \tfrac{1}{\sqrt2}\left(|00\rangle + |11\rangle\right) ,
\label{eq:I-bell}
\end{equation}
which is pure: the global state is known exactly. Ask what qubit $A$
alone is in. There is no answer of the form $|\chi\rangle_A$, and
\S\ref{ssec:I-ptrace} shows that the honest answer is a matrix. Even a
perfectly known world can have imperfectly describable parts, and that is
not a limitation of the observer.

\paragraph{The properties, and why they are the right ones.} Whatever
its origin, $\rho$ satisfies three conditions:
\begin{equation}
\rho^\dagger = \rho ,
\qquad \rho \ge 0 ,
\qquad \tr \rho = 1 .
\label{eq:I-axioms}
\end{equation}
Hermiticity makes $\langle A\rangle$ real for Hermitian $A$; positivity
makes $\langle A \rangle \ge 0$ for positive $A$, which is what stops
probabilities going negative; the trace condition normalizes them. A
useful scalar is the purity
\begin{equation}
P(\rho) \;=\; \tr \rho^2 ,
\qquad \tfrac{1}{d} \;\le\; P \;\le\; 1 ,
\label{eq:I-purity}
\end{equation}
with $P = 1$ if and only if $\rho = |\psi\rangle\!\langle \psi|$ for some
vector, and $P = 1/d$ for the maximally mixed state $\rho = \tfrac{1}{d}\mathbf{1}$.
Closed evolution is the von Neumann equation
\begin{equation}
\ii\,\frac{\dd \rho}{\dd t} \;=\; [H,\rho] ,
\qquad
\rho(t) = U(t)\,\rho(0)\,U(t)^\dagger ,
\quad U = e^{-\ii H t} ,
\label{eq:I-vonneumann}
\end{equation}
which preserves the purity exactly --- unitary conjugation cannot change
$\tr\rho^2$. Remember that: it is the fact that the rest of the appendix
breaks.

\paragraph{A warning that will be used twice.} The decomposition in
Eq.~\eqref{eq:I-rho} is not unique. For a single qubit,
\begin{equation}
\tfrac12 \mathbf{1}
 \;=\; \tfrac12 |0\rangle\!\langle 0| + \tfrac12 |1\rangle\!\langle 1|
 \;=\; \tfrac12 |+\rangle\!\langle +| + \tfrac12 |-\rangle\!\langle -| ,
\label{eq:I-nonunique}
\end{equation}
with $|\pm\rangle = (|0\rangle\pm|1\rangle)/\sqrt2$. These are physically
different preparations --- a coin flip between two orthogonal states, and
a coin flip between two \emph{other} orthogonal states --- and by
Eq.~\eqref{eq:I-expect} no measurement whatsoever distinguishes them. The
information about which preparation occurred is not hidden in $\rho$; it
is gone. That is the first appearance in this appendix of the fact Part~III
is built on, and it appears before any dynamics has been written down.

\subsection{Step 1 (continued) --- the partial trace, worked}
\label{ssec:I-ptrace}

Let the world be $\mathcal{H}_A \otimes \mathcal{H}_B$ and let an
observer have access only to $A$. What matrix on $\mathcal{H}_A$
reproduces every measurement that observer can make?

\paragraph{Derivation.} The observer's observables are $A \otimes
\mathbf{1}_B$. Demand
\begin{equation}
\tr_A\!\left(\rho_A\, A\right)
 \;=\; \tr\!\left(\rho \,(A \otimes \mathbf{1}_B)\right)
 \qquad \text{for every } A ,
\label{eq:I-ptrace-def}
\end{equation}
and expand the right side in a product basis
$|i\rangle_A \otimes |k\rangle_B$:
\begin{equation}
\tr\!\left(\rho\,(A\otimes\mathbf{1})\right)
 = \sum_{i,j,k} \rho_{ik,\,jk}\, A_{ji}
 = \sum_{i,j}\Big(\sum_k \rho_{ik,\,jk}\Big) A_{ji} .
\label{eq:I-ptrace-expand}
\end{equation}
Comparing with the left side, and since $A$ is arbitrary, the matrix is
forced:
\begin{equation}
(\rho_A)_{ij} \;=\; \sum_k \rho_{ik,\,jk}
 \;\equiv\; \left(\tr_B \rho\right)_{ij} .
\label{eq:I-ptrace}
\end{equation}
The partial trace is not a modeling choice. It is the unique map
consistent with Eq.~\eqref{eq:I-ptrace-def}, and any other prescription
would make some accessible measurement come out wrong.

\paragraph{Worked example, the entangled case.} For
Eq.~\eqref{eq:I-bell}, in the basis $(|00\rangle,|01\rangle,|10\rangle,
|11\rangle)$,
\begin{equation}
\rho = |\Phi^+\rangle\!\langle \Phi^+|
 = \frac{1}{2}
 \begin{pmatrix}
 1 & 0 & 0 & 1 \\
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 \\
 1 & 0 & 0 & 1
 \end{pmatrix} ,
\qquad
\rho_A = \tr_B \rho = \frac{1}{2}
 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} .
\label{eq:I-bell-reduced}
\end{equation}
Apply Eq.~\eqref{eq:I-ptrace} entry by entry: $(\rho_A)_{00} =
\rho_{00,00} + \rho_{01,01} = \tfrac12 + 0$; $(\rho_A)_{01} =
\rho_{00,10} + \rho_{01,11} = 0 + 0$; and similarly for the rest. The
off-diagonal entries of $\rho$ that carried all the interference sit in
positions the sum in Eq.~\eqref{eq:I-ptrace} never visits.

The numbers to keep: the global purity is $\tr\rho^2 = 1$, the reduced
purity is $\tr\rho_A^2 = \tfrac12$, the minimum for a qubit, and the von
Neumann entropy $-\tr(\rho_A \ln \rho_A) = \ln 2$, exactly one bit.

\paragraph{Worked example, the product case.} For
$|0\rangle_A \otimes |+\rangle_B$ the same computation gives $\rho_A =
|0\rangle\!\langle 0|$, purity $1$, entropy zero. Nothing was lost.

\emph{Physics intuition.} Comparing the two examples isolates the
culprit. Marginalizing a product state loses nothing; marginalizing an
entangled state loses everything about the correlation. Entanglement is
what makes forgetting a subsystem expensive, and since interaction
generically produces entanglement, any subsystem that has ever interacted
with its surroundings has paid.

\subsection{Step 1 (continued) --- channels, and why positivity is not enough}
\label{ssec:I-cp}

What is the most general transformation a state can undergo? Write it as
a map $\mathcal{E}$ on matrices and list what it must do. It should be
\emph{linear}, because probabilistic mixtures must be respected: a
coin flip between two preparations followed by the process is the same
as the process applied to each and then the flip. It should be
\emph{trace preserving}, so that probabilities still sum to one. And it
should be \emph{positive}: $\rho \ge 0 \Rightarrow
\mathcal{E}(\rho) \ge 0$, so that no probability comes out negative.

Those three conditions look complete. They are not, and the failure is
sharp enough to compute.

\paragraph{The counterexample.} Let $T$ be matrix transposition in a
fixed basis, $T(\rho) = \rho^{\mathsf{T}}$. It is linear, trace
preserving, and positive: transposition leaves the spectrum unchanged, so
a positive matrix goes to a positive matrix. Now let $T$ act on qubit $B$
only, with qubit $A$ untouched, and apply it to the Bell state of
Eq.~\eqref{eq:I-bell-reduced}. The partial transpose exchanges the
entries $\rho_{ik,jl} \to \rho_{il,jk}$, giving
\begin{equation}
(\mathbf{1}\otimes T)\,\rho
 \;=\; \frac{1}{2}
 \begin{pmatrix}
 1 & 0 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 0 & 1
 \end{pmatrix} ,
\qquad
\text{eigenvalues } \left\{
 \tfrac12,\ \tfrac12,\ \tfrac12,\ -\tfrac12 \right\} .
\label{eq:I-pt}
\end{equation}
A negative eigenvalue of a would-be density matrix is a negative
probability. So $T$ is positive but $\mathbf{1}\otimes T$ is not, and $T$
is not a physical process.

\paragraph{The fix, and why it is the right one.} Demand
\emph{complete} positivity: $\mathcal{E}$ is CP if
$\mathrm{id}_n \otimes \mathcal{E}$ is positive for every $n$. The
physical reading is exactly the counterexample. The system on which the
process acts may be entangled with a spectator that the process never
touches, and the joint state must remain a state. A criterion that
examines the system alone cannot see this, because
Eq.~\eqref{eq:I-bell-reduced} showed that the system alone looks like
$\tfrac12\mathbf{1}$ either way.

A completely positive trace-preserving map is called a \emph{channel},
or CPTP map. It is the correct notion of ``what can happen to a state'',
and it is strictly larger than unitary evolution and strictly smaller
than the positive maps.

\subsection{Step 1 (continued) --- the Kraus representation}
\label{ssec:I-kraus}

\paragraph{Statement.} A linear map $\mathcal{E}$ on $d \times d$
matrices is CPTP if and only if there exist operators $K_k$ with
\begin{equation}
\mathcal{E}(\rho) \;=\; \sum_{k} K_k\, \rho\, K_k^\dagger ,
\qquad
\sum_k K_k^\dagger K_k \;=\; \mathbf{1} ,
\label{eq:I-kraus}
\end{equation}
and at most $d^2$ of them are needed --- four for a qubit.

\paragraph{Proof sketch.} The efficient route is the Choi matrix. Define
\begin{equation}
C(\mathcal{E}) \;=\; \sum_{i,j}
 |i\rangle\!\langle j| \otimes \mathcal{E}\!\left(|i\rangle\!\langle j|\right)
 \;=\; (\mathrm{id}_d \otimes \mathcal{E})
   \left(d \, |\Omega\rangle\!\langle \Omega|\right) ,
\label{eq:I-choi}
\end{equation}
where $|\Omega\rangle = d^{-1/2}\sum_i |ii\rangle$ is the maximally
entangled state. Two facts do the work. First, $\mathcal{E}$ is CP if and
only if $C(\mathcal{E}) \ge 0$: the ``only if'' is immediate, since
$C$ is $\mathcal{E}$ applied to one half of a state; the ``if'' follows
because $C$ determines $\mathcal{E}$ completely and one may run the
argument backwards. Second, a positive matrix can be written as
$C = \sum_k |v_k\rangle\!\langle v_k|$ with $|v_k\rangle$ the eigenvectors
scaled by the square roots of the eigenvalues, and $C$ is a $d^2 \times
d^2$ matrix, so there are at most $d^2$ terms. Reshaping each vector
$|v_k\rangle \in \mathbb{C}^d \otimes \mathbb{C}^d$ into a $d\times d$
matrix $K_k$ turns that sum into Eq.~\eqref{eq:I-kraus}, and trace
preservation becomes the stated completeness relation. \qed

\paragraph{The structural corollary: Stinespring.} Every channel can be
written as
\begin{equation}
\mathcal{E}(\rho) \;=\;
 \tr_{E}\!\left[\, U \left(\rho \otimes |0\rangle\!\langle 0|_E\right)
 U^\dagger \,\right] ,
\label{eq:I-stinespring}
\end{equation}
for some environment $E$ and some unitary $U$ on the enlarged space: put
the index $k$ of Eq.~\eqref{eq:I-kraus} into an ancilla, and the Kraus
sum is a partial trace. So a channel is not an alternative to unitary
evolution. \emph{A channel is a unitary evolution seen through a partial
trace} --- the projection of \S\ref{ssec:I-ptrace} applied to
Eq.~\eqref{eq:I-vonneumann}. Step 3 is going to lean on this sentence
hard, so it is worth being clear that it is a theorem of ordinary
quantum mechanics with nothing added.

\subsection{Step 1 (continued) --- the Born--Markov derivation, step by step}
\label{ssec:I-lindblad}

Equation~\eqref{eq:I-kraus} describes one jump: state in, state out. For
a continuous-time evolution one wants a differential equation whose
solution is a channel at every instant. Here is the standard derivation,
with each approximation named where it is made rather than collected in a
footnote.

\emph{Step 0, the setup.} Total Hamiltonian $H = H_S + H_B + \lambda H_I$
with $H_I = \sum_a A_a \otimes B_a$, the $A_a$ acting on the system and
the $B_a$ on the bath. Take $\tr_B(B_a \rho_B) = 0$; if not, shift the
term into $H_S$.

\emph{Step 1, interaction picture.} Move to the frame rotating with
$H_S + H_B$. The exact equation of motion is
$\dd\rho_{\text{tot}}/\dd t = -\ii\lambda[H_I(t), \rho_{\text{tot}}(t)]$,
still exact.

\emph{Step 2, integrate once and resubstitute.} Writing
$\rho_{\text{tot}}(t) = \rho_{\text{tot}}(0) - \ii\lambda\int_0^t \dd s\,
[H_I(s),\rho_{\text{tot}}(s)]$ and putting it back gives an equation
correct to second order in $\lambda$:
\begin{equation}
\frac{\dd \rho_{\text{tot}}}{\dd t}
 = -\ii\lambda [H_I(t), \rho_{\text{tot}}(0)]
 - \lambda^2 \int_0^t \!\dd s\,
   \big[H_I(t),[H_I(s), \rho_{\text{tot}}(s)]\big] .
\label{eq:I-secondorder}
\end{equation}

\emph{Step 3, trace out the bath.} The first term of
Eq.~\eqref{eq:I-secondorder} dies by the centring assumption of Step 0.

\emph{Step 4, the Born approximation.} Replace
$\rho_{\text{tot}}(s) \to \rho(s) \otimes \rho_B$. Two things are being
assumed at once and they should be distinguished: that the initial state
\emph{was} a product, and that the coupling is weak enough that the
correlations built up since then are negligible at order $\lambda^2$.
The result is the Redfield equation,
\begin{equation}
\frac{\dd \rho}{\dd t}
 = -\lambda^2 \int_0^t \!\dd s\;
   \tr_B\!\big[H_I(t),[H_I(s),\rho(s)\otimes\rho_B]\big] ,
\label{eq:I-redfield}
\end{equation}
which is not yet a semigroup: $\rho(s)$ under the integral remembers.

\emph{Step 5, the Markov approximation.} Replace $\rho(s) \to \rho(t)$.
This is legitimate when the bath correlation functions
$C_{ab}(t-s) = \tr_B[B_a(t)B_b(s)\rho_B]$ decay over a time $\tau_B$ much
shorter than the relaxation time $\tau_R$ over which $\rho$ changes.

\emph{Step 6, extend the limit.} Substitute $u = t - s$ and send the
upper limit to infinity, which costs nothing once $t \gg \tau_B$ since
the integrand has already died.

\emph{Step 7, the secular approximation.} Decompose the system operators
into eigenoperators of $H_S$: $A_a = \sum_\omega A_a(\omega)$ with
$[H_S, A_a(\omega)] = -\omega A_a(\omega)$. Terms with $\omega \ne
\omega'$ oscillate as $e^{\ii(\omega'-\omega)t}$ and average away
provided the level spacings are large compared with the relaxation rate.
Dropping them is what guarantees the final generator has the right sign
structure --- Redfield without this step can produce negative
probabilities.

\emph{Step 8, the result.} What survives is the
Gorini--Kossakowski--Sudarshan--Lindblad equation
\citep{Lindblad1976,GoriniKossakowskiSudarshan1976}:
\begin{equation}
\frac{\dd\rho}{\dd t}
 \;=\; -\ii\left[H_S + H_{\text{LS}},\ \rho\right]
 \;+\; \sum_k \gamma_k \left(
   L_k \rho L_k^\dagger
   - \tfrac12 \left\{ L_k^\dagger L_k,\ \rho \right\} \right) ,
\qquad \gamma_k \ge 0 .
\label{eq:I-lindblad}
\end{equation}
$H_{\text{LS}}$ is the Lamb shift, a bath-induced correction to the
system's own energies; the $L_k$ are the jump operators; the $\gamma_k$
are rates read off the Fourier transforms of the bath correlation
functions.

\paragraph{The converse, which is the important half.} Read the other
way, Eq.~\eqref{eq:I-lindblad} is not an approximation scheme at all. It
is a classification: \emph{any} generator of a one-parameter family of
CPTP maps satisfying $\Lambda_t \Lambda_s = \Lambda_{t+s}$ has this form
and no other. Each ingredient is forced --- Hermiticity of $\rho$ forces
the structure of the commutator and anticommutator terms, trace
preservation forces the $-\tfrac12\{L^\dagger L,\rho\}$ against the
$L\rho L^\dagger$, and complete positivity forces $\gamma_k \ge 0$. Take
one $\gamma_k$ negative and the evolution still preserves the trace and
Hermiticity, and still fails to be a channel.

\subsection{Step 1 (concluded) --- the dephasing qubit, solved}
\label{ssec:I-dephasing}

Take a two-level system with $H_S = \tfrac{\omega_0}{2}\sigma_z$ and a
single jump operator $L = \sigma_z$ at rate $\gamma$. Since
$\sigma_z^\dagger \sigma_z = \mathbf{1}$, the anticommutator term is just
$-\rho$ and Eq.~\eqref{eq:I-lindblad} collapses to
\begin{equation}
\frac{\dd\rho}{\dd t}
 = -\ii\frac{\omega_0}{2}\left[\sigma_z, \rho\right]
 + \gamma\left(\sigma_z \rho \sigma_z - \rho\right) .
\label{eq:I-dephase-eq}
\end{equation}
Write $\rho$ in components. On the diagonal, $\sigma_z \rho \sigma_z$
leaves $\rho_{00}$ and $\rho_{11}$ alone and the commutator vanishes, so
\begin{equation}
\dot\rho_{00} = \dot\rho_{11} = 0 .
\label{eq:I-populations}
\end{equation}
Off the diagonal, $(\sigma_z\rho\sigma_z)_{01} = -\rho_{01}$, so the
dissipator contributes $\gamma(-\rho_{01}-\rho_{01}) = -2\gamma
\rho_{01}$, and
\begin{equation}
\rho_{01}(t) \;=\; \rho_{01}(0)\,
 e^{-\ii\omega_0 t}\, e^{-2\gamma t} .
\label{eq:I-dephase-sol}
\end{equation}
On the Bloch ball, $z$ is frozen while $x$ and $y$ contract by
$e^{-2\gamma t}$: the ball is squashed onto the $z$ axis. The
decoherence time is $T_2 = 1/(2\gamma)$, and the energy
$\langle H_S \rangle$ is exactly constant --- nothing is exchanged, and
the state decoheres anyway.

\paragraph{The same channel in Kraus form.} Set
$q(t) = \tfrac12\left(1 - e^{-2\gamma t}\right)$ and take
\begin{equation}
K_0 = \sqrt{1-q}\;\mathbf{1} ,
\qquad
K_1 = \sqrt{q}\;\sigma_z ,
\qquad
K_0^\dagger K_0 + K_1^\dagger K_1
 = (1-q)\mathbf{1} + q\,\mathbf{1} = \mathbf{1} .
\label{eq:I-dephase-kraus}
\end{equation}
Then $\mathcal{E}(\rho) = (1-q)\rho + q\,\sigma_z\rho\sigma_z$ multiplies
the off-diagonal entry by $1 - 2q = e^{-2\gamma t}$, reproducing
Eq.~\eqref{eq:I-dephase-sol} with the free phase removed. Two Kraus
operators suffice, well below the bound of four.

\paragraph{Where the invertibility goes.} Regard $\mathcal{E}$ as a
linear map on the four-dimensional space of $2\times2$ matrices. Its
determinant in that representation is $(1-2q)^2 = e^{-4\gamma t}$, which
is non-zero for every finite $t$ and tends to zero as $t \to \infty$. In
the limit the rank of the map falls from four to two: $|+\rangle$ and
$|-\rangle$ are both sent to $\tfrac12\mathbf{1}$, and by
Eq.~\eqref{eq:I-nonunique} nothing downstream can tell them apart.

That is worth stating twice. At finite time the map has an inverse as a
\emph{linear} map. What it does not have, at any positive time, is an
inverse that is itself a channel --- and \S\ref{ssec:I-step2} shows that
this is not a peculiarity of dephasing.

\subsection{Step 2 --- what was assumed, and what survives losing it}
\label{ssec:I-step2}

\paragraph{The assumptions, listed.} Equation~\eqref{eq:I-lindblad} was
obtained using: a factorized initial state; weak coupling, truncating at
$\lambda^2$; a stationary bath; a separation of timescales
$\tau_B \ll \tau_R$; and the secular approximation. Every one of them can
fail in practice. Initial correlations occur; structured environments
have long memories; strongly coupled systems do not admit the truncation;
nearly degenerate levels defeat the secular step. When they fail the
exact reduced dynamics is still well defined --- it is a memory-kernel
equation of Nakajima--Zwanzig type, of which
Eq.~\eqref{eq:I-redfield} is the second-order truncation --- but it is not
a semigroup, and the map at time $t$ need not be divisible into maps at
intermediate times.

\paragraph{What does not depend on any of them.} Now the structural
point, and it is the reason this appendix exists.

Whatever the coupling strength and whatever the memory, the reduced
dynamics of a subsystem of a unitarily evolving whole, from a product
initial state, is a CPTP map: Eq.~\eqref{eq:I-stinespring} read
backwards. And a one-parameter family of channels composing as
\begin{equation}
\Lambda_t \circ \Lambda_s = \Lambda_{t+s} ,
\qquad \Lambda_0 = \mathrm{id} ,
\qquad t,s \ge 0 ,
\label{eq:I-semigroup}
\end{equation}
is a \emph{semigroup} and not a group. The restriction $t \ge 0$ is not
an omission awaiting repair. There is no $\Lambda_{-t}$.

\paragraph{Why there is no inverse: the theorem.} Suppose a channel
$\Lambda$ has an inverse $\Lambda^{-1}$ that is also a channel. Every
CPTP map contracts the trace distance,
\begin{equation}
\left\| \Lambda(\rho) - \Lambda(\sigma) \right\|_1
 \;\le\; \left\| \rho - \sigma \right\|_1 ,
\label{eq:I-contraction}
\end{equation}
which is the statement that a physical process cannot make two states
easier to tell apart. Apply Eq.~\eqref{eq:I-contraction} to $\Lambda$ and
then to $\Lambda^{-1}$: both inequalities must be equalities, so
$\Lambda$ preserves trace distance exactly. A bijection of the state
space preserving trace distance is implemented by conjugation with a
unitary. Hence: \emph{the only invertible channels are the unitary ones}.
Anything that genuinely loses information is irreversible, with no
exception and no small parameter to blame.

\paragraph{So the arrow here is structural, not statistical.} No
coarse-graining over a large number of particles was invoked, no
thermodynamic limit, no typicality argument. Equations
\eqref{eq:I-semigroup} and \eqref{eq:I-contraction} follow from complete
positivity and trace preservation, which follow in turn from the single
assumption that one is describing a \emph{part}. Reversibility is what
the whole has. Directedness is what a part has.

\paragraph{What the Lindblad form does not settle.} Two things, and both
must be said plainly because it is easy to let the machinery imply more
than it delivers.

\emph{It does not say which outcome occurs.} Equation~\eqref{eq:I-lindblad}
evolves $\rho$, and $\rho$ is a description of an ensemble. A dephasing
channel takes a superposition to a diagonal matrix, which is often
described as ``the measurement happening'', but by
Eq.~\eqref{eq:I-nonunique} a diagonal $\rho$ is exactly as agnostic about
which state the system is in as the original was about which basis it was
prepared in. Decoherence explains why interference stops being
observable. It does not produce a single result, and no equation of this
type can, since every one of them is linear in $\rho$.

\emph{It does not derive the measure.} The weights are put in as the
$p_i$ of Eq.~\eqref{eq:I-rho}. The sharpest general statement of how far
one can get without further input is Gleason's theorem
\citep{Gleason1957}: on a Hilbert space of dimension at least three, any
assignment of probabilities to projections that is additive on orthogonal
families and does not depend on which complete set the projection is read
in must take the form $p(P) = \tr(\rho P)$ for some density matrix
$\rho$. That is a strong result --- the Born \emph{form} is not an
independent postulate --- and it is also a limited one. It is a
statement about consistent assignments, not about dynamics; it assumes
noncontextuality; it says nothing about how a particular run comes to
have a particular value; and it fails in dimension two. Where the
program of Part~III stands on that boundary is stated in
Chapter~\ref{sec:oggep} and owned by \citet{ArisakaA8QM}.

\subsection{Step 3 --- Part III's reading: observation as the projection}
\label{ssec:I-step3}

Part~III makes one move, and this appendix has supplied its entire
technical content. The move is to identify the partial trace of
\S\ref{ssec:I-ptrace} with observation itself.

\paragraph{The identification.} \GGEP{} (\AX{2}) asserts a single
geometric object on a six-dimensional bulk. \OGGEP{} (\AX{3}) asserts
that such a theory is not physically defined until it supplies an
operational dictionary: a map from bulk structure to registered boundary
observables, together with the evolutions between registrations
(Chapter~\ref{sec:oggep}). Concretely the map erases the two non-spatial
fiber coordinates --- a scale interval and a compact phase circle --- and
delivers a boundary Hilbert space, a state, instruments, and channels.

Now read Eq.~\eqref{eq:I-stinespring} with that dictionary in place. A
bulk evolution that is unitary, restricted to what a boundary observer
can address, \emph{is} a channel; not approximately, not in a limit, but
by the theorem of \S\ref{ssec:I-kraus}. The openness of the boundary
dynamics is therefore not an extra postulate laid on top of the axiom. It
is what the axiom says, expressed in the language of this appendix.

\paragraph{What follows, and at which grade.} \GDOS{} takes it from
there \citep[see][]{ArisakaA5GDOS}. The reduction is CPTP; the exact
boundary evolution is a memory-kernel equation with no approximation; and
only in a stated regime --- the one whose assumptions
\S\ref{ssec:I-step2} listed --- does it take the semigroup form
\eqref{eq:I-lindblad}. The staging matters, and the honest register
requires repeating what Chapter~\ref{sec:oggep} says: the arrow of time
in this program is \emph{constitutional}, located in \AX{3}, and not
derived. What is derived is everything downstream --- that the semigroup
has no inverse, that a monotone increases, that the second law reads as
information loss rather than as a new postulate.

\paragraph{The cascade.} Chapter~\ref{sec:grip} adds the dynamics on top
of the structure: \GRIP{} descends the \GFF{} landscape through
\DGI{}, \GSSB{} and \GA{}, and the descent is monotone
(\citealp{ArisakaA6Gi4}). It is worth noticing what that requires. A
Lyapunov functional is a statement about a semigroup; on a group it would
be vacuous, since anything that decreases forward increases backward.
The landscape picture of Chapter~\ref{sec:grip} is available only because
Eq.~\eqref{eq:I-contraction} holds, and it would be meaningless in a
closed unitary description.

\paragraph{The central structural claim.} Here it is, in the vocabulary
of this appendix.

\begin{quote}\itshape
An exact symmetry of the bulk can be broken in the boundary reading,
because the projection that produces the boundary has no inverse.
\end{quote}

Nothing exotic is required to see how that is possible. Take a bulk of
two qubits with the exchange Hamiltonian
$H = \tfrac{J}{2}(\sigma_x\sigma_x + \sigma_y\sigma_y)$, which commutes
exactly with the total charge $Q = \sigma_z \otimes \mathbf{1} +
\mathbf{1}\otimes\sigma_z$. Start in $|1\rangle_A|0\rangle_B$ and let a
boundary observer see qubit $A$ alone. Then
\begin{equation}
\langle \sigma_z \rangle_A (t) = \cos 2Jt ,
\qquad
\rho_A(t) = \mathrm{diag}\!\left(\cos^2 Jt,\ \sin^2 Jt\right) ,
\label{eq:I-toy}
\end{equation}
so the boundary charge is not conserved at all, while the bulk charge is
conserved exactly. At $Jt = \pi/4$ the boundary state is $\tfrac12\mathbf{1}$
and the initial condition is unrecoverable from it: the two distinct
starting states $|1\rangle_A|0\rangle_B$ and $|0\rangle_A|1\rangle_B$ have been
mapped to the same boundary state, which is the rank collapse of
\S\ref{ssec:I-dephasing} in its simplest form.

Two lessons, and they are the two halves of Part~III. First, the boundary
conservation law has not been repealed; it has become an audit, with the
missing charge appearing as flux through the seam --- the extra term of
Chapter~\ref{sec:oggep}'s open Ward identity, which
Chapter~\ref{sec:ward} supplies in closed form. Second, the asymmetry the
boundary observer reports is real for that observer and is not a
violation of anything in the bulk. A projection with no inverse is
exactly the license that permits an exact symmetry above and a broken
one below.

\paragraph{And the fourth move.} Chapter~\ref{sec:qi4} extends the same
architecture --- a landscape, a monotone, a forward semigroup --- beyond
particle physics, and the review states it without defending it. The
relevant point for this appendix is narrow --- Table~\ref{tab:I-closed-open}
puts the closed and open columns on the same page, and the whole of Part~III
lives in the differences between them --- and it is this: whatever one
thinks of the
extension, the machinery it uses is the machinery of Step 1, and the
grade of any claim built on it can be no better than the grade of its
premise.

\begin{table}[H]
\centering
\caption[Closed and open dynamics compared.]{The two columns, on the same
Hilbert space. Only the second and third rows are usually presented as
the difference; the fourth is the one Part~III uses, and the last is the
pivot of Chapter~\ref{sec:oggep} written in this appendix's language. Two columns on the same Hilbert space, seven rows: the state, the evolution, composition, invertibility, purity, the generator, and what a conservation law says in each setting. The table is set out in full rather than at the two rows usually shown, because the rows usually left out are the ones this paper needs.}
\label{tab:I-closed-open}\small
\begin{tabular}{@{}P{0.20\textwidth}P{0.33\textwidth}P{0.35\textwidth}@{}}
\toprule
 & \textbf{closed system} & \textbf{open system} \\
\midrule
state & vector $|\psi\rangle$, or pure $\rho$ &
density matrix, generically mixed \\
evolution & unitary, Eq.~\eqref{eq:I-vonneumann} &
channel, Eq.~\eqref{eq:I-kraus} \\
composition & group: $U_t U_s = U_{t+s}$, all $t,s$ &
semigroup: Eq.~\eqref{eq:I-semigroup}, $t,s \ge 0$ \\
inverse & always, $U^{-1} = U^\dagger$ &
only if the channel is unitary \\
purity & exactly conserved & non-increasing \\
generator & $-\ii[H,\cdot\,]$ &
Eq.~\eqref{eq:I-lindblad}, with $\gamma_k \ge 0$ \\
a conservation law says & what cannot leave &
what the books must show when it does \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Problems and worked solutions}
\label{ssec:I-problems}

\medskip\noindent\textbf{I.1. For the Bell state of
Eq.~\eqref{eq:I-bell}, compute $\rho$, $\rho_A$, both purities and the
entropy of $\rho_A$. Then do the same for
$|0\rangle_A\otimes|+\rangle_B$ and say what distinguishes the two cases.}

\noindent\emph{Solution.} \emph{Step 1.} $\rho$ is the $4\times4$ matrix
of Eq.~\eqref{eq:I-bell-reduced}, with $\tr\rho^2 = 1$ since it is a
projector onto a single vector. \emph{Step 2.} Equation~\eqref{eq:I-ptrace}
gives $\rho_A = \tfrac12\mathbf{1}$, so $\tr\rho_A^2 = \tfrac12$ and
$S = -2 \times \tfrac12\ln\tfrac12 = \ln 2$, one bit. \emph{Step 3.} For
the product state $\rho_A = |0\rangle\!\langle 0|$, purity $1$, entropy $0$.
\emph{Physics intuition:} the difference is entanglement, and it shows up
as the gap between a purity of $1$ for the whole and $\tfrac12$ for the
part. A pure global state with mixed parts is the ordinary situation, not
a pathology, and it is why the density matrix is unavoidable even for an
observer who knows everything knowable about the world.

\medskip\noindent\textbf{I.2. Show that transposition is a positive map,
then compute the eigenvalues in Eq.~\eqref{eq:I-pt} and state exactly
which of the three requirements on a channel it violates.}

\noindent\emph{Solution.} \emph{Step 1.} $\rho^{\mathsf{T}}$ has the same
characteristic polynomial as $\rho$, since
$\det(\rho^{\mathsf{T}} - \lambda\mathbf{1}) =
\det(\rho-\lambda\mathbf{1})^{\mathsf{T}}$; so positivity is preserved,
and the trace is too. \emph{Step 2.} The partial transpose of the Bell
state is $\tfrac12$ times the swap operator, whose eigenvalues are $+1$
on the three-dimensional symmetric subspace and $-1$ on the
one-dimensional antisymmetric one, giving
$\{\tfrac12,\tfrac12,\tfrac12,-\tfrac12\}$ as in
Eq.~\eqref{eq:I-pt}. \emph{Step 3.} Linearity and trace preservation hold;
what fails is complete positivity --- positivity of
$\mathrm{id}\otimes T$. \emph{Physics intuition:} the counterexample is
built from a spectator the map never touches, and that is the whole
content of the word ``complete''. A criterion tested only on the system
is blind to what the system is correlated with, and by
Eq.~\eqref{eq:I-bell-reduced} the system alone looks the same either way.

\medskip\noindent\textbf{I.3. Give two different preparations of a qubit
that produce $\rho = \tfrac12\mathbf{1}$, and prove that no measurement
distinguishes them. What does this say about running the preparation
backwards?}

\noindent\emph{Solution.} \emph{Step 1.} The two decompositions in
Eq.~\eqref{eq:I-nonunique} --- an even mixture of
$|0\rangle,|1\rangle$ and an even mixture of $|+\rangle,|-\rangle$.
\emph{Step 2.} By Eq.~\eqref{eq:I-expect} every prediction is
$\tr(\rho A)$, and $\rho$ is the same matrix, so every prediction is the
same for every $A$; the statement covers positive-operator measurements
too, since those are also linear functionals of $\rho$. \emph{Step 3.}
The map from preparation to state is therefore not injective, and no
process --- physical or mathematical --- can undo it, since the
information required is not present in the output. \emph{Physics
intuition:} this is irreversibility before any dynamics has been
introduced. It is a fact about the state space, and it is the reason the
irreversibility of \S\ref{ssec:I-step2} needs no thermodynamic limit to
justify it.

\medskip\noindent\textbf{I.4. Solve Eq.~\eqref{eq:I-dephase-eq}, find the
Kraus operators, check the completeness relation, and compute the
determinant of the channel as a linear map on $2\times2$ matrices.}

\noindent\emph{Solution.} \emph{Step 1.} The diagonal entries are
untouched by both terms, Eq.~\eqref{eq:I-populations}; the off-diagonal
entry picks up $-\ii\omega_0$ from the commutator and $-2\gamma$ from the
dissipator, giving Eq.~\eqref{eq:I-dephase-sol}. \emph{Step 2.} With
$q = \tfrac12(1-e^{-2\gamma t})$ the operators of
Eq.~\eqref{eq:I-dephase-kraus} satisfy $K_0^\dagger K_0 + K_1^\dagger
K_1 = (1-q)\mathbf{1} + q\mathbf{1} = \mathbf{1}$, and the map they
generate multiplies $\rho_{01}$ by $1-2q = e^{-2\gamma t}$.
\emph{Step 3.} In the basis
$\{|0\rangle\!\langle 0|,|0\rangle\!\langle 1|,|1\rangle\!\langle 0|,|1\rangle\!\langle 1|\}$
the map is diagonal with entries $1, 1-2q, 1-2q, 1$, so the determinant
is $(1-2q)^2 = e^{-4\gamma t}$. \emph{Physics intuition:} the determinant
is the volume of the Bloch ball, and it is shrinking exponentially with
nothing being exchanged --- the energy is constant throughout. Losing
information and losing energy are different transactions, and this model
performs only the first.

\medskip\noindent\textbf{I.5. Prove that a channel whose inverse is also
a channel must be unitary, and explain why this makes
Eq.~\eqref{eq:I-semigroup} a semigroup rather than a group.}

\noindent\emph{Solution.} \emph{Step 1.} Every CPTP map satisfies the
contraction inequality \eqref{eq:I-contraction}. \emph{Step 2.} If
$\Lambda^{-1}$ is CPTP as well, then
$\|\rho - \sigma\|_1 = \|\Lambda^{-1}\Lambda(\rho) -
\Lambda^{-1}\Lambda(\sigma)\|_1 \le \|\Lambda(\rho)-\Lambda(\sigma)\|_1
\le \|\rho-\sigma\|_1$, so both inequalities are equalities and
$\Lambda$ preserves trace distance. \emph{Step 3.} A linear bijection of
the state space preserving trace distance is conjugation by a unitary;
hence any channel that loses anything at all has no channel inverse, and
the family \eqref{eq:I-semigroup} cannot be extended to negative $t$.
\emph{Physics intuition:} the theorem is why one may not read the
semigroup as an approximation waiting to be improved. The forward-only
structure is not the residue of a coarse-graining; it is what being a
part of something larger costs, exactly.

\medskip\noindent\textbf{I.6. Verify Eq.~\eqref{eq:I-toy} for the
two-qubit exchange model, confirm that the bulk charge is conserved, and
identify the time at which the boundary state stops remembering which
qubit was excited.}

\noindent\emph{Solution.} \emph{Step 1.} $H$ acts on the
two-dimensional span of $|10\rangle$ and $|01\rangle$ as $J$ times the
off-diagonal exchange, so $|10\rangle \mapsto \cos Jt\,|10\rangle -
\ii \sin Jt\,|01\rangle$, and $[H,Q] = 0$ because $H$ moves an excitation
without creating or destroying one. \emph{Step 2.} Tracing out $B$ by
Eq.~\eqref{eq:I-ptrace} gives $\rho_A = \mathrm{diag}(\cos^2 Jt, \sin^2
Jt)$ and hence $\langle\sigma_z\rangle_A = \cos^2 - \sin^2 = \cos 2Jt$,
while $\langle Q\rangle = 0$ for all $t$. \emph{Step 3.} At $Jt = \pi/4$
one has $\rho_A = \tfrac12\mathbf{1}$; the same state is reached from
$|0\rangle_A|1\rangle_B$, so the boundary description no longer determines
the initial condition. \emph{Physics intuition:} the bulk symmetry was
never touched --- the exchange Hamiltonian commutes with $Q$ exactly, at
every instant. What changed is who is reading. That is the whole of the
review's central structural claim, in four lines of two-qubit algebra,
and everything Part~III adds is the proposal that the world's observed
sector stands in this relation to a six-dimensional bulk.

\GGtakeaway{\emph{Standard result:} a subsystem's state is a density
matrix obtained by partial trace; its evolution is a completely positive
trace-preserving map, equivalently a Kraus sum, equivalently a unitary on
a larger space followed by a partial trace; and if the memory time of the
environment is short the generator takes the Lindblad form, which is the
unique form a CPTP semigroup generator can take. \emph{What it assumed:}
for the Lindblad equation specifically --- a factorized initial state,
weak coupling, a short bath memory, and the secular step. \emph{What
survives dropping all of them:} complete positivity, trace preservation,
the composition law $\Lambda_t\Lambda_s = \Lambda_{t+s}$ for $t,s \ge 0$,
and the theorem that the only invertible channels are unitary --- so the
forward direction is structural, not statistical. \emph{What it does not
settle:} which outcome occurs, and where the measure comes from;
Gleason's theorem marks how far one gets without further input.
\emph{Continue in:} Chapter~\ref{sec:oggep} for the axiom that reads
observation as this projection, Chapter~\ref{sec:grip} for the descent it
makes possible, and Chapter~\ref{sec:qi4} for the extension the review
states and fences.}

\section{Counting the parameters: what a constant of nature is}
\label{app:params}
\GGteachbox{What a ``free parameter'' of a quantum field theory actually
is --- not a number but a triple of value, scheme and scale; the
flavor-counting theorem derived from orbit-stabilizer rather than
quoted, with the quark sector worked to the last phase ($36$ real
entries, $27-1 = 26$ removed, $10$ left); the mixing matrix counted
twice, once as a coset and once by rephasing; the leptonic sector done
for massless, Dirac and Majorana neutrinos so that the two extra phases
appear where they come from; the full twenty-eight-item tally assembled
sector by sector; and running worked in closed form for one coupling,
until it is obvious why a quoted value without a scheme and a scale is
not a measurement. The pay-off is Chapter~\ref{sec:zero-parameters} and
the sheet of Chapter~\ref{sec:sm-sheet}.}

\GGreadalong{\citet{PeskinSchroeder1995}, \S11.4 and Ch.~12, for
renormalization conditions and the running coupling done slowly;
\citet{ChengLi1984}, Ch.~11--12, for the flavor sector with the
counting spelled out; \citet{KobayashiMaskawa1973} for the three-line
argument that three families are the minimum;
\citet{Weinberg1996}, Vol.~II, Ch.~18, for the renormalization group as
a statement about a space of theories; and \citet{PDG2024} for what the
numbers are this year, with their schemes attached.}

\subsection{Step 1 --- what a free parameter is}
\label{ssec:J-step1}

Start with the thing everybody skips. A Lagrangian is a formula
containing symbols, and a symbol is not a number until somebody says
which measurement fixes it. The step from one to the other is the
subject of this section, and it is more interesting than it looks.

Write the Standard Model in blocks, each carrying its own symbols:
\begin{align}
\mathcal{L} \;=\;
&-\tfrac14 \sum_{i=1}^{3} \frac{1}{g_i^2}\,
  \tr F^{(i)}_{\mu\nu} F^{(i)\,\mu\nu}
 \;+\; \sum_{i} \theta_i \,
  \tr F^{(i)} \wedge F^{(i)}
\nonumber\\
&+\; |D_\mu H|^2 \;-\; \mu^2 |H|^2 \;-\; \lambda |H|^4
\;+\; \sum_{\psi} \bar\psi\, \ii \slashed{D} \psi
\nonumber\\
&-\; \Big(
 \bar Q\, Y_u\, \widetilde H\, u_R
+\bar Q\, Y_d\, H\, d_R
+\bar L\, Y_e\, H\, e_R
+ \text{h.c.} \Big)
\;+\; \frac{c_{ij}}{\Lambda}\,(L_i H)(L_j H) \;+\;\cdots
\label{eq:J-lagrangian}
\end{align}
The fermion kinetic term carries no parameters at all: any coefficient
in front of it can be scaled to one by redefining the field, and a
redefinition of a field is not a change of theory. That is the first
appearance of the principle this whole appendix runs on. \emph{What is
counted is not symbols but symbols modulo redefinitions.}

\paragraph{A parameter is a renormalization condition.} Take the
simplest coupling in Eq.~\eqref{eq:J-lagrangian}, the electric charge.
The symbol $e$ in the classical Lagrangian is not measurable. In the
quantum theory the amplitude for two electrons to scatter is a sum over
diagrams whose loop corrections diverge, and the divergence is absorbed
by declaring that the symbol in the Lagrangian was never the physical
charge. What one declares instead is a \emph{renormalization condition}:
``the four-point amplitude, evaluated at this specified kinematic point,
equals this number, and that number is what I call the coupling.''

Two choices have been made and both matter. \emph{The scheme} is the
recipe for which finite part to keep alongside the divergence: minimal
subtraction with $\ln 4\pi - \gamma_E$ absorbed --- the
$\overline{\rm MS}$ scheme --- keeps one, and the on-shell scheme, which
imposes the condition at a physical threshold, keeps another; they
differ by a calculable non-zero amount. \emph{The scale} is the
kinematic point at which the condition is imposed: zero momentum
transfer gives one number, $q^2 = M_Z^2$ gives another.

So a free parameter of a quantum field theory is not a real number. It
is a triple,
\begin{equation}
\text{parameter} \;=\; \big(\text{value},\ \text{scheme},\ \text{scale}\big) ,
\label{eq:J-triple}
\end{equation}
and any two of the three without the third is an incomplete statement.
The fine-structure constant makes the point in one line: the same
theory, the same universe, the same coupling gives
\begin{equation}
\alpha^{-1}(0) \;=\; 137.035999177 ,
\qquad
\alpha^{-1}(M_Z)\big|_{\overline{\rm MS}} \;=\; 127.930(8) ,
\label{eq:J-alpha-two}
\end{equation}
and the reader who has been told that $\alpha$ ``is'' $1/137$ has been
told the value at one particular scale in one particular convention
without being told which.

\paragraph{What is being counted, precisely.} Fix the gauge group and
the representation content. The renormalizable Lagrangians with that
content form a manifold; two points differing by a field redefinition
are the same physics; and moving along an RG trajectory is not moving to
a different theory but describing the same theory at a different scale.
So:
\begin{quote}\itshape
The number of free parameters is the dimension of the space of
renormalizable theories with a given field content, quotiented by field
redefinitions, and counted at one fixed scale in one fixed scheme.
\end{quote}
That is a statement about a quotient, and the consequence is that the
answer is an invariant even though every individual value is
convention-laden. The value of $\alpha$ depends on the scheme. The
statement ``there are three gauge couplings'' does not.

\subsection{Step 1 (continued) --- the counting theorem, and the quark
sector worked in full}
\label{ssec:J-yukawa}

Everything in the flavor sector follows from one counting principle,
so the principle is derived once and then used four times.

\paragraph{The principle.} Let the couplings live in a real vector space
$V$ of dimension $n_V$, and let a group $G$ act on $V$ by field
redefinitions that leave the rest of the Lagrangian alone. Pick a
generic point $Y \in V$. Its orbit $G \cdot Y$ has dimension
$\dim G - \dim H$, where $H \subset G$ is the stabilizer of $Y$ --- the
subgroup that moves nothing. Points on one orbit are the same theory.
Hence
\begin{equation}
n_{\text{phys}} \;=\; n_V \;-\; \big(\dim G - \dim H\big) .
\label{eq:J-orbit}
\end{equation}
The stabilizer term is the one that gets forgotten, and it is exactly
where baryon number is going to appear.

\paragraph{The group.} The fermion kinetic terms of
Eq.~\eqref{eq:J-lagrangian} do not know about flavor: they are sums
over the three copies of each gauge representation, and any unitary
rotation among the copies leaves them alone. For the quarks the copies
are $Q_i$ (left-handed doublets), $u_{R\,i}$ and $d_{R\,i}$, so
\begin{equation}
G_F^{\text{quark}} \;=\; U(3)_Q \times U(3)_u \times U(3)_d ,
\qquad
\dim G_F^{\text{quark}} \;=\; 3 \times 9 \;=\; 27 ,
\label{eq:J-gf-quark}
\end{equation}
acting on the Yukawa matrices as
\begin{equation}
Y_u \;\longmapsto\; V_Q\, Y_u\, V_u^\dagger ,
\qquad
Y_d \;\longmapsto\; V_Q\, Y_d\, V_d^\dagger .
\label{eq:J-yaction}
\end{equation}
Note that $V_Q$ appears in both, because $Q$ is one field. That single
fact is the origin of quark mixing: were the two transformations
independent, both matrices could be diagonalized separately and there
would be no CKM matrix at all.

\paragraph{The count.} $Y_u$ and $Y_d$ are each $3\times3$ complex,
hence $18$ real entries apiece:
\begin{equation}
n_V \;=\; 2 \times 2 \times 3^2 \;=\; 36 .
\label{eq:J-nv36}
\end{equation}
Now the stabilizer. Choose a basis in which $Y_u = D_u$ is diagonal with
distinct positive entries and $Y_d = V D_d$ with $D_d$ diagonal positive
and $V$ a generic unitary matrix with no vanishing entry. An element
$(V_Q, V_u, V_d)$ fixes $D_u$ only if $V_Q D_u V_u^\dagger = D_u$, and
with distinct entries this forces $V_Q = V_u =
\mathrm{diag}(e^{\ii\beta_1}, e^{\ii\beta_2}, e^{\ii\beta_3})$. Feeding
that into the second condition, $V_Q V D_d V_d^\dagger = V D_d$ requires
$V^\dagger V_Q V$ to be diagonal, which for generic $V$ happens only
when $V_Q$ is a multiple of the identity. So
\begin{equation}
H \;=\; \big\{\, V_Q = V_u = V_d = e^{\ii\beta}\mathbf{1} \,\big\}
 \;\cong\; U(1)_B ,
\qquad \dim H = 1 .
\label{eq:J-stab}
\end{equation}
The stabilizer is baryon number, and it appears here not as a physical
insight bolted on but as the answer to a linear-algebra question.
Equation~\eqref{eq:J-orbit} then gives
\begin{equation}
n_{\text{phys}}^{\text{quark}}
 \;=\; 36 \;-\; (27 - 1) \;=\; 36 - 26 \;=\; 10 .
\label{eq:J-quark10}
\end{equation}

\paragraph{The same ten, counted the other way.} The second route says
what the ten \emph{are}. Any complex matrix has a singular-value
decomposition $Y = U_L D U_R^\dagger$ with $D$ diagonal, real and
non-negative. Use $U(3)_u$ and part of $U(3)_Q$ to set $Y_u = D_u$; use
$U(3)_d$ and the rest of $U(3)_Q$ to bring $Y_d$ to $V D_d$. What is
left is
\begin{equation}
\underbrace{3}_{\text{up masses}}
+\underbrace{3}_{\text{down masses}}
+\underbrace{4}_{V_{\rm CKM}} \;=\; 10 ,
\label{eq:J-quark-decomp}
\end{equation}
and the next section explains why the residual matrix $V$ carries four
and not nine.

\paragraph{The general case, since it costs nothing.} For $n$
generations the same three steps give $n_V = 4n^2$,
$\dim G_F = 3n^2$, $\dim H = 1$, so
\begin{equation}
n_{\text{phys}}^{\text{quark}}(n)
 \;=\; 4n^2 - 3n^2 + 1 \;=\; n^2 + 1 ,
\label{eq:J-quark-n}
\end{equation}
which is $2$ for one generation, $5$ for two, $10$ for three and $17$
for four. The decomposition tracks it: $2n$ masses, $n(n-1)/2$ angles
and $(n-1)(n-2)/2$ phases, summing to $n^2+1$ for every $n$.

\subsection{Step 1 (continued) --- angles, phases, and why three
generations}
\label{ssec:J-mixing}

The mixing matrix $V$ is unitary, so it has $n^2$ real parameters. Split
them. The real orthogonal matrices form a subgroup $O(n) \subset U(n)$
of dimension $n(n-1)/2$; call those the \emph{angles}. What is left,
\begin{equation}
n^2 - \frac{n(n-1)}{2} \;=\; \frac{n(n+1)}{2} ,
\label{eq:J-phasecount}
\end{equation}
are the \emph{phases}: $3$ angles and $6$ phases for $n = 3$.

Most of those phases are not physics. The quark fields still carry
individual phase freedoms that the diagonalization did not use up:
$u_i \to e^{\ii\phi_i} u_i$ and $d_j \to e^{\ii\chi_j} d_j$ leave the
diagonal mass matrices alone and act on $V$ as
\begin{equation}
V_{ij} \;\longmapsto\; e^{-\ii\phi_i}\, V_{ij}\, e^{\ii\chi_j} .
\label{eq:J-rephase}
\end{equation}
That is $2n$ parameters, one combination of which --- the common phase
$\phi_i = \chi_j = $ constant --- does nothing at all, so $2n-1$ phases
can be removed. Hence
\begin{equation}
n_{\text{phase}}
 \;=\; \frac{n(n+1)}{2} - (2n - 1)
 \;=\; \frac{(n-1)(n-2)}{2} ,
\label{eq:J-physphase}
\end{equation}
which is $0$ for $n = 1$, $0$ for $n = 2$, and $1$ for $n = 3$.

This is the Kobayashi--Maskawa argument \citep{KobayashiMaskawa1973},
and it deserves to be seen for what it is. It is a counting exercise, it
takes ten minutes, and it says that a world with two families cannot
violate $CP$ through quark mixing however the couplings are chosen ---
the phase is not small there, it is absent, removable by a change of
variables. In 1973 two families were all that had been seen and the
$CP$ violation of \citet{Christenson1964} was nine years old and
unexplained. So the CKM matrix carries $3 + 1 = 4$ parameters, which
completes Eq.~\eqref{eq:J-quark-decomp}.

\paragraph{The invariant version.} A skeptic can object that phases are
basis-dependent and that a count of them is a count of conventions. The
answer is that the physical content can be written without choosing a
basis at all. The Jarlskog invariant
\begin{equation}
J_{\rm CP} \;=\; \frac{\mathrm{Im}\,
 \det\!\big[\, Y_u Y_u^\dagger,\ Y_d Y_d^\dagger \,\big]}
 {\text{(mass factors)}}
\label{eq:J-jarlskog}
\end{equation}
is built from traces and determinants of the Yukawa matrices and is
therefore invariant under Eq.~\eqref{eq:J-yaction}. It vanishes if and
only if $CP$ is conserved, and it vanishes automatically when any two
up-type or any two down-type masses coincide. Nothing about a basis
enters. The count of \emph{phases} is a convenient bookkeeping; the
existence of one non-removable $CP$-odd quantity is a fact.

\subsection{Step 1 (continued) --- the leptons, and where two extra
phases come from}
\label{ssec:J-leptons}

The lepton sector is where the count depends on a physical question that
is still open, so it is worth doing three times.

\paragraph{Massless neutrinos: the classic thirteen.} Drop the last term
of Eq.~\eqref{eq:J-lagrangian}. The couplings are $Y_u, Y_d, Y_e$, so
$n_V = 3 \times 18 = 54$. The flavor group is
$U(3)^5 = U(3)_Q \times U(3)_u \times U(3)_d \times U(3)_L \times
U(3)_e$, of dimension $45$. The stabilizer: baryon number as before,
and --- because $Y_e$ can be made diagonal, leaving each charged lepton
free to rephase together with its own neutrino --- the three individual
lepton numbers separately. So $\dim H = 4$ and
\begin{equation}
n_{\text{phys}} \;=\; 54 - (45 - 4) \;=\; 54 - 41 \;=\; 13
 \;=\; \underbrace{9}_{\text{masses}}
 +\underbrace{3}_{\text{angles}}
 +\underbrace{1}_{\text{phase}} .
\label{eq:J-thirteen}
\end{equation}
The often-repeated slogan is that $U(3)^5$ is broken to
$U(1)_B \times U(1)_L$. That is the statement for \emph{massive} Dirac
neutrinos; with massless ones the three lepton numbers survive
individually, and getting the stabilizer wrong here changes the answer
by two. Two things belong with it. The residual $U(1)_B$ and $U(1)_L$
are accidental --- nobody imposed them, they are what was left over ---
and they are anomalous: the electroweak instanton violates $B$ and $L$
while preserving $B - L$ (Appendix~\ref{app:anomalies}), so the exactly
conserved combination is one, not two. A symmetry that emerged from a
counting argument is partly destroyed by a quantum effect.

\paragraph{Dirac neutrinos: add seven.} Give the neutrinos a Yukawa
coupling to right-handed singlets. Repeating the count in the same way
adds three neutrino masses and a lepton mixing matrix with the same
structure as the CKM matrix --- three angles and one phase by
Eq.~\eqref{eq:J-physphase}. Seven new numbers, and the total goes from
$19$ to $26$ once the gauge and Higgs sectors are included below.

\paragraph{Majorana neutrinos: the last two phases are new in kind.}
Keep the dimension-five operator of Eq.~\eqref{eq:J-lagrangian}, whose
coefficient $c_{ij}$ is a \emph{symmetric} $3\times3$ complex matrix ---
symmetric because $(L_iH)(L_jH)$ is symmetric under $i \leftrightarrow
j$ --- and therefore has $6$ complex, hence $12$ real, entries. With
$Y_e$'s $18$ this is $n_V = 30$, and the available group is only
$U(3)_L \times U(3)_e$, of dimension $18$. The stabilizer is now
\emph{trivial}: a Majorana mass term $\nu^{\mathsf{T}} C \nu$ is not
invariant under $\nu \to e^{\ii\alpha}\nu$ for any $\alpha \ne 0$, so
lepton number is gone and nothing is left to fix. Hence
\begin{equation}
n_{\text{phys}}^{\text{lepton}} \;=\; 30 - 18 \;=\; 12
 \;=\; \underbrace{3}_{\text{charged}}
 +\underbrace{3}_{\nu\ \text{masses}}
 +\underbrace{3}_{\text{angles}}
 +\underbrace{3}_{\text{phases}} .
\label{eq:J-twelve}
\end{equation}
Three phases, where the Dirac case had one, and the mixing-matrix route
shows where the extra two live. In the Dirac case
Eq.~\eqref{eq:J-rephase} applies on both sides of the mixing matrix,
removing $2n-1 = 5$ phases from the $6$. In the Majorana case the
neutrino fields cannot be rephased at all, so only the $n = 3$
charged-lepton phases are available and $6 - 3 = 3$ survive. The
difference, $5 - 3 = 2$, is the two \emph{Majorana phases} --- not a
modeling choice but the two rephasings the theory does not own, because
the neutrino is its own antiparticle. That is why the honest count is
twenty-eight: counting twenty-six quietly assumes the answer to an
unsettled experimental question.

\subsection{Step 1 (continued) --- the full tally}
\label{ssec:J-tally}

Assemble the sectors. Table~\ref{tab:J-tally} is the whole list.

\begin{table}[H]
\centering
\caption[The twenty-eight parameters, by sector.]{The twenty-eight
parameters of the Standard Model with Majorana neutrino masses, sector by
sector. Dropping the neutrino sector entirely gives the pre-1998 count of
nineteen; adding Dirac neutrino masses gives twenty-six; the last two
are the Majorana phases of \S\ref{ssec:J-leptons}. The electroweak
$\theta$ angle is absent because it can be rotated away using the
anomalous $B+L$ current, and the color one cannot.}
\label{tab:J-tally}\small
\begin{tabular}{@{}P{0.20\textwidth}P{0.08\textwidth}P{0.30\textwidth}P{0.30\textwidth}@{}}
\toprule
\textbf{sector} & \textbf{count} & \textbf{what they are} &
\textbf{where the count comes from} \\
\midrule
gauge & $3$ & $g_1, g_2, g_3$ &
one per simple factor of the gauge group \\
Higgs & $2$ & $\mu^2, \lambda$; equivalently $v$ and $m_h$ &
the only renormalizable potential for one doublet \\
fermion masses & $12$ &
$6$ quarks, $3$ charged leptons, $3$ neutrinos &
singular values of $Y_u, Y_d, Y_e$ and of $c_{ij}$ \\
CKM & $4$ & $3$ angles, $1$ phase &
Eqs.~\eqref{eq:J-phasecount} and \eqref{eq:J-physphase} at $n=3$ \\
PMNS & $6$ & $3$ angles, $1$ Dirac phase, $2$ Majorana phases &
Eq.~\eqref{eq:J-twelve}: no neutrino rephasing available \\
strong $CP$ & $1$ & $\bar\theta$ &
$\theta_{\rm QCD} + \arg\det(Y_u Y_d)$, the one invariant combination \\
\midrule
\textbf{total} & $\mathbf{28}$ & & \\
\bottomrule
\end{tabular}
\end{table}

Three arithmetic checks, all elementary, all worth doing:
\begin{equation}
3+2+12+4+6+1 = 28 ,
\qquad
3+2+9+4+1 = 19 ,
\qquad
19 + 3 + 4 = 26 .
\label{eq:J-tallies}
\end{equation}

\paragraph{Why $\bar\theta$ is one number and not two.} The QCD
Lagrangian admits a term $\theta\,\tr F\wedge F$: a total derivative
with no effect at any order in perturbation theory and a large effect
non-perturbatively. It is not by itself observable, because a chiral
rotation of the quark fields shifts $\theta$ while rotating the phase of
the quark mass matrix the other way. Only the combination
\begin{equation}
\bar\theta \;=\; \theta_{\rm QCD} \;+\; \arg\det\!\left(Y_u Y_d\right)
\label{eq:J-thetabar}
\end{equation}
is invariant, so exactly one parameter survives --- and it survives only
because every quark is massive. A single massless quark would let the
chiral rotation set $\bar\theta$ to zero for free and the parameter
would leave the list. That conditional matters in Step 2.

\paragraph{And a subtraction usually left implicit.} Of the
twenty-eight, fourteen are dimensionful --- $v$, $m_h$ and the twelve
masses --- and fourteen are dimensionless: three gauge couplings, four
CKM entries, six PMNS entries and $\bar\theta$. Fourteen dimensionful
quantities carry thirteen independent ratios plus one overall scale, and
the overall scale is not physics but the choice of what a kilogram is.
So the honest reading is $27$ dimensionless numbers plus one unit
convention. The distinction is pedantic here and becomes load-bearing in
Step 3, where a program that claims to derive everything still has to
choose units.

\subsection{Step 1 (concluded) --- running, and why a value without a
scheme is not a value}
\label{ssec:J-running}

Renormalization left a scale $\mu$ in the definition of every coupling,
but no physical amplitude may depend on a choice. Both statements are
true, and the reconciliation is that the couplings move with $\mu$ in
exactly the way needed to cancel the explicit $\mu$-dependence. That is
the renormalization group, and for one coupling it is
\begin{equation}
\mu \frac{\dd g}{\dd \mu} \;=\; \beta(g)
 \;=\; -\frac{b_0}{16\pi^2}\, g^3 \;+\; O(g^5) ,
\label{eq:J-beta}
\end{equation}
whose one-loop solution is a straight line in the right variables:
\begin{equation}
\alpha^{-1}(\mu) \;=\; \alpha^{-1}(\mu_0)
 \;+\; \frac{b_0}{2\pi}\, \ln\frac{\mu}{\mu_0} ,
\qquad
\alpha \;\equiv\; \frac{g^2}{4\pi} .
\label{eq:J-running}
\end{equation}
For a non-abelian group with fermions and scalars,
$b_0 = \tfrac{11}{3}C_2(G) - \tfrac43 \sum_f T(R_f)
- \tfrac16 \sum_s T(R_s)$; for QCD with $n_f$ light flavors this is
$11 - 2n_f/3$, positive, and the coupling falls at short distance. For
QED, where the gauge boson does not carry charge, only the fermion term
survives and $b_0 = -\tfrac43 \sum_f Q_f^2$, negative: the coupling
grows.

\paragraph{Doing it with numbers.} Take QED between the electron mass
and $M_Z$, counting the electron alone. With $Q = -1$ and one Dirac
fermion, Eq.~\eqref{eq:J-running} gives
\begin{equation}
\Delta \alpha^{-1}
 \;=\; -\frac{2}{3\pi}\, \ln\frac{M_Z}{m_e}
 \;=\; -2.566 ,
\label{eq:J-alpha-run}
\end{equation}
using $M_Z = 91.1876$ GeV and $m_e = 0.51099895$ MeV. The measured gap
is $137.036 - 127.930 = 9.106$, so the electron supplies $28\%$ of the
running and the muon, the tau, and the hadronic states supply the rest.
The check is not decoration. It says the movement in
Eq.~\eqref{eq:J-alpha-two} is not a convention: it is a sum over every
charged particle light enough to be created in the loop, and it is
predicted, not fitted.

\paragraph{Why the scheme is not a footnote either.} Two published
values of the weak mixing angle:
\begin{equation}
\sin^2\theta_W^{\rm eff,\,lept} = 0.23155(4) ,
\qquad
\sin^2\hat\theta_W(M_Z)\big|_{\overline{\rm MS}} = 0.23122(4) .
\label{eq:J-sw-two}
\end{equation}
They differ by $3.3\times10^{-4}$, which is more than eight of their own
standard deviations. Both are correct. They are different quantities:
one is defined by the ratio of couplings in the $Z$ boson's effective
leptonic vertex, the other by a subtraction prescription at a fixed
scale. Any claim of the form ``theory predicts
$\sin^2\theta_W = X$'' that does not say which of the two it means has
not yet made a testable statement, and Step 3 will run straight into
this.

\subsection{Step 2 --- what the count assumes, and what it does not
undermine}
\label{ssec:J-step2}

\paragraph{The count is convention-dependent in three specific ways, and
no others.} It depends on the field content assumed --- adding a
right-handed neutrino changes it. It depends on the answer to a physical
question --- Dirac or Majorana, worth two. And it depends on whether one
counts quantities that are pure unit conventions, which is the
fourteen-versus-thirteen distinction of \S\ref{ssec:J-tally}. Each is a
well-posed statement about which theory is under discussion, not a
looseness in the counting.

What is \emph{not} convention-dependent is anything the count is used
for. The dimension of a quotient does not depend on coordinates. Whether
$CP$ can be violated does not depend on a phase convention, because
Eq.~\eqref{eq:J-jarlskog} is written without one. So a reader who
concludes ``the count is conventional, therefore the puzzle is not
real'' has drawn the wrong inference from a true premise. The puzzle is
not that the number is twenty-eight. It is that the theory contains a
twenty-eight-dimensional space of possibilities and says nothing about
which point of it we occupy.

\paragraph{``No prediction'' is precise, not a complaint.} This is often
heard as a lament about unfinished work. It is not. A renormalizable
quantum field theory is, by construction, a map from a
finite-dimensional parameter space to the set of all its predictions,
and the parameters are the \emph{boundary conditions} of the
renormalization-group flow. Asking the Standard Model to explain the
electron mass is asking a differential equation to explain its own
initial data. Nothing has gone wrong; the question is outside the
object. Any answer must come from a structure that determines the
boundary conditions --- which is a statement about what kind of theory
could possibly help, not about how hard anyone has tried.

\paragraph{The naturalness expectation, and what happened to it.} For
thirty years there was a widely shared principle for guessing which
parameters need explaining. \citet{tHooft1980} gave it its sharp form: a
small dimensionless parameter is \emph{technically natural} if setting
it to zero enlarges the symmetry of the theory, because then quantum
corrections to it must be proportional to the parameter itself and
cannot regenerate it. Run the criterion over the sheet:

\emph{The electron mass is natural.} Set $y_e = 0$ and a chiral symmetry
$e_L \to e^{\ii\alpha}e_L$ appears. Corrections to $m_e$ are therefore
proportional to $m_e$ and only logarithmically enhanced. Its smallness
is stable, and needs no explanation in this sense.

\emph{The strong-$CP$ angle is not.} Set $\bar\theta = 0$ and no
symmetry appears, because every quark is massive --- exactly the
conditional flagged after Eq.~\eqref{eq:J-thetabar}. So a bound of
$|\bar\theta| \lesssim 7.5\times10^{-11}$, following from the neutron
electric dipole moment limit of $1.8\times10^{-26}$ $e$~cm together with
the estimate $d_n \approx 2.4\times10^{-16}\,\bar\theta$ $e$~cm, is a
number with
no known reason to be small. That is the strong-$CP$ problem, and it is
the sharpest naturalness question that remains genuinely open.

\emph{The Higgs mass is not either.} Setting $m_h = 0$ restores no
symmetry of the quantum theory --- scale invariance is broken by the
anomaly --- so the criterion predicts that the electroweak scale should
be dragged to whatever the next physical threshold is.

That last prediction is the one with a verdict. It motivated an
expectation of new states near the TeV scale, specific enough to be
tested; the states were not found. The correct lesson is neither that
naturalness was foolish nor that it is vindicated. It is a heuristic
about how theories are typically distributed, with an excellent record
where it could be checked --- the charged-neutral pion mass splitting
pointed at the $\rho$, the $K_L$--$K_S$ mass difference pointed at the
charm quark, and both pointings were right --- alongside one prominent
failure. A program that removes the parameters altogether, rather than
explaining why they sit where they do, is answering a different question
and must be judged by different evidence. That is Step 3.

\subsection{Step 3 --- Part IV's reading: no dial to turn}
\label{ssec:J-step3}

\paragraph{What the axiom forbids.} The program this review is
examining does not try to make twenty-eight numbers natural. Its fourth
axiom, \MSMF{} (\AX{4}) --- maximum symmetry, minimum freedom --- forbids
continuous freedom outright. Chapter~\ref{sec:zero-parameters} states
the resulting discipline as a closure theorem
\citep{ArisakaA4Euclid}: every dimensionless constant is a ledger
integer, or arithmetic built from ledger integers, or a ratio of
conversion factors; every dimensionful quantity is a conversion factor
times ledger integers and integer powers of $e$ and $\pi$. The list of
admitted primitives is closed, and that --- not its shortness --- is the
claim.

A closed form in this framework therefore has a restricted grammar. It
may be an integer or rational partition of the ledger $(M,N,L;J,K) =
(1,3,5;7,35)$, with $J = 2N+1$ and $K = LJ = M^2+N^2+L^2$; or an
exponential of ledger integers and $\pi$. It may not be a fitted
decimal, and it may not carry a continuous coefficient adjusted after
the fact.

\paragraph{Two examples, spelled out.} The first is the fine-structure
constant in the infrared:
\begin{equation}
\alpha^{-1}(0) \;=\; 2^{J} + N^{2} \;=\; 128 + 9 \;=\; 137 ,
\label{eq:J-alpha137}
\end{equation}
against the measured $137.035999177$, a fractional deviation of
$-2.6\times10^{-4}$. At the matching scale the same expression drops its
second term: $\alpha^{-1}(M_Z) = 2^J = 128$ against $127.930(8)$, a
deviation of $+5.5\times10^{-4}$. The integers $J = 7$ and $N = 3$ were
fixed by an anomaly identity \citep[see][]{ArisakaA3BI6} before the
fine-structure constant was consulted, which is the ordering that gives
the claim whatever force it has.

The second is the weak mixing angle:
\begin{equation}
\sin^2\theta_W \;=\; \frac{L^2}{4N^3}
 \;=\; \frac{25}{108} \;=\; 0.2314815 ,
\label{eq:J-sw}
\end{equation}
a ratio of two ledger integers and nothing else. Against the effective
leptonic value of Eq.~\eqref{eq:J-sw-two} the deviation is
$-3.0\times10^{-4}$: agreement at the $0.03\%$ level for a quantity with
no adjustable content. And now \S\ref{ssec:J-running} collects its debt.
Against the $\overline{\rm MS}$ value in the same equation the deviation
is $+1.1\times10^{-3}$, which is $+6.5$ of that measurement's own
standard deviations. One closed form, a striking success in one scheme
and a serious tension in another. The owning paper prints both
\citep{ArisakaA9SM}, Chapter~\ref{sec:sm-sheet} prints both, and this
appendix insists on it, because Eq.~\eqref{eq:J-triple} was not a
pedantic preamble: a closed form that has not said which member of the
triple it predicts has not finished its work.

\paragraph{The dimensionful clause.} The subtraction of
\S\ref{ssec:J-tally} is where the second half of the closure theorem
works. The electroweak scale is not offered as a dimensionless miracle
but as a conversion factor times ledger arithmetic:
\begin{equation}
M_{\rm lock} = \tfrac12 M_{\rm Pl}\, e^{-K} = 3.849\ \text{TeV} ,
\qquad
v = M_{\rm lock}\left(\frac{2}{L}\right)^{\!N} = 246.33\ \text{GeV} ,
\label{eq:J-vev}
\end{equation}
against a measured $246.21964$ GeV, a deviation of
$+4.5\times10^{-4}$. Read the structure rather than the agreement:
$e^{-35}$ is an integer power of $e$ with a ledger integer in the
exponent, $(2/L)^N$ is a rational power, and $M_{\rm Pl}$ is one of the
admitted conversion factors. Nothing continuous appears anywhere in the
expression --- which, believed or not, is the shape the grammar
demanded.

\paragraph{The honest statement of what has been bought.} The
twenty-eight numbers of Table~\ref{tab:J-tally} were, in the Standard
Model, twenty-eight independent boundary conditions. If the program's
arithmetic holds they are readings of five integers satisfying two
relations --- a real reduction in the dimension of the space of
possibilities, from twenty-eight continuous coordinates to a discrete
choice. But it converts the question ``why these values?'' into the
question ``why this ledger?'', and that is a different question, not
obviously an easier one. Chapter~\ref{sec:zero-parameters} argues the
ledger is forced by four steps that consume no integer a later step
concludes; one of those steps is a minimality axiom rather than a
theorem.

Two items belong in the same breath. Eight of the twenty-eight closed
forms are graded \emph{asserted} by the owning paper itself
\citep[the charged-fermion mass sheet of][]{ArisakaA9SM}: which ledger
integer plays which role is a working hypothesis there, and the
first-order quark deviations run to thirty percent. And the global
trials factor of the codebook has not been computed --- a small alphabet
of integers with $e$ and $\pi$ can express a great many numbers, and
until somebody counts the targets the alphabet had to hit, the closed
forms are an honest ansatz. On the other side stands the ordering: the
same five integers must serve the gauge sector, the mass sheet, two
mixing matrices and the contents of the universe without being re-chosen
between sectors, and a ledger that must hit a dozen independent targets
is a different object from a formula fitted to one. Which of the two
this is remains an open question with a computable answer that nobody
has computed.

\emph{Physics intuition.} A student meeting the Standard Model is told
its parameters are ``measured, not predicted'', and hears this as a
temporary embarrassment. It is neither temporary nor an embarrassment;
it is what a quantum field theory is. The interesting question is not
whether the parameters can be predicted from inside --- they provably
cannot --- but what kind of structure outside would have to exist for
them to be predicted at all. Only two kinds have been proposed
seriously: a structure that determines them, and an ensemble in which
they vary. Part~IV is about a candidate of the first kind, and about
what such a candidate must expose in order to be tested.

\subsection{Problems and worked solutions}
\label{ssec:J-problems}

\medskip\noindent\textbf{J.1. Derive Eq.~\eqref{eq:J-quark-n} for $n$
generations, evaluate it for $n = 1,2,3,4$, and check in each case that
the decomposition into masses, angles and phases sums to the same
number.}

\noindent\emph{Solution.} \emph{Step 1.} Two $n\times n$ complex
matrices carry $2 \times 2n^2 = 4n^2$ real entries; the flavor group
$U(n)^3$ has dimension $3n^2$; the stabilizer at a generic point is the
single overall phase of Eq.~\eqref{eq:J-stab}. Equation~\eqref{eq:J-orbit}
gives $4n^2 - 3n^2 + 1 = n^2 + 1$. \emph{Step 2.} That is $2, 5, 10, 17$
for $n = 1,2,3,4$. \emph{Step 3.} The decomposition is $2n$ masses,
$n(n-1)/2$ angles, $(n-1)(n-2)/2$ phases; summing,
$2n + \tfrac12(n^2-n) + \tfrac12(n^2-3n+2) = n^2 + 1$, so the two routes
agree identically and not just at $n = 3$. \emph{Physics intuition:} the
$+1$ is baryon number, sitting in the count as a subgroup that acts on
nothing. A conservation law appearing as a bookkeeping term in a
dimension formula is a good reminder that the flavor symmetries of the
Standard Model were never imposed --- they are what the kinetic terms
happened to leave over.

\medskip\noindent\textbf{J.2. Redo the count for the full three-family
Standard Model with massless neutrinos, and then with the Weinberg
operator switched on. Identify precisely which step changes and by how
much.}

\noindent\emph{Solution.} \emph{Step 1.} Massless neutrinos:
$n_V = 3\times18 = 54$, $\dim G_F = \dim U(3)^5 = 45$, and the
stabilizer is $U(1)_B$ together with the three separate lepton numbers,
$\dim H = 4$. So $54 - 41 = 13$, as in Eq.~\eqref{eq:J-thirteen}.
\emph{Step 2.} Switch on $c_{ij}/\Lambda$. In the lepton sector alone,
$n_V$ goes from $18$ to $18 + 12 = 30$ because a symmetric $3\times3$
complex matrix has $12$ real entries, while the available group is still
$U(3)_L \times U(3)_e$ of dimension $18$. \emph{Step 3.} The stabilizer
collapses: a Majorana mass forbids every neutrino rephasing, so
$\dim H = 0$ in the lepton sector and $30 - 18 = 12$ survive, which is
Eq.~\eqref{eq:J-twelve}. \emph{Physics intuition:} the whole difference
between twenty-six and twenty-eight is the disappearance of a
stabilizer. Two numbers enter the sheet not because a field was added
but because a symmetry that used to remove them stopped existing --- so
the Dirac-or-Majorana question is a question about the size of the
parameter space, not only about a decay mode.

\medskip\noindent\textbf{J.3. Show that an $n\times n$ unitary matrix
carries $n(n-1)/2$ angles and $n(n+1)/2$ phases, and derive
Eq.~\eqref{eq:J-physphase}. Then explain what goes wrong with the
argument for Majorana neutrinos.}

\noindent\emph{Solution.} \emph{Step 1.} $U(n)$ has $n^2$ real
parameters. Its real points form $O(n)$, of dimension $n(n-1)/2$, and
those are the angles; the remainder, $n^2 - n(n-1)/2 = n(n+1)/2$, are
phases. \emph{Step 2.} Equation~\eqref{eq:J-rephase} supplies $2n$
rephasings of which the diagonal one acts trivially, so $2n-1$ phases can
be removed, leaving $n(n+1)/2 - 2n + 1 = (n-1)(n-2)/2$. At $n = 2$ this
is zero and at $n = 3$ it is one. \emph{Step 3.} For Majorana neutrinos
the right-hand rephasings in Eq.~\eqref{eq:J-rephase} do not exist,
because $\nu^{\mathsf{T}}C\nu$ is not phase-invariant; only $n$
rephasings are available, and $n(n+1)/2 - n = n(n-1)/2 = 3$ phases
survive at $n = 3$. \emph{Physics intuition:} the counting difference is
exactly $(2n-1) - n = n - 1 = 2$, the two Majorana phases --- not extra
physics inserted by hand, but the phases the Dirac theory could throw
away and the Majorana theory cannot afford to.

\medskip\noindent\textbf{J.4. Use Eq.~\eqref{eq:J-running} to compute
the electron's contribution to the running of $\alpha^{-1}$ from $m_e$
to $M_Z$, compare with the measured change, and say what makes up the
difference.}

\noindent\emph{Solution.} \emph{Step 1.} For QED with one fermion of
charge $Q$, $b_0 = -\tfrac43 Q^2$, so
$\alpha^{-1}(\mu) = \alpha^{-1}(\mu_0) - (2Q^2/3\pi)\ln(\mu/\mu_0)$.
\emph{Step 2.} With $Q = -1$, $\mu_0 = m_e = 0.51099895$ MeV and
$\mu = M_Z = 91.1876$ GeV, the logarithm is $\ln(1.78\times10^5) =
12.09$ and $\Delta\alpha^{-1} = -(2/3\pi)(12.09) = -2.566$, which is
Eq.~\eqref{eq:J-alpha-run}. \emph{Step 3.} The measured change is
$127.930 - 137.036 = -9.106$, so the electron accounts for $28\%$; the
muon, the tau and the hadronic states --- the last taken from data
rather than computed perturbatively --- supply the remaining $-6.54$.
\emph{Physics intuition:} every charged particle light enough to appear
in a vacuum-polarization loop screens the charge, so $127.930$ is a
census of the particle content below $M_Z$. That the census comes out
right is among the least advertised successes of the theory, and it is
why Eq.~\eqref{eq:J-triple} cannot be waved away.

\medskip\noindent\textbf{J.5. Apply 't Hooft's naturalness criterion to
$m_e$, to $\bar\theta$ and to $m_h$, and state in each case what symmetry
is or is not restored.}

\noindent\emph{Solution.} \emph{Step 1.} $m_e \to 0$ restores a chiral
symmetry acting on the electron alone, so corrections to $m_e$ are
proportional to $m_e$: technically natural \citep{tHooft1980}.
\emph{Step 2.} $\bar\theta \to 0$ restores nothing, because
Eq.~\eqref{eq:J-thetabar} could only be rotated away by a chiral rotation
of a massless quark and there is none: not natural, and the bound is
$|\bar\theta| \lesssim 10^{-10}$. \emph{Step 3.} $m_h \to 0$ restores
scale invariance only classically; the anomaly breaks it, so the Higgs
mass is not protected, and the criterion predicts new structure near the
electroweak scale. \emph{Physics intuition:} the criterion is about what
quantum corrections can and cannot generate, not about aesthetics, and
it is checkable. Two of its three verdicts here are uncontroversial; the
third made a prediction that has not been borne out, which is worth
stating plainly rather than defending or mocking, since it is the only
place in this appendix where a widely held theoretical expectation met
an experiment and lost.

\medskip\noindent\textbf{J.6. Verify Eqs.~\eqref{eq:J-alpha137},
\eqref{eq:J-sw} and \eqref{eq:J-vev} arithmetically, compute all three
deviations, and then state the scheme problem in
Eq.~\eqref{eq:J-sw} precisely.}

\noindent\emph{Solution.} \emph{Step 1.} $2^7 + 3^2 = 128 + 9 = 137$;
against $137.035999177$ the deviation is $-2.63\times10^{-4}$. And
$L^2/4N^3 = 25/108 = 0.2314815$. \emph{Step 2.}
$M_{\rm lock} = \tfrac12 M_{\rm Pl}e^{-35}$ with
$M_{\rm Pl} = 1.22089\times10^{19}$ GeV gives $3848.9$ GeV, and
$(2/5)^3 = 0.064$, so $v = 3848.9 \times 0.064 = 246.33$ GeV against
$246.21964$: $+4.5\times10^{-4}$. The ledger closes too, $2N+1 = 7$ and
$LJ = 35 = 1 + 9 + 25$. \emph{Step 3.} The scheme problem:
$0.2314815$ sits $-3.0\times10^{-4}$ from the effective leptonic
$0.23155(4)$, which is $1.7$ of that measurement's standard deviations,
and $+1.1\times10^{-3}$ from the $\overline{\rm MS}$ $0.23122(4)$, which
is $6.5$ of \emph{its} standard deviations. \emph{Physics intuition:}
one closed form, two verdicts, and the difference between them is not
experimental error but the definition of the quantity. Until the
prediction says which definition it is a prediction for, the $0.03\%$
agreement and the $6.5\sigma$ tension are the same statement read two
ways --- which is the discipline Eq.~\eqref{eq:J-triple} demanded of
everybody, textbook and program alike.

\GGtakeaway{\emph{Standard result:} a free parameter of a quantum field
theory is a renormalization condition --- a value, a scheme and a scale
--- and the number of them is the dimension of a space of theories
modulo field redefinitions. In the quark sector, $36$ real Yukawa
entries minus $27-1 = 26$ removable by $U(3)^3$ leaves $10 = 6$ masses
$+\,3$ angles $+\,1$ phase, with the $-1$ being baryon number as a
stabilizer; the phase count $(n-1)(n-2)/2$ is why three families are the
minimum for $CP$ violation. With Majorana neutrinos the full tally is
$3+2+12+4+6+1 = 28$. \emph{What it assumed:} a field content, an answer
to the Dirac-or-Majorana question, and a convention about counting unit
choices --- and nothing else; the dimension of a quotient does not depend
on coordinates. \emph{What it leaves open:} everything about the values.
A renormalizable theory cannot predict its own boundary conditions, so
``the Standard Model does not explain its constants'' is a structural
fact, not a shortfall of effort; and the naturalness heuristic that told
us which of them most needed explaining has one open case
($\bar\theta$) and one failed prediction (the Higgs mass).
\emph{Continue in:} Chapter~\ref{sec:zero-parameters} for a proposal that
forbids continuous freedom outright, Chapter~\ref{sec:sm-sheet} for the
twenty-eight closed forms and their grades, and
Chapter~\ref{sec:discrete} together with Appendix~\ref{app:discrete} for
the two entries on the sheet that are discrete-symmetry data rather than
masses or angles.}

\section{The three mirrors: $C$, $P$, $T$, and the theorem that binds
them}
\label{app:discrete}
\GGteachbox{What a discrete symmetry is when there is no generator to
differentiate, and the fork Wigner's theorem leaves open; the proof ---
twice over, from the canonical commutator and from the demand that
energies stay positive --- that time reversal has to be antiunitary; the
three operators written out explicitly on the Dirac field, with the
matrices derived rather than quoted; the sixteen bilinears organized
into five Lorentz species and their behavior under $C$, $P$, $T$, $CP$
and $CPT$ tabulated, with three of the five rows derived by hand and a
fourth left as a problem; the $CPT$ theorem's
three hypotheses and the analytic-continuation argument that gets from
them to the result; and the experimental record from the 1956 cobalt to
the current bounds. The pay-off is Chapter~\ref{sec:discrete}, and the
two entries of Chapter~\ref{sec:sm-sheet} that are discrete-symmetry
data rather than masses.}

\GGreadalong{\citet{PeskinSchroeder1995}, \S3.6, for the Dirac-field
operators and the bilinear table in the conventions used here;
\citet{Weinberg1996}, \S2.6 for Wigner's theorem and \S5.8 for the
$CPT$ theorem done properly; \citet{ChengLi1984}, Ch.~11, for the
flavor sector and the Jarlskog construction; and the four experimental
papers themselves --- \citet{LeeYang1956}, \citet{Wu1957},
\citet{Garwin1957}, \citet{Christenson1964} --- which are short, and
which read better than any summary of them.}

\subsection{Step 1 --- what a discrete symmetry is, and the fork Wigner
leaves open}
\label{ssec:K-step1}

Every symmetry in Part~II came with a generator. You could
differentiate it, and what you got was a conserved current. A discrete
symmetry has no generator: there is no ``infinitesimal parity''. So
before anything is computed it is worth being clear about what such a
symmetry can possibly \emph{say}, since Noether's route to a conserved
quantity is closed at the first step.

The answer is that a discrete symmetry gives a \emph{multiplicative}
quantum number and a selection rule rather than an additive conserved
charge. If $S$ is an operator commuting with the Hamiltonian and
$S^2 = \mathbf{1}$, its eigenvalues are $\pm 1$, states organize into
even and odd, and the product of the eigenvalues on either side of a
reaction must match. That is a weaker statement than a conservation law
--- it constrains which processes may happen, not how much of anything
flows --- and it is the only kind of statement available. It is also,
historically, the kind of statement that has been overturned twice.

\paragraph{What the three mirrors are.} Fix the operations before
representing them.

\emph{Parity} $P$ reverses the spatial coordinates,
$(t, \mathbf{x}) \mapsto (t, -\mathbf{x})$, leaving time alone.

\emph{Time reversal} $T$ reverses the time coordinate,
$(t, \mathbf{x}) \mapsto (-t, \mathbf{x})$, leaving space alone.

\emph{Charge conjugation} $C$ exchanges every particle for its
antiparticle, flipping all internal charges and touching no spacetime
coordinate at all.

Two observations before the machinery. First, $C$ is of a different
species from the other two: it acts on the internal labels, not on the
manifold, and there is no classical field configuration you can point at
and call ``the charge-conjugated coordinates''. That $C$ nonetheless
sits in the same table as $P$ and $T$ is a fact that wants explaining,
and Step 3 is where the explanation lives. Second, $P$ is disconnected
from the identity only because space has an odd number of dimensions.
The matrix $-\mathbf{1}_3$ has determinant $(-1)^3 = -1$, so no rotation
reaches it; in two spatial dimensions $-\mathbf{1}_2$ has determinant
$+1$ and \emph{is} a rotation, by $\pi$. Parity is a genuinely new
operation in our world for a reason as cheap as the parity of the
number~$3$, and any argument about $P$ that does not use the dimension
somewhere is suspect.

\paragraph{Wigner's fork.} A symmetry of quantum mechanics is not a map
on state vectors but a map on \emph{rays}, since $|\psi\rangle$ and
$e^{\ii\alpha}|\psi\rangle$ describe the same state. The only thing that
must be preserved is the transition probability,
\begin{equation}
\bigl|\langle \phi' | \psi' \rangle\bigr|^2
 \;=\; \bigl|\langle \phi | \psi \rangle\bigr|^2 ,
\label{eq:K-wigner}
\end{equation}
and Wigner's theorem says that any such map may be realized on vectors
in exactly one of two ways: by a \emph{unitary, linear} operator, for
which $\langle \phi'|\psi'\rangle = \langle\phi|\psi\rangle$; or by an
\emph{antiunitary, antilinear} one, for which
$\langle\phi'|\psi'\rangle = \langle\phi|\psi\rangle^*$. Both preserve
the modulus, which is all Eq.~\eqref{eq:K-wigner} asks. Antilinear means
$A(c|\psi\rangle) = c^*A|\psi\rangle$: the operator conjugates the
c-numbers it passes.

The theorem gives a fork and does not say which tine to take. The
standard course announces the answer --- $C$ and $P$ unitary, $T$
antiunitary --- and the announcement is usually the only content
delivered. It deserves a proof, and it has two.

\subsection{Step 1 (continued) --- why time reversal must be
antiunitary}
\label{ssec:K-antiunitary}

\paragraph{First proof: the canonical commutator.} Time reversal should
leave positions alone and reverse momenta, because momentum is mass
times a velocity and a velocity is a displacement per unit time:
\begin{equation}
T\, \hat{x}\, T^{-1} \;=\; \hat{x} ,
\qquad
T\, \hat{p}\, T^{-1} \;=\; -\hat{p} .
\label{eq:K-txp}
\end{equation}
Now conjugate the canonical commutator $[\hat{x}, \hat{p}] = \ii\hbar$
by $T$. The left-hand side becomes
$[T\hat{x}T^{-1}, T\hat{p}T^{-1}] = [\hat{x}, -\hat{p}] = -\ii\hbar$.
The right-hand side becomes $T (\ii\hbar) T^{-1}$, and here everything
turns on whether $T$ is linear. If it is, the number $\ii\hbar$ passes
straight through and the right-hand side is $+\ii\hbar$, giving
\begin{equation}
-\ii\hbar \;=\; +\ii\hbar ,
\label{eq:K-contradiction}
\end{equation}
which is false. If instead $T$ is antilinear, it conjugates the number:
$T(\ii\hbar)T^{-1} = -\ii\hbar$, and the two sides agree. There is no
third possibility, so $T$ is antilinear, and by Wigner's theorem
antiunitary. \qed

That is a complete proof and it takes four lines. Notice what it uses:
nothing about relativity, nothing about fields, nothing about the
Hamiltonian --- only Eq.~\eqref{eq:K-txp} and the commutator that
defines quantum mechanics. The antiunitarity of $T$ is a statement about
the kinematics of quantum theory, not about any particular dynamics.

\paragraph{Second proof: the spectrum must stay bounded below.} The
second proof is worth having because it explains the physical stake. A
time-reversal operation should turn evolution forward into evolution
backward:
\begin{equation}
T\, e^{-\ii \hat{H} t / \hbar}\, T^{-1}
 \;=\; e^{+\ii \hat{H} t / \hbar} ,
\label{eq:K-evolution}
\end{equation}
which is the same as saying that running the film backwards and then
evolving forward is the same as evolving backward. Suppose $T$ were
linear. Then it passes the $\ii$ and acts only on $\hat{H}$:
$T e^{-\ii\hat{H}t/\hbar}T^{-1} = e^{-\ii (T\hat{H}T^{-1})t/\hbar}$, and
matching against Eq.~\eqref{eq:K-evolution} forces
\begin{equation}
T \hat{H} T^{-1} \;=\; -\hat{H} \qquad (\text{if } T \text{ linear}) .
\label{eq:K-badH}
\end{equation}
This is a catastrophe. It says that for every state of energy $E$ there
is a partner state of energy $-E$: the spectrum is symmetric about zero,
unbounded below, and there is no ground state. Matter would have no
stable configuration at all. If instead $T$ is antilinear, it conjugates
the $\ii$ as well, the two sign flips cancel, and matching
Eq.~\eqref{eq:K-evolution} gives
\begin{equation}
T \hat{H} T^{-1} \;=\; +\hat{H} \qquad (\text{if } T \text{ antilinear}) ,
\label{eq:K-goodH}
\end{equation}
which is what a symmetry ought to say and which leaves the spectrum
where it was. \qed

\emph{Physics intuition.} Put the two proofs side by side. The first says
$T$ must be antiunitary or the commutator breaks; the second says $T$
must be antiunitary or the world has no ground state. They are the same
fact seen from two ends, because the $\ii$ in $[\hat{x},\hat{p}]$ and
the $\ii$ in $e^{-\ii\hat{H}t}$ are the same $\ii$. Everything peculiar
about time reversal traces back to a single imaginary unit, and that
observation is exactly the hinge Step 3 turns on.

\paragraph{Two corollaries, free.} First, $T$ carries no conserved
quantum number. An antilinear operator has no rephasing-invariant
eigenvalue: if $T|\psi\rangle = \lambda |\psi\rangle$ then on
$|\psi'\rangle = e^{\ii\alpha}|\psi\rangle$ one finds
$T|\psi'\rangle = e^{-2\ii\alpha}\lambda\,|\psi'\rangle$, so the
``eigenvalue'' can be rotated to anything of the same modulus. There is
no $T$-parity in the sense that there is a $P$-parity, and a reader who
has wondered why no table of particle properties lists one now knows.

Second, \emph{Kramers' degeneracy}. For a system of half-integer total
angular momentum, $T^2 = -\mathbf{1}$. Then no state can be
$T$-invariant up to a phase --- if $T|\psi\rangle = c|\psi\rangle$ then
$T^2|\psi\rangle = c^* c |\psi\rangle = |c|^2|\psi\rangle$, positive,
contradicting $T^2 = -1$ --- so if $[\hat{H},T] = 0$ every level is at
least doubly degenerate. That degeneracy survives any electric field, of
any strength and any shape, and is lifted only by a magnetic field. It
is a selection rule with real laboratory teeth, derived from a sign.

\subsection{Step 1 (continued) --- the three operators on the Dirac
field}
\label{ssec:K-operators}

Conventions, once, so that every sign below can be checked. Metric
$g^{\mu\nu} = \mathrm{diag}(+,-,-,-)$; Clifford algebra
$\{\gamma^\mu, \gamma^\nu\} = 2 g^{\mu\nu}\mathbf{1}$, so
$(\gamma^0)^2 = +\mathbf{1}$ and $(\gamma^i)^2 = -\mathbf{1}$;
$\gamma^5 = \ii \gamma^0\gamma^1\gamma^2\gamma^3$, which anticommutes
with all four $\gamma^\mu$ and squares to $+\mathbf{1}$;
$\bar\psi = \psi^\dagger \gamma^0$; and
$\sigma^{\mu\nu} = \tfrac{\ii}{2}[\gamma^\mu,\gamma^\nu]$. It is
convenient to abbreviate the sign that distinguishes a time index from a
space index,
\begin{equation}
(-1)^\mu \;\equiv\;
\begin{cases}
 +1 , & \mu = 0 ,\\
 -1 , & \mu = 1,2,3 ,
\end{cases}
\label{eq:K-signconv}
\end{equation}
with no summation implied; every entry of the table below is built from
it.

\paragraph{Parity, derived.} We want $P\psi(t,\mathbf{x})P^{-1} =
\eta_P\, S\, \psi(t,-\mathbf{x})$ for some constant matrix $S$, fixed by
demanding that the Dirac equation keep its form. Start from
$(\ii\gamma^\mu\partial_\mu - m)\psi = 0$. Under $\mathbf{x} \to
-\mathbf{x}$ the derivatives transform as $\partial_0 \to \partial_0$,
$\partial_i \to -\partial_i$, so the equation for the reflected field
reads $(\ii\gamma^0\partial_0 - \ii\gamma^i\partial_i - m)\,S^{-1}\psi'
= 0$. Multiplying on the left by $S$, the form is recovered provided
\begin{equation}
S \gamma^0 S^{-1} = \gamma^0 ,
\qquad
S \gamma^i S^{-1} = -\gamma^i .
\label{eq:K-parity-req}
\end{equation}
The matrix that commutes with $\gamma^0$ and anticommutes with all three
$\gamma^i$ is $\gamma^0$ itself. Hence
\begin{equation}
P\, \psi(t,\mathbf{x})\, P^{-1}
 \;=\; \eta_P\, \gamma^0\, \psi(t,-\mathbf{x}) ,
\label{eq:K-Pdef}
\end{equation}
with $\eta_P$ a phase. Because $(\gamma^0)^2 = \mathbf{1}$, applying $P$
twice gives $\eta_P^2$, and for a Dirac field $\eta_P^4 = 1$ so
$\eta_P \in \{\pm 1, \pm\ii\}$; for a Majorana field the choice is
forced to $\pm\ii$. The physically meaningful statement is not $\eta_P$
for one field but the \emph{relative} intrinsic parity of a fermion and
its antifermion, which Eq.~\eqref{eq:K-Pdef} makes $-1$ --- because
$\gamma^0$ has eigenvalue $+1$ on the upper (particle) components and
$-1$ on the lower ones. That minus sign is measured, in positronium: the
ground state $^1S_0$ decays to two photons and not three.

\paragraph{Charge conjugation, derived.} $C$ must turn a solution of the
Dirac equation minimally coupled to $A_\mu$ with charge $+e$ into a
solution with charge $-e$. Take the complex conjugate of
$(\ii\gamma^\mu(\partial_\mu + \ii e A_\mu) - m)\psi = 0$:
\begin{equation}
\left(-\ii\gamma^{\mu *}(\partial_\mu - \ii e A_\mu) - m\right)\psi^* = 0 .
\label{eq:K-conj}
\end{equation}
If we can find a matrix $\mathcal{C}$ with
$\mathcal{C}\gamma^{\mu T}\mathcal{C}^{-1} = -\gamma^\mu$, then
$\psi^c \equiv \eta_C\, \mathcal{C}\bar\psi^{\,T}$ satisfies the Dirac
equation with $e \to -e$, which is exactly what was wanted. In the
Dirac representation the matrix is
\begin{equation}
\mathcal{C} \;=\; \ii \gamma^2 \gamma^0 ,
\qquad
\mathcal{C}^{-1} \;=\; \ii \gamma^0 \gamma^2 .
\label{eq:K-Cmatrix}
\end{equation}
The inverse is worth checking rather than believing:
$(\gamma^2\gamma^0)(\gamma^0\gamma^2) = (\gamma^2)^2 = -\mathbf{1}$, so
$(\gamma^2\gamma^0)^{-1} = -\gamma^0\gamma^2$ and
$(\ii\gamma^2\gamma^0)^{-1} = -\ii(-\gamma^0\gamma^2) =
\ii\gamma^0\gamma^2$. And the defining property, verified on $\mu = 0$
where $\gamma^{0T} = \gamma^0$:
\begin{equation}
\mathcal{C}\gamma^{0T}\mathcal{C}^{-1}
 = (\ii\gamma^2\gamma^0)\,\gamma^0\,(\ii\gamma^0\gamma^2)
 = -\gamma^2\gamma^0\gamma^2
 = +\gamma^0 (\gamma^2)^2
 = -\gamma^0 ,
\label{eq:K-Ccheck}
\end{equation}
using $(\gamma^0)^2 = \mathbf{1}$ and then anticommuting $\gamma^0$
past $\gamma^2$ once. So
\begin{equation}
C\, \psi(x)\, C^{-1}
 \;=\; \eta_C\, \mathcal{C}\, \bar\psi^{\,T}(x)
 \;=\; -\ii\,\eta_C\, \gamma^2 \psi^*(x) .
\label{eq:K-Cdef}
\end{equation}
$C$ is unitary and linear: the complex conjugation in
Eq.~\eqref{eq:K-Cdef} is conjugation of the \emph{field}, which for a
quantum field operator is the transpose-and-dagger of an operator, not
antilinearity of the map.

\paragraph{Time reversal, derived.} Now the antilinearity has to be
carried by the operator itself. Write $T = U_T \mathcal{K}$ with
$\mathcal{K}$ complex conjugation. Requiring the Dirac equation to hold
for $\psi'(t,\mathbf{x}) = U_T \psi^*(-t,\mathbf{x})$ gives the
condition $U_T \gamma^{\mu *} U_T^{-1} = (-1)^\mu \gamma^\mu$, whose
solution in the Dirac representation --- where $\gamma^0, \gamma^1,
\gamma^3$ are real and $\gamma^2$ is pure imaginary --- is
$U_T = \gamma^1\gamma^3$:
\begin{equation}
T\, \psi(t,\mathbf{x})\, T^{-1}
 \;=\; \eta_T\, \gamma^1 \gamma^3\, \psi(-t, \mathbf{x}) ,
\qquad T \text{ antiunitary} .
\label{eq:K-Tdef}
\end{equation}
Squaring it reproduces Kramers: $(\gamma^1\gamma^3)(\gamma^1\gamma^3)^*
= (\gamma^1\gamma^3)^2 = -\gamma^1\gamma^1\gamma^3\gamma^3 =
-(-1)(-1)\mathbf{1} = -\mathbf{1}$ once the anticommutation is used, so
$T^2 = -\mathbf{1}$ on a spinor, as promised in
\S\ref{ssec:K-antiunitary}.

\subsection{Step 1 (continued) --- the five bilinears, derived and
tabulated}
\label{ssec:K-bilinears}

Any $4\times 4$ complex matrix is a combination of sixteen basis
matrices, and they sort themselves by Lorentz behavior into five
species:
\begin{equation}
\underbrace{\mathbf{1}}_{1}
\;\cup\; \underbrace{\gamma^\mu}_{4}
\;\cup\; \underbrace{\sigma^{\mu\nu}}_{6}
\;\cup\; \underbrace{\gamma^\mu\gamma^5}_{4}
\;\cup\; \underbrace{\gamma^5}_{1} ,
\qquad 1+4+6+4+1 = 16 ,
\label{eq:K-sixteen}
\end{equation}
giving the scalar $S$, vector $V$, tensor $T$, axial vector $A$ and
pseudoscalar $P$ bilinears. \emph{Every} local operator you can build
from one Dirac field and one conjugate is a combination of these five,
so a table of how the five behave under the three mirrors is a table of
how \emph{every} interaction behaves. That is why the exercise is worth
the pages.

Three rows derived by hand first; the rest follow by the same two moves
(push a $\gamma^0$ or a $\gamma^1\gamma^3$ through, and count
anticommutations) and are collected in
Table~\ref{tab:K-bilinears}.

\paragraph{Table row 1: the scalar under $P$.} Using
Eq.~\eqref{eq:K-Pdef} with $|\eta_P|^2 = 1$,
\begin{equation}
\bar\psi\psi
 \;\longmapsto\;
 (\gamma^0\psi)^\dagger \gamma^0 (\gamma^0\psi)
 \;=\; \psi^\dagger \gamma^0 \gamma^0 \gamma^0 \psi
 \;=\; \psi^\dagger\gamma^0\psi
 \;=\; \bar\psi\psi ,
\label{eq:K-Sp}
\end{equation}
using $(\gamma^0)^\dagger = \gamma^0$ and $(\gamma^0)^2 = \mathbf{1}$,
with the spatial argument reflected throughout. Even. The mass term of
the Dirac Lagrangian is $-m\bar\psi\psi$, so a mass never violates
parity --- which is why the parity violation of the weak interaction had
to be looked for in the couplings, not in the spectrum.

\paragraph{Table row 2: the pseudoscalar under $P$.} The same computation with
$\ii\gamma^5$ inserted:
\begin{equation}
\bar\psi\, \ii\gamma^5 \psi
 \;\longmapsto\;
 \psi^\dagger\gamma^0\gamma^0\,\ii\gamma^5\,\gamma^0\psi
 \;=\; \ii\,\psi^\dagger \gamma^5\gamma^0\psi
 \;=\; -\,\ii\,\psi^\dagger\gamma^0\gamma^5\psi
 \;=\; -\,\bar\psi\,\ii\gamma^5\psi ,
\label{eq:K-Pp}
\end{equation}
the single sign coming from $\{\gamma^5,\gamma^0\} = 0$. Odd. That one
anticommutator is the whole difference between a scalar and a
pseudoscalar, and it is why the pion --- which is a $\bar q \ii\gamma^5
q$ state --- has intrinsic parity $-1$.

\paragraph{Table row 3: the vector under $P$, and under $C$.} Under $P$,
\begin{equation}
\bar\psi\gamma^\mu\psi
 \;\longmapsto\;
 \bar\psi\,\gamma^0\gamma^\mu\gamma^0\,\psi
 \;=\; (-1)^\mu\, \bar\psi\gamma^\mu\psi ,
\label{eq:K-Vp}
\end{equation}
because $\gamma^0\gamma^0\gamma^0 = \gamma^0$ while
$\gamma^0\gamma^i\gamma^0 = -\gamma^i$. The time component is even and
the space components odd, which is exactly how a charge density and a
current density should behave under a mirror.

Under $C$ it pays to derive the general rule once. With
$\psi \mapsto \mathcal{C}\bar\psi^{\,T}$ and
$\bar\psi \mapsto -\psi^T\mathcal{C}^{-1}$, a bilinear
$\bar\psi\,\Gamma\,\psi$ becomes
$-\psi^T\mathcal{C}^{-1}\Gamma\,\mathcal{C}\bar\psi^{\,T}$. It is a
single number, so transpose it; transposing exchanges the two Grassmann
fields and costs a second minus sign, which cancels the first, and using
$\mathcal{C}^T = -\mathcal{C}$ twice the two extra signs cancel again.
The result is clean:
\begin{equation}
\bar\psi\,\Gamma\,\psi
 \;\longmapsto\;
 \bar\psi\,\bigl(\mathcal{C}\,\Gamma^{T}\mathcal{C}^{-1}\bigr)\,\psi
\qquad\text{under } C .
\label{eq:K-Crule}
\end{equation}
Feed it $\Gamma = \gamma^\mu$ and the defining property
$\mathcal{C}\gamma^{\mu T}\mathcal{C}^{-1} = -\gamma^\mu$ verified in
Eq.~\eqref{eq:K-Ccheck} gives
\begin{equation}
\bar\psi\gamma^\mu\psi
 \;\longmapsto\;
 -\,\bar\psi\gamma^\mu\psi .
\label{eq:K-Vc}
\end{equation}
Odd. Equation~\eqref{eq:K-Crule} is worth keeping: it turns every
remaining $C$ entry into a one-line computation with the transpose, and
Problem K.2 uses it. This is the single most consequential entry in the table: the
electromagnetic current is $C$-odd, therefore the photon is $C$-odd,
therefore a state of $n$ photons has $C = (-1)^n$, therefore $\pi^0 \to
\gamma\gamma$ is allowed and $\pi^0 \to \gamma\gamma\gamma$ is not.
Charge conjugation invariance of electromagnetism is tested to better
than one part in $10^{5}$ in exactly that branching ratio.

\begin{table}[H]
\centering
\caption[The five Dirac bilinears under the three mirrors.]{The five
Dirac bilinears under $C$, $P$, $T$ and their products, in the
conventions of \S\ref{ssec:K-operators}, with $(-1)^\mu$ defined by
Eq.~\eqref{eq:K-signconv} and no sum implied. The spacetime argument is
reflected as the operation requires. The first three rows are derived in the
text and the axial row is Problem K.2. The last column obeys one rule with no exceptions: $CPT$
gives $(-1)^{n}$ where $n$ is the number of free Lorentz vector indices
--- $+1$ for $S$, $P$ and $T$, and $-1$ for $V$ and $A$. That the column
depends on nothing but an index count is the first sign that $CPT$ is a
different kind of statement from the other three.}
\label{tab:K-bilinears}\small
\begin{tabular}{@{}P{0.24\textwidth}ccccc@{}}
\toprule
\textbf{bilinear} & $P$ & $C$ & $T$ & $CP$ & $CPT$ \\
\midrule
$\bar\psi\psi$ \ (scalar $S$)
 & $+1$ & $+1$ & $+1$ & $+1$ & $+1$ \\
$\bar\psi\,\ii\gamma^5\psi$ \ (pseudoscalar $P$)
 & $-1$ & $+1$ & $-1$ & $-1$ & $+1$ \\
$\bar\psi\gamma^\mu\psi$ \ (vector $V$)
 & $(-1)^\mu$ & $-1$ & $(-1)^\mu$ & $-(-1)^\mu$ & $-1$ \\
$\bar\psi\gamma^\mu\gamma^5\psi$ \ (axial $A$)
 & $-(-1)^\mu$ & $+1$ & $(-1)^\mu$ & $-(-1)^\mu$ & $-1$ \\
$\bar\psi\sigma^{\mu\nu}\psi$ \ (tensor $T$)
 & $(-1)^\mu(-1)^\nu$ & $-1$ & $-(-1)^\mu(-1)^\nu$
 & $-(-1)^\mu(-1)^\nu$ & $+1$ \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Three things the table tells you at once.} First, look at the
$CP$ column for $V$ and $A$: both give $-(-1)^\mu$, the same entry.
That is why the weak charged current, which is the particular
combination $\bar\psi\gamma^\mu(1-\gamma^5)\psi$, transforms
\emph{coherently} under $CP$ even though $V$ and $A$ have opposite
parities. The $V-A$ structure violates $P$ maximally and $C$ maximally,
and preserves $CP$ --- so $CP$ violation cannot come from the Lorentz
structure of the current. It has to come from somewhere else, and the
only place left is a complex number in the coupling constants.

Second, consider the operator responsible for an electric dipole moment,
$d\, \bar\psi \sigma^{\mu\nu}\gamma^5\psi F_{\mu\nu}$. Take the
electric-field part. The dipole interaction is $-\mathbf{d}\cdot
\mathbf{E}$, with $\mathbf{d}$ necessarily along the only available
vector, the spin $\mathbf{s}$. Under $P$, $\mathbf{E}$ is odd and
$\mathbf{s}$ is even, so the product is odd; under $T$, $\mathbf{E}$ is
even and $\mathbf{s}$ is odd, so the product is odd again. A permanent
electric dipole moment of a non-degenerate state therefore violates both
$P$ and $T$ --- and by the $CPT$ theorem, violates $CP$. That is why the
neutron electric dipole moment is the single sharpest laboratory probe
of $CP$ violation available, and why \S\ref{ssec:K-experiment} quotes
its bound.

Third, the $CPT$ column is trivial in a way the others are not. It does
not depend on which bilinear, only on how many vector indices it
carries. No choice of coupling constants can make a Lorentz-invariant
hermitian interaction $CPT$-odd, because summing a $CPT$-odd bilinear
against a $CPT$-odd field strength always produces an even total. That
observation, made precise, \emph{is} the $CPT$ theorem.

\subsection{Step 1 (continued) --- the $CPT$ theorem: three hypotheses,
and the proof in outline}
\label{ssec:K-cpt}

\paragraph{The hypotheses.} The theorem is not a symmetry principle
imposed on a theory. It is a consequence of three assumptions each of
which most physicists would make without noticing:

\emph{(i) Lorentz invariance.} The fields transform in finite-dimensional
representations of the proper orthochronous Lorentz group, and the
vacuum and the $S$-matrix are invariant under it.

\emph{(ii) Locality.} Fields at spacelike separation commute (bosons) or
anticommute (fermions); interactions are polynomials in the fields and
finitely many derivatives at a single point.

\emph{(iii) A hermitian Hamiltonian with a spectrum bounded below,} and
the normal spin-statistics connection --- integer spin quantized with
commutators, half-integer with anticommutators.

Drop any one and the theorem goes. Nothing else is assumed: no
particular gauge group, no particular particle content, no perturbative
expansion.

\paragraph{The proof, in outline.} The classic argument is Jost's, and
its logic runs through analyticity rather than through any algebraic
identity. Consider the vacuum expectation values of products of fields,
the Wightman functions
$W(x_1,\ldots,x_n) = \langle 0 | \phi(x_1)\cdots\phi(x_n)|0\rangle$.
Four steps:

\emph{Step (a): positivity of energy buys analyticity.} Insert a
complete set of states between the fields and use hypothesis (iii). The
energy of every intermediate state is non-negative, so each factor
$e^{-\ii p \cdot (x_j - x_{j+1})}$ decays rather than grows when the
difference variable is given an imaginary part inside the forward light
cone. Hence $W$ extends to a function analytic on the \emph{forward
tube}: the difference variables $\zeta_j = x_j - x_{j+1}$ complexified
with $\mathrm{Im}\,\zeta_j$ in the open forward cone.

\emph{Step (b): Lorentz invariance enlarges the domain.} $W$ is
invariant under real Lorentz transformations. By the
Bargmann--Hall--Wightman theorem that invariance extends holomorphically
to the \emph{complex} Lorentz group $L_+(\mathbb{C})$, enlarging the
domain to the \emph{extended tube}, which --- crucially --- contains
real points. Those are the Jost points: real configurations in which
every partial sum of the difference vectors is spacelike.

\emph{Step (c): the total inversion is a complex rotation.} Here is the
one genuinely surprising ingredient. In four spacetime dimensions the
total inversion $x \mapsto -x$ --- which is precisely $PT$ --- lies in
the \emph{identity component} of $L_+(\mathbb{C})$. To see it, rotate by
a complex angle $\ii\pi$ in the $(0,1)$ plane and simultaneously by
$\pi$ in the $(2,3)$ plane; the composition is $-\mathbf{1}_4$, and the
path to it is continuous through complex parameters. This works because
the number of spacetime dimensions is \emph{even}: the reversals pair
up. In odd spacetime dimension $-\mathbf{1}$ has determinant $-1$ and no
such path exists, and the theorem fails. So the theorem knows the
dimension of spacetime, in exactly the way \S\ref{ssec:K-step1} noted
that parity knows the dimension of space.

\emph{Step (d): locality closes the loop.} At a Jost point all
separations are spacelike, so hypothesis (ii) permits the fields to be
reordered freely (up to the statistics sign, which is where
spin-statistics enters and where the $C$ of $CPT$ is generated). Setting
the reordered function equal to the analytically continued one and using
step (c) gives
\begin{equation}
W(x_1,\ldots,x_n)
 \;=\; W(-x_n,\ldots,-x_1) ,
\label{eq:K-cpt-wightman}
\end{equation}
which by the reconstruction theorem is precisely the statement that the
theory is invariant under $\Theta = CPT$. \qed

\paragraph{What it predicts.} $CPT$ invariance forces, for every
particle and its antiparticle: equal masses; equal total lifetimes;
equal magnitudes and opposite signs for all charges; equal magnitudes
and opposite signs for magnetic moments; and equal gyromagnetic ratios.
It does \emph{not} force equal partial widths into individual channels
--- those may differ, and their differences are exactly where $CP$
violation shows up in decays, with the total constrained to compensate.
That distinction is one of the most useful facts in experimental
flavor physics and one of the most frequently misremembered.

\paragraph{One converse worth knowing.} Greenberg's theorem states that
in an interacting local field theory, $CPT$ violation implies Lorentz
violation. The three hypotheses are therefore not independent hurdles: a
$CPT$-violating signal would not be a small repair to the table but the
collapse of the spacetime symmetry the whole framework is built on.

\subsection{Step 1 (concluded) --- what the experiments actually found}
\label{ssec:K-experiment}

\paragraph{1956--57: parity.} \citet{LeeYang1956} did something more
useful than proposing a violation --- they \emph{audited the literature}
and found that while parity conservation had been tested to high
precision in strong and electromagnetic processes, it had never been
tested at all in weak decays. The assumption had been carried across by
habit. They then listed experiments that would settle it. Within months,
\citet{Wu1957} polarized ${}^{60}\mathrm{Co}$ nuclei at
$\sim 0.01\,\mathrm{K}$ and found the emitted electrons preferentially
\emph{opposite} to the nuclear spin. Since spin is an axial vector (even
under $P$) and momentum a vector (odd), a non-zero
$\langle \mathbf{s}\cdot\mathbf{p}\rangle$ is a pseudoscalar and its
observation is parity violation, full stop --- no model required.
\citet{Garwin1957} confirmed it within days in the
$\pi \to \mu \to e$ chain, where the same asymmetry appears twice.

The violation turned out to be \emph{maximal}: only left-handed
neutrinos participate. $C$ is violated maximally by the same fact, since
$C$ applied to a left-handed neutrino gives a left-handed antineutrino,
which does not exist. But the two maximal violations conspire: $CP$
takes a left-handed neutrino to a right-handed antineutrino, which
\emph{does} exist. So $CP$ survived 1957 intact, and for seven years
looked like the true mirror.

\paragraph{1964: $CP$.} \citet{Christenson1964} looked for the decay
$K_L \to \pi^+\pi^-$, forbidden if $CP$ were exact, and found it at a
branching ratio near $2 \times 10^{-3}$ of the charged decays --- about
forty-five events in a sample of twenty-two thousand. The effect is
small, which is the point: unlike parity, $CP$ is violated \emph{barely},
and that smallness is not explained by any principle. It is a number.

\paragraph{1973: where the number can live.} \citet{KobayashiMaskawa1973}
asked where in a renormalizable gauge theory a $CP$-violating phase can
hide, and gave the counting: an $n \times n$ unitary mixing matrix has
$n(n-1)/2$ angles and $(n-1)(n-2)/2$ physical phases after rephasing.
For $n = 2$ that is one angle and \emph{zero} phases; for $n = 3$, three
angles and exactly one phase. $CP$ violation in the quark sector
therefore requires at least three generations --- a prediction of the
third family, from a symmetry argument, at a time when the fourth quark
had not been found. The measure of the effect is the Jarlskog invariant
$J_{CP}$, the rephasing-invariant combination whose value is
approximately $3.1 \times 10^{-5}$ \citep{PDG2024}; Problem K.5 computes
it.

\paragraph{Now: the $CPT$ bounds.} $CPT$ is the most precisely tested
symmetry in physics, by a wide margin. Three representative numbers
\citep{PDG2024}: the fractional neutral-kaon mass difference
$|m_{K^0} - m_{\bar{K}^0}|/m_{K^0} < 6\times 10^{-19}$, an absolute
bound of about $3 \times 10^{-10}\,\mathrm{eV}$; the electron and
positron $g$-factors agreeing to
$(g_{e^+} - g_{e^-})/g_{\rm avg} = (-0.5 \pm 2.1)\times 10^{-12}$; and
the antiproton-to-proton charge-to-mass ratio agreeing at the level of
parts in $10^{11}$ or better. Nothing has cracked.

\paragraph{And the one that has not moved.} The strong-$CP$ angle
$\bar\theta$ of Eq.~\eqref{eq:K-thetabar} below is bounded by the
neutron electric dipole moment, currently
$|d_n| < 1.8\times 10^{-26}\,e\,\mathrm{cm}$. With the standard hadronic
estimate $d_n \approx 2.4\times 10^{-16}\,\bar\theta\;e\,\mathrm{cm}$
this gives $|\bar\theta| \lesssim 8\times 10^{-11}$: below $10^{-10}$,
with nothing in the theory requiring it. Problem K.4 does the
arithmetic.

\subsection{Step 2 --- what the theorem assumes, and the puzzle it
leaves standing}
\label{ssec:K-step2}

\paragraph{Each hypothesis is a place a violation could enter.} Take
them in turn, because the usual summary --- ``$CPT$ is safe'' --- hides
the structure.

\emph{Locality.} A genuinely non-local theory --- string field theory,
or any framework with a minimal length implemented as a smearing --- is
not covered by the proof. Whether it violates $CPT$ is then a question
about that theory, not settled by Jost.

\emph{Lorentz invariance.} If the vacuum has a preferred frame, step (b)
fails and there is no extended tube. This is the entry point most
actively searched for, and by Greenberg's converse it is the only one
that can appear alone.

\emph{Hermiticity and positivity.} This is the hypothesis most often
violated in practice, and usually on purpose. Any \emph{open} system ---
a subsystem exchanging information with an environment, described by the
Lindblad generator of Appendix~\ref{app:opensys}
(\citealp{Lindblad1976}; \citealp{GoriniKossakowskiSudarshan1976}) ---
does not have a hermitian Hamiltonian generating its evolution. Its
dynamics is not time-reversal invariant, and no theorem says it should
be. The lesson is not that $CPT$ fails in open systems but that the
theorem is a statement about \emph{closed, complete} descriptions, and
that the arrow of time we experience is not evidence against it. This is
worth saying plainly because the confusion is common: microscopic $T$
violation, at the $10^{-3}$ level in kaons, and macroscopic
irreversibility, at the level of every cup of coffee, are unrelated
phenomena with unrelated explanations.

\paragraph{What a violation would mean.} Not a correction. A
$CPT$-violating result --- a mass splitting between a particle and its
antiparticle, say --- would take down Lorentz invariance with it, and
with Lorentz invariance goes the entire representation-theoretic
apparatus by which particles have definite spins and masses at all.
There is no small version of this failure. That is why the bounds are
pushed so hard, and why nobody expects them to break.

\paragraph{The strong-$CP$ puzzle, stated without decoration.} The QCD
Lagrangian admits the term
$\theta_{\rm QCD}\,\tr F \wedge F$: a total derivative, invisible to
every order of perturbation theory, and physical non-perturbatively. It
is not separately observable, because a chiral rotation of the quark
fields shifts $\theta_{\rm QCD}$ while rotating the phase of the quark
mass matrix the other way. Only the combination
\begin{equation}
\bar\theta \;=\; \theta_{\rm QCD}
 \;+\; \arg\det\!\left(Y_u Y_d\right)
\label{eq:K-thetabar}
\end{equation}
is invariant, and it is one of the twenty-eight parameters of
\S\ref{sec:sm-sheet}. It is $P$-odd and $T$-odd, hence by the theorem
$CP$-odd; it generates a neutron electric dipole moment; and experiment
says it is smaller than $10^{-10}$.

Now the puzzle, sharply. There is no symmetry of the Standard Model that
sets $\bar\theta$ to zero. A single massless quark would do it --- the
chiral rotation would then be free and $\bar\theta$ would leave the
parameter list entirely --- but no quark is massless. Nothing else in
QCD prefers zero. And the two terms in Eq.~\eqref{eq:K-thetabar} come
from utterly different sectors: one from the topology of the color
vacuum, one from the phases of the Yukawa matrices, whose $CP$-violating
phase is emphatically \emph{not} zero. Two unrelated quantities cancel
to ten decimal places. By \citet{tHooft1980}'s criterion the value
$\bar\theta = 0$ is technically natural once achieved, since setting it
to zero restores $CP$ in the strong sector; what is unnatural is that it
should have been achieved, given that the boundary condition was free
and nothing selected it.

That is the honest statement of the problem, and it is a problem about
\emph{a parameter}, which is why this appendix and
Appendix~\ref{app:params} are two halves of one question.

\subsection{Step 3 --- Part~IV's reading: three
orientations, and one total}
\label{ssec:K-step3}

Everything above treats the table of \S\ref{ssec:K-bilinears} as data.
$C$ and $P$ are unitary, $T$ is antiunitary, and Wigner's theorem is
consulted to confirm that both options were available. What is not
explained anywhere in the standard treatment is \emph{why the fork was
resolved the way it was}. The choice is assigned.
Chapter~\ref{sec:discrete} reports a program that derives it, and the
derivation is short enough to state in full here.

\paragraph{The premise.} The bulk geometry is real and oriented. It
carries no imaginary unit; every $\ii$ that appears in physics is the
boundary's bookkeeping for a real rotation in a real plane. And the bulk
has exactly three orientable factors: a compact fiber circle carrying a
field's phase, a spatial slice, and a causal clock. One mirror per
orientation --- $C$ reverses the fiber, $P$ reverses the slice, $T$
reverses the clock --- with no room for a fourth and no way to have
fewer.

\paragraph{The derivation.} A representation is antiunitary exactly when
it reverses the boundary's quarter-turn, the operation called
``multiply by $\ii$''. That quarter-turn is built from one specific
orientation: the clock's. So test the three reversals against it.
Reverse the spatial slice: neither the clock nor the fiber is touched,
so $\ii$ is untouched, so $P$ is unitary. Reverse the fiber: the charge
label flips together with the rotation it generates, the two minus signs
cancel, $\ii$ survives, so $C$ is unitary. Reverse the clock: ``a
quarter-period ahead'' becomes ``a quarter-period behind'', so
$\ii \mapsto -\ii$, which is antilinearity by definition, so $T$ is
antiunitary. The general rule follows at once: \emph{a discrete symmetry
is antiunitary if and only if it contains $T$}. The fork that Wigner
leaves open is closed by a one-line computation about which reversal
touches the source of $\ii$.

Compare that with \S\ref{ssec:K-antiunitary}. The textbook proofs are
correct and this reading does not replace them; what it adds is an
answer to a question they do not ask. Both textbook proofs turn on the
$\ii$ in $[\hat{x},\hat{p}]$ and in $e^{-\ii\hat{H}t}$, and both stop
there. The orientation reading says what that $\ii$ \emph{is}, and the
antiunitarity of $T$ then follows without being postulated. Kramers'
degeneracy and the absence of a $T$-parity fall out of the same
sentence, exactly as they did in \S\ref{ssec:K-antiunitary}.

\paragraph{$CPT$ as the total orientation.} Now the asymmetry that
matters. $CPT$ reverses all three orientations at once, which is the
total orientation of the geometry. A coupling built from real geometric
data --- magnitudes, which are even, and fiber angles, which are merely
relabeled --- cannot detect that reversal, so no texture in the
couplings, however contrived, can break $CPT$; whereas each individual
factor can be broken by a texture that treats one orientation
asymmetrically. Three mirrors breakable, their product not. That is the
same conclusion as Table~\ref{tab:K-bilinears}'s last column, reached
from the opposite direction: there, $CPT$ was blind to everything but an
index count; here, it is blind to everything but the total orientation,
which no coupling carries.

The honest comparison should travel with the claim.
Section~\ref{ssec:K-cpt}'s theorem reaches interacting fields, rigorously
--- and says nothing whatever about \emph{what} the three mirrors are or
why exactly three exist. The orientation reading names the anatomy and
explains the count, and proves the interacting case only conditionally.
They are complementary, and the experimental stakes are identical:
Greenberg's converse applies either way.

\paragraph{Where the $CP$ violation then comes from.} Section
\ref{ssec:K-bilinears} established that $CP$ violation cannot come from
the Lorentz structure of a current, since $V$ and $A$ share a $CP$
column. It must come from a complex coupling. In the reading of
Chapter~\ref{sec:discrete} the couplings are wires on a graph carrying
twists on the fiber circle, and the physical content is the loop
holonomy --- a flavor-space Aharonov--Bohm phase
\citep{AharonovBohm1959}, invariant because the rephasing freedom
telescopes around any closed loop. $CP$ is a symmetry exactly when every
loop holonomy is $0$ or $\pi$; $CP$ violation is the statement that some
loop sits off the real axis. What the ledger $(M,N,L;J,K) = (1,3,5;7,35)$
adds is the \emph{alphabet}: with one compact circle and block integer
$L = 5$ the first-order holonomies live on the lattice $(\pi/5)\mathbb{Z}$,
the Higgs sector's fraction $2/L$ of the block forces a branch phase
$\pi(1 - 2/L) = 3\pi/5 = 108^\circ$, and $3\pi/5$ is not a multiple of
$\pi$. So the pentagon is chiral, and $CP$ violation was not an option
the construction could have declined. Chapter~\ref{sec:discrete} carries
the resulting closed forms and their grades.

The lepton sector is read on the same lattice with the other ledger
integer. With $J = 7$,
\begin{equation}
\delta_{\rm PMNS} \;=\; \pi + 2\arcsin\!\left(\tfrac{1}{7}\right)
 \;=\; 3.42829\ \mathrm{rad} \;=\; 196.4^\circ ,
\label{eq:K-dpmns}
\end{equation}
against a global-fit central value near $1.08\pi = 194.4^\circ$ for
normal ordering --- agreement at the one-percent level for a quantity
with no continuous content, on a measurement whose own uncertainty is
several times larger than the gap. And $\bar\theta$ of
Eq.~\eqref{eq:K-thetabar} is given as zero up to a residual leakage
$\bar\theta \lesssim 5\times10^{-18}$ by a determinant-phase
cancellation, graded derived-conditional in
Appendix~\ref{app:confine}: it is falsified by any neutron electric
dipole moment above that envelope. Chapter~\ref{sec:triangle}
carries that argument and grades it not-yet-proved end to end.

\emph{Physics intuition.} The whole of Part~II was about
continuous symmetry, and the reader may reasonably ask why three
discrete mirrors get a chapter and an appendix of their own. Because
they are the only place where the framework's central move --- that a
symmetry is a statement about geometry rather than an assumption about
dynamics --- can be checked against a table that was written down
seventy years ago and has not changed since. If the mirrors are
orientation reversals, then their count, their unitarity assignments,
and the robustness of their product are all forced. If they are not,
the table remains what it has always been: correct, useful, and
unexplained.

\subsection{Problems and worked solutions}
\label{ssec:K-problems}

\medskip\noindent\textbf{K.1. Prove that $T$ must be antiunitary in two
independent ways, and then show that the two proofs are the same proof.}

\noindent\emph{Solution.} \emph{Step 1.} Conjugate
$[\hat{x},\hat{p}] = \ii\hbar$ by $T$ and use
Eq.~\eqref{eq:K-txp}. The left side gives $[\hat{x},-\hat{p}] =
-\ii\hbar$; a linear $T$ would leave the right side at $+\ii\hbar$,
which is Eq.~\eqref{eq:K-contradiction} and false. \emph{Step 2.}
Independently, demand Eq.~\eqref{eq:K-evolution}. A linear $T$ forces
$T\hat{H}T^{-1} = -\hat{H}$, so for every state of energy $E$ there is
one of energy $-E$: the spectrum is unbounded below and no ground state
exists. An antilinear $T$ gives $T\hat{H}T^{-1} = +\hat{H}$ and no such
disaster. \emph{Step 3.} The two are one because the $\ii$ in the
commutator and the $\ii$ in the propagator are the same $\ii$:
$e^{-\ii\hat{H}t/\hbar}$ generates translations in $t$ exactly as
$e^{-\ii\hat{p}a/\hbar}$ generates translations in $x$, and both
exponentials carry the imaginary unit that
$[\hat{x},\hat{p}] = \ii\hbar$ installed. Reversing a coordinate flips
the sign in the exponent; whether that flip is absorbed by the operator
or by the number is precisely the linear/antilinear fork.
\emph{Physics intuition:} students often read antiunitarity as a
technical wrinkle. It is the opposite --- it is what allows time
reversal to exist \emph{at all} without destroying the stability of
matter. The alternative is not a slightly different formalism; the
alternative is a world with no lowest energy state.

\medskip\noindent\textbf{K.2. Derive the $P$, $C$ and $T$ entries for
the axial current $\bar\psi\gamma^\mu\gamma^5\psi$, confirm its $CPT$
entry against the index-count rule, and explain using
Table~\ref{tab:K-bilinears} why a $V-A$ interaction violates $P$ and $C$
maximally yet conserves $CP$.}

\noindent\emph{Solution.} \emph{Step 1.} Under $P$, insert
$\gamma^0\gamma^0 = \mathbf{1}$ in the middle and split the product:
$\gamma^0\gamma^\mu\gamma^5\gamma^0
= (\gamma^0\gamma^\mu\gamma^0)(\gamma^0\gamma^5\gamma^0)
= \bigl((-1)^\mu\gamma^\mu\bigr)\bigl(-\gamma^5\bigr)
= -(-1)^\mu\,\gamma^\mu\gamma^5$, using Eq.~\eqref{eq:K-Vp} for the
first factor and $\{\gamma^5,\gamma^0\} = 0$ for the second. So the $P$
entry is $-(-1)^\mu$. Under $C$ use the rule
Eq.~\eqref{eq:K-Crule}: $\Gamma^T = (\gamma^\mu\gamma^5)^T =
\gamma^{5T}\gamma^{\mu T}$, and conjugating each factor separately gives
$(\mathcal{C}\gamma^{5T}\mathcal{C}^{-1})
(\mathcal{C}\gamma^{\mu T}\mathcal{C}^{-1})
= (+\gamma^5)(-\gamma^\mu) = -\gamma^5\gamma^\mu = +\gamma^\mu\gamma^5$,
the last step one anticommutation. So the $C$ entry is $+1$ --- the
transpose reversed the order, and the reversal supplied the sign that
distinguishes the axial current from the vector one. (That
$\mathcal{C}\gamma^{5T}\mathcal{C}^{-1} = +\gamma^5$ follows from the
same rule applied four times to
$\gamma^5 = \ii\gamma^0\gamma^1\gamma^2\gamma^3$: four minus signs, and
reversing the order of four distinct anticommuting matrices costs
$(-1)^{6} = +1$.) Under $T$ the entry is $(-1)^\mu$.
\emph{Step 2.} Multiply the
three: $-(-1)^\mu \times (+1) \times (-1)^\mu = -\bigl((-1)^\mu\bigr)^2
= -1$. The rule says $(-1)^n$ with $n = 1$ free vector index, so $-1$.
They agree. \emph{Step 3.} $P$: the $V$ row is $(-1)^\mu$ and the $A$
row is $-(-1)^\mu$, opposite, so the interference term in
$|V - A|^2$ is $P$-odd --- maximal violation. $C$: the rows are $-1$
and $+1$, again opposite, so $C$ is maximally violated too. $CP$: the
two rows are \emph{identical}, both $-(-1)^\mu$, so the whole current
transforms by one overall factor and every observable built from
$|V-A|^2$ is $CP$-even. \emph{Physics intuition:} nature violated the
two mirrors as hard as it is possible to violate them, and then, by
choosing the one combination whose $CP$ column is degenerate, arranged
that the product looked exact. Seven years passed before anyone found
the crack. The lesson is that a maximal violation and a conspiracy can
be the same algebraic fact.

\medskip\noindent\textbf{K.3. Compute $T^2$ for a Dirac spinor from
Eq.~\eqref{eq:K-Tdef}, generalize to arbitrary spin, and state the
degeneracy that follows for total angular momentum
$j = \tfrac12, 1, \tfrac32, 2$.}

\noindent\emph{Solution.} \emph{Step 1.} $T = \eta_T \gamma^1\gamma^3
\mathcal{K}$, so $T^2 = \eta_T \gamma^1\gamma^3 \mathcal{K}\eta_T
\gamma^1\gamma^3\mathcal{K} = |\eta_T|^2 (\gamma^1\gamma^3)
(\gamma^1\gamma^3)^*$. In the Dirac representation $\gamma^1$ and
$\gamma^3$ are real, so the conjugation is idle and
$T^2 = (\gamma^1\gamma^3)^2 = -\gamma^1\gamma^1\gamma^3\gamma^3
= -(-\mathbf{1})(-\mathbf{1}) = -\mathbf{1}$, the middle step using
$\{\gamma^1,\gamma^3\} = 0$ and the last $(\gamma^i)^2 = -\mathbf{1}$.
\emph{Step 2.} In general $T = e^{-\ii\pi J_y/\hbar}\mathcal{K}$ up to a
phase, and a rotation by $2\pi$ acting on angular momentum $j$ gives
$(-1)^{2j}$. So $T^2 = (-1)^{2j}\mathbf{1}$: $-\mathbf{1}$ for
half-integer $j$, $+\mathbf{1}$ for integer $j$. \emph{Step 3.} When
$T^2 = -\mathbf{1}$ and $[\hat{H},T] = 0$, no level can be
$T$-invariant, so levels come in pairs: $j = \tfrac12$ and
$j = \tfrac32$ are at least doubly degenerate, and no static electric
field of any shape can split them; $j = 1$ and $j = 2$ carry no such
protection. \emph{Physics intuition:} this is a genuine protection
theorem of the kind Part~II is about --- a degeneracy
that survives arbitrary perturbations of a whole class, guaranteed by a
sign rather than by fine-tuning, and destroyed only by a perturbation
that breaks the symmetry itself. A magnetic field is $T$-odd, so it
lifts the doublet; an electric field is $T$-even, so it cannot, no
matter how strong. Kramers is the crystal-field chemist's version of the
same idea that makes a Chern number an integer.

\medskip\noindent\textbf{K.4. From the neutron electric dipole moment
bound $|d_n| < 1.8\times10^{-26}\,e\,\mathrm{cm}$ and the hadronic
estimate $d_n \approx 2.4\times10^{-16}\,\bar\theta\;e\,\mathrm{cm}$,
extract the bound on $\bar\theta$. Then express $|d_n|$ as a charge
separation and compare it to the size of the neutron.}

\noindent\emph{Solution.} \emph{Step 1.} Divide:
$|\bar\theta| < 1.8\times10^{-26} / 2.4\times10^{-16}
= 7.5\times10^{-11}$, comfortably below $10^{-10}$. \emph{Step 2.} A
dipole moment $d = q\,r$ with $q = e$ gives a separation
$r < 1.8\times10^{-26}\,\mathrm{cm}$. The neutron's charge radius is
about $0.8\times10^{-13}\,\mathrm{cm}$, so the fractional displacement
is $1.8\times10^{-26} / 0.8\times10^{-13} = 2.3\times10^{-13}$.
\emph{Step 3.} Scale it up for intuition. Inflate the neutron to the
size of the Earth, radius $6.4\times10^{8}\,\mathrm{cm}$; the permitted
charge displacement scales to $6.4\times10^{8} \times 2.3\times10^{-13}
\approx 1.4\times10^{-4}\,\mathrm{cm}$, roughly a micron --- a hundredth
of a hair's width on a planet. \emph{Physics intuition:} that is what
``$\bar\theta$ is small'' means in the laboratory, and it is worth
converting once because the exponent alone does not communicate it. A
free parameter of the theory, permitted by every symmetry to be of order
one, is measured to be under $10^{-10}$. Either something forces it to
zero, or the world drew a number out of a hat and got ten leading
zeros.

\medskip\noindent\textbf{K.5. Compute the Jarlskog invariant
$J_{CP} = c_{12}s_{12}c_{23}s_{23}c_{13}^2 s_{13}\sin\delta$ from the
measured CKM angles $s_{12} = 0.22500$, $s_{23} = 0.04182$,
$s_{13} = 0.003707$ and $\delta = 1.144$ rad. Separately, count the
rephasing-invariant loops of the flavor graph for $N$ generations and
reconcile that count with the number of physical $CP$ phases.}

\noindent\emph{Solution.} \emph{Step 1.} The cosines are
$c_{12} = 0.97436$, $c_{23} = 0.99913$, $c_{13} = 0.99999$, and
$\sin(1.144) = 0.91030$. Multiplying,
$J_{CP} = 3.09\times10^{-5}$, matching the quoted
$\simeq 3.1\times10^{-5}$ \citep{PDG2024}. \emph{Step 2.} Note how the
value is built: it is the product of \emph{every} angle, so it vanishes
if any two quarks of the same charge are degenerate or if any single
mixing angle vanishes. $CP$ violation in the quark sector is fragile in
a way no single phase suggests. \emph{Step 3.} The flavor graph of
Chapter~\ref{sec:discrete} is complete bipartite on $N + N$ vertices
with $N^2$ edges; the independent plaquette holonomies number
$N^2 - (2N - 1) = (N-1)^2$, giving $0, 1, 4, 9$ for $N = 1,2,3,4$.
Imposing unitarity of the mixing matrix then collapses these to the
physical count $(N-1)(N-2)/2 = 0, 0, 1, 3$: the orthogonality relations
$\sum_k V_{ik}V^{*}_{jk} = 0$, one for each of the $\binom{N}{2}$ pairs
of rows, each remove one phase, and
$(N-1)^2 - \binom{N}{2} = (N-1)(N-2)/2$ identically. For $N = 2$ the single loop
survives as an invariant but unitarity forces its holonomy to $0$ or
$\pi$ --- real, hence $CP$-conserving. For $N = 3$ the four loops are
tied by unitarity to one physical angle, and $J_{CP}$ is that angle's
invariant measure. \emph{Physics intuition:} the two counts answer
different questions --- how many loop phases are rephasing-invariant,
and how many are dynamically independent --- and confusing them is the
commonest error in this subject. Both are needed: the first says where a
phase could live, the second says how many can be measured.

\medskip\noindent\textbf{K.6. (a) Convert the fractional bound
$|m_{K^0} - m_{\bar{K}^0}|/m_{K^0} < 6\times10^{-19}$ into an absolute
energy, using $m_{K^0} = 497.611\,\mathrm{MeV}$. (b) Evaluate
Eq.~\eqref{eq:K-dpmns} in degrees and compare with the global fit. (c)
Say what each of the two numbers would take down if it failed.}

\noindent\emph{Solution.} \emph{Step 1.} (a)
$497.611\times10^{6}\,\mathrm{eV} \times 6\times10^{-19}
= 3.0\times10^{-10}\,\mathrm{eV}$. For scale, that is about $10^{-19}$
of the kaon's own rest energy and some four and a half orders of
magnitude below the $K_S$ decay width --- the equality of a particle and
antiparticle mass is the most accurately verified statement in physics.
\emph{Step 2.} (b) $\arcsin(1/7) = 0.143348\,\mathrm{rad} =
8.2132^\circ$, so
$\delta_{\rm PMNS} = 180^\circ + 16.426^\circ = 196.43^\circ$, against a
normal-ordering global-fit central value near $1.08\pi = 194.4^\circ$: a
gap of about $2^\circ$, or one percent, on a measurement whose own
uncertainty spans some $20^\circ$. The comparison is favorable and is
not yet a test; it becomes one when the experimental error shrinks by an
order of magnitude. \emph{Step 3.} (c) The two failures are of entirely
different weight. A non-zero kaon mass splitting would violate $CPT$,
hence by Greenberg's converse violate Lorentz invariance, hence remove
the framework in which particles have definite masses --- it would not
be a correction to anything. A $\delta_{\rm PMNS}$ measured away from
Eq.~\eqref{eq:K-dpmns} would falsify one closed form in one program
and leave every theorem in this appendix untouched.
\emph{Physics intuition:} keep those two registers separate. A theorem
with three hypotheses, tested to $10^{-19}$, is not the same kind of
object as an arithmetic ansatz agreeing at one percent, and a review
that presented them in the same voice would be misleading its reader
about what is at stake in each.

\GGtakeaway{\emph{Standard result:} $C$ and $P$ are unitary and $T$ is
antiunitary --- the last provably so, from $[\hat{x},\hat{p}] =
\ii\hbar$ and from the demand that the energy spectrum stay bounded
below. On the Dirac field the three operators are $\gamma^0$,
$\ii\gamma^2\mathcal{K}_{\rm field}$ and $\gamma^1\gamma^3\mathcal{K}$,
and the five bilinears of Eq.~\eqref{eq:K-sixteen} transform as
Table~\ref{tab:K-bilinears}. $P$ and $C$ are maximally violated by the
weak interaction; $CP$ is violated barely; $CPT$ is not violated at all,
and cannot be, given locality, Lorentz invariance and a hermitian
Hamiltonian bounded below --- the proof running through analytic
continuation to the complex Lorentz group, where the total inversion
lies in the identity component because spacetime has an even number of
dimensions. \emph{What it assumed:} exactly those three hypotheses, and
by Greenberg's converse they stand or fall together. \emph{What it
leaves open:} why the fork was resolved as it was --- the assignment is
made, not derived --- and why $\bar\theta$, permitted by every symmetry
to be of order one, is below $10^{-10}$. \emph{Continue in:}
Chapter~\ref{sec:discrete} for the three mirrors read as orientation
reversals and $CPT$ as the total orientation, Chapter~\ref{sec:triangle}
for the claim that $\bar\theta$ vanishes exactly,
Appendix~\ref{app:params} for the parameter this appendix could not
explain, and Appendix~\ref{app:opensys} for the one hypothesis that is
routinely and deliberately given up.}

\section{Confinement, the color triangle, and strong $CP$}
\label{app:confine}

\GGteachbox{The Wilson loop defined from what it measures, with its two
possible behaviors --- area law and perimeter law --- read off a static
potential rather than announced; the strong-coupling lattice calculation
in full, with the tiling argument that produces
$\langle W\rangle \sim e^{-\sigma A}$ and a closed form for the string
tension; the Cornell potential with its color factor derived, and the
tension converted into a force a reader can weigh; the center
$\mathbb{Z}_3$ and the Polyakov loop, with the confined and deconfined
phases distinguished in the direction that surprises everyone the first
time; Regge trajectories from a rotating string, which delivers $\alpha'$
from $\sigma$ with no new input; the representation theory that says why
three quarks close and two do not; and the $\theta$ term, its topological
origin, and the bound that makes it a problem. The pay-off is
Chapter~\ref{sec:triangle}.}

\GGreadalong{\citet{Wilson1974} itself, still the cleanest statement of
the criterion; \citet{Zee2010}, Part~VII, for the strong-coupling
expansion and the $\theta$ vacuum told with the physics kept in front;
\citet{PeskinSchroeder1995}, Ch.~17, for asymptotic freedom;
\citet{ChengLi1984}, Ch.~16, for the $U(1)$ problem and the strong-$CP$
angle in their historical setting.}

\subsection{Step 1 --- the Wilson loop, and what an area law is a
statement about}
\label{ssec:L-step1}

Start with the question, not the apparatus. Hold a quark and an
antiquark a distance $R$ apart and ask what energy that costs. In
electrodynamics the answer is Coulomb's: the energy tends to a constant,
which is why $13.6$~eV buys you a free proton and a free electron. No one
has ever walked away with a free quark. If the energy instead grows
without bound as $R$ grows, that fact is explained --- and the object
that measures whether it does is the Wilson loop, built from the one
thing gauge theory lets you measure without choosing a gauge.
Chapter~\ref{sec:wilson} and Appendix~\ref{app:wilson} established it:
the holonomy of a connection around a closed curve $C$,
\begin{equation}
W(C) \;=\; \tr \, \mathcal{P} \exp\!\left(
 \ii g \oint_C A_\mu \, \dd x^\mu \right) ,
\label{eq:L-wilson}
\end{equation}
with $\mathcal{P}$ the path ordering that non-commuting matrices force
on us and the trace supplying the last piece of gauge invariance --- a
gauge transformation conjugates the holonomy, and the trace does not
care.

\paragraph{Why the loop measures a potential.} Take $C$ to be a rectangle
in the $(t,z)$ plane, $R$ wide in space and $T$ long in time, with $T \gg
R$. The two timelike sides are the world-lines of an infinitely heavy
color source and its conjugate held apart at distance $R$; the two
spacelike sides create the pair and destroy it. So $\langle W \rangle$ is
the amplitude for a static pair to be created, to sit at separation $R$
for a time $T$, and to be annihilated. Insert a complete set of energy
eigenstates, as in any Euclidean path-integral argument, and the
long-time behavior is dominated by the lowest state of that two-source
system:
\begin{equation}
\langle W(R,T) \rangle \;\xrightarrow[T \to \infty]{}\;
 c(R) \, e^{-V(R)\,T} .
\label{eq:L-transfer}
\end{equation}
That is the whole reason the Wilson loop is interesting. It is not an
abstract gauge invariant; it is a machine that returns the static
potential. Now the two behaviors it can have carry two different
physical worlds.

\emph{Perimeter law:} $\langle W \rangle \sim e^{-c\,(2R + 2T)}$. Then
Eq.~\eqref{eq:L-transfer} gives $V(R) \to \text{const}$ at large $R$. The
sources can be separated at finite cost. This is what electrodynamics
does, and what any theory with screening does.

\emph{Area law:} $\langle W \rangle \sim e^{-\sigma\,RT} = e^{-\sigma A}$,
with $A = RT$ the area the loop bounds. Then
\begin{equation}
V(R) \;=\; \sigma R ,
\label{eq:L-linear}
\end{equation}
a potential that rises linearly forever. The constant $\sigma$ has
dimensions of energy per unit length --- it is a tension, and the picture
it suggests is a string of that tension joining the two sources. This is
Wilson's criterion \citep{Wilson1974}: \emph{confinement is an area law
for the fundamental-representation Wilson loop.}

The criterion is worth pausing on for what it does \emph{not} say. It
says nothing about quarks trapped inside a bag; it says the energy of a
separated pair grows without bound. Only the second is something a
calculation can be asked to produce.

\subsection{Step 1 (continued) --- the strong-coupling lattice
derivation, done properly}
\label{ssec:L-strong}

Here is the one place where the area law is not a hope but a
computation. Put the theory on a hypercubic lattice of spacing $a$. The
degrees of freedom are not the algebra-valued $A_\mu$ but the
group-valued \emph{link variables},
\begin{equation}
U_\mu(n) \;=\; \exp\!\left( \ii\, a\, g\, A_\mu(n) \right)
 \;\in\; \mathrm{SU}(N) ,
\label{eq:L-link}
\end{equation}
one for each nearest-neighbor bond, understood as the parallel
transport along it --- the lattice statement of the fact
Appendix~\ref{app:connections} makes in the continuum, that a connection
is something you integrate along a path and its integral is a group
element.

The smallest closed curve on the lattice is the elementary square, the
\emph{plaquette}, and its holonomy is
\begin{equation}
U_p \;=\; U_\mu(n)\, U_\nu(n+\hat\mu)\,
 U_\mu^\dagger(n+\hat\nu)\, U_\nu^\dagger(n) .
\label{eq:L-plaquette}
\end{equation}
Wilson's action is built from these and nothing else:
\begin{equation}
S \;=\; \beta \sum_p \left(
 1 - \frac{1}{N}\,\mathrm{Re}\,\tr\, U_p \right) ,
\qquad
\beta \;=\; \frac{2N}{g^2} .
\label{eq:L-wilsonaction}
\end{equation}
Expanding Eq.~\eqref{eq:L-plaquette} to leading order in $a$ recovers
$\tfrac14 F_{\mu\nu}^a F^{a\,\mu\nu}$, which is the check that this is
Yang--Mills. What matters here is that Eq.~\eqref{eq:L-wilsonaction} is
exactly gauge invariant at finite $a$, because a plaquette is a closed
loop.

\paragraph{The integrals that do all the work.} On a compact group the
link integrals use the invariant Haar measure, normalized so that
$\int \dd U = 1$. Two identities carry the entire calculation:
\begin{equation}
\int \dd U \; U_{ij} \;=\; 0 ,
\qquad
\int \dd U \; U_{ij}\, (U^\dagger)_{kl}
 \;=\; \frac{1}{N}\, \delta_{il}\, \delta_{jk} .
\label{eq:L-haar}
\end{equation}
The first says an unpaired link averages to nothing: a lone parallel
transport, with no return trip, carries no gauge-invariant information.
It is true because the measure is invariant under $U \to VU$ for any $V$,
so the integral must equal itself multiplied by an arbitrary group
element. The second says a link and its conjugate can annihilate, at the
price of $1/N$. Everything below is bookkeeping on those two facts.

\paragraph{The tiling.} Take $\beta$ small (that is, $g$ large: strong
coupling), and expand the Boltzmann weight,
\begin{equation}
e^{-S} \;=\; \prod_p \left[
 1 + \frac{\beta}{2N}\left(\tr U_p + \tr U_p^\dagger\right)
 + O(\beta^2) \right] .
\label{eq:L-expand}
\end{equation}
Now compute $\langle W(C) \rangle$ for a rectangular loop of area
$A = R T$, whose minimal surface holds $n = A/a^2$ elementary plaquettes.
The loop contributes one $U$ per link along $C$, every one of them
unpaired, and by the first of Eqs.~\eqref{eq:L-haar} an unpaired link
kills the whole term. The only source of partners is the expansion
\eqref{eq:L-expand}.

Bring down one plaquette lying just inside $C$ and sharing one link with
it. That link is now paired and integrates away by the second of
Eqs.~\eqref{eq:L-haar}. But the plaquette had four links: three new
unpaired ones have appeared, one step further in. Bring down the next
plaquette and again three appear. The process cannot stop until the
unpaired boundary has been pushed inward to nothing --- until the entire
minimal surface is tiled. There is no cheaper way.

Each tile costs a factor $\beta/(2N)$ from Eq.~\eqref{eq:L-expand} and a
factor $1/N$ from the pairing integral, so the leading result is
\begin{equation}
\langle W(C) \rangle \;\simeq\;
 \left( \frac{\beta}{2N^2} \right)^{\!A/a^2}
 \;=\; \exp\!\left[ -\frac{A}{a^2}\,
 \ln\!\left( \frac{2N^2}{\beta} \right) \right]
 \;=\; e^{-\sigma A} ,
\label{eq:L-arealaw}
\end{equation}
with, on substituting $\beta = 2N/g^2$,
\begin{equation}
\sigma \, a^2 \;=\; \ln\!\left( N g^2 \right) .
\label{eq:L-sigma-sc}
\end{equation}
An area law, with a string tension in closed form. For $\mathrm{SU}(3)$
at $g^2 = 1$ this is $\sigma a^2 = \ln 3 = 1.0986$; at $g^2 = 2$,
$\ln 6 = 1.792$. The tension is positive for any $g^2 > 1/N$, and it
grows with the coupling.

\emph{Physics intuition.} The area appeared because the boundary had to
be filled in, and the boundary had to be filled in because a lone
parallel transport averages to zero on a compact group. Notice what
\emph{failed} to happen: no gluon exchanged, no potential summed, no
diagram drawn. Confinement in the strong-coupling lattice theory is a
statement about measure theory on a compact group, and its central
feature --- area rather than perimeter --- is combinatorial.

\subsection{Step 1 (continued) --- the linear potential, the Cornell
form, and the force in tonnes}
\label{ssec:L-cornell}

Real QCD is asymptotically free, so at short distance the potential is
Coulombic with a running coupling. Empirically --- from quarkonium
spectroscopy and from lattice measurements of
Eq.~\eqref{eq:L-transfer} --- the static potential is described over the
whole range, to remarkable accuracy, by the two-term \emph{Cornell}
form
\begin{equation}
V(r) \;=\; -\,\frac{4}{3}\,\frac{\alpha_s}{r} \;+\; \sigma\, r .
\label{eq:L-cornell}
\end{equation}
Both coefficients deserve their derivation.

\paragraph{The color factor $4/3$.} The one-gluon-exchange potential
between two color sources carries the group factor
$\langle T_1^a T_2^a \rangle$, where $T^a$ are the generators in the
sources' representations. Get it from a Casimir identity rather than from
a table. If the pair sits in a total representation $R$, then
$(T_1 + T_2)^2 = T_1^2 + T_2^2 + 2\,T_1 \!\cdot\! T_2$, so
\begin{equation}
T_1 \!\cdot\! T_2 \;=\;
 \tfrac12 \left[ C_2(R) - C_2(R_1) - C_2(R_2) \right] .
\label{eq:L-casimir}
\end{equation}
For $\mathrm{SU}(3)$ the quadratic Casimir of the representation with
Dynkin labels $(p,q)$ is $C_2 = (p^2 + q^2 + pq + 3p + 3q)/3$, giving
$C_2(\mathbf{3}) = C_2(\bar{\mathbf{3}}) = 4/3$, $C_2(\mathbf{6}) =
10/3$, $C_2(\mathbf{8}) = 3$, $C_2(\mathbf{10}) = 6$. A quark and an
antiquark in the color \emph{singlet} therefore have
\begin{equation}
T_1 \!\cdot\! T_2 \;=\; \tfrac12\left[ 0 - \tfrac43 - \tfrac43 \right]
 \;=\; -\,\tfrac43 ,
\label{eq:L-cf}
\end{equation}
which is where the $4/3$ in Eq.~\eqref{eq:L-cornell} comes from, the
minus sign being why the singlet channel is attractive. The octet channel
gives $\tfrac12(3 - \tfrac83) = +\tfrac16$: repulsive. This is the
sharpest short statement of why hadrons are singlets before topology is
mentioned --- the singlet is the channel one-gluon exchange binds
hardest.

\paragraph{The tension, weighed.} The lattice and quarkonium value is
$\sqrt\sigma = 440$~MeV:
\begin{equation}
\sigma \;=\; (0.440)^2 \;=\; 0.1936~\mathrm{GeV}^2 .
\label{eq:L-sigmaval}
\end{equation}
With $\hbar c = 0.19733$~GeV$\cdot$fm this is $0.9811$~GeV/fm --- about
one GeV per fermi, the number to memorize. Convert again, all the way to
the machine shop: one GeV is $1.602\times10^{-10}$~J and one fermi is
$10^{-15}$~m, so
\begin{equation}
\sigma \;=\; 1.572\times10^{5}~\mathrm{N}
 \;=\; 16.0~\text{tonnes weight} .
\label{eq:L-tonnes}
\end{equation}
A quark and an antiquark are joined by something that pulls with the
weight of a loaded lorry, and --- this part has no analogue anywhere else
in physics --- pulls with that weight \emph{no matter how far apart they
are}.

What happens instead of runaway energy is worth stating. Pull the pair
apart by one fermi and you have spent $0.98$~GeV, more than enough to
make a light quark--antiquark pair out of the vacuum; the string snaps in
the middle and caps itself with the new pair, giving two mesons and not
one separated quark. So Eq.~\eqref{eq:L-linear} is a statement about the
\emph{static} sources of the Wilson loop, where no light quarks are
available to break the string. With dynamical quarks the potential
flattens once $\sigma r$ reaches roughly twice a constituent mass --- and
that is what makes hadrons finite objects rather than infinitely long
strings.

\subsection{Step 1 (continued) --- the center, the Polyakov loop, and
which phase is the symmetric one}
\label{ssec:L-centre}

The Wilson loop criterion is a statement about a long rectangle in
infinite volume at zero temperature. At finite temperature $T$, Euclidean
time is a circle of circumference $1/T$, and a new gauge-invariant object
becomes available: a loop that \emph{wraps} that circle rather than
bounding a surface. This is the Polyakov loop,
\begin{equation}
L(\mathbf{x}) \;=\; \frac{1}{N}\,
 \tr\, \mathcal{P} \exp\!\left(
 \ii g \int_0^{1/T} \! A_0(\mathbf{x}, \tau)\, \dd \tau \right) ,
\label{eq:L-polyakov}
\end{equation}
and by the same transfer-matrix argument that gave
Eq.~\eqref{eq:L-transfer} it measures the free energy of a \emph{single}
static quark:
\begin{equation}
\langle L \rangle \;=\; e^{-F_q / T} .
\label{eq:L-freeenergy}
\end{equation}

\paragraph{The center symmetry.} The center of $\mathrm{SU}(N)$ is the
subgroup that commutes with everything,
\begin{equation}
Z(\mathrm{SU}(3)) \;=\; \{\,\mathbf{1},\ \omega \mathbf{1},\
 \omega^2 \mathbf{1}\,\} \;\cong\; \mathbb{Z}_3 ,
\qquad \omega = e^{2\pi \ii/3} ,
\label{eq:L-centre}
\end{equation}
and it is not part of the Lie algebra: no generator differentiates to it,
so nothing local sees it --- a discrete symmetry of the kind
Appendix~\ref{app:discrete} deals with, hiding inside a continuous group.

At finite temperature, multiply every temporal link crossing one fixed
time slice by a center element $z$. Any plaquette crosses that slice
either not at all or twice --- once with $U_0$, once with $U_0^\dagger$
--- so its holonomy picks up $z z^* = 1$ and the action
\eqref{eq:L-wilsonaction} is untouched. This is a genuine symmetry of the
pure gauge theory, and not a gauge transformation, because it is not
periodic in Euclidean time. But the Polyakov loop crosses the slice
exactly once, so
\begin{equation}
L \;\longmapsto\; z\, L .
\label{eq:L-centre-transf}
\end{equation}
An order parameter, in the textbook sense: an object that transforms
non-trivially under a symmetry of the action.

\paragraph{The direction that surprises everyone.} Read
Eqs.~\eqref{eq:L-freeenergy} and \eqref{eq:L-centre-transf} together. If
the center symmetry is \emph{unbroken}, $\langle L \rangle$ must equal
$z \langle L \rangle$ for all three $z$, so $\langle L \rangle = 0$ and
$F_q = \infty$: it costs infinite free energy to put one quark into the
world by itself. \emph{Unbroken symmetry means confinement.} If the
center symmetry is \emph{spontaneously broken}, $\langle L \rangle \neq
0$, $F_q$ is finite, and a single quark can be thermally liberated.
\emph{Broken symmetry means deconfinement.}

This runs opposite to every intuition trained on magnets and Higgs
fields, where the broken phase is the cold ordered one. Here the cold
phase is the symmetric one. Pure $\mathrm{SU}(3)$ gauge theory has a
genuine first-order deconfinement transition at $T_c \approx
0.63\sqrt{\sigma} \approx 277$~MeV, a real phase transition precisely
because $\mathbb{Z}_3$ is an exact symmetry to break.

\paragraph{The caveat that must travel with it.} Add quarks in the
fundamental representation and the center symmetry is broken
\emph{explicitly}: a quark world-line wrapping the time circle picks up
$z$ uncompensated. So in QCD as it actually is, $\langle L \rangle$ is
never exactly zero, the Polyakov loop is not an order parameter, and
there is no phase transition --- lattice calculations find a smooth
crossover near $155$~MeV. Confinement in the real world is a statement
about which states exist, not about a broken symmetry.

\subsection{Step 1 (continued) --- Regge trajectories from a rotating
string}
\label{ssec:L-regge}

If the linear potential really is a string, the string should spin, and
its spectrum should be checkable against the hadrons. This is the one
place in hadron physics where a two-line calculation predicts a measured
slope.

Model a light-quark meson as a straight relativistic string of tension
$\sigma$, length $2R$, rotating rigidly about its center with its
massless endpoints moving at the speed of light. Then the transverse
speed at distance $r$ from the center is $v(r) = r/R$ (units $c=1$), and
the two integrals are elementary:
\begin{align}
E &\;=\; 2\int_0^R \frac{\sigma \, \dd r}{\sqrt{1 - v^2}}
 \;=\; 2\sigma R \int_0^1 \frac{\dd x}{\sqrt{1-x^2}}
 \;=\; \pi \sigma R ,
\label{eq:L-regge-E} \\[2pt]
\mathcal{J} &\;=\; 2\int_0^R
 \frac{\sigma \, r \, v(r)\, \dd r}{\sqrt{1 - v^2}}
 \;=\; 2\sigma R^2 \int_0^1 \frac{x^2 \, \dd x}{\sqrt{1-x^2}}
 \;=\; \frac{\pi \sigma R^2}{2} ,
\label{eq:L-regge-J}
\end{align}
using $\int_0^1 (1-x^2)^{-1/2}\dd x = \pi/2$ and
$\int_0^1 x^2 (1-x^2)^{-1/2} \dd x = \pi/4$. (The spin is written
$\mathcal{J}$ throughout, because in this cohort the plain letter $J$ is
the ledger integer $J = 7$ and is spoken for.) Eliminate $R$ between the
two:
\begin{equation}
\mathcal{J} \;=\; \frac{E^2}{2\pi\sigma} \;=\; \alpha' E^2 ,
\qquad
\alpha' \;=\; \frac{1}{2\pi\sigma} .
\label{eq:L-regge}
\end{equation}
A linear Regge trajectory --- spin proportional to mass squared --- with
a slope fixed entirely by the tension. Put in
Eq.~\eqref{eq:L-sigmaval}:
\begin{equation}
\alpha' \;=\; \frac{1}{2\pi \times 0.1936~\mathrm{GeV}^2}
 \;=\; 0.822~\mathrm{GeV}^{-2} .
\label{eq:L-alphaprime}
\end{equation}

Now the test. The light meson trajectory
$\rho(775) \to a_2(1318) \to \rho_3(1689) \to a_4(1996)$ carries
$\mathcal{J} = 1,2,3,4$; a least-squares fit of $\mathcal{J}$ against
$M^2$ gives slope $0.888$~GeV$^{-2}$ and intercept $0.464$
\citep{PDG2024}. The prediction \eqref{eq:L-alphaprime} is $7.4\%$ low;
the tension that would fit exactly is $423$~MeV rather than $440$.

Seven percent, from a rigid straight string with massless ends and no
quantum corrections, using a tension measured in an entirely different
experiment: not a precision test, but far better than a model this crude
has any right to. What it establishes is the \emph{form} --- that
$\mathcal{J}$ is linear in $M^2$ with a slope of one over the tension ---
and the form is what confinement predicts and no short-range potential
model can.

\subsection{Step 1 (continued) --- why singlets need triples}
\label{ssec:L-singlets}

The color-singlet rule is usually taught as an empirical fact with a
dynamical excuse. It is in fact a theorem of $\mathrm{SU}(3)$
representation theory, and the theorem takes three lines.

An $\mathrm{SU}(3)$ tensor with $n$ upper and $\bar n$ lower indices can
be reduced only by contracting with invariant tensors, and
$\mathrm{SU}(3)$ has exactly two kinds: $\delta^i_j$, removing one upper
and one lower index, and $\epsilon^{ijk}$ or $\epsilon_{ijk}$, removing
three of a kind. Define the \emph{triality}
\begin{equation}
t \;=\; (n - \bar n) \bmod 3 .
\label{eq:L-triality}
\end{equation}
Contraction with $\delta$ changes $n - \bar n$ by $0$, with $\epsilon$ by
$\pm 3$; neither changes $t$. Triality is therefore a conserved label,
additive modulo $3$, and since the singlet has $t = 0$:
\begin{equation}
\text{a color singlet can be built}
\quad\Longleftrightarrow\quad
\sum_i t_i \;\equiv\; 0 \pmod 3 .
\label{eq:L-singletrule}
\end{equation}
Every consequence follows by arithmetic. A quark has $t = +1$, an
antiquark $t = -1 \equiv 2$, a gluon (adjoint, one upper and one lower
index) $t = 0$.

\begin{table}[H]
\centering
\caption[Triality and the color-singlet rule.]{Triality and the color-singlet rule. Dimensions check:
$3\times3 = 9 = 6+3$, no singlet; $3\times3\times3 = 27 = 10+8+8+1$,
exactly one. The rule counts triality, not constituents, which is why
tetraquarks and pentaquarks are permitted --- and observed. Four columns: the quark content, its triality sum, the product decomposition, and whether a singlet appears. The arithmetic is shown rather than asserted so that a reader can check the rule against a case it does not obviously cover before deciding whether to believe it.}
\label{tab:L-triality}\small
\begin{tabular}{@{}P{0.17\textwidth}P{0.15\textwidth}P{0.42\textwidth}P{0.11\textwidth}@{}}
\toprule
\textbf{content} & \textbf{triality} $\sum t_i$ & \textbf{product} &
\textbf{singlet?} \\
\midrule
$q$ & $1$ & $\mathbf{3}$ & no \\
$qq$ & $2$ & $\mathbf{3}\otimes\mathbf{3}
 = \mathbf{6}\oplus\bar{\mathbf{3}}$ & no \\
$q\bar q$ & $0$ & $\mathbf{3}\otimes\bar{\mathbf{3}}
 = \mathbf{8}\oplus\mathbf{1}$ & yes \\
$qqq$ & $0$ & $\mathbf{3}^{\otimes 3}
 = \mathbf{10}\oplus\mathbf{8}\oplus\mathbf{8}\oplus\mathbf{1}$ & yes \\
$qq\bar q\bar q$ & $0$ & (contains a singlet) & yes \\
$qqqq\bar q$ & $0$ & (contains a singlet) & yes \\
$gg$ (adjoint) & $0$ & $\mathbf{8}\otimes\mathbf{8}
 \supset \mathbf{1}$ & yes \\
\bottomrule
\end{tabular}
\end{table}

Two entries carry the argument. Two quarks cannot make a singlet:
$\mathbf{3}\otimes\mathbf{3}$ has dimension nine and decomposes as
$\mathbf{6}\oplus\bar{\mathbf{3}}$, with no singlet anywhere, because
$2 \not\equiv 0 \bmod 3$. Three quarks can: $\epsilon_{ijk} q^i q^j q^k$
is the invariant, $27 = 10 + 8 + 8 + 1$, exactly one singlet.
\emph{Three is the smallest number of quarks that closes, and the reason
is that the invariant antisymmetric tensor of $\mathrm{SU}(3)$ has three
indices.} Nothing about forces enters. The baryon has three quarks
because $3$ is $3$.

Notice what the rule permits: it counts triality, not quarks, so
$qq\bar q\bar q$, $qqqq\bar q$ and pure-gluon states all pass, and nature
has obliged on the first two. And notice what it does not deliver. The
adjoint has $t = 0$ and passes cleanly, yet no free gluon has ever been
seen. Triality zero is necessary and not sufficient; what removes the
gluon is the mass gap, which is dynamics, and which Step~2 places.

\subsection{Step 1 (concluded) --- the $\theta$ term and the bound that
makes it a problem}
\label{ssec:L-theta}

One more term is allowed in the QCD Lagrangian by every symmetry the
theory has, and it is this appendix's second subject:
\begin{equation}
\mathcal{L}_\theta \;=\; \theta \,
 \frac{g^2}{32\pi^2}\, F^a_{\mu\nu} \tilde F^{a\,\mu\nu} ,
\qquad
\tilde F^{a\,\mu\nu} = \tfrac12 \epsilon^{\mu\nu\rho\lambda}
 F^a_{\rho\lambda} .
\label{eq:L-theta}
\end{equation}
It is Lorentz invariant, gauge invariant, and dimension four, so it
belongs; and it is $P$-odd and $T$-odd, hence $CP$-odd
(Appendix~\ref{app:discrete}).

\paragraph{Why it is not obviously physical.} $F\tilde F$ is a total
derivative, $F^a_{\mu\nu}\tilde F^{a\,\mu\nu} = \partial_\mu K^\mu$ with
$K^\mu$ the Chern--Simons current
(Appendices~\ref{app:diffcohom} and \ref{app:anomalies}), and a total
derivative does not change the classical equations of motion. In an
abelian theory with trivial topology that would be the end of it.

\paragraph{Why it is physical anyway.} Because the integral of a total
derivative need not vanish when the field configurations are classified
by topology. For finite-action Euclidean configurations,
\begin{equation}
\nu \;=\; \int \dd^4x \;
 \frac{g^2}{32\pi^2}\, F^a_{\mu\nu} \tilde F^{a\,\mu\nu}
 \;\in\; \mathbb{Z} ,
\label{eq:L-pontryagin}
\end{equation}
the Pontryagin index, an integer counting the winding of the gauge field
at infinity --- an instance of the characteristic-class arithmetic of
\citet{MilnorStasheff1974} and, in physics dress,
\citet{Nakahara2003}, and the same phenomenon as the integrality of a
Chern number in Appendix~\ref{app:connections}. So the $\theta$ term
contributes a sector-dependent phase $e^{\ii\theta\nu}$ to the path
integral. It is physical, it is periodic under $\theta \to \theta + 2\pi$
because $\nu$ is an integer, and the instanton solutions
\citep{Bogomolny1976} populate the sectors with $\nu \neq 0$.

\paragraph{What is actually observable.} Not $\theta$. A chiral rotation
of a quark field, $q \to e^{\ii\alpha\gamma^5}q$, is not a symmetry of
the measure --- this is Fujikawa's derivation of the anomaly
\citep{Fujikawa1979}, and the same fact discovered diagrammatically by
\citet{Adler1969} and \citet{BellJackiw1969}. The measure shifts
$\theta$, and the rotation also changes the phase of the quark mass
matrix. Only the combination survives:
\begin{equation}
\bar\theta \;=\; \theta \;+\; \arg\det\!\left(M_u M_d\right) .
\label{eq:L-thetabar}
\end{equation}
This is the physical strong-$CP$ angle. Two corollaries. If any quark
were exactly massless, $\arg\det M$ would be undefined and $\bar\theta$
could be rotated away --- the ``massless up quark'' solution, now
excluded by lattice determinations of $m_u$. And $\bar\theta$ mixes the
gauge sector with the flavor sector, so a solution must address both
terms, not just the first.

\paragraph{The bound.} A $CP$-odd, $P$-odd strong interaction gives the
neutron a permanent electric dipole moment. Hadronic estimates give
$d_n \approx 2.4\times10^{-16}\,\bar\theta\; e\,$cm against an
experimental limit $|d_n| < 1.8\times10^{-26}\,e\,$cm
\citep{PDG2024}, so
\begin{equation}
|\bar\theta| \;<\; \frac{1.8\times10^{-26}}{2.4\times10^{-16}}
 \;=\; 7.5\times10^{-11} .
\label{eq:L-thetabound}
\end{equation}
An angle free to be anything on a circle of circumference $2\pi$ is
measured to lie within one part in $10^{11}$ of zero.
Appendix~\ref{app:discrete} converts that into a laboratory picture
(problem~K.4); here the point is the shape of the number. This is not a
small coupling to be explained by a small mass. It is a \emph{pure
angle}, dimensionless, with no scale attached, and no symmetry of the
Standard Model would set it to zero if restored.

\subsection{Step 2 --- what is proved, and what everyone knows is not}
\label{ssec:L-step2}

Step~1 has a computation for the area law and a theorem for the singlet
rule. Be exact about how far each reaches.

\paragraph{The Clay problem is not confinement.} The Millennium Prize
problem asks for two things: a construction of quantum Yang--Mills theory
on $\mathbb{R}^4$ satisfying the standard axioms, and a proof that it has
a \emph{mass gap} $\Delta > 0$. Existence plus a gap. It is not the area
law, not the linear potential, and not the absence of free quarks; the
three are believed to travel together and none of the implications is
established. The distinction is worth keeping, because a framework that
says something about which states exist has said nothing about the gap.

\paragraph{The strong-coupling derivation does not survive the continuum
limit --- and here is the counterexample.} Equation~\eqref{eq:L-arealaw}
was derived by expanding in small $\beta$, that is, large $g$. But
Yang--Mills theory is asymptotically free: taking $a \to 0$ at fixed
physical $\sigma$ requires $g \to 0$, exactly where the expansion fails.
Is the area law then an artifact of the lattice? For compact
$\mathrm{U}(1)$ the answer is \emph{yes}: the same tiling argument gives
an area law at strong coupling, and the theory nevertheless has a
deconfined weak-coupling phase whose continuum limit is ordinary
photodynamics. So the strong-coupling area law by itself proves nothing
about the continuum. What supports confinement in continuum
$\mathrm{SU}(3)$ is numerical evidence that the area law persists through
the scaling window --- evidence, not proof, which is the status the Clay
problem records.

\paragraph{No local order parameter, and no phase boundary either.} Two
theorems close the obvious ways of tidying this up. Elitzur's theorem
\citep{Elitzur1975} says a local gauge symmetry cannot break
spontaneously, so no local gauge-variant field can serve as an order
parameter. And the Fradkin--Shenker analysis \citep{FradkinShenker1979}
shows that with matter in the fundamental representation the
``confining'' and ``Higgs'' regions of the phase diagram are analytically
connected: no singularity separates them. There is therefore no order
parameter distinguishing confinement from the Higgs phase when
fundamental matter is present, and the center symmetry of
\S\ref{ssec:L-centre} is explicitly broken by the quarks in any case.
``Confinement'' in the real world is a statement about which asymptotic
states exist, not about a phase in the Landau sense.

\paragraph{The rotating string is a model.} Equations
\eqref{eq:L-regge-E}--\eqref{eq:L-regge} came from a classical rigid
string, not from QCD; the $7.4\%$ agreement is better than the derivation
deserves, and the intercept is not predicted at all.

\paragraph{And the strong-$CP$ problem has no Standard-Model answer.}
Within the Standard Model $\bar\theta$ is a free parameter. It is
dimensionless and radiatively stable --- the CKM phase feeds into it only
at high loop order, at the level of $10^{-17}$ --- so it is
\emph{technically natural} to set it small once you have set it small.
Nothing explains why it \emph{is} small. The known options each add
structure: a massless up quark (excluded); a Peccei--Quinn symmetry whose
breaking produces an axion, costing a new global symmetry, a new scale
and a particle not yet found; spontaneous $CP$ violation with a mechanism
to keep the induced $\bar\theta$ small; anthropic selection. The problem
is not that no solution exists, but that every solution replaces the
parameter needing explanation with a parameter of the explanation.

\subsection{Step 3 --- Chapter~\ref{sec:triangle}'s reading}
\label{ssec:L-step3}

The body reads all of this off one structure: a compact non-spatial cycle
whose holonomy takes values in the color center,
Eq.~\eqref{eq:L-centre}. Three claims follow, of quite different
weights, and they should be graded separately.

\paragraph{The triangle that closes.} The descent breaks the unified
group by a Wilson line on the compact cycle --- the Hosotani mechanism
\citep{Hosotani1983}, whose content is that a holonomy on a
non-simply-connected space is a physical, gauge-invariant datum with no
local field to carry it. The setting that survives the spectrum's
requirements is a center element, $z = \mathrm{diag}(\omega, \omega,
\omega, 1, 1)$: three detents on one abelian dial, the vertices of an
equilateral triangle inscribed in the cycle. Baryon number is then the
winding of that phase, $B = t/3$, with $t$ the triality of
Eq.~\eqref{eq:L-triality}. The identity itself is not new --- it has been
visible in unified models since 1974. What the reading adds is the
promotion of the phase to a winding number of a real cycle, which is what
makes it an integer and therefore protected.

\paragraph{Color singlets as a registration rule.} Read that way, the
singlet rule of Eq.~\eqref{eq:L-singletrule} becomes an admissibility
condition on what a boundary can register rather than a consequence of a
force law: a readout must close on the cycle, a curve closes when its
winding is an integer, $B \in \mathbb{Z}$ is $\sum_i t_i \equiv 0 \bmod
3$, and that is the singlet rule. The structural contrast with the
standard account should be stated as a contrast and not as a correction.
There the rule is an \emph{output} of strong dynamics, which also
supplies $\sigma$, the mass gap and the spectrum; here it is \emph{prior}
to the dynamics, and the dynamics supplies the rest. The two agree on
every state ever observed --- Table~\ref{tab:L-triality} is common
property --- and differ in what is taken as given.

Two fences go with it, and the body erects both. Triality zero is
necessary and not sufficient: the gluon passes and is still absent, and
that absence is the mass gap's work, which this reading does not do. And
nothing here prices $\sigma$: the $16$ tonnes of
Eq.~\eqref{eq:L-tonnes} are not derived by a winding argument and are not
claimed to be.

\paragraph{Charge quantization as a protected integer.} Run the same
fragment table against electric charge and one finds $Q + B \in
\mathbb{Z}$ on every fragment, so a color singlet has integer electric
charge automatically. This is the ladder's sixth-rung claim: the quantity
is exact not because a symmetry forbids its drift but because there is no
continuous path off an integer. Three classic routes reach charge
quantization, and it is worth seeing how this one differs. Dirac's
\citep{Dirac1931} makes $eg = 2\pi n$ conditional on a monopole existing
somewhere; the Wu--Yang bundle formulation \citep{WuYang1975} identifies
the same $n$ as a Chern number over the sphere
(Appendix~\ref{app:connections}); anomaly cancellation fixes the hypercharges
given the fermion content \citep{ArisakaA9SM}. Here the integer is a
winding number of a real compact cycle, needing neither monopole nor
anomaly input.

The exactness is measured, and measured very well: levitation experiments
on macroscopic samples of ordinary matter bound any residual fractional
charge at $10^{-21}$ of $e$ per nucleon \citep{MarinelliMorpurgo1984}.
That number is the empirical face of the integer, and it is also the
falsifier --- a single confirmed isolated fractional charge would be a
free triality, an unclosed winding presented as closed, and the reading
would be finished.

\paragraph{And $\bar\theta$.} The strong-$CP$ angle of
Eq.~\eqref{eq:L-thetabar} is read through the same closure, with its two
terms taking different grades. That the projected four-dimensional seam
admits no independent continuous $\theta$-source, because the color
holonomy is a quantized center element rather than a dial, is derived
within the framework's own axioms (\OGGEP{}; \citealp{ArisakaA1Constitution}).
That the quark determinant phases cancel is graded
\emph{derived-conditional} --- the codebook is real at first order, and
the argument is not closed end to end. The standing prediction is
therefore not a hard zero but a residual leakage $\bar\theta \lesssim
5\times10^{-18}$, about seven orders below the present bound
\eqref{eq:L-thetabound} and a factor $1.5\times10^{7}$ beyond current
dipole sensitivity. A measured neutron dipole anywhere above that
envelope kills the reading --- which is the only property that matters
for Chapter~\ref{sec:falsifiers}.

\subsection{Problems and worked solutions}
\label{ssec:L-problems}

\medskip\noindent\textbf{L.1. From $\sqrt\sigma = 440$~MeV, compute the
string tension as a force in tonnes weight; compute the energy stored in
separating a static pair by $1$~fm; and explain, quantitatively, why that
energy means the linear potential is never actually observed to run
away.}

\noindent\emph{Solution.} \emph{Step 1.} $\sigma = (0.440)^2 =
0.1936$~GeV$^2$; divide by $\hbar c = 0.19733$~GeV$\cdot$fm to get
$0.981$~GeV/fm. Then $0.981 \times 1.602\times10^{-10}$~J per
$10^{-15}$~m $= 1.572\times10^{5}$~N, and dividing by $g =
9.807$~m/s$^2$ weighs it: $1.60\times10^{4}$~kg, or $16.0$~tonnes.
\emph{Step 2.} The energy to separate by $r = 1$~fm is $\sigma r =
0.981$~GeV. Compare with what the vacuum can make: a light $q\bar q$ pair
at roughly $0.3$~GeV a quark costs about $0.6$~GeV, and a pion pair
$0.28$~GeV. \emph{Step 3.} So before the string has been stretched by
even one fermi it is cheaper to create a pair in the middle and cap the
two halves: two mesons, never one free quark. \emph{Physics intuition:}
the linear potential is a statement about \emph{static} sources, and the
world's light quarks are why it flattens at about $1.3$~fm instead of
rising forever. Both facts are needed --- without the rise there would be
free quarks, and without the breaking there would be infinitely long
strings.

\medskip\noindent\textbf{L.2. In the strong-coupling expansion, evaluate
the leading contribution to a $2\times1$ Wilson loop and to a $1\times1$
loop, confirm the area scaling, and evaluate $\sigma a^2$ for
$\mathrm{SU}(3)$ at $g^2 = 1$ and $g^2 = 2$. Then explain what would have
to be true of the group integral for a perimeter law to appear instead.}

\noindent\emph{Solution.} \emph{Step 1.} A $1\times1$ loop is a single
plaquette, one tile, leading term $\beta/(2N^2)$ by
Eq.~\eqref{eq:L-arealaw}; a $2\times1$ loop needs two tiles,
$(\beta/(2N^2))^2$. The exponent doubles with the area, whereas the
perimeters are $4$ and $6$, a ratio of $1.5$ rather than the observed
$2$.
\emph{Step 2.} From Eq.~\eqref{eq:L-sigma-sc} with $N=3$: at $g^2 = 1$,
$\sigma a^2 = \ln 3 = 1.099$; at $g^2 = 2$, $\sigma a^2 = \ln 6 =
1.792$. The tension grows with the coupling, as it must in an expansion
whose small parameter is $1/g^2$.
\emph{Step 3.} The area law came from the vanishing of $\int \dd U\,
U_{ij}$, which forced every boundary link to be paired and so forced the
surface to be filled. Were the group non-compact, or the measure not
invariant, a single unpaired link would give a nonzero result; each link
of $C$ could then be saturated independently, the cost would be one
factor per \emph{link}, and $\ln\langle W \rangle$ would scale with the
perimeter. \emph{Physics intuition:} compactness of the gauge group is
the hinge --- the same compactness that makes the $\mathrm{U}(1)$ charge
an integer in Appendix~\ref{app:connections} and that puts an integer on the
left of Eq.~\eqref{eq:L-pontryagin}.

\medskip\noindent\textbf{L.3. Derive Eqs.~\eqref{eq:L-regge-E} and
\eqref{eq:L-regge-J} for the rotating string, obtain $\alpha' =
1/(2\pi\sigma)$, evaluate it, and compare with the measured light-meson
slope. What does the discrepancy tell you, and what does the agreement?}

\noindent\emph{Solution.} \emph{Step 1.} With ends at the speed of light,
$v(r) = r/R$; each element $\sigma\,\dd r$ carries energy
$\sigma\,\dd r/\sqrt{1-v^2}$ and angular momentum
$\sigma r v\,\dd r/\sqrt{1-v^2}$. Substituting $x = r/R$ and using
$\int_0^1 (1-x^2)^{-1/2}\dd x = \pi/2$, $\int_0^1 x^2(1-x^2)^{-1/2}\dd x
= \pi/4$ gives $E = \pi\sigma R$, $\mathcal{J} = \pi\sigma R^2/2$, and
eliminating $R$ gives $\mathcal{J} = E^2/(2\pi\sigma)$.
\emph{Step 2.} With $\sigma = 0.1936$~GeV$^2$, $\alpha' = 1/(2\pi \times
0.1936) = 0.822$~GeV$^{-2}$. The four-state fit to
$\rho, a_2, \rho_3, a_4$ gives $0.888$~GeV$^{-2}$: the prediction is
$7.4\%$ low, and the tension that would fit exactly is $423$~MeV.
\emph{Step 3.} The discrepancy says the model is a rigid classical string
with massless ends, missing quark masses, thickness and quantum
corrections at the several-percent level --- exactly the size of the
miss. The agreement says more: $\mathcal{J}$ is \emph{linear} in $M^2$
over four states spanning a factor of seven, with a slope set by an
independently measured tension. \emph{Physics intuition:} no short-range
potential model produces linear Regge trajectories; a string does
automatically, because the two integrals differ by one power of $r$.
Getting the \emph{form} right from a one-line calculation beats getting a
number right by fitting.

\medskip\noindent\textbf{L.4. Show from invariant tensors that
$\mathbf{3}\otimes\mathbf{3}$ contains no singlet while
$\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3}$ contains exactly one.
Verify by dimension counting. Then classify $qq\bar q\bar q$, $qqqq\bar
q$ and the adjoint, and say why the last is nevertheless absent from the
free spectrum.}

\noindent\emph{Solution.} \emph{Step 1.} The invariant tensors of
$\mathrm{SU}(3)$ are $\delta^i_j$, $\epsilon^{ijk}$ and $\epsilon_{ijk}$.
A singlet is a full contraction of the product with invariants. For
$q^i q^j$ there is no invariant with two upper indices, so no singlet:
$t = 1 + 1 = 2 \not\equiv 0$. For $q^i q^j q^k$ there is exactly one,
$\epsilon_{ijk}$, so exactly one singlet: $t = 3 \equiv 0$.
\emph{Step 2.} Dimensions: $3 \times 3 = 9 = 6 + 3$, and neither $6$ nor
$3$ is $1$. And $3\times3\times3 = 27 = 10 + 8 + 8 + 1$: one singlet,
with multiplicity exactly one, matching the count of invariant tensors.
\emph{Step 3.} $qq\bar q \bar q$ has $t = 0$, $qqqq\bar q$ has
$t = 3 \equiv 0$, and the adjoint has $t = 0$: all three permitted. The
first two are seen; the third is not. \emph{Physics intuition:} the rule
is \emph{necessary} --- a statement about which states can be built, not
about which are bound. The gluon is absent for a dynamical reason, the
mass gap, that no counting argument addresses, and that is exactly the
boundary between what Chapter~\ref{sec:triangle} claims and what it
leaves to the strong dynamics. A rule that forbids is not a rule that
creates.

\medskip\noindent\textbf{L.5. Using the Casimir identity
\eqref{eq:L-casimir}, compute $T_1\!\cdot\!T_2$ in all four channels of
$q\bar q$ and $qq$, and interpret the signs. Which diquark channel is
attractive, and how strongly compared with the meson?}

\noindent\emph{Solution.} \emph{Step 1.} $C_2(\mathbf{3}) =
C_2(\bar{\mathbf{3}}) = 4/3$, $C_2(\mathbf{6}) = 10/3$, $C_2(\mathbf{8})
= 3$, $C_2(\mathbf{1}) = 0$, all from $C_2 = (p^2+q^2+pq+3p+3q)/3$.
\emph{Step 2.} For $q\bar q$: singlet, $\tfrac12(0 - \tfrac43 -
\tfrac43) = -\tfrac43$; octet, $\tfrac12(3 - \tfrac83) = +\tfrac16$. For
$qq$: $\bar{\mathbf{3}}$, $\tfrac12(\tfrac43 - \tfrac83) = -\tfrac23$;
$\mathbf{6}$, $\tfrac12(\tfrac{10}{3} - \tfrac83) = +\tfrac13$.
\emph{Step 3.} Negative is attractive: one-gluon exchange binds the meson
singlet with strength $4/3$ and the antisymmetric $\bar{\mathbf{3}}$
diquark with strength $2/3$, exactly half, and repels in the
$\mathbf{8}$ and $\mathbf{6}$. Check on the baryon: with all three quarks
in the total singlet, $(\sum_i T_i)^2 = 0$ gives $3 \times \tfrac43 +
2\sum_{i<j} T_i\!\cdot\!T_j = 0$, so the three pairs share $-2$, that is
$-\tfrac23$ each --- the diquark value, as it must be, since any two
quarks in a singlet baryon sit in the $\bar{\mathbf{3}}$. \emph{Physics
intuition:} the factor of two is why the diquark is a useful effective
degree of freedom and not a bound state on its own. Note also that this
argument, unlike \S\ref{ssec:L-strong}, is perturbative: it says which
channel the force prefers, not which states exist.

\medskip\noindent\textbf{L.6. (a) Show that $\theta$ and $\theta + 2\pi$
give the same physics. (b) Extract the bound on $\bar\theta$ from the
neutron dipole limit and express it as a fraction of a full turn. (c)
State the improvement factor needed to reach the envelope
$\bar\theta \lesssim 5\times10^{-18}$, and say what that means for the
falsifier.}

\noindent\emph{Solution.} \emph{Step 1.} (a) The $\theta$ term
contributes $e^{\ii\theta\nu}$ to the Euclidean weight of a
configuration, with $\nu \in \mathbb{Z}$ by
Eq.~\eqref{eq:L-pontryagin}. Shifting $\theta \to \theta + 2\pi$
multiplies every weight by $e^{2\pi \ii \nu} = 1$. So $\theta$ is an
angle on a circle, and any statement about its size is a statement about
its distance from zero \emph{modulo} $2\pi$.
\emph{Step 2.} (b) From $|d_n| < 1.8\times10^{-26}\,e\,$cm and $d_n
\approx 2.4\times10^{-16}\,\bar\theta\,e\,$cm, $|\bar\theta| <
7.5\times10^{-11}$. As a fraction of a full turn that is
$7.5\times10^{-11}/2\pi = 1.19\times10^{-11}$: about one part in $10^{11}$
of the circle.
\emph{Step 3.} (c) To probe $5\times10^{-18}$ the dipole sensitivity must
improve by $7.5\times10^{-11}/5\times10^{-18} = 1.5\times10^{7}$. Current
programs aim at one to two orders of magnitude this decade, so the
envelope is not reachable soon, and its role is as a \emph{one-sided}
falsifier: a dipole \emph{detected} anywhere between the present bound
and that envelope kills the closure claim of \S\ref{ssec:L-step3}, while
null results confirm nothing in particular. \emph{Physics intuition:}
say the asymmetry plainly. Predicting a very small number is cheap when
the experiment cannot reach it; what makes this one meaningful is that
the whole interval from $10^{-10}$ down to $10^{-17}$ is forbidden by the
claim and permitted by the Standard Model --- and that interval is where
the experiments are about to look.

\GGtakeaway{\emph{Standard result:} confinement is the area law
$\langle W\rangle \sim e^{-\sigma A}$, equivalent by
Eq.~\eqref{eq:L-transfer} to a linear potential $V = \sigma r$, and
derived exactly in strong-coupling lattice gauge theory by tiling the
minimal surface, Eq.~\eqref{eq:L-arealaw}, with $\sigma a^2 = \ln(Ng^2)$.
With $\sqrt\sigma = 440$~MeV the tension is $16$ tonnes weight. The
center $\mathbb{Z}_3$ and the Polyakov loop give an order parameter whose
\emph{unbroken} phase is the confined one. A rotating relativistic string
gives $\mathcal{J} = \alpha' E^2$ with $\alpha' = 1/(2\pi\sigma) =
0.822$~GeV$^{-2}$ against a measured $0.888$. Triality modulo three, not
quark counting, decides which states can be built. \emph{What it
assumed:} strong coupling and finite lattice spacing for the area law; a
classical rigid string for the trajectories; and, for $\bar\theta$,
nothing at all --- it is a free parameter bounded at $7.5\times10^{-11}$.
\emph{What it leaves open:} the Clay problem, which is existence and the
mass gap, not confinement; the continuum limit of the area law, where
compact $\mathrm{U}(1)$ is the standing counterexample; the absence of a
phase boundary once fundamental matter is present
\citep{FradkinShenker1979}; and the smallness of $\bar\theta$.
\emph{Continue in:} Chapter~\ref{sec:triangle} for the closed triangle
and the claim that $\bar\theta$ has no residue left to be;
Appendix~\ref{app:wilson} for the holonomy;
Appendix~\ref{app:anomalies} for the chiral rotation that makes only
$\bar\theta$ physical; and Appendix~\ref{app:bianchi} for the protected
integers this appendix kept meeting.}

\section{Baryogenesis and leptogenesis}
\label{app:lepto}

\GGteachbox{The one number this whole subject is about, and the two
independent measurements that fix it; Sakharov's three conditions with
each argued rather than listed, including the proof from $CPT$ that
thermal equilibrium alone forces the baryon asymmetry to zero; the two
quantitative failures of the Standard Model, with the ten-order
shortfall of the quark phase estimated explicitly; the electroweak
sphaleron, the anomaly that lets it change $B{+}L$ while conserving
$B{-}L$, and where the conversion factor $28/79$ comes from; the seesaw
mechanism and what it costs; and then leptogenesis assembled with every
standard factor named --- the $CP$ asymmetry, the washout efficiency,
the equilibrium abundance, the entropy dilution and the entropy-to-photon
conversion --- and run end to end until a number falls out that can be
compared with the measurement. The pay-off is
Chapter~\ref{sec:leptogenesis}.}

\GGreadalong{\citet{Sakharov1967} itself, which is two pages long and
contains the whole logical skeleton; \citet{ChengLi1984}, Ch.~13 and 15,
for the anomaly and for neutrino masses in the textbook presentation;
\citet{Weinberg1996}, \S23.5, for anomalies and topological vacua done
carefully; and \citet{PeskinSchroeder1995}, \S19, for the chiral anomaly
in the form used below.}

\subsection{Step 1 --- the number, and how it is measured twice}
\label{ssec:M-step1}

Everything in this appendix is about one dimensionless number. Count the
baryons in a comoving volume and divide by the photons in the same
volume:
\begin{equation}
\eta_B \;\equiv\; \frac{n_B - n_{\bar B}}{n_\gamma}
 \;=\; (6.12 \pm 0.04) \times 10^{-10} .
\label{eq:M-etaB}
\end{equation}
Roughly one baryon per $1.6$ billion photons. Before anything is
explained it is worth understanding what that number \emph{is}, because
the naive reading --- ``the universe is mostly light'' --- misses the
point entirely.

At temperatures above a GeV, quarks and antiquarks were as abundant as
photons; $n_q \approx n_{\bar q} \approx n_\gamma$. The measured $\eta_B$
therefore says that at that epoch the fractional imbalance was
\begin{equation}
\frac{n_q - n_{\bar q}}{n_q + n_{\bar q}}
 \;\approx\; \frac{\eta_B}{2} \;\approx\; 3 \times 10^{-10} ,
\label{eq:M-imbalance}
\end{equation}
about three parts in ten billion: for every $1\,000\,000\,000$
antiquarks there were $1\,000\,000\,003$ quarks. Then the universe cooled,
the pairs annihilated into photons, and the three survivors per ten
billion are the entire material contents of the world. Every galaxy,
every atom, and every reader of this appendix is the rounding error in
that subtraction.

And the alternative is not merely aesthetically unappealing; it is
quantitatively excluded. If $\eta_B$ had been exactly zero, nucleons and
antinucleons would have annihilated until the remaining density was too
low to find partners, freezing out at $n_B/n_\gamma \sim 10^{-19}$ ---
nine orders below what is observed, and far too little to make a single
galaxy. A symmetric universe is a nearly empty one. Something made the
three.

\paragraph{The first measurement: the microwave background.} Baryons
carry inertia and photons carry pressure, and the acoustic oscillations
in the pre-recombination plasma are driven by the competition. Increasing
the baryon density deepens the compressions relative to the rarefactions,
which raises the odd-numbered acoustic peaks relative to the even ones.
That asymmetry is a clean, geometric measurement of $\Omega_b h^2$, and
the conversion to $\eta_B$ involves only measured constants:
\begin{equation}
\eta_B \;=\; 2.74 \times 10^{-8}\; \Omega_b h^2 .
\label{eq:M-etaomega}
\end{equation}
With $\Omega_b h^2 = 0.02237$ \citep{Planck2018} this gives $\eta_B =
6.13\times10^{-10}$.

\paragraph{The second measurement: the light elements.} At $T \sim
0.1$~MeV, nuclear reactions froze out and the primordial abundances of
$^2$H, $^3$He, $^4$He and $^7$Li were set. Each depends on $\eta_B$ in a
different way --- deuterium is the sharpest, because a higher baryon
density burns deuterium more efficiently into helium, giving $\mathrm{D/H}
\propto \eta_B^{-1.6}$. The measured primordial ratio $\mathrm{D/H} =
2.53\times10^{-5}$ implies $\eta_B \approx 6.1\times10^{-10}$.

Two determinations, from physics separated by four hundred thousand years
and by an entirely different set of nuclear and atomic inputs, agree to
within a percent. That agreement is what makes $\eta_B$ a datum rather
than a model-dependent inference, and it is the reason a theory of the
early universe can be held to it.

\subsection{Step 1 (continued) --- Sakharov's three conditions, and why
each one is necessary}
\label{ssec:M-sakharov}

In 1967 Sakharov wrote down what any dynamical origin of $\eta_B$ must
contain \citep{Sakharov1967}. The list is short, and each entry is
genuinely necessary; the necessity arguments are worth having in full,
because two of them are more subtle than they look.

\paragraph{(i) Baryon number must be violated.} This one is immediate. If
$B$ is exactly conserved, then $B$ today equals $B$ at any earlier time,
and either the universe began with a baryon asymmetry --- which is an
initial condition, not an explanation --- or it has none now. The
condition is the statement that we would like an explanation rather than
an initial condition.

\paragraph{(ii) Both $C$ and $CP$ must be violated.} Here the argument
has two stages, and the second is the one usually skipped.

Suppose $C$ were a symmetry. Then for every process $X \to \text{final}$
with baryon number $\Delta B$, the charge-conjugate process $\bar X \to
\overline{\text{final}}$ has $-\Delta B$ and exactly the same rate. Sum
over the two and the net production of $B$ vanishes. So $C$ must be
broken.

But breaking $C$ is not enough. Suppose $C$ is broken and $CP$ is exact.
$CP$ relates a process to its conjugate with all momenta and helicities
reversed. Summing rates over final-state spins and integrating over
angles --- which is what a thermal bath does --- the $P$ part washes out,
and $CP$ invariance forces the \emph{summed} rate for $X \to f$ to equal
that for $\bar X \to \bar f$. The net $B$ production vanishes again. This
is why the weak interaction's maximal $C$ violation
(\citealp{LeeYang1956}; \citealp{Wu1957}) was not by itself an answer,
and why 1964 \citep{Christenson1964} mattered so much: $CP$ violation is
the condition, and it was found only three years before Sakharov used it.

\paragraph{(iii) There must be a departure from thermal equilibrium.}
This is the one with a real proof, and it is short enough to give. Let
$\Theta$ be the antiunitary $CPT$ operator (Appendix~\ref{app:discrete}).
$CPT$ invariance means $\Theta H \Theta^{-1} = H$, and the baryon number
operator is $CPT$-odd, $\Theta \hat{B} \Theta^{-1} = -\hat{B}$. In full
thermal equilibrium the density matrix is $\rho = e^{-\hat H/T}/Z$, which
depends on the Hamiltonian alone. Then
\begin{equation}
\langle \hat B \rangle_{\rm eq}
 \;=\; \frac{1}{Z}\,\mathrm{Tr}\!\left[ e^{-\hat H/T} \hat B \right]
 \;=\; \frac{1}{Z}\,\mathrm{Tr}\!\left[
   \Theta^{-1} e^{-\hat H/T} \hat B\, \Theta \right]
 \;=\; -\,\langle \hat B \rangle_{\rm eq} ,
\label{eq:M-cpt}
\end{equation}
so $\langle \hat B \rangle_{\rm eq} = 0$. \emph{Thermal equilibrium
erases the baryon asymmetry no matter what the microphysics is}, provided
only that $CPT$ holds and $B$ is not conserved. Notice how strong the
statement is: it does not depend on couplings, on the mechanism, or on
the temperature. If $B$ is violated at all, equilibrium drives the
asymmetry to zero; the asymmetry can only be a relic of a period when the
relevant reactions were \emph{out} of equilibrium.

\emph{Physics intuition.} The three conditions have a memorable shape. The
first says there must be a door. The second says something must push
preferentially through it in one direction. The third says the door must
close before the room can equalize again. Miss any one and the room ends
up empty.

\subsection{Step 1 (continued) --- why the Standard Model fails, and by
how much}
\label{ssec:M-smfails}

Remarkably, the Standard Model owns a version of all three conditions. It
fails anyway, and it fails quantitatively on two of them --- which is a
much more interesting kind of failure than owning none.

\paragraph{The out-of-equilibrium condition fails.} The Standard Model's
candidate departure from equilibrium is the electroweak phase transition,
where the Higgs field acquires its expectation value. If that transition
were strongly first order, bubbles of the broken phase would nucleate and
expand, and their walls would provide exactly the non-equilibrium
environment Sakharov requires. Whether the transition is first order
depends on the Higgs mass: lattice studies of the electroweak theory find
a first-order line ending at a critical Higgs mass of about $72$--$80$~GeV,
above which the transition becomes a smooth \emph{crossover}. The Higgs is
$125$~GeV. There are no bubbles, no walls, and no departure from
equilibrium: the Higgs expectation value turns on continuously as the
universe cools, and the sphaleron rate falls smoothly through the
freeze-out at $T \approx 132$~GeV. The measurement of the Higgs mass in
2012 closed this route.

\paragraph{The $CP$ condition fails by ten orders of magnitude.} The
Standard Model's $CP$ violation is the single Kobayashi--Maskawa phase
\citep{KobayashiMaskawa1973}, and its rephasing-invariant measure is the
Jarlskog determinant $J_{CP} = 3.1\times10^{-5}$
(Appendix~\ref{app:discrete}). But $J_{CP}$ is not the whole story,
because $CP$ violation in the quark sector vanishes if any two quarks of
the same charge are degenerate. The lowest-dimension $CP$-odd invariant
therefore carries every quark mass splitting:
\begin{equation}
d_{CP} \;=\; J_{CP}\,
 \frac{(m_t^2 - m_c^2)(m_t^2 - m_u^2)(m_c^2 - m_u^2)
 (m_b^2 - m_s^2)(m_b^2 - m_d^2)(m_s^2 - m_d^2)}{T^{12}} .
\label{eq:M-gavela}
\end{equation}
At $T \approx 100$~GeV, with running masses at that scale, this evaluates
to $d_{CP} \approx 1\times10^{-21}$. The required asymmetry is
$Y_B = n_B/s \approx 8.7\times10^{-11}$. The shortfall is about ten orders
of magnitude --- the conclusion Gavela, Hern\'andez, Orloff and P\`ene
established with the transport calculation done properly, and which
Eq.~\eqref{eq:M-gavela} reproduces at the level of an estimate. The
reason is structural rather than accidental: at $100$~GeV the light-quark
masses are negligible compared with the temperature, so the very
splittings that $CP$ violation requires are the ones the thermal bath
cannot resolve.

\emph{Physics intuition.} The phase that the $B$ factories measure to
several decimal places is real, is large by the standards of a phase, and
is irrelevant to why matter exists. Those three facts sit together, and
the reason is dimensional analysis: a phase is dimensionless, but its
\emph{effect} must be built from mass differences, and at
electroweak temperatures the quarks are effectively massless.

\subsection{Step 1 (continued) --- the sphaleron, the anomaly, and the
$28/79$}
\label{ssec:M-sphaleron}

The Standard Model's baryon-number violation is real, subtle, and
essential to everything that follows.

\paragraph{The anomaly.} Classically, $B$ and $L$ are each conserved:
they are accidental global symmetries of the renormalizable Lagrangian,
of exactly the kind Appendix~\ref{app:noether} produces. Quantum
mechanically they are not. The triangle anomaly
(\citealp{Adler1969}; \citealp{BellJackiw1969}) applied to the weak
$\mathrm{SU}(2)_L$ current gives, per generation,
\begin{equation}
\partial_\mu j_B^\mu \;=\; \partial_\mu j_L^\mu
 \;=\; \frac{g^2}{32\pi^2}\, W^a_{\mu\nu}\tilde W^{a\,\mu\nu} ,
\label{eq:M-anomaly}
\end{equation}
where the right-hand side is the topological density of
Appendix~\ref{app:anomalies}. Two consequences follow immediately and
they organize the whole subject. Since $B$ and $L$ have the \emph{same}
anomaly, the difference is exactly conserved:
\begin{equation}
\partial_\mu \left( j_B^\mu - j_L^\mu \right) \;=\; 0 ,
\qquad \text{while } B+L \text{ is anomalous.}
\label{eq:M-bminusl}
\end{equation}
And since the integral of the right-hand side of
Eq.~\eqref{eq:M-anomaly} over a transition is an integer --- the change
in Chern--Simons number --- the change in baryon number comes in units of
one per generation:
\begin{equation}
\Delta B \;=\; \Delta L \;=\; N_f \, \Delta N_{CS} \;=\; 3\,n ,
\qquad n \in \mathbb{Z} .
\label{eq:M-deltaB}
\end{equation}
Baryon number changes in threes. This is not a coincidence with the
color triangle of Chapter~\ref{sec:triangle}; it is the same
counting seen from the weak side, and it is why the lightest
$B$-violating process the Standard Model permits destroys three baryons
at once rather than one.

\paragraph{The barrier, cold and hot.} Transitions between topological
sectors were first analyzed by \citet{tHooft1971,tHooft1980}, who
computed the tunneling amplitude and found it suppressed by the
instanton action:
\begin{equation}
\Gamma_{\rm tunnel} \;\sim\; e^{-4\pi/\alpha_W}
 \;=\; e^{-372} \;\approx\; 10^{-162} ,
\qquad \alpha_W = \frac{g^2}{4\pi} \approx \frac{1}{30} .
\label{eq:M-tunnel}
\end{equation}
The word ``suppressed'' does not do justice to $10^{-162}$. No proton has
decayed by this channel in the history of the universe, and none will:
the vacuum's own $B$-violation is, at zero temperature, exactly as
unobservable as it is real.

But there is a static, unstable, finite-energy solution sitting at the
top of the barrier between adjacent sectors --- the \emph{sphaleron} of
Klinkhamer and Manton, with energy
\begin{equation}
E_{\rm sph} \;=\; \frac{4\pi v}{g}\, B(\lambda/g^2)
 \;\approx\; 9~\mathrm{TeV} ,
\label{eq:M-sphenergy}
\end{equation}
using $v = 246$~GeV, $g = 0.65$, and $B \approx 1.9$. At temperatures
comparable to and above this scale, the barrier can be climbed thermally
rather than tunneled through, and the transition rate per unit volume
becomes unsuppressed, $\Gamma/V \sim \alpha_W^5 T^4$. This is the
Kuzmin--Rubakov--Shaposhnikov observation, and it changes the picture
completely: for $132~\mathrm{GeV} \lesssim T \lesssim 10^{12}$~GeV,
sphalerons are in full thermal equilibrium and $B+L$ is not a good
quantum number at all, while $B-L$ remains exactly conserved by
Eq.~\eqref{eq:M-bminusl}.

\paragraph{The conversion factor.} Now the practical question. Suppose
some earlier process created an asymmetry in $B-L$. The sphalerons then
process it. What fraction ends up as baryons?

The answer follows from chemical equilibrium, not from any dynamics. In a
relativistic plasma each species $i$ carries a chemical potential
$\mu_i$, and to linear order the number asymmetry is $n_i - \bar n_i =
(g_i/6)\mu_i T^2$ for a fermion and $(g_i/3)\mu_i T^2$ for a boson. The
chemical potentials are then tied together by every reaction fast enough
to be in equilibrium:

\begin{itemize}
\item Yukawa interactions relate left-handed doublets, right-handed
singlets and the Higgs, e.g.\ $\mu_{q_L} = \mu_H + \mu_{u_R}$;
\item the electroweak sphaleron itself enforces
$\sum_{\rm gen}(3\mu_{q_L} + \mu_{\ell_L}) = 0$;
\item the QCD sphaleron enforces its own analogous relation;
\item and the universe is electrically neutral and hypercharge neutral,
which is one more linear constraint.
\end{itemize}

\noindent Solving that linear system for $N_f$ generations and $N_H$
Higgs doublets gives the classic Harvey--Turner result
\begin{equation}
B \;=\; \frac{8N_f + 4N_H}{22N_f + 13N_H}\,(B-L)
\;\;\xrightarrow[N_f=3,\;N_H=1]{}\;\;
B \;=\; \frac{28}{79}\,(B-L) \;=\; 0.3544\,(B-L) ,
\label{eq:M-2879}
\end{equation}
and correspondingly $L = -({51}/{79})(B-L)$. The sign is positive: a
positive $B-L$ asymmetry yields a positive baryon asymmetry. Note that
the whole factor is arithmetic --- counting degrees of freedom and
imposing equilibrium --- with no free parameter and no coupling
constant in it.

\emph{Physics intuition.} The sphaleron is a valve that lets $B$ and $L$
slosh into each other while their difference is held fixed. If the early
universe hands it a stock of $B-L$, it returns about a third of it as
baryons and the rest as leptons, and the ratio is fixed by counting
alone. This is why so much of baryogenesis is really the problem of
making $B-L$: once you have it, the reprocessing is free.

\subsection{Step 1 (continued) --- the seesaw}
\label{ssec:M-seesaw}

The one sector the Standard Model left obviously incomplete is the
neutrinos. They have mass, which the minimal model forbids, and the
cheapest way to give them mass is also the one that violates lepton
number.

Add a heavy right-handed neutrino $N$, a gauge singlet. Because it is a
singlet, nothing forbids a Majorana mass $\tfrac12 M \bar N^c N$, and
this term violates lepton number by two units. The Dirac mass $m_D =
Y_\nu v/\sqrt2$ comes from the usual Yukawa coupling to the Higgs. In the
basis $(\nu_L, N^c)$ the mass matrix is
\begin{equation}
\mathcal{M} \;=\;
\begin{pmatrix} 0 & m_D \\ m_D & M \end{pmatrix} ,
\label{eq:M-seesawmatrix}
\end{equation}
with eigenvalues $M$ and $-m_D^2/M$ for $M \gg m_D$. The light mass is
\begin{equation}
m_\nu \;\simeq\; \frac{m_D^2}{M} .
\label{eq:M-seesaw}
\end{equation}
The name is exact: push one end down and the other rises. The whole
appeal is that the smallness of $m_\nu$ needs no small coupling, only a
large $M$.

Put numbers to it with $m_\nu = 0.05$~eV, the atmospheric scale. If $M =
10^{15}$~GeV, then $m_D = 224$~GeV and $Y_\nu = 1.28$ --- a top-like
coupling, and the reason the GUT scale is the traditional home of the
seesaw. If $M = 10^{10}$~GeV, then $m_D = 0.71$~GeV and $Y_\nu =
4.1\times10^{-3}$, about the charm quark's. If $M$ is only a few TeV,
$Y_\nu \sim 10^{-6}$, comparable to the electron's, which is small but
not absurd.

Two things must be said about the price. First, $N$ is not observable at
any of these masses, so the seesaw is an explanation whose mechanism
cannot be examined directly. Second, and more usefully, the seesaw
\emph{predicts} that neutrinos are Majorana particles --- their own
antiparticles --- because the term that does the work is a $\Delta L = 2$
mass. That prediction is testable, and \S\ref{ssec:M-step2} returns to
who tests it.

\subsection{Step 1 (concluded) --- leptogenesis assembled, and evaluated}
\label{ssec:M-chain}

Now put the pieces together, because $N$ satisfies all three Sakharov
conditions in the lepton sector.

\paragraph{The $CP$ asymmetry.} The heavy $N_1$ decays to $\ell H$ and to
$\bar\ell \bar H$. If the Yukawa matrix carries complex phases, the
interference between the tree amplitude and the one-loop vertex and
self-energy corrections makes the two rates differ:
\begin{equation}
\epsilon_1 \;=\;
\frac{\Gamma(N_1 \to \ell H) - \Gamma(N_1 \to \bar\ell \bar H)}
     {\Gamma(N_1 \to \ell H) + \Gamma(N_1 \to \bar\ell \bar H)} .
\label{eq:M-epsilon}
\end{equation}
For hierarchical heavy masses this quantity is bounded above --- the
Davidson--Ibarra bound \citep{DavidsonIbarra2002},
$\epsilon_1 \le (3/16\pi)\, M_1 m_3/v^2$ with $v =
174$~GeV. At $M_1 = 10^{10}$~GeV and $m_3 = 0.05$~eV that ceiling is
$\epsilon_1 \le 9.9\times10^{-7}$, and it is the reason hierarchical
leptogenesis needs $M_1 \gtrsim 10^{9}$~GeV: below that, the asymmetry
simply cannot be large enough.

\paragraph{Washout.} The same Yukawa couplings that let $N_1$ decay let
the inverse decay and the $\Delta L = 2$ scatterings destroy what it
made. The competition is measured by the decay parameter $K =
\Gamma_1/H(T{=}M_1)$, which can be written entirely in terms of masses as
$K = \widetilde m/m_\star$, where $\widetilde m$ is the effective light
mass coupling to $N_1$ and
\begin{equation}
m_\star \;=\; \frac{16\pi^{5/2}\sqrt{g_\star}}{3\sqrt5}\,
 \frac{(v/\sqrt2)^2}{M_{\rm Pl}} \;=\; 1.07 \times 10^{-3}~\mathrm{eV} .
\label{eq:M-mstar}
\end{equation}
This is a striking little quantity: it contains no parameter of the
seesaw at all --- only $v$, $M_{\rm Pl}$, and the relativistic
degree-of-freedom count --- and yet it decides whether washout is strong
or weak. Since the measured neutrino mass splittings put $\widetilde m$
in the range $10^{-3}$ to $10^{-1}$~eV, nature sits at $K$ of order a few
to a few tens: the moderate-to-strong washout regime. The efficiency
factor $\kappa(K)$, obtained by integrating the Boltzmann equations,
falls off roughly as $1/K$ there:
$\kappa(1) = 0.62$, $\kappa(2.8) = 0.18$, $\kappa(4.7) = 0.088$,
$\kappa(9.4) = 0.035$, $\kappa(47) = 0.005$.

Strong washout has a virtue that deserves naming: it erases all memory of
initial conditions. Whatever abundance of $N_1$ the universe started
with, and whatever lepton asymmetry preceded it, the final answer depends
only on the decay parameters. A mechanism whose prediction is independent
of what came before is worth more than one that is merely successful.

\paragraph{The abundance, the entropy, and the conversions.} The
asymmetry is generated per decaying $N_1$, so it must be normalized to
how many $N_1$ there were. In relativistic equilibrium a Majorana fermion
with $g = 2$ has $n_N = \tfrac32 \zeta(3) T^3/\pi^2$, and the entropy
density is $s = (2\pi^2/45)\,g_\star T^3$ with the full Standard-Model
count
\begin{equation}
g_\star \;=\; \underbrace{28}_{\text{bosons}}
 + \tfrac78 \underbrace{90}_{\text{fermions}} \;=\; 106.75
\label{eq:M-gstar}
\end{equation}
(bosons: $8\times2$ gluons, $3\times2$ weak, $1\times2$ hypercharge,
$4$ real Higgs components; fermions: $6\times3\times2\times2 = 72$ quark
states, $3\times2\times2 = 12$ charged leptons, $3\times1\times2 = 6$
neutrinos). Hence
\begin{equation}
Y_N^{\rm eq} \;\equiv\; \frac{n_N}{s}
 \;=\; \frac{(3/2)\zeta(3)/\pi^2}{(2\pi^2/45)\, g_\star}
 \;=\; 3.90\times10^{-3}
 \;=\; \frac{0.416}{g_\star} .
\label{eq:M-YNeq}
\end{equation}
Finally, the comparison with Eq.~\eqref{eq:M-etaB} is per photon, not per
unit entropy, so one conversion remains. Today $g_{\star s} = 2 +
\tfrac78 \times 6 \times \tfrac{4}{11} = 3.909$, and
\begin{equation}
\frac{s}{n_\gamma}
 \;=\; \frac{(2\pi^2/45)\, g_{\star s}}{2\zeta(3)/\pi^2}
 \;=\; 7.04 .
\label{eq:M-sng}
\end{equation}

\paragraph{The chain, end to end.} Assemble:
\begin{equation}
\eta_B \;=\; \frac{s}{n_\gamma} \times \frac{28}{79}
 \times \epsilon_1 \times \kappa \times Y_N^{\rm eq} .
\label{eq:M-chain}
\end{equation}
Every factor has now been derived or defined. Evaluate it on a
representative hierarchical case: $M_1 = 10^{10}$~GeV, $\epsilon_1 =
7\times10^{-7}$ (below the Davidson--Ibarra ceiling), and $\widetilde m =
5$~meV, giving $K = 4.7$ and $\kappa = 0.088$.

\begin{table}[H]
\centering
\caption[Hierarchical leptogenesis, evaluated end to end.]{Standard hierarchical leptogenesis evaluated end to end, with
every factor named. The chain lands $2\%$ from the measured value. What
that does \emph{not} establish is discussed in \S\ref{ssec:M-step2}: two
of the three inputs were chosen, and choosing them differently moves the
answer over many orders of magnitude.}
\label{tab:M-chain}\small
\begin{tabular}{@{}P{0.14\textwidth}P{0.50\textwidth}P{0.21\textwidth}@{}}
\toprule
\textbf{factor} & \textbf{origin} & \textbf{value} \\
\midrule
$\epsilon_1$ & $CP$ asymmetry in $N_1$ decay,
 Eq.~\eqref{eq:M-epsilon} & $7.0\times10^{-7}$ \\
$\kappa$ & washout efficiency at $K = 4.7$ &
 $8.8\times10^{-2}$ \\
$Y_N^{\rm eq}$ & equilibrium abundance,
 Eq.~\eqref{eq:M-YNeq} & $3.90\times10^{-3}$ \\
$Y_{B-L}$ & product of the three above &
 $2.40\times10^{-10}$ \\
$28/79$ & sphaleron reprocessing,
 Eq.~\eqref{eq:M-2879} & $0.3544$ \\
$Y_B$ & baryon asymmetry per unit entropy &
 $8.52\times10^{-11}$ \\
$s/n_\gamma$ & entropy-to-photon conversion,
 Eq.~\eqref{eq:M-sng} & $7.04$ \\
\midrule
$\eta_B$ & the chain, Eq.~\eqref{eq:M-chain} &
 $6.00\times10^{-10}$ \\
measured & Eq.~\eqref{eq:M-etaB} &
 $6.12\times10^{-10}$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent The result, $6.00\times10^{-10}$ against a measured
$6.12\times10^{-10}$, is a two percent agreement. It should be read with
the caveat printed in the caption, and \S\ref{ssec:M-step2} makes that
caveat precise. But the exercise has done its job: leptogenesis is not a
sketch. It is a calculation whose every factor can be written down,
evaluated, and audited, and which delivers a number of the right size
using a heavy mass and a $CP$ asymmetry that are individually reasonable.

\subsection{Step 2 --- what leptogenesis assumes, and what it leaves
open}
\label{ssec:M-step2}

\paragraph{It is a framework, not a prediction.} Count the parameters. A
type-I seesaw with three right-handed neutrinos has, after all
rephasings, $18$ physical parameters. Low-energy neutrino data can
determine at most $9$ of them: three light masses, three mixing angles,
one Dirac phase and two Majorana phases. The remaining $9$ --- the three
heavy masses and the three complex angles of the Casas--Ibarra $R$-matrix
--- are unconstrained by any measurement now or in prospect. And it is
exactly those $9$ that control $\epsilon_1$.

The consequence should be stated plainly rather than apologetically.
Table~\ref{tab:M-chain} chose $M_1$ and $\epsilon_1$; the choice was
reasonable, and a different reasonable choice gives an $\eta_B$ different
by many orders of magnitude. The observed value is therefore a
\emph{constraint surface} in a nine-dimensional space, not an output. In
that sense leptogenesis is like a good coordinate system: it tells you
what has to be true, it removes the possibility of getting the answer for
free, and it does not by itself tell you why the answer is what it is.

\paragraph{The Majorana prerequisite, and its arbiter.} The whole
mechanism rests on lepton-number violation by two units, which requires
neutrinos to be Majorana particles. If they are Dirac, the $\Delta L = 2$
term in Eq.~\eqref{eq:M-seesawmatrix} is absent, standard type-I
leptogenesis is not merely disfavored but unavailable, and one must
start over.

There is exactly one practical arbiter: neutrinoless double beta decay.
The process $(A,Z) \to (A,Z+2) + 2e^-$ violates $L$ by two units and can
proceed only if the neutrino is its own antiparticle. Its rate is
controlled by the effective mass $m_{\beta\beta} = |\sum_i U_{ei}^2
m_i|$, in which the Majorana phases appear and can cause cancellations.
Current limits sit in the tens of meV; the next generation aims an order
of magnitude lower. An \emph{observation} would establish the Majorana
nature and with it the seesaw's central assumption. A null result at the
sensitivity of the coming decade would not refute leptogenesis --- the
phases can cancel --- but it would remove its only foothold in a
laboratory.

\paragraph{And the scale is unreachable.} A heavy mass of $10^{9}$ to
$10^{15}$~GeV is not merely beyond present accelerators; it is beyond any
accelerator that could be built. The mechanism's central actor is
permanently invisible. That is the honest description of the field's
default scenario, and it is why proposals that place the heavy states at
accessible energies are worth taking seriously even when the rest of
their apparatus is unfamiliar.

\subsection{Step 3 --- Chapter~\ref{sec:leptogenesis}'s reading}
\label{ssec:M-step3}

The body assembles the same chain on inherited quantities and reads three
things off a discrete structure. Each carries a different grade, and the
grades matter more here than anywhere else in this appendix.

\paragraph{A lattice of phases, and where the letters land.} The flavor
sector of Chapter~\ref{sec:discrete} values first-order $CP$ phases on
the discrete lattice $(\pi/5)\mathbb{Z}$ --- ten letters on one circle,
of which the two on the real axis, $\{0, \pi\}$, are achiral and the
other eight chiral in mirror pairs. Two lattice values enter the birth of
matter. The leptonic Dirac phase is read as an anchor plus a deviation,
\begin{equation}
\delta_{\rm PMNS} \;=\; \pi + 2\arcsin\!\left(\tfrac17\right)
 \;=\; 180^\circ + 16.43^\circ \;=\; 196.4^\circ ,
\label{eq:M-dpmns}
\end{equation}
with $7$ the ledger integer $J$; and the Majorana parity
$\alpha_{31} = \pi$, with $\alpha_{21} = 0$.

The surprise the chapter reports is that the driver is $\alpha_{31}$, an
\emph{achiral} letter that the naive criterion would call
$CP$-conserving. The resolution is worth carrying because it is a genuine
piece of physics: a Majorana phase enters amplitudes at half angle, as
$e^{\ii\alpha/2}$, so $\alpha_{31} = \pi$ acts as $e^{\ii\pi/2} = \ii$
--- the maximally imaginary insertion a discrete letter can supply. What
the Lagrangian's bookkeeping records as a parity, an interference
experiences as a quarter turn. And at a resonance, where the asymmetry
saturates and the Yukawa magnitude cancels, the phases are no longer
asked ``how much'' but only ``which way'' --- and a parity is precisely
an answer to ``which way''.

\paragraph{The magnitude, and the exponent that is postulated.} The
chapter's throughput is the product of asymmetry and efficiency, and it
is claimed to reconstruct the ledger's throat exponential:
\begin{equation}
\epsilon\,\kappa \;\simeq\;
 (2.5\times10^{-6})\times(1.0\times10^{-2})
 \;=\; 2.5\times10^{-8} \;=\; e^{-K/2} ,
\qquad K = 35 .
\label{eq:M-throughput}
\end{equation}
\textbf{The exponent $K/2$ is postulated, not derived.} It is
geometrically suggestive --- it is the square root of the lock-scale
suppression $e^{-K}$ that fixes $M_{\rm lock}$ elsewhere in the same
framework --- but no derivation of the factor of one half exists, and the
parent treatment carries it as an open work order rather than a result.
Feeding it through the standard reprocessing of \S\ref{ssec:M-chain},
\begin{equation}
\eta_B \;=\; 7.04 \times \frac{28}{79}
 \times \frac{1.25}{g_\star} \times e^{-K/2}
 \;=\; 7.34\times10^{-10}
\qquad \text{against} \qquad 6.12\times10^{-10} ,
\label{eq:M-etaframework}
\end{equation}
an excess of $+19.9\%$ with no fitted parameter. \emph{Again, plainly: the
$+19.9\%$ is the output of a chain one of whose inputs, the exponent
$K/2$, is a postulate.} A twenty-percent agreement obtained by postulating
one exponent is a different object from a twenty-percent agreement obtained
by deriving it, and the difference should not be allowed to blur.

\paragraph{Two joints that are open, and one $O(1)$ that is absorbed.}
Honesty about this chain requires naming three things.

First, the chain runs in the \emph{resonant} regime --- the heavy states
sit at the lock with percent-level splittings, and the asymmetry is
enhanced by the ratio of splitting to width. The on-resonance condition
is the framework's own named input, carried in public as a conjecture
with an identified but unclosed candidate mechanism
\citep{ArisakaA10Cosmos}.

Second, the exponent. It is postulated. This appendix has now said so
twice, which is the appropriate number of times.

Third, an $O(1)$ that a careful reader will find and that should not be
found by surprise. Equation~\eqref{eq:M-etaframework} uses $1/g_\star$
where Eq.~\eqref{eq:M-YNeq} gives the relativistic equilibrium abundance
as $0.416/g_\star$. The compressed formula therefore carries the species count $3$
relative to the per-species normalization of the chain --- inside the
declared factor-of-$\sim$3 band of uncomputed $O(1)$'s, but a real
difference, and the kind that a percent-level quotation can hide. The
correct summary is that the framework reproduces $\eta_B$ to within the
band it declares, and that the displayed $+19.9\%$ should be read as a
central value inside that band rather than as a precision.

\paragraph{What the reading does claim.} Stripped of the numerology, the
structural claim is a division of labor: the pentagon's odd block
integer makes $CP$ violation \emph{exist} (a chiral texture, with no dial
that could have declined it); a Majorana parity fixes its \emph{sign};
and the ledger integer sets the \emph{size}. Three jobs, three
structures, and --- if the postulated exponent is granted --- no
adjustable parameter among them. Contrast this structurally, and only
structurally, with \S\ref{ssec:M-step2}: there the observed $\eta_B$
selects a surface in a nine-dimensional space; here it is claimed to be
determined. The two accounts are not in competition over any measured
quantity; they differ in how many things must be chosen, and that
difference is exactly what \S\ref{ssec:M-step2}'s parameter count and
this section's grades are for.

\paragraph{A rhyme at another scale, and not a claim.} Life on Earth uses
L-amino acids and D-sugars almost exclusively, and the mechanisms by
which a small initial enantiomeric excess can be amplified by
autocatalysis into complete dominance are now well studied
\citep{Blackmond2010}. The structural rhyme with this chapter is real: a
tiny bias, plus an amplifying and irreversible step, plus the absence of
a route back, yields total selection. It is worth noticing because it
shows the pattern is not exotic. It is \emph{not} a claim that the
framework of this review explains biological homochirality, and nothing
in the cosmological chain bears on the biochemistry --- the energy
scales, the mechanisms, and the amplifying dynamics are unrelated. Two
one-way selections at scales twenty orders apart are two one-way
selections, and the resemblance is a resemblance.

\subsection{Problems and worked solutions}
\label{ssec:M-problems}

\medskip\noindent\textbf{M.1. From $\Omega_b h^2 = 0.02237$, compute
$\eta_B$; convert it into the fractional quark--antiquark imbalance at
$T > 1$~GeV; and state what the universe would contain if that imbalance
had been zero.}

\noindent\emph{Solution.} \emph{Step 1.} By
Eq.~\eqref{eq:M-etaomega}, $\eta_B = 2.74\times10^{-8} \times 0.02237 =
6.13\times10^{-10}$, agreeing with the direct determination
\eqref{eq:M-etaB} to within its error. \emph{Step 2.} At $T$ above a
GeV, $n_q \approx n_{\bar q} \approx n_\gamma$, so the fractional
imbalance is $\eta_B/2 = 3.1\times10^{-10}$: about three extra quarks per
ten billion pairs. \emph{Step 3.} With zero imbalance, nucleons and
antinucleons annihilate until the survivors are too dilute to find
partners; the freeze-out leaves $n_B/n_\gamma \sim 10^{-19}$, nine orders
below the observed value and far below what any structure formation
requires. \emph{Physics intuition:} the interesting fact about
Eq.~\eqref{eq:M-etaB} is not that it is small but that it is not zero,
and not zero by an amount that is itself extremely specific. Ten orders
of magnitude of annihilation is the most efficient destruction process in
the history of the universe, and the entire material world is what
survived it.

\medskip\noindent\textbf{M.2. Prove that the equilibrium expectation
value of baryon number vanishes, stating precisely which properties of
$CPT$ and of $\hat B$ the proof uses. Then explain why the argument does
not equally forbid a net electric charge.}

\noindent\emph{Solution.} \emph{Step 1.} Let $\Theta$ implement $CPT$.
The proof uses two facts: $\Theta \hat H \Theta^{-1} = \hat H$, which is
the $CPT$ theorem for a Lorentz-invariant local hermitian Hamiltonian
(Appendix~\ref{app:discrete}); and $\Theta \hat B \Theta^{-1} = -\hat B$,
because $B$ is odd under $C$ and even under $P$ and $T$.
\emph{Step 2.} In equilibrium $\rho \propto e^{-\hat H/T}$ depends only
on $\hat H$, so inserting $\Theta^{-1}\Theta$ inside the trace and using
the invariance of the trace under a similarity transformation gives
Eq.~\eqref{eq:M-cpt}: $\langle \hat B\rangle = -\langle\hat B\rangle$,
hence zero. \emph{Step 3.} The argument does not forbid a net electric
charge because charge is exactly conserved. If $[\hat Q, \hat H] = 0$,
the equilibrium ensemble is not $e^{-\hat H/T}$ but the grand canonical
$e^{-(\hat H - \mu_Q \hat Q)/T}$, which is \emph{not} $CPT$ invariant for
$\mu_Q \neq 0$, and the derivation breaks at its first line. That is the
whole content: a conserved charge may be carried as an initial condition
through a chemical potential, and an unconserved one may not.
\emph{Physics intuition:} this is why the third Sakharov condition is
inseparable from the first. If $B$ were conserved, equilibrium would
happily hold a baryon asymmetry forever --- and would explain nothing.
The moment you make $B$ violable in order to explain it, you make
equilibrium fatal to it.

\medskip\noindent\textbf{M.3. Derive $g_\star = 106.75$ for the Standard
Model above the electroweak scale, and derive $s/n_\gamma = 7.04$ today.
Why do the two calculations use different degree-of-freedom counts?}

\noindent\emph{Solution.} \emph{Step 1.} Bosons: gluons $8 \times 2 =
16$; $\mathrm{SU}(2)$ gauge bosons $3\times2 = 6$; hypercharge $1\times2
= 2$; the complex Higgs doublet, four real scalars. Total $28$. Fermions:
quarks $6\;\text{flavors} \times 3\;\text{colors} \times
2\;\text{spins} \times 2\;(\text{particle/antiparticle}) = 72$; charged
leptons $3\times2\times2 = 12$; neutrinos $3\times1\times2 = 6$. Total
$90$. Then $g_\star = 28 + \tfrac78(90) = 28 + 78.75 = 106.75$.
\emph{Step 2.} Today only photons and the decoupled neutrinos
contribute to entropy, and the neutrinos are colder by $(4/11)^{1/3}$
after $e^+e^-$ annihilation, so $g_{\star s} = 2 + \tfrac78 \times 6
\times \tfrac{4}{11} = 3.909$. With $s = (2\pi^2/45)g_{\star s}T^3$ and
$n_\gamma = (2\zeta(3)/\pi^2)T^3$, the ratio is $(0.43865 \times
3.909)/(0.24359) = 7.04$. \emph{Step 3.} The counts differ because
$g_\star$ counts what is relativistic \emph{then}, at $T \sim
10^{10}$~GeV, when everything in the Standard Model is massless compared
with the temperature; $g_{\star s}$ counts what is relativistic
\emph{now}. \emph{Physics intuition:} the ratio $n_B/s$ is the quantity
that is actually conserved through the expansion, because entropy per
comoving volume is conserved when nothing dumps heat. So a mechanism
predicts $Y_B = n_B/s$, and the factor $7.04$ is what translates a
theorist's conserved quantity into an observer's $\eta_B$. Getting it
backwards is a factor-of-seven error, which is more than the whole
disagreement anyone in this subject argues about.

\medskip\noindent\textbf{M.4. For $m_\nu = 0.05$~eV, tabulate the Dirac
mass and Yukawa coupling required by the seesaw at $M = 10^{6}$,
$10^{10}$, $10^{15}$~GeV, and compare each Yukawa with a charged-fermion
one. What does the comparison suggest, and what does it not establish?}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:M-seesaw}, $m_D = \sqrt{m_\nu M}$, and $Y_\nu =
m_D/(v/\sqrt2)$ with $v/\sqrt2 = 174$~GeV. At $M = 10^{6}$~GeV: $m_D =
7.1$~MeV, $Y_\nu = 4.1\times10^{-5}$. At $10^{10}$~GeV: $m_D =
0.71$~GeV, $Y_\nu = 4.1\times10^{-3}$. At $10^{15}$~GeV: $m_D = 224$~GeV,
$Y_\nu = 1.28$. \emph{Step 2.} Compare: the electron's Yukawa is
$2.9\times10^{-6}$, the charm's $3.6\times10^{-3}$, the top's $0.98$. So
the three cases correspond to a neutrino Yukawa between the electron's
and the muon's, one essentially equal to the charm's, and one slightly
above the top's. \emph{Step 3.} The comparison suggests that the GUT-scale
seesaw is the one that makes $Y_\nu$ ``natural'' in the sense of being
$O(1)$, which is the traditional argument for $M \sim 10^{15}$~GeV. It
establishes nothing, because the charged-fermion Yukawas themselves span
six orders of magnitude with no explanation, so ``natural'' here means
only ``resembling the largest of a set of numbers we cannot account
for''. \emph{Physics intuition:} keep track of which arguments are
explanations and which are aesthetic preferences among unexplained
numbers. The seesaw's real achievement is that it makes $m_\nu$ small
without a small coupling; the choice of $M$ is then a taste, and tastes
should be labeled.

\medskip\noindent\textbf{M.5. Evaluate the full chain
\eqref{eq:M-chain} for $\epsilon_1 = 7\times10^{-7}$ and
$\widetilde m = 5$~meV, showing each intermediate quantity, and compare
with the measurement. Then find what $\epsilon_1$ would be needed if
$\widetilde m$ were $50$~meV instead.}

\noindent\emph{Solution.} \emph{Step 1.} $K = \widetilde m/m_\star =
5\times10^{-3}/1.07\times10^{-3} = 4.7$, giving $\kappa = 0.088$. Then
$Y_{B-L} = \epsilon_1 \kappa Y_N^{\rm eq} = (7\times10^{-7})(0.088)
(3.90\times10^{-3}) = 2.40\times10^{-10}$.
\emph{Step 2.} $Y_B = (28/79) Y_{B-L} = 8.52\times10^{-11}$, and
$\eta_B = 7.04\, Y_B = 6.00\times10^{-10}$, which is $2\%$ below the
measured $6.12\times10^{-10}$.
\emph{Step 3.} At $\widetilde m = 50$~meV, $K = 47$ and $\kappa$ falls to
$5.2\times10^{-3}$, a factor $17$ smaller. To hold $\eta_B$ fixed,
$\epsilon_1$ must rise by the same factor, to $1.2\times10^{-5}$ --- which
exceeds the Davidson--Ibarra ceiling at $M_1 = 10^{10}$~GeV by more than
an order of magnitude, so this case would require either a heavier $M_1$
or a resonant enhancement. \emph{Physics intuition:} this is where the
whole subject's structure shows itself. Heavier neutrinos mean stronger
washout, stronger washout means a larger asymmetry is needed, and a larger
asymmetry means either a higher scale or a degeneracy. The measured
neutrino masses sit close enough to $m_\star$ that the trade is live
rather than academic, which is why $m_\star$ --- a quantity built from
$v$, $M_{\rm Pl}$ and $g_\star$ alone --- turns out to be one of the more
consequential numbers in cosmology.

\medskip\noindent\textbf{M.6. (a) Evaluate the compressed chain
$\eta_B = 7.04 \times (28/79) \times (1.25/g_\star) \times e^{-K/2}$ for
$K = 35$ and state its deviation. (b) Identify the $O(1)$ that the earlier,
silent form $1/g_\star$ had absorbed, by comparison with
Eq.~\eqref{eq:M-chain}. (c) Show that a
Majorana phase $\alpha = \pi$ acts as $\ii$ in an interference, and
explain how a letter the naive criterion calls $CP$-conserving can drive
an asymmetry.}

\noindent\emph{Solution.} \emph{Step 1.} (a) $e^{-17.5} =
2.511\times10^{-8}$. Then $7.04 \times 0.35443 = 2.4952$; times
$2.511\times10^{-8}$ gives $6.266\times10^{-8}$; times the triplet yield
factor $1.25$ gives $7.833\times10^{-8}$; divided by $106.75$
gives $\eta_B = 7.34\times10^{-10}$. Against $6.12\times10^{-10}$ the
deviation is $+19.9\%$.
\emph{Step 2.} (b) Equation~\eqref{eq:M-chain} carries $Y_N^{\rm eq} =
0.416/g_\star$ per species, and the displayed form carries $1.25/g_\star$
summed. The silent form $1/g_\star$ used in earlier printings therefore
absorbed a factor $1.25/0.416 = 3$ --- the species count itself. It sits
inside the declared factor-of-$\sim$3 band, but it is real, and a reader
who takes a percentage at face value without it will over-read the
result. Note also that the whole chain rests on the postulated exponent
$K/2$; the quality of the number is bounded by the quality of that
postulate and not by the arithmetic.
\emph{Step 3.} (c) A Majorana mass term $\tfrac12 m e^{\ii\alpha}
\bar\nu^c \nu$ can be brought to real positive form by $\nu \to
e^{-\ii\alpha/2}\nu$, so the phase enters every amplitude at half angle.
For $\alpha = \pi$ this is $e^{\ii\pi/2} = \ii$. An interference term
$2\,\mathrm{Re}(A_1 A_2^*)$ with a relative factor $\ii$ becomes
$2\,\mathrm{Im}(A_1 \tilde A_2^*)$ --- maximally shifted, not zero.
\emph{Physics intuition:} the criterion ``a phase is $CP$-violating only
if it is not a multiple of $\pi$'' is a statement about \emph{Dirac}
phases, where the phase enters at full angle. Majorana phases enter at
half angle, so the achiral letters of the full-angle lattice land on the
imaginary axis at half angle. The lesson is general and worth more than
this application: before asking whether a phase is physical, ask at what
angle it enters the amplitude.

\GGtakeaway{\emph{Standard result:} $\eta_B = 6.12\times10^{-10}$,
measured twice over --- acoustic peaks and light elements --- and
representing a three-in-ten-billion quark excess before annihilation.
Sakharov's three conditions are each necessary, the third provably so
from $CPT$ alone, Eq.~\eqref{eq:M-cpt}. The Standard Model fails
quantitatively twice: no first-order transition at a $125$~GeV Higgs, and
a quark-phase throughput ten orders too small,
Eq.~\eqref{eq:M-gavela}. Electroweak sphalerons violate $B{+}L$ in units
of three while conserving $B{-}L$ exactly, and equilibrium chemistry fixes
the reprocessing at $B = (28/79)(B{-}L)$. The seesaw gives $m_\nu =
m_D^2/M$ at the price of Majorana neutrinos, and leptogenesis assembled
from $\epsilon_1$, $\kappa$, $Y_N^{\rm eq}$, $28/79$ and $s/n_\gamma =
7.04$ lands on $6.0\times10^{-10}$ for reasonable inputs.
\emph{What it assumed:} nine unmeasured seesaw parameters, of which two
were chosen to make Table~\ref{tab:M-chain}; and the Majorana nature of
the neutrino, whose only arbiter is neutrinoless double beta decay.
\emph{What it leaves open:} why $\eta_B$ has the value it has rather than
another --- the observed number is a constraint surface, not an output ---
and any prospect of examining the heavy sector directly.
\emph{Continue in:} Chapter~\ref{sec:leptogenesis} for the reading in
which the existence, the sign and the size of the excess come from three
different discrete structures, with the exponent $K/2$ carried as a
postulate and the on-resonance condition as a named open input;
Appendix~\ref{app:discrete} for the mirrors themselves;
Appendix~\ref{app:anomalies} for the anomaly behind
Eq.~\eqref{eq:M-anomaly}; and Chapter~\ref{sec:falsifiers} for what
would take the whole account down.}

\section{Chronology, entropy, and the arrow of time}
\label{app:time}
\GGteachbox{Boltzmann's $H$-theorem derived in full, with the one step
that inserts the arrow marked as it is taken; Loschmidt's and Zermelo's
objections stated properly and conceded --- both are correct against the
original derivation; the Poincare recurrence time computed for a
macroscopic gas; coarse-graining, and why it relocates the difficulty
instead of removing it; the past hypothesis, with Penrose's estimate of
its improbability; Landauer's principle derived twice and evaluated at
room temperature; the Hamiltonian constraint of canonical quantum
gravity, which leaves the fundamental equation with no time in it; and
the radiation-era time--temperature law derived from the Friedmann
equation. The pay-off is Chapter~\ref{sec:what-is-time}.}

\GGreadalong{\citet{MisnerThorneWheeler1973}, Ch.~21, for the
Arnowitt--Deser--Misner split and the constraint that becomes the
problem of time; \citet{Weinberg1996}, Ch.~7, for the constrained
Hamiltonian machinery in field-theory language;
\citet{Planck2018} for the measured decoupling redshift, which this
appendix is careful never to confuse with a topological one; and
\citet{ArisakaA10Cosmos} for the descent whose radial coordinate the
last section maps onto clock time.}

\subsection{Step 1 --- the $H$-theorem, derived, and the step that inserts the arrow}
\label{ssec:N-step1}

The question in its rawest form: every microscopic law we have ---
Newton's, Maxwell's, Schrodinger's --- is invariant under $t \to -t$
(with the accompanying reversals Chapter~\ref{sec:discrete} handles),
yet nothing in daily life is. Cream mixes into coffee and does not
unmix. Either the arrow is in the microscopic laws and we have missed
it, or it is somewhere else. Boltzmann's 1872 paper was the first
serious attempt to put it in the laws, and the most instructive thing
about it is exactly where it fails.

\paragraph{The kinetic equation.} Let $f(\mathbf{x},\mathbf{v},t)$ be
the one-particle distribution: $f\,\dd^3x\,\dd^3v$ is the number of
molecules in a phase-space cell. Between collisions the molecules
stream, so $f$ is carried along the flow; collisions redistribute
velocities. Writing the streaming as a total derivative,
\begin{equation}
\left(
\frac{\partial}{\partial t}
+ \mathbf{v}\cdot\nabla_{\mathbf{x}}
+ \frac{\mathbf{F}}{m}\cdot\nabla_{\mathbf{v}}
\right) f
\;=\; C[f] ,
\label{eq:N-boltzmann}
\end{equation}
where $C[f]$ counts the gain and loss of molecules at velocity
$\mathbf{v}$ from binary collisions. To write $C$ one must say how often
a molecule of velocity $\mathbf{v}$ meets one of velocity
$\mathbf{v}_1$. The number of such encounters per unit time and volume
is proportional to the flux $|\mathbf{v} - \mathbf{v}_1|$, to the
differential cross-section $\sigma(\Omega)$, and to the
\emph{two}-particle distribution $f_2(\mathbf{v},\mathbf{v}_1)$
evaluated just before the encounter.

\paragraph{The assumption, named as it is made.} Boltzmann now sets
\begin{equation}
f_2(\mathbf{v},\mathbf{v}_1;\,t)
\;=\; f(\mathbf{v},t)\, f(\mathbf{v}_1,t)
\qquad\text{for pairs \emph{about to} collide.}
\label{eq:N-stossz}
\end{equation}
This is the \emph{Stosszahlansatz}, the ``collision-number
hypothesis'': the velocities of two molecules that are about to meet
are statistically independent. Mark this equation. It is the only place
in the entire derivation where anything time-asymmetric enters, and
everything that follows is honest bookkeeping. The asymmetry is in the
words \emph{about to}. Nothing is assumed about the pair \emph{after}
they collide --- and indeed after a collision the two velocities are
strongly correlated, since each was produced from the other.

With Eq.~\eqref{eq:N-stossz} the collision term becomes
\begin{equation}
C[f] \;=\; \int \dd^3 v_1 \int \dd\Omega \;
 |\mathbf{v} - \mathbf{v}_1|\, \sigma(\Omega)
 \left[ f' f_1' - f f_1 \right] ,
\label{eq:N-collision}
\end{equation}
where $f = f(\mathbf{v})$, $f_1 = f(\mathbf{v}_1)$, and primes denote
the post-collision velocities $\mathbf{v}',\mathbf{v}_1'$ determined by
$\mathbf{v},\mathbf{v}_1,\Omega$ through energy and momentum
conservation. The gain term $f'f_1'$ uses microscopic reversibility ---
the cross-section for $(\mathbf{v}',\mathbf{v}_1') \to
(\mathbf{v},\mathbf{v}_1)$ equals that for the forward process --- and
the Jacobian of $(\mathbf{v},\mathbf{v}_1) \mapsto
(\mathbf{v}',\mathbf{v}_1')$ is unity.

\paragraph{The functional.} Define, for a spatially uniform gas with no
external force,
\begin{equation}
H(t) \;=\; \int \dd^3 v \; f(\mathbf{v},t) \, \ln f(\mathbf{v},t) ,
\label{eq:N-H}
\end{equation}
which up to a sign and a constant is the entropy per unit volume,
$s = -k_B H$.

\paragraph{Theorem ($H$-theorem).} With Eqs.~\eqref{eq:N-boltzmann} and
\eqref{eq:N-stossz},
\begin{equation}
\frac{\dd H}{\dd t} \;\le\; 0 ,
\end{equation}
with equality if and only if $f$ is Maxwellian.

\paragraph{Proof.} Differentiate Eq.~\eqref{eq:N-H} and use
Eq.~\eqref{eq:N-boltzmann}:
\begin{equation}
\frac{\dd H}{\dd t}
 = \int \dd^3 v \,(1 + \ln f)\,\frac{\partial f}{\partial t}
 = \int \dd^3 v \,(1 + \ln f)\, C[f] ,
\label{eq:N-dHdt-raw}
\end{equation}
the $1$ dropping out because $\int C[f]\,\dd^3v = 0$ (collisions
conserve particle number). Now symmetrize. The integrand of
Eq.~\eqref{eq:N-dHdt-raw} with Eq.~\eqref{eq:N-collision} inserted is
invariant under three relabellings: exchange
$\mathbf{v}\leftrightarrow\mathbf{v}_1$; exchange
$(\mathbf{v},\mathbf{v}_1)\leftrightarrow(\mathbf{v}',\mathbf{v}_1')$,
which flips the sign of the bracket; and the composition of the two.
Averaging the four copies,
\begin{equation}
\frac{\dd H}{\dd t}
 = -\frac{1}{4}\int\!\dd^3v \int\!\dd^3 v_1 \int\!\dd\Omega \;
 |\mathbf{v}-\mathbf{v}_1|\,\sigma(\Omega)\,
 \left( f'f_1' - f f_1\right)
 \ln\!\frac{f' f_1'}{f f_1} .
\label{eq:N-dHdt}
\end{equation}
Every factor in front of the last two is non-negative. And for any
positive $x,y$ the product $(x - y)(\ln x - \ln y) \ge 0$, because the
logarithm is increasing: the two factors always carry the same sign.
Hence the integral is non-negative and $\dd H/\dd t \le 0$. \qed

Equality requires $f' f_1' = f f_1$ for all collisions, that is,
$\ln f$ must be conserved in every collision, so $\ln f$ is a linear
combination of the collision invariants $1, \mathbf{v}, v^2$:
\begin{equation}
\ln f = a + \mathbf{b}\cdot\mathbf{v} + c\,v^2
\quad\Longrightarrow\quad
f \propto \exp\!\left[-\frac{m(\mathbf{v} - \mathbf{u})^2}{2 k_B T}\right] ,
\label{eq:N-maxwell}
\end{equation}
the Maxwell--Boltzmann distribution, obtained here as the unique
stationary point rather than assumed.

\emph{Physics intuition.} This is a beautiful theorem and it does
genuine work: it explains why a gas relaxes to a Maxwellian, it gives
the relaxation rate, and it is the foundation of transport theory. What
it does not do --- despite a century of textbooks implying otherwise ---
is derive the arrow of time from mechanics. The arrow went in at
Eq.~\eqref{eq:N-stossz}, and the next two sections are the proof that it
did.

\subsection{Step 1 (continued) --- Loschmidt and Zermelo, stated properly, and conceded}
\label{ssec:N-objections}

Two objections were raised within Boltzmann's lifetime. Both are
correct. It is worth being blunt about that, because the usual telling
treats them as puzzles that were subsequently cleared up, and they were
not.

\paragraph{Loschmidt's reversibility objection (1876).} Let
$\Gamma = (\mathbf{x}_i, \mathbf{p}_i)$ be a microstate of the gas and
let $R\Gamma = (\mathbf{x}_i, -\mathbf{p}_i)$ be its velocity reversal.
Hamiltonian dynamics with a time-even potential satisfies
\begin{equation}
\phi_{-t} \;=\; R \circ \phi_t \circ R ,
\qquad R^2 = \mathrm{id} ,
\label{eq:N-loschmidt}
\end{equation}
where $\phi_t$ is the flow. So for every solution there is a
time-reversed solution, obtained by an operation that is available at
every instant and costs nothing in principle. Take a trajectory along
which $H$ falls from $H_0$ at time $0$ to $H_1 < H_0$ at time $\tau$.
Apply $R$ at time $\tau$. The resulting trajectory is a legitimate
solution of the microscopic equations along which $H$ \emph{rises} from
$H_1$ back to $H_0$, since $H$ depends on velocities only through
$f(\mathbf{v})$ and the reversal is a relabeling of an isotropic
functional.

The conclusion is inescapable: a statement holding for every solution of
the microscopic equations cannot be time-asymmetric, because the
solution set is symmetric. So $\dd H/\dd t \le 0$ is not a consequence
of the mechanics; something else was used, and it was
Eq.~\eqref{eq:N-stossz} --- the reversed trajectory has, by
construction, incoming pairs whose velocities are correlated.

\paragraph{Zermelo's recurrence objection (1896).} Poincare's theorem
of 1890 says that a measure-preserving flow on a phase-space region of
finite measure returns almost every point arbitrarily close to where it
started. The gas in a box qualifies: Liouville's theorem gives the
measure preservation, and fixed energy and fixed volume give the
finiteness. Hence $H(t)$, a continuous function on phase space, must
return arbitrarily close to $H(0)$ for almost every initial condition.
A quantity that decreases monotonically forever is therefore impossible.

\paragraph{Proof of the recurrence theorem.} It takes four lines and is
worth having. Let $\phi$ be a measure-preserving map of a space $X$ with
$\mu(X) < \infty$, and let $A \subseteq X$ have $\mu(A) > 0$. Let
$A_0 \subseteq A$ be the set of points that never return to $A$:
$\phi^n(a) \notin A$ for all $n \ge 1$. Then the sets $A_0,
\phi(A_0), \phi^2(A_0),\dots$ are pairwise disjoint --- if
$\phi^m(A_0) \cap \phi^n(A_0) \ne \emptyset$ with $m < n$, then
applying $\phi^{-m}$ gives a point of $A_0$ returning to $A_0
\subseteq A$ after $n - m \ge 1$ steps, contradiction. All have measure
$\mu(A_0)$. If $\mu(A_0) > 0$ their union has infinite measure, which
$X$ does not have. So $\mu(A_0) = 0$: almost every point of $A$
returns. \qed

\paragraph{The status of the two objections.} Both are valid. Neither
was ever refuted; both were \emph{absorbed}, by the admission that the
$H$-theorem is a statement about typical behavior of an ensemble with a
particular kind of initial correlation structure, not a theorem of
mechanics. Boltzmann himself came to say something close to this. The
honest summary is: \emph{the $H$-theorem is a theorem about the
Boltzmann equation, and the Boltzmann equation is not the mechanics.}

\subsection{Step 1 (continued) --- how long is a recurrence, really}
\label{ssec:N-recurrence}

Zermelo's objection is correct in principle and irrelevant in practice,
and it is worth seeing exactly how irrelevant, because the size of the
number is itself an argument.

Estimate the return time as the inverse of the fraction of accessible
phase-space volume occupied by the target state, times a microscopic
timescale $\tau$ at which the system samples a fresh configuration:
\begin{equation}
t_{\text{rec}} \;\sim\; \tau \, e^{\Delta S / k_B} ,
\label{eq:N-trec}
\end{equation}
with $\Delta S$ the entropy difference between the equilibrium
macrostate and the target. Take the cleanest target: all $N$ molecules
of a gas found in the left half of their box. The probability of a given
molecule being on the left is $1/2$, so the volume fraction is $2^{-N}$
and $\Delta S = N k_B \ln 2$.

Take $\tau \simeq 10^{-9}$~s (a generous mean free time) and one cubic
centimeter of air at room temperature and pressure, $N = 2.687 \times
10^{19}$. Then
\begin{equation}
\log_{10} t_{\text{rec}}[\mathrm{s}]
 \;=\; N \log_{10} 2 - 9
 \;=\; 8.09 \times 10^{18} ,
\label{eq:N-trec-num}
\end{equation}
so $t_{\text{rec}} \sim 10^{8\times 10^{18}}$ seconds. The subtraction
of $9$ is invisible. Using the full Sackur--Tetrode entropy of the same
sample, $S/k_B \simeq 18.6\,N = 5.0\times 10^{20}$, gives a full
phase-space recurrence at $10^{2.2\times 10^{20}}$~s instead --- a
different number, the same verdict.

For scale, the age of the universe is $4.35\times 10^{17}$~s, that is
$10^{17.6}$~s. Even a system of only $N = 100$ particles has
$2^{100} \times 10^{-9}\,\mathrm{s} = 1.27\times 10^{21}$~s, about
$2900$ times the age of the universe. Recurrence is real and it is not
an experimental concern.

\emph{Physics intuition.} Notice what this did and did not do. It made
Zermelo harmless; it did not make him wrong. The $H$-theorem as stated
--- $\dd H/\dd t \le 0$, always --- remains false. What survives is
$\dd H/\dd t \le 0$ for times short against $10^{10^{19}}$ seconds: a
much weaker statement, and one still silent about which of the two
temporal directions the decrease runs in.

\subsection{Step 1 (continued) --- coarse-graining, and what it moves rather than solves}
\label{ssec:N-coarse}

The modern repair begins from Gibbs. Define the fine-grained entropy of
a distribution $\rho(\Gamma)$ on the full phase space,
\begin{equation}
S_{\text{fg}} = -k_B \int \dd\Gamma \; \rho \ln \rho .
\label{eq:N-gibbs}
\end{equation}
Liouville's theorem says $\rho$ is carried unchanged along the flow, so
$S_{\text{fg}}$ is \emph{exactly constant} --- not approximately, not
usually, exactly. The fine-grained entropy of an isolated system never
changes. This is not a failure of the definition; it is a statement that
information is not destroyed by Hamiltonian evolution.

Now partition phase space into cells $C_\alpha$ of some finite size and
define the coarse-grained distribution $\bar\rho$ as constant on each
cell, equal to the cell average. The coarse-grained entropy
\begin{equation}
S_{\text{cg}} = -k_B \sum_\alpha
 P_\alpha \ln \frac{P_\alpha}{\Omega_\alpha} ,
\qquad P_\alpha = \int_{C_\alpha} \rho \, \dd\Gamma ,
\label{eq:N-coarse}
\end{equation}
satisfies $S_{\text{cg}} \ge S_{\text{fg}}$ and grows as the exact
$\rho$ develops filaments the cells cannot resolve --- ink stirred into
water until every macroscopic region looks gray, while microscopically
the ink and the water remain entirely unmixed.

This is correct and it is genuinely illuminating. It is also not a
solution, for two reasons that must be said plainly.

\emph{First, the partition is a choice.} Nothing in the dynamics
supplies the cells. Different partitions give different $S_{\text{cg}}$
and different growth rates, and one can always construct a partition
adapted to the state at a chosen instant, for which the entropy is
momentarily minimal and grows in \emph{both} temporal directions.

\emph{Second --- and this is the fatal one --- the argument is
time-symmetric.} Filamentation is generic in both directions of time.
Given a state of intermediate entropy and no further information, the
same reasoning that predicts higher entropy in the future
\emph{retrodicts} higher entropy in the past. Since we do not believe
the past had higher entropy, the coarse-graining argument has to be
supplemented by an assumption about the past. That is not a repair of
the $H$-theorem's gap; it is the same gap, relocated from
Eq.~\eqref{eq:N-stossz} to the initial condition.

\subsection{Step 1 (continued) --- the past hypothesis, and what it costs}
\label{ssec:N-past}

The standard cosmological repair, then, is to impose what is now called
the \emph{past hypothesis}: the universe began in a state of very low
entropy. Given that, coarse-graining does the rest --- entropy increases
toward the future because it started low and there is far more room
above than below.

This works. It is also, as an explanation, a boundary condition rather
than a law, and it inherits the question of why that boundary condition
holds. Penrose's contribution was to quantify how strong an assumption
it is, and the estimate is worth reproducing because it is short.

The maximum entropy available to the matter in the observable universe
is achieved by putting it all into a single black hole, since the
Bekenstein--Hawking entropy
\begin{equation}
\frac{S_{\text{BH}}}{k_B} \;=\; \frac{4\pi G M^2}{\hbar c}
\label{eq:N-bh}
\end{equation}
grows as $M^2$ and therefore beats any extensive entropy at large mass.
With $M \sim 1.5\times 10^{53}$~kg for the observable universe,
Eq.~\eqref{eq:N-bh} gives $S_{\text{BH}}/k_B \simeq 6\times 10^{122}$,
that is $\sim 10^{123}$. The observed entropy is far smaller ---
dominated by the cosmic microwave background at $\sim 10^{88} k_B$ ---
and the initial entropy smaller still.

The corresponding phase-space volume ratio is $e^{-S_{\text{max}}/k_B}$,
which is where Penrose's figure of one part in $10^{10^{123}}$ comes
from. One caution about that number, in the interest of the honesty this
appendix is trying to model: converting $e^{10^{123}}$ to a power of ten
gives $10^{4.3\times 10^{122}}$, so the middle exponent is $122.6$
rather than $123$. At this scale the distinction is cosmetic --- but it
is the kind of thing worth checking rather than repeating, and the
computation takes one line.

\emph{Physics intuition.} A number like $10^{10^{123}}$ measures
nothing; it says the initial condition was extraordinarily special by
the only measure we know how to write down. Whether ``special by the
Liouville measure'' is the right notion of improbability for a boundary
condition of the whole universe is exactly what the past hypothesis
leaves open, since that measure was built for subsystems of a universe,
not for universes.

\subsection{Step 1 (continued) --- Landauer's principle, derived twice}
\label{ssec:N-landauer}

The arrow shows up in information processing as a hard energy cost, and
the derivation is short enough to do twice from different starting
points. This matters for the body's reading, because it is the one place
where the standard account already locates irreversibility in a map that
has no inverse rather than in a probability.

\paragraph{Derivation one: phase-space compression.} A one-bit memory is
a physical system whose accessible phase space is divided into two
regions, $L$ and $R$, of equal volume $V$. ``Erase'' means: whatever the
state, end in $L$. The map therefore takes an accessible volume $2V$ to
an accessible volume $V$. But Liouville's theorem forbids a Hamiltonian
map from compressing phase-space volume. The volume must go somewhere:
it goes into the environment. The memory's entropy drops by
\begin{equation}
\Delta S_{\text{mem}} = -k_B \ln 2 ,
\end{equation}
so the environment's must rise by at least $k_B \ln 2$, and if the
environment is a bath at temperature $T$ the heat delivered to it obeys
\begin{equation}
Q \;\ge\; T \,\Delta S_{\text{env}} \;=\; k_B T \ln 2 .
\label{eq:N-landauer}
\end{equation}

\paragraph{Derivation two: the Szilard cycle.} Take Szilard's 1929
engine: a single molecule in a box of volume $V$ at temperature $T$.
Insert a partition; observe which side the molecule is on; attach the
partition to a weight through a pulley; let the molecule push the
partition isothermally to the far wall. The work extracted is
\begin{equation}
W = \int_{V/2}^{V} p \,\dd V
 = \int_{V/2}^{V} \frac{k_B T}{V'} \,\dd V'
 = k_B T \ln 2 .
\label{eq:N-szilard}
\end{equation}
Remove the partition and the engine is back where it started ---
apparently a cyclic device converting heat from a single bath into work,
which the second law forbids. The resolution, due to
\citet{Bennett1982} following \citet{Landauer1961}, is that the engine
is \emph{not} back where
it started: the memory still records which side the molecule was on.
Completing the cycle requires erasing that record, and by
Eq.~\eqref{eq:N-landauer} erasure costs at least $k_B T \ln 2$, exactly
canceling Eq.~\eqref{eq:N-szilard}. Measurement can be free;
forgetting cannot.

\paragraph{The number.} At $T = 300$~K,
\begin{equation}
k_B T \ln 2
 = (1.380649\times 10^{-23}\,\mathrm{J\,K^{-1}})(300\,\mathrm{K})(0.693147)
 = 2.87\times 10^{-21}\ \mathrm{J} ,
\label{eq:N-landauer-num}
\end{equation}
which is $0.0179$~eV, about $0.7\,k_B T$, and a little under half the
translational thermal energy $\tfrac32 k_B T$ of a single air molecule. Erasing one gigabyte
($8.59\times 10^9$ bits) costs $2.47\times 10^{-11}$~J --- enough to
lift a grain of sand nowhere at all.

\emph{Physics intuition.} Landauer's bound is not about the energy of
representing a bit; it is about the energy of \emph{destroying} the
distinction between two states. That is why the bound attaches to
erasure and not to writing, and why it is the cleanest thermodynamic
statement in physics about a many-to-one map.

\subsection{Step 1 (continued) --- the problem of time in canonical quantum gravity}
\label{ssec:N-wdw}

There is one more standard result that a chapter on time must have on
the table, and it points the opposite way from everything above:
in the deepest formulation we have of gravity, there is no time
variable at all.

Split spacetime into spatial slices, following Arnowitt, Deser and
Misner. Write the metric as
\begin{equation}
\dd s^2 = -\left(N^2 - N_i N^i\right)\dd t^2
 + 2 N_i \,\dd x^i \dd t
 + h_{ij}\,\dd x^i \dd x^j ,
\label{eq:N-adm}
\end{equation}
with $h_{ij}$ the three-metric, $N$ the lapse and $N^i$ the shift. The
Einstein--Hilbert action in these variables has no time derivatives of
$N$ or $N^i$: they are Lagrange multipliers. Varying with respect to
them therefore produces constraints rather than evolution equations, and
the total Hamiltonian is a sum of those constraints,
\begin{equation}
H = \int \dd^3 x \left( N \mathcal{H}_\perp + N^i \mathcal{H}_i \right),
\qquad
\mathcal{H}_\perp \approx 0 , \quad \mathcal{H}_i \approx 0 .
\label{eq:N-constraints}
\end{equation}
This is Noether's second theorem in Hamiltonian dress: general
covariance is a symmetry with four arbitrary functions, so by
Appendix~\ref{app:noether} it delivers identities and constraints and no
charge. The consequence is that the ``Hamiltonian'' of general
relativity generates a gauge transformation --- a relabeling of the
slicing --- rather than a physical motion.

Quantize. The constraint becomes an operator equation on the wave
functional of three-geometries,
\begin{equation}
\hat{\mathcal{H}}_\perp \, \Psi[h_{ij}] \;=\; 0 ,
\label{eq:N-wdw}
\end{equation}
the Wheeler--DeWitt equation. Compare it with the Schrodinger equation
$\ii\,\partial_t \Psi = \hat H \Psi$. The right-hand side is zero and
there is no $\partial_t$. The fundamental equation of canonical quantum
gravity contains no time parameter.

The standard response is that time must be \emph{relational}: pick some
internal degree of freedom --- a scalar field value, a volume, the phase
of a matter mode --- and ask how everything else correlates with it.
That works semiclassically and is beset in general by the obvious
difficulty of which variable to pick. The point needed here is narrower
and safe: \emph{time is not a primitive of the deepest formalism we
have.} Whatever it is, it is recovered.

\subsection{Step 1 (concluded) --- the time--temperature relation, from Friedmann}
\label{ssec:N-frw}

Finally, the piece of standard cosmology that Chapter~\ref{sec:what-is-time}
quotes. It is elementary and it is exact within its regime.

The Friedmann equation for a spatially flat universe is
\begin{equation}
H^2 \;=\; \left(\frac{\dot a}{a}\right)^2
 \;=\; \frac{8\pi G}{3}\,\rho .
\label{eq:N-friedmann}
\end{equation}
In the radiation era the energy density is that of a relativistic gas,
\begin{equation}
\rho = \frac{\pi^2}{30}\, g_*(T)\, T^4 ,
\label{eq:N-rho-rad}
\end{equation}
with $g_*$ the effective number of relativistic degrees of freedom.
Substituting Eq.~\eqref{eq:N-rho-rad} into Eq.~\eqref{eq:N-friedmann}
and using $G = M_{\rm Pl}^{-2}$,
\begin{equation}
H \;=\; \sqrt{\frac{8\pi^3}{90}}\,\sqrt{g_*}\,\frac{T^2}{M_{\rm Pl}}
 \;=\; 1.660\,\sqrt{g_*}\,\frac{T^2}{M_{\rm Pl}} .
\label{eq:N-H-rad}
\end{equation}
With $\rho \propto a^{-4}$ Eq.~\eqref{eq:N-friedmann} integrates to
$a \propto t^{1/2}$, hence $H = 1/(2t)$, hence
\begin{equation}
t \;=\; \frac{1}{2H}
 \;=\; \frac{M_{\rm Pl}}{3.32\,\sqrt{g_*}\;T^2}
 \;=\; \frac{2.42}{\sqrt{g_*}}
 \left(\frac{T}{\mathrm{MeV}}\right)^{-2} \ \mathrm{s} ,
\label{eq:N-tT}
\end{equation}
using $M_{\rm Pl} = 1.221\times10^{19}$~GeV and
$\hbar = 6.582\times10^{-25}$~GeV\,s. At $T = 1$~MeV with
$g_* = 10.75$ this gives $t = 0.738$~s, the standard
big-bang-nucleosynthesis clock reading.

The structural content of Eq.~\eqref{eq:N-tT} is one line:
\begin{equation}
t \propto T^{-2}
\qquad\Longleftrightarrow\qquad
\frac{\dd \ln t}{\dd \ln T} = -2
\qquad (\text{radiation}) .
\label{eq:N-slope-T}
\end{equation}
In the matter era $a \propto t^{2/3}$ and the same steps give
$t \propto T^{-3/2}$, so the exponent changes from $2$ to $3/2$ at
matter--radiation equality.

\paragraph{Rewriting in the descent's coordinate.} Introduce the radial
coordinate used throughout the descent,
\begin{equation}
r \;=\; \ln\!\frac{E_{\rm GUT}}{\mu} ,
\label{eq:N-rdef}
\end{equation}
where $\mu$ is the renormalization scale; in the thermal universe the
relevant scale is the temperature, $\mu = T$. Then
$\dd r = -\dd\ln T$, and Eq.~\eqref{eq:N-slope-T} becomes
\begin{equation}
\boxed{\ \frac{\dd \ln t}{\dd r} \;=\; 2\ }
\qquad (\text{radiation}) ,
\qquad
\frac{\dd \ln t}{\dd r} = \frac{3}{2}
\qquad (\text{matter}) .
\label{eq:N-slope}
\end{equation}
Each unit of $r$ multiplies the clock by $e^2 = 7.389$. One caution
about the word ``exactly'': the exponent $2$ is exact wherever $g_*$ is
constant, since $g_*$ enters Eq.~\eqref{eq:N-tT} only through the
prefactor. At a threshold --- the QCD transition, electron--positron
annihilation --- $g_*$ drops and the intercept steps, without bending
the line. So the slope law is exact and the ledger built from it carries
the small, computable threshold offsets.

\subsection{Step 2 --- what the standard account settles, and what it leaves open}
\label{ssec:N-step2}

The standard account is not confused, and it should not be caricatured.
It settles a great deal:

\emph{It settles the mechanism of relaxation.} The Boltzmann equation
predicts transport coefficients, relaxation rates, and the approach to
Eq.~\eqref{eq:N-maxwell}, and it is quantitatively right.

\emph{It settles the bookkeeping of information.} Equations
\eqref{eq:N-gibbs} and \eqref{eq:N-landauer} together say something
sharp and true: nothing is lost by Hamiltonian evolution, and any
apparent loss is a transfer, with a thermodynamic price attached.

\emph{It settles the cosmological chronology.} Equation
\eqref{eq:N-tT} is not in dispute.

What it does not settle is the direction itself, and the failure is
structural rather than technical. Every route to a monotone quantity
inserts an asymmetry by hand at some identifiable point, and
Table~\ref{tab:N-arrows} lists them side by side.

\begin{table}[H]
\centering
\caption[Where the arrow of time is inserted.]{Four standard routes to
an arrow, and the point at which each inserts one. The first three
insert an assumption about states; only the last inserts a property of
a \emph{map}, and \S\ref{ssec:N-step3} takes that as the clue. Three columns: the route to an arrow, the point at which that route inserts an asymmetry, and the status of the input it inserts. The third column is what makes the comparison fair --- every route on the list inserts something, and the question the table asks is what kind of thing each one inserts, not whether it inserts anything.}
\label{tab:N-arrows}\small
\begin{tabular}{@{}P{0.24\textwidth}P{0.40\textwidth}P{0.26\textwidth}@{}}
\toprule
\textbf{route} & \textbf{where the asymmetry enters} &
\textbf{status of the input} \\
\midrule
$H$-theorem &
Stosszahlansatz, Eq.~\eqref{eq:N-stossz}: independence assumed for
incoming pairs only &
an assumption about correlations, false along reversed trajectories \\
coarse-graining &
choice of partition in Eq.~\eqref{eq:N-coarse}, plus a condition on the
past to break the retrodiction symmetry &
a modeling choice plus a boundary condition \\
past hypothesis &
low entropy imposed at the initial slice &
a boundary condition, not a law \\
Landauer erasure &
the erasure map is many-to-one, Eq.~\eqref{eq:N-landauer} &
a property of the map itself \\
\bottomrule
\end{tabular}
\end{table}

None of these are errors. A boundary condition is a respectable
ingredient of a physical explanation --- the solar system's planarity is
explained by initial conditions too --- and the objection is not that
the past hypothesis is illegitimate but that it is often presented as
though the arrow had been \emph{derived}, when what was shown is that
the arrow follows from a low-entropy past plus dynamics. And the
difficulty is not confined to statistical mechanics: by
Eq.~\eqref{eq:N-wdw}, at the level where gravity is quantized the
parameter the arrow is supposed to point along has itself gone
missing.

The open question, then, is not ``why does entropy increase'' --- given a
low-entropy past, that is answered. It is: \emph{is there any structure
in physics from which one-wayness follows without an assumption about
initial states?} Table~\ref{tab:N-arrows}'s last row is the hint, and
Appendix~\ref{app:opensys} has already supplied the machinery.

\subsection{Step 3 --- the tick and the arrow, carried on different fibers}
\label{ssec:N-step3}

Chapter~\ref{sec:what-is-time}'s reading begins from an observation
about vocabulary rather than about physics. The word ``time'' is doing
two jobs, and they are not the same job.

\paragraph{The tick is a return; the arrow is a loss.} A clock does two
things. It \emph{repeats} --- the pendulum comes back, the caesium phase
advances by $2\pi$ --- and it \emph{accumulates} --- the escapement
ratchets, the count goes up and never down. Those are different physical
facts about different parts of the apparatus, and that they travel
together in one device is engineering, not time.

The repetition is a \emph{holonomy}. Appendix~\ref{app:connections}
defines it: transport a state around a closed loop and compare it with
where it started; the comparison is a group element attached to the
loop, and it is generically not the identity. A period is precisely that
--- a closed-loop return with a nontrivial phase, which is why $E =
\hbar\omega$ reads energy as a winding rate, and why the geometric phase
of \citet{Berry1984} is measurable at all. Now note what a holonomy on a
circle cannot do: its windings come in both signs, $\pm n$, and the
inverse loop is as available as the loop. A circle repeats and cannot
point. Conversely, a fiber with no closed loops has nothing to be
periodic about: an interval points and cannot repeat.

\paragraph{The arrow is the projection, and it has no inverse.}
Appendix~\ref{app:opensys} established --- without any statistics,
without a thermodynamic limit, without a large number of anything ---
that a completely positive trace-preserving map admits a channel
inverse if and only if it is unitary. A projection that loses anything
at all is therefore not merely unlikely to reverse. It has no inverse in
the algebra of physical maps. Chapters~\ref{sec:ggep} and
\ref{sec:oggep} read observation as exactly such a projection, and
\citet{ArisakaA8QM} owns the argument.

Run Loschmidt and Zermelo against that structure and watch both fail,
for the same reason and in different words.

\emph{Loschmidt} needs Eq.~\eqref{eq:N-loschmidt}: an involution $R$
such that $\phi_{-t} = R \phi_t R$. That construction requires
$\phi_{-t}$ to exist. Against a semigroup $\Lambda_t$ with $t \ge 0$
there is no $\Lambda_{-t}$ to construct, so the objection has no
premise. This is a stronger statement than ``the reversed trajectory is
improbable''. It is that the reversed map is not an element of the set.

\emph{Zermelo} needs three things for the recurrence theorem proved in
\S\ref{ssec:N-objections}: an invertible map, a preserved measure, and a
finite total measure. A non-unitary channel supplies none of them. The
proof used $\phi^{-m}$ explicitly, at the one step where disjointness
was established, and there is nothing to substitute for it.

So the two objections that are correct against Boltzmann are not
objections at all against a projection --- and the difference is not one
of degree. This is the payoff of doing Appendix~\ref{app:opensys}
before this one.

\paragraph{What this does not claim.} Honesty about the boundary of the
claim is more useful here than enthusiasm. The semigroup reading
supplies \emph{directedness}: it says why there is no inverse, without
recourse to typicality. It does not by itself supply the \emph{gradient}
--- the observed rate of entropy production, or the smallness of the
early-universe entropy. Those still ride on the boundary data of the
descent, which is where Eq.~\eqref{eq:N-ledger} below gets its numbers
and which \citet{ArisakaA10Cosmos} owns. What has changed is the
location of the residue: it is no longer ``why is one-wayness possible''
but ``what fixed the initial datum'', and the two questions have
different characters.

\paragraph{The chronology, and the slope.} Equation~\eqref{eq:N-slope}
was derived above from nothing but Friedmann and $\mu = T$: in the
radiation era $\dd \ln t/\dd r = 2$ exactly, kinking to $3/2$ at
matter--radiation equality. Chapter~\ref{sec:what-is-time} reads this as
the statement that boundary time is the exponential shadow of a linear
descent --- the read head advances at unit rate in $r$, and the clock
multiplies by $e^2 = 7.389$ per unit. The endpoint of that integration
is an audited sum,
\begin{equation}
\ln\frac{t_0}{t_{\rm Pl}} \;=\; 140.24
 \;=\; \underbrace{36.11}_{\text{launch}}
 \;+\; \underbrace{91.60}_{\text{radiation}}
 \;+\; \underbrace{12.53}_{\text{matter}+\Lambda} ,
\label{eq:N-ledger}
\end{equation}
whose three segments sum to $140.24$ arithmetically and whose total
agrees with the direct evaluation from measured quantities:
$t_0 = 13.787\,\mathrm{Gyr} = 4.351\times10^{17}$~s and
$t_{\rm Pl} = 5.391\times10^{-44}$~s give
$\ln(t_0/t_{\rm Pl}) = 140.243$. That is a real check and it passes; it
is done as problem~N.6.

\paragraph{One coincidence, flagged.} With the ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ --- where the ledger integer is $N = 3$,
not the letter of this appendix --- one has $4K = 140$, and
\begin{equation}
\frac{140.24}{4K} \;=\; \frac{140.24}{140} \;=\; 1.0017 .
\label{eq:N-4K}
\end{equation}
This is \textbf{FLAGGED}: arithmetic without a mechanism. No derivation
connects $4K$ to the integrated chronology, and the observation is
recorded at that grade and no higher. It is worth adding one number that
makes the flag concrete rather than ritual. Agreement to $0.17\%$ here
is agreement in a \emph{logarithm}: the underlying ratio $t_0/t_{\rm Pl}$
differs from $e^{4K}$ by a factor $e^{0.243} = 1.275$. To make
Eq.~\eqref{eq:N-4K} exact the age of the universe would have to be
$10.8$~Gyr rather than $13.8$~Gyr. A near-miss in a log of order $140$
is a much weaker constraint than the two-parts-in-a-thousand phrasing
suggests, and saying so is the point of flagging it.

\paragraph{Two quantities, one symbol --- the discipline stated.} The
same chapter carries a case where the discipline of keeping quantities
apart is doing visible work, and it belongs here because it is the habit
this whole ladder argues for.

\emph{The corridor switch} sits at $\ln(1 + z_{\rm sw}) = J = 7$
exactly, so $1 + z_{\rm sw} = e^7 = 1096.633$. It is topological: an
integer of the ledger, exact by construction, with no measurement in it.

\emph{The atomic decoupling redshift} is $z_{\rm dec} \simeq 1090$
\citep{Planck2018}. It is an observable, derived through the atomic
recombination chain and measured from the microwave background.

These are two different quantities that happen to land near each other,
and they must never be quoted as one number. Their separation is
\begin{equation}
\ln(1096.6) - \ln(1090) = 0.00604
\quad\text{e-folds}
\;=\; 0.086\% \ \text{of } J = 7 ,
\label{eq:N-zsep}
\end{equation}
or $0.61\%$ in $z$ itself. Two things follow, and both matter.
First, the closeness is a striking fact, not an identity, and it is
reported as such. Second --- this is the part that gets garbled if the
two are merged --- the relevant falsifier (Chapter~\ref{sec:falsifiers})
tests $z_{\rm dec}$ against the \emph{atomic chain}, at the $0.5\%$
level, and is silent about $z_{\rm sw}$. The $0.61\%$ gap in
Eq.~\eqref{eq:N-zsep} is therefore not a $0.61\%$ discrepancy in
anything: it is the distance between a topological integer and a
measured atomic quantity, and a distance between different quantities
cannot be an error in either.

\emph{Physics intuition.} The habit is the one running through
Chapter~\ref{sec:noether} and Chapter~\ref{sec:bianchi}: keep the
identity and the law in different columns. An exact ledger integer and a
measured redshift differ in kind as $\dd F = 0$ differs from
$\dd \star F = \star J$, and the temptation to collapse them --- because
they agree to half a percent, because it makes a better sentence --- is
the one temptation a zero-parameter claim
(Chapter~\ref{sec:zero-parameters}) cannot afford even once.

\subsection{Problems and worked solutions}
\label{ssec:N-problems}

\medskip\noindent\textbf{N.1. Estimate the Poincare recurrence time for
one cubic centimeter of air at room temperature and pressure, taking as
the target state ``all molecules in the left half of the box''. Compare
with the age of the universe, and then repeat for a system of $100$
particles.}

\noindent\emph{Solution.} \emph{Step 1.} At standard conditions the
number density is $n = p/k_BT = 2.687\times 10^{19}\ \mathrm{cm^{-3}}$,
so $N = 2.687\times10^{19}$. The target's phase-space fraction is
$2^{-N}$, giving $\Delta S/k_B = N \ln 2$. \emph{Step 2.}
Equation~\eqref{eq:N-trec} with $\tau = 10^{-9}$~s gives
$\log_{10} t_{\text{rec}}[\mathrm{s}] = N\log_{10}2 - 9 =
8.09\times10^{18} - 9 \simeq 8.09\times 10^{18}$, so
$t_{\text{rec}} \sim 10^{8\times10^{18}}$~s against an age of the
universe of $10^{17.6}$~s. The exponent of the answer exceeds the
answer's competitor by $18$ orders of magnitude \emph{in the exponent};
no choice of $\tau$ within thirty orders of magnitude changes the
verdict. \emph{Step 3.} For $N = 100$, $2^{100}\times10^{-9}\,
\mathrm{s} = 1.27\times10^{21}$~s $= 2900$ ages of the universe --- so
even a hundred particles are already beyond reach.
\emph{Physics intuition:} the insensitivity to $\tau$ is the lesson.
When the answer is a double exponential the prefactor is not a detail
one must get right; it is a detail one must notice cannot matter --- and
that is why Zermelo's objection, though logically unanswerable, never
had experimental teeth.

\medskip\noindent\textbf{N.2. Evaluate the Landauer bound at $300$~K.
Then estimate the Landauer power of a processor running at $4$~GHz with
$10^{10}$ transistors of which one in ten switches per cycle, and
compare with a thermal design power of $125$~W.}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:N-landauer-num},
$k_BT\ln 2 = 1.380649\times10^{-23}\times300\times0.693147
= 2.871\times10^{-21}$~J, or $0.0179$~eV. \emph{Step 2.} The bit-erasure
rate is $4\times10^{9}\ \mathrm{s^{-1}} \times 10^{10} \times 0.1 =
4\times10^{18}$ erasures per second, so the Landauer floor is
$4\times10^{18}\times2.871\times10^{-21} = 1.15\times10^{-2}$~W, about
$11$~mW. \emph{Step 3.} The ratio to $125$~W is $1.1\times10^{4}$: real
silicon dissipates roughly ten thousand times the thermodynamic minimum.
\emph{Physics intuition:} the gap is the good news and the bad news at
once. Four orders of magnitude of headroom remain before thermodynamics
stops the shrinking --- but the floor is real, it does not scale away,
and it is set by exactly the same many-to-one structure that
Appendix~\ref{app:opensys} shows has no inverse. The heat from your
laptop is, in the last analysis, the price of forgetting.

\medskip\noindent\textbf{N.3. Derive $t \propto T^{-2}$ in the radiation
era from the Friedmann equation, evaluate $t$ at $T = 1$~MeV with
$g_* = 10.75$, and verify that $\dd\ln t/\dd r = 2$ with
$r = \ln(E_{\rm GUT}/T)$. What is the slope in the matter era?}

\noindent\emph{Solution.} \emph{Step 1.} With $\rho \propto a^{-4}$,
Eq.~\eqref{eq:N-friedmann} gives $\dot a\, a = \text{const}$, hence
$a \propto t^{1/2}$ and $H = 1/(2t)$. Since $T \propto a^{-1}$ at fixed
$g_{*s}$, $T \propto t^{-1/2}$, so $t \propto T^{-2}$. \emph{Step 2.}
Quantitatively, Eq.~\eqref{eq:N-tT} gives $t = 2.4205\,g_*^{-1/2}
(T/\mathrm{MeV})^{-2}$~s; at $T = 1$~MeV and $g_* = 10.75$,
$t = 2.4205/\sqrt{10.75} = 0.738$~s. \emph{Step 3.} From
$t \propto T^{-2}$, $\dd\ln t/\dd\ln T = -2$; and $r = \ln E_{\rm GUT} -
\ln T$ gives $\dd r = -\dd \ln T$, so $\dd\ln t/\dd r = +2$, which is
Eq.~\eqref{eq:N-slope}. In the matter era $a \propto t^{2/3}$ gives
$T \propto t^{-2/3}$, hence $t \propto T^{-3/2}$ and slope $3/2$.
\emph{Physics intuition:} the slope is a pure consequence of the
equation of state --- $2$ for $w = 1/3$, $3/2$ for $w = 0$ --- and
carries no free constant. Everything model-dependent, including all of
$g_*$, sits in the intercept. That is why a plot of $\ln t$ against $r$
is a straight line with one kink and small steps at thresholds, and why
the kink locates equality without any fitting.

\medskip\noindent\textbf{N.4. The corridor switch sits at
$\ln(1+z_{\rm sw}) = J = 7$ exactly; the atomic decoupling redshift is
measured at $z_{\rm dec} \simeq 1090$. Compute the separation in
e-folds, express it as a percentage of $J$, and say why this is not a
test of anything.}

\noindent\emph{Solution.} \emph{Step 1.} $e^7 = 1096.633$, so the
switch sits at $1096.6$ in $(1+z)$; the measured decoupling is $1090$.
\emph{Step 2.} The separation is
$\ln(1096.6) - \ln(1090) = 6.99997 - 6.99393 = 0.006037$ e-folds, which
is $0.006037/7 = 0.0862\%$ of $J$, and $0.61\%$ measured in $z$ itself.
\emph{Step 3.} It is not a test because the two numbers are not the same
quantity. One is a ledger integer, exact by construction and carrying no
error bar; the other is an observable produced by the atomic
recombination chain and measured against the microwave background. The
falsifier of Chapter~\ref{sec:falsifiers} compares the measured
$z_{\rm dec}$ with the \emph{atomic chain} at the $0.5\%$ level ---
observable against observable --- and says nothing about $z_{\rm sw}$.
\emph{Physics intuition:} a difference between two quantities of
different kinds cannot be an error in either, and naming which is which
every time is what stops a striking coincidence being promoted, by
repetition alone, into a claimed agreement.

\medskip\noindent\textbf{N.5. Show explicitly that the $H$-theorem's
conclusion is not invariant under Loschmidt's reversal, and identify
which equation of \S\ref{ssec:N-step1} fails along the reversed
solution.}

\noindent\emph{Solution.} \emph{Step 1.} Let $f(\mathbf{v},t)$ solve the
mechanics with $H(t)$ decreasing on $[0,\tau]$. Apply $R$ at $t = \tau$:
by Eq.~\eqref{eq:N-loschmidt} the reversed configuration evolves through
the mirror sequence of microstates, so its one-particle distribution at
time $\tau + s$ equals the original's at $\tau - s$ up to
$\mathbf{v}\to-\mathbf{v}$. \emph{Step 2.} $H$ in
Eq.~\eqref{eq:N-H} is invariant under $\mathbf{v}\to-\mathbf{v}$, being
an integral over all velocities of a function of $f$ alone. Hence
$H(\tau + s) = H(\tau - s)$: along the reversed solution $H$
\emph{increases}, back to $H(0)$. \emph{Step 3.} Since both are exact
solutions of the mechanics, the $H$-theorem cannot follow from the
mechanics. The step that fails is Eq.~\eqref{eq:N-stossz}: in the
reversed solution the incoming pairs are precisely the outgoing pairs of
the original, and those are correlated, so $f_2 \ne f f_1$ for pairs
about to collide.
\emph{Physics intuition:} the theorem is true of the Boltzmann equation
and the Boltzmann equation is not true of the reversed gas. Everything
that looks like a paradox here is the difference between a model and the
thing modeled --- and \S\ref{ssec:N-step3}'s move is to ask what kind
of object could carry the asymmetry \emph{as a property of itself}
rather than as a property of the states fed to it.

\medskip\noindent\textbf{N.6. Compute $\ln(t_0/t_{\rm Pl})$ from
measured values, check it against the ledger
Eq.~\eqref{eq:N-ledger}, and then assess the coincidence
$140.24 = 4K \times 1.0017$ by asking what age of the universe would
make it exact.}

\noindent\emph{Solution.} \emph{Step 1.} Take $t_0 = 13.787$~Gyr
$= 4.3508\times10^{17}$~s \citep{Planck2018} and
$t_{\rm Pl} = 5.3912\times10^{-44}$~s \citep{PDG2024}. The ratio is
$8.070\times10^{60}$, so $\ln(t_0/t_{\rm Pl}) = 140.243$. \emph{Step 2.}
The ledger's segments sum to $36.11 + 91.60 + 12.53 = 140.24$, agreeing
with the direct evaluation to the quoted precision --- an independent
check on the chronology, since the segments were computed by
integrating Eq.~\eqref{eq:N-slope} through the epochs and the direct
evaluation uses only two measured times. \emph{Step 3.} With $K = 35$,
$4K = 140$ and $140.24/140 = 1.0017$. To make that exact one needs
$\ln(t_0/t_{\rm Pl}) = 140$, that is $t_0$ smaller by
$e^{0.243} = 1.275$, namely $t_0 = 10.8$~Gyr --- excluded by
observation many times over. \emph{Physics intuition:} this is why the
$4K$ observation is carried as \textbf{FLAGGED} and not as a result. A
logarithm compresses; agreement to two parts in a thousand \emph{in the
log} is disagreement by twenty-eight percent in the quantity, and a
mechanism that predicted $4K$ exactly would be in tension with the
measured age rather than in support of it. What would change the grade is a derivation of the $O(1)$ factor; until
one exists, the right thing to do with a suggestive integer is to write
it down, label it, and refuse to lean on it.

\GGtakeaway{\emph{Standard result:} the $H$-theorem, derived in full
from Eqs.~\eqref{eq:N-boltzmann}--\eqref{eq:N-dHdt}; the Maxwellian as
its unique stationary point; Landauer's bound $k_BT\ln 2 =
2.87\times10^{-21}$~J at $300$~K, derived from Liouville and again from
the Szilard cycle; and the radiation-era chronology $t \propto T^{-2}$
of Eq.~\eqref{eq:N-tT}. \emph{What it assumed:} the Stosszahlansatz,
Eq.~\eqref{eq:N-stossz} --- and Loschmidt and Zermelo are both
\emph{correct} against the derivation without it, so the arrow was
inserted rather than derived; coarse-graining relocates that insertion
to the choice of partition and to a condition on the past, and the past
hypothesis is a boundary condition whose improbability Penrose estimates
at one part in $10^{10^{123}}$. \emph{The reading this paper takes:} the
tick and the arrow are two things under one word --- the tick is a
closed-loop return, a holonomy (Appendix~\ref{app:connections}), and the
arrow is the non-invertibility of the projection
(Appendix~\ref{app:opensys}) --- Table~\ref{tab:H-dictionary} collects
every such translation in one place, with the grade each carries --- so
that Loschmidt's reversal and Zermelo's recurrence both fail because the
required inverse map does not exist in
the algebra, not because it is improbable; the chronology follows from
$\dd\ln t/\dd r = 2$, Eq.~\eqref{eq:N-slope}, and closes at
Eq.~\eqref{eq:N-ledger}, with $140.24 = 4K \times 1.0017$ carried as
FLAGGED arithmetic and the two-quantity discipline of
Eq.~\eqref{eq:N-zsep} kept explicit. \emph{Continue in:}
Chapter~\ref{sec:what-is-time} for the four endowments and the clock
ladder, Appendix~\ref{app:opensys} for the semigroup this reading rests
on, and Chapter~\ref{sec:falsifiers} for what would break it.}

\section{The \GGTheory{} dictionary and the grade ledger}
\label{app:dictionary}
\GGteachbox{Every acronym used in this paper with a one-line definition
and its home paper; the complete grade ledger, giving for each imported
claim its status, the premises that status depends on, and what would be
lost if a premise failed; and an explicit statement of what this review
imports and never re-derives. This appendix is a reference, not a
narrative, and it is the one to consult when checking a claim made in
Part~III.}

\subsection{Dictionary}
\label{ssec:H-dictionary}

Terms are listed alphabetically. The third column gives the paper that
defines the term and is the authority for it.

\begingroup\small
\begin{longtable}{@{}P{0.16\textwidth}P{0.55\textwidth}P{0.22\textwidth}@{}}
\caption[The dictionary.]{\textbf{The dictionary.} Every program term
used in this paper, with the member paper that defines it and is the
authority for it. Three columns: the term, a one-line definition, and the member paper that defines it and is the authority for it. The third column is the operative one. This review defines nothing of its own, so a term whose home is named here can be checked at its home, and a term with no home would be this review overreaching.}
\label{tab:H-dictionary}\\
\toprule
\textbf{term} & \textbf{one-line definition} & \textbf{home} \\
\midrule
\endfirsthead
\toprule
\textbf{term} & \textbf{one-line definition} & \textbf{home} \\
\midrule
\endhead
\bottomrule
\endfoot
\AXkw{}-1 (\LC{}) & Locality and causality; the first axiom, presupposed
by all the others & \citet{ArisakaA1Constitution} \\
\AXkw{}-2 (\GGEP{}) & Geometry--gauge equivalence: a single composite
connection carries both gravitational and gauge structure &
\citet{ArisakaA1Constitution} \\
\AXkw{}-3 (\OGGEP{}) & The operational bulk-to-boundary dictionary;
observation is a projection & \citet{ArisakaA1Constitution} \\
\AXkw{}-4 (\MSMF{}) & Maximum symmetry, minimum freedom; the minimality
axiom identified in \S\ref{ssec:exposure} as the most exposed premise &
\citet{ArisakaA1Constitution} \\
\BIsix{} & The six-dimensional modified Bianchi identity
$\dd H_3 = X_4$, with the coefficient vector
$(1,-1,-\tfrac25,-\tfrac25)$ & \citet{ArisakaA3BI6} \\
\DGI{} & Dynamical gauge invariance: the top of the relaxation cascade,
the critical point of maximal Morse index & \citet{ArisakaA6Gi4} \\
\GA{} & Geometric attractor: the cascade's endpoint, a Morse minimum;
the cohort's candidate for what a particle is &
\citet{ArisakaA6Gi4} \\
\GDOS{} & Geometric dynamics of open systems: the boundary's completely
positive, trace-preserving forward evolution & \citet{ArisakaA5GDOS} \\
\GFF{} & The geometric free-energy functional on boundary moduli space;
a Morse function, and the landscape the cascade descends &
\citet{ArisakaA6Gi4} \\
\GRIP{} & Gauge-runtime invariance pipeline: the three-stage descent
$\DGI{} \to \GSSB{} \to \GA{}$ of the \GFF{} landscape. Note that
\GFF{} is a sibling pillar, not a stage of \GRIP{} &
\citet{ArisakaA6Gi4} \\
\GSSB{} & Geometric spontaneous symmetry breaking: the intermediate
saddles of the descent, connected by instanton trajectories &
\citet{ArisakaA6Gi4} \\
holonomy & The phase acquired on parallel transport around a closed
loop; the third datum, not determined by curvature where loops fail to
bound & \S\ref{ssec:C-holonomy} \\
$\kappa$-bridge & The theorem that the \HS{} pushforward of the \BIsix{} data equals the boundary's Berry line bundle on the benchmark background & \citet[Appendix~H]{ArisakaP2N3} \\
\OGNfive{} & The ledger $(M,N,L;J,K) = (1,3,5;7,35)$, satisfying
$K = LJ = M^2+N^2+L^2$ and $J = (M^2+N^2+L^2)/L$ & \citet{ArisakaA2GG6} \\
$q(x)$ & The quadratic refinement $x(x-1)/2$, at characteristic element
$a = 1$; appears three times in the program &
\S\ref{ssec:qx}, \S\ref{ssec:G-refinement} \\
\QiFour{} & The second group of pillars, extending the architecture
beyond physics; cited once in this paper and defended nowhere in it &
\citet{ArisakaA7Qi4} \\
toll ladder & The instanton cost $\pi k$ of passing from Morse index $2k$ to $2(k-1)$, summing to $\pi N_g(N_g-1)/2$ &
\citet{ArisakaA6Gi4}, \S\ref{ssec:solvency} \\
Wu class & The class $\nu_j$ defined by
$\mathrm{Sq}^j(x) = \nu_j \smile x$; vanishes for $2j > n$ &
\S\ref{ssec:G-wu} \\
\end{longtable}
\endgroup

\subsection{The grade ledger}
\label{ssec:H-ledger}

Table~\ref{tab:H-ledger} is the full version of Table~\ref{tab:grades}.
The fourth column is the one the body's table omits: what a reader loses
if the premise fails.

\begingroup\small
\begin{longtable}{@{}P{0.20\textwidth}P{0.15\textwidth}P{0.28\textwidth}P{0.28\textwidth}@{}}
\caption[The grade ledger, with dependencies.]{\textbf{The grade
ledger.} The full version of Table~\ref{tab:grades}. The fourth column is
the one the body's table omits: what a reader loses if the premise in the
third column is withdrawn. Four columns: the claim, its grade, what it depends on, and what a reader loses if that dependency is withdrawn. The fourth column is the reason this longer version exists. A grade tells you how much to believe a claim; only the last column tells you what else goes with it if the belief turns out to be misplaced.}
\label{tab:H-ledger}\\
\toprule
\textbf{claim} & \textbf{grade} & \textbf{depends on} &
\textbf{if the premise fails} \\
\midrule
\endfirsthead
\toprule
\textbf{claim} & \textbf{grade} & \textbf{depends on} &
\textbf{if the premise fails} \\
\midrule
\endhead
\bottomrule
\endfoot
\GGEP{} & axiom & \AXkw{}-1 & All of Part~III fails; Part~II is
untouched \\
\OGGEP{} & axiom & \AXkw{}-1, \AXkw{}-2 & The open-system turn fails;
the ladder of Part~II stands \\
\OGNfive{} ledger & theorem & the closure identities, arithmetic only &
Nothing; the arithmetic is checkable in a line \\
\BIsix{} closure and coefficients & theorem &
\GGEP{}, the intended background class & The trace-ratio lock and
everything downstream of it \\
Selection of $k = 1$ & derived-conditional & \AXkw{}-4 (minimality) &
The whole family $(5k,3k)$ returns; $\SUfive{}$ loses its privilege \\
\GDOS{} boundary law & theorem-level & \OGGEP{}; the semigroup form
additionally needs a stated Markov regime & The open-system dynamics;
the exact memory-kernel form survives \\
Arrow of time & \textbf{axiom} & \AXkw{}-3 & --- (it is assumed, not
derived; see \S\ref{ssec:gdos}) \\
\GRIP{} cascade & framework & \GDOS{}, Morse positivity on the relevant
strata & The toll ladder and hence the upper bound on $N_g$ \\
Born rule (dynamical realization) & derived-conditional & four named
premises; one analytic residue open & The Born \emph{form} survives via
Gleason; the dynamical account does not \\
$N_g = 3$ & derived-conditional & an empirical input ($CP$ violation), a coupling clause, one topological residue & The upper bound; the lower bound $N_g \ge 3$ is independent \\
$\kappa$-bridge & theorem on a benchmark & the benchmark background;
boundary quantum mechanics as \emph{input} & Generality off the
benchmark; the benchmark result stands \\
\end{longtable}
\endgroup

\subsection{The verification harness}
\label{ssec:H-harness}

Every integer, coefficient, date and identity typeset in this paper was
checked by an independent computation before it was typeset. The checks
live in a single program, run against the source file at every build; it
both recomputes the physics and scans the typeset text. At the version
shipped it reports \textbf{166 checks passing, none failing}, together
with 25 standing notes recorded for review. The blocks are listed in
Table~\ref{tab:H-blocks}:

\begingroup\small
\begin{longtable}{@{}P{0.08\textwidth}P{0.86\textwidth}@{}}
\caption[The verification blocks.]{\textbf{The verification blocks.}
What each block of the checking program recomputes. The blocks are
independent of one another and of the text: each recomputes its own
quantities from scratch and then scans the source for the values
typeset, so a stale number fails rather than passes.}
\label{tab:H-blocks}\\
\toprule
\textbf{block} & \textbf{what it checks} \\
\midrule
\endfirsthead
\toprule
\textbf{block} & \textbf{what it checks} \\
\midrule
\endhead
\bottomrule
\endfoot
A & Noether's two theorems symbolically: the free particle; the
complex-scalar current and the on-shell condition;
$\partial_\mu\partial_\nu F^{\mu\nu} \equiv 0$ for arbitrary $A_\mu$ \\
B & The gauge-group arithmetic: $\dim \SU(n) = n^2-1$; the
Standard-Model count $8+3+1$; the $\SUfive{}$ branching; the electroweak
state count \\
C & The Bianchi identities: $\dd F = 0$ over all $64$ index triples for
arbitrary $A$; the Maxwell split; $\nabla_\mu G^{\mu\nu} = 0$ on a
cosmological metric with arbitrary scale factor \\
D & The \GS{} and \BIsix{} arithmetic: the ledger and its dual lock; the
coefficient vector; the trace-ratio value; primitivity and the factor
$5$ \\
E & Dirac quantization and the flux quanta; $q(x) = x(x-1)/2$, its
functional equation, its characteristic element and its reflection; the
\HS{} typing in six dimensions; the Wu vanishing \\
F & The imports used in Part~III as counting statements: the capacity
against the toll ladder; the channel charge $c = L-N$; the identity of
the three appearances of $q$ \\
G & The typeset source itself: abstract length against the ceiling; the
natbib two-Arisaka-keys rule; internal-history language; duplicate
labels; unresolved references; orphan bibliography entries; and the
register scan that confines the first person to
\S\ref{sec:concluding} \\
H & The numbers and dates of Parts~I, IV and V: the periodic-table block
widths; the charge-equality bound quoted as a level; the naturalness
tuning scaling; the ordering of every date typeset \\
I & Part~II's derivations: the Landau-level coefficient; the four
Standard-Model anomaly identities as integer sums; the dimension $496$
with its caution; the Witten factor; the three Wus; the Bianchi priority
ordering \\
J & Every answer asserted by a Check-yourself question, verified before
the question was typeset \\
K & The worked examples added to Part~II: the oscillator energy; the
covariant-derivative repair; $F$-invariance in all sixteen components;
the rank count; the index triples; the Aharonov--Bohm period \\
L & The exhaustive search establishing that no chiral anomaly-free
solution exists below five fermions in the stated range \\
M & The Part~III imports checked against their home papers: the toll
ladder and its closed form; both readings of the bound; the \GiFour{}
structure; the Morse indices; the Born chain; the axiom order \\
P & The appendix derivations: the $\binom{n}{k}$ counts; $\dd\dd = 0$;
holonomy as enclosed solid angle; the descent degrees; the two exact
sequences \\
\end{longtable}
\endgroup

\noindent
Three of the notes the harness carries are worth repeating here, because
each records a place where the text was made weaker than a first draft
had it: that dimension $496$ is necessary and not sufficient; that the
five-fermion chirality result is verified only within a stated search
range; and that the selection of $k = 1$ in \S\ref{ssec:bi6} is an axiom
rather than arithmetic.

\subsection{What this paper imports and never re-derives}
\label{ssec:H-imports}

Stated once, flatly, so that no reader need reconstruct it.

Part~II is a review of standard results and re-derives them freely, at
the level of Appendices~\ref{app:forms}--\ref{app:diffcohom}. Nothing in
Part~II is original except the reading.

Part~III derives \emph{nothing}. Every result in it is imported. The
paper's contribution there is exposition, grading, and the placement of
those results at the end of the ladder Part~II climbs. Specifically:

\begin{itemize}[leftmargin=1.4em,itemsep=0.2em]
\item the axioms are quoted from \citet{ArisakaA1Constitution};
\item the six-dimensional geometry and the ledger from
  \citet{ArisakaA2GG6};
\item the modified identity, its coefficients and its refinement from
  \citet{ArisakaA3BI6};
\item the boundary law from \citet{ArisakaA5GDOS};
\item the cascade, the Morse structure and the tolls from
  \citet{ArisakaA6Gi4};
\item the Born-rule status from \citet{ArisakaA8QM};
\item the family-number pincer, the capacity, and the pushforward result from \citet{ArisakaP2N3}.
\end{itemize}

Where this review's phrasing differs from a member paper's --- as
\S\ref{ssec:gssb} notes for the word ``productive'', and as
\S\ref{sec:grip} notes for the composition of \GRIP{} --- the difference
is stated at the point of use, and the member paper is the authority.

\GGtakeaway{\emph{What this appendix is:} the reference apparatus ---
every term with its home, every imported claim with its grade and its
dependency, and an explicit list of what is imported. \emph{How to use
it:} when Part~III makes a claim, find it in
Table~\ref{tab:H-ledger}, read the third column to see what it rests on,
and read the fourth to see what survives if that premise is withdrawn.
\emph{The one row to read first:} the arrow of time is an axiom, not a
theorem. \emph{Continue in:} Part~\ref{part:qa}, which answers the
questions this apparatus most often provokes.}



\subsection*{Part~VII in one paragraph, and what comes next}
That is the toolkit. Fifteen appendices, each built from its own textbook
starting point and stopped exactly where the body stopped needing it:
forms, Noether's two theorems, connections and holonomy, Wilson lines, Ward
identities, anomalies and differential cohomology for the seven rungs; the
Bianchi identity the ladder leans on and no other appendix derived; the
open-system machinery --- density matrices, channels, the Lindblad
generator --- on which the whole of Part~\ref{part:turn} rests; and five
more for the chapters of Part~\ref{part:ledger}, with the dictionary and
its grade ledger closing the Part. Eighty-two problems carry full worked
solutions, because a derivation one has not used is a derivation one has
not learned.

Part~\ref{part:qa} is the other way in. It asks the questions a reader
actually arrives with --- one hundred and two of them, from what a
conservation law is to whether any of this can be wrong --- and answers
each at the level of the body, naming the section it came from, so that a
question can always be turned back into a chapter.

%% ===================================================================
%%  PART VIII --- QUESTIONS AND ANSWERS
%% ===================================================================
\GGPart[Questions and answers]{Questions and Answers}{One hundred and two questions an attentive reader will
actually have --- from the plainly basic to the frankly profound ---
organized by topic and answered at the level of the body}
\label{part:qa}
%% The appendices set \thesection to \Alph{section}; with fifteen of
%% them the Q&A Part's fourteen sections would run past Z and overflow
%% the alphabetic counter.  Number them Q1..Q14 instead, which also
%% stops Q&A sections from looking like appendices.
\renewcommand{\thesection}{Q\arabic{section}}
\setcounter{section}{0}

This Part answers one hundred and two questions, organized in sixteen
groups. They are not rhetorical: they are the questions this material
actually provokes --- some plainly basic, some as deep as any in
physics --- sharpened, and where the honest answer is ``not settled''
the response says so and names what would settle it.

The first group is different from the rest and comes first
deliberately: twenty questions from the \emph{curious reader}, basic in
form and philosophically deep in substance, about what symmetry means
at all, where the antimatter went, whether the constants could have
been otherwise, and whether a framework that explains this much can be
trusted. Popular science has settled several of them in directions this
review does not take, and a reader is owed plain answers before the
specialists get a turn. The eight groups after it follow the seven
rungs and the turn in the order of the paper, and the ninth takes the
ledger of Part~\ref{part:ledger} chapter by chapter. The last six range
wider --- to the program as a whole, to the multiverse, to the
numerology worry, to method, and to the questions a reader carries home
after the physics is done.

Every group opens with a paragraph naming the hardest question in it,
so that a reader who wants the strongest challenge rather than the
gentlest can find it in one pass. Nothing is deflected; several
responses concede outright.

The register is the body's rather than Part~\ref{part:appendices}'s --- a reader who has
read the chapters but not the appendices should find every answer here
self-contained.

\newenvironment{qabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt
   \setlength{\parskip}{0.45em}\relax}%
  {\par\endgroup}

\section{For the curious reader: the big questions,
plainly answered}
\label{sec:qa-curious}

The twenty questions that follow are not a specialist's. They are the
ones a thoughtful reader without a physics degree actually carries into
a book like this: what symmetry even means, where the antimatter went,
whether the constants could have been otherwise, whether a universe
this well arranged wants a designer or a multiverse behind it, and
whether a small pile of integers explaining a great deal ought to be
believed at all. Several of them have been settled in popular science
in directions this review does not take, and a reader is owed plain
answers before the technical clusters begin.

They are met first, and plainly. Where a claim rests on a step that is
not closed, the response says so and names the step; where the honest
answer is that the framework does not reach the question, it says that
too. The hardest of the twenty is the seventh --- why
$2^{7} + 3^{2} = 137$ should be believed rather than filed with
Eddington --- and the response concedes the worry outright before
saying what would settle it. The specialist clusters that follow take
the same ground again, in the technical register, in the order of the
paper.

\bigskip
\noindent\textbf{Question 1 (Q-W1-1) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Everyone says symmetry is the deepest idea in modern physics. Your title suggests you disagree. Do you?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Both halves of that are true, and keeping them apart is
what this paper is for. Symmetry is the greatest organizing idea physics
has had. \citet{Yang1980} compressed a century into three words ---
symmetry dictates interaction --- and he was right: turn your apparatus
and angular momentum is conserved; demand that the arbitrary phase of a
quantum field be choosable independently at every point, and
electromagnetism and the nuclear forces come back at you without further
input. Nothing in this paper takes any of that back.

The disagreement is about how far up it reaches. The paper climbs seven
steps in the order history took them, from Noether in 1918 to the
topological quantization of electric charge in 2005, and at each step
asks one question: what would it actually take to violate this? Near the
bottom the answer is ordinary --- write a different equation of motion
and the conservation law goes away. Near the top it stops being ordinary.
At the seventh step the conserved thing is an \emph{integer}, and an
integer is safe because it has nowhere continuous to go: nothing that
varies smoothly can nudge a $3$ into a $4$ (\S\ref{ssec:top-of-ladder}).
No symmetry appears anywhere in that argument.

So the honest answer is a distinction, not a contradiction. Symmetry is
the best organizing idea of the twentieth century, and the most
unbreakable statements in physics turn out not to be symmetries at all.
The title is about the second clause.
\end{qabody}

\bigskip
\noindent\textbf{Question 2 (Q-W1-2) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What does a physicist even mean by `symmetry'? I thought it meant left--right, like a butterfly.''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The butterfly is a real example and the right place to
start. Reflect it left for right and you get the same butterfly: a change
that leaves something unchanged. Everything after that is a
generalization of the word \emph{change}.

First generalization: change the whole experiment rather than the object.
Rotate the apparatus, carry it across the room, run it tomorrow instead
of today. Each time nature returns the same answer, and each of those is
a claim about the world that could have failed and does not
(\S\ref{ssec:symmetry}).

Second generalization, and the strange one: change nothing physical at
all --- only the bookkeeping. Where you put the zero of potential energy.
Which coordinates you use. The overall phase of a quantum state. Nobody
expects a change of convention to be detectable, so of the three this
looks like the one that says nothing.

It is the one that turned out to be the workhorse. Insist that the
convention be choosable \emph{independently at every point of space and
time}, and the insistence becomes so severe that it fixes what the forces
are. That is what physicists call gauge symmetry, and the whole Standard
Model runs on it. Meanwhile the butterfly's own symmetry --- mirror
reflection --- is precisely the one nature was caught breaking, in 1957,
by a cobalt sample in a magnetic field \citep{Wu1957}. The change nobody
thought was a symmetry rules the subject; the change everybody thought
was obvious is false.
\end{qabody}

\bigskip
\noindent\textbf{Question 3 (Q-W1-3) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If the laws are symmetric and the world plainly is not, where did the symmetry go?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Nowhere. The laws still have it; the world simply
declined to sit in the symmetric state. A pencil balanced on its point
obeys equations with no preferred direction, and it falls one way anyway
--- the direction is not in the law, it is in the outcome. Physicists call
this spontaneous breaking, and it is the ordinary answer.

The paper's answer goes one step further, and the chapter
``Symmetry, breaking, and the honest record'' is where it is argued. The
world did not merely happen to break the symmetry. It had to, or there
would be no world to ask about. If matter and antimatter had entered on
exactly equal terms they would have annihilated into radiation and there
would be no atoms. If space were exactly homogeneous there would be no
galaxies, because there would be no denser place for anything to fall
toward. If reflection were an exact symmetry of chemistry, both mirror
forms of every sugar and every amino acid would be equally common, and
the biochemistry that runs you would not work as it does
(\S\ref{ssec:empty}).

So the usual register --- symmetry is the beautiful thing, breaking is
the damage --- is backwards. Breaking is the productive step: the
operation by which a world with things in it is selected out of a world
with nothing in particular in it (\S\ref{ssec:breaking-source}). The
question this leaves is how such a thoroughly broken world can still
contain quantities known to twenty-one decimal places, and that is what
the seven steps are for.
\end{qabody}

\bigskip
\noindent\textbf{Question 4 (Q-W1-4) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why are there conservation laws at all? Why can't energy just appear out of nowhere?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because of one theorem, and it is worth stating plainly
because it is usually made to sound more mysterious than it is. In 1918
Emmy Noether proved that a conservation law is not an extra rule bolted
onto the physics --- it is what it \emph{means} for the rules not to
depend on when or where you apply them \citep{Noether1918}. If the rules
of the game are the same tomorrow as today, energy is conserved. If they
are the same here as across the room, momentum is. If they do not care
which way you are facing, angular momentum is. Same theorem, three
readings.

Turn the question round with that in hand. ``Could energy appear from
nowhere?'' becomes ``could the rules themselves change from one moment to
the next?'' No laboratory has ever seen that happen. But the universe as
a whole is a different case: an expanding cosmos has no
time-translation symmetry to offer, and the energy carried by light
really does decrease as space stretches. There is no violation, because
there was never a conservation law there to violate.

Then the twist this paper is built around. Not every conservation
statement comes from a symmetry. Two of Maxwell's four equations follow
from what a field \emph{is} --- from the fact that it was written as the
curvature of a potential --- and no symmetry appears in the derivation
(\S\ref{ssec:not-a-law}). Those turn out to be the sturdier half.
\end{qabody}

\bigskip
\noindent\textbf{Question 5 (Q-W1-5) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You say some conservation laws are mere bookkeeping. Doesn't that make them less true?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It makes them \emph{more} robust, and the inversion is
one of the central observations of the paper.

Take the statement that magnetic field lines never end --- no north pole
without its south. That looks like a law of nature waiting to be tested,
and people have tested it for a century. But it follows from having
written the magnetic field as the curvature of a potential, and once you
have done that, the statement is a piece of arithmetic. To overturn it
you would not need a new particle in a detector. You would need to
decline to describe magnetism that way at all. There is no experiment
whose result could come back the other way, which is exactly what makes
it an identity rather than a law (\S\ref{ssec:not-a-law}).

Compare that with a genuine Noether conservation law. It is true because
the world happens to have a certain symmetry, and a different world ---
or a different term added to the same world's equations --- would not
have it. Contingent, testable, breakable.

There is a price, and the paper insists on naming it. The identity buys
its immunity by saying nothing about how anything moves. It constrains
the description, not the motion. That trade --- less content, more
rigidity --- is the axis the whole ladder is measured along
(\S\ref{ssec:axis}), and it runs from Noether's contingent currents at
the bottom to protected integers at the top.
\end{qabody}

\bigskip
\noindent\textbf{Question 6 (Q-W1-6) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Every book I open says gauge symmetry is not a real symmetry --- and then uses it for everything. Which is it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Both, and the tension dissolves as soon as you say what
is redundant.

Two field configurations related by a gauge transformation are not two
states of the world. They are one state written down twice, in two
notations. That is not a slogan; it is a theorem. Elitzur showed that a
local gauge symmetry cannot break, for the plain reason that breaking
means choosing between distinguishable alternatives and here there are
none to choose between \citep{Elitzur1975}. So ``symmetry'' is the wrong
noun (\S\ref{ssec:elitzur}).

Why does a mere redundancy still predict? Because it has to be
consistent --- locally, at every point, to all orders --- and almost no
theory manages that. The demand is not on nature but on the description,
and it is severe enough to determine the forces.

Here is the concrete case that settles what is real. You cannot measure
the potential: its value at a point is a convention you fixed yourself.
Send electrons around both sides of a long solenoid and they travel
through regions where the magnetic field is exactly zero --- yet their
interference fringes shift, by an amount set by the flux they enclosed
but never touched \citep{AharonovBohm1959}. The local potential is
convention; the local field is zero; what is physical is the phase picked
up around the closed loop. Not the notation, not nothing --- the loop
(\S\ref{ssec:C-ab}).
\end{qabody}

\bigskip
\noindent\textbf{Question 7 (Q-W1-7) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Popular books say nobody knows why the fine-structure constant is $1/137$. You write $2^7 + 3^2$. Why isn't that just a coincidence?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It might be. Here is the claim, and then the reasons to
be careful, in that order.

The framework fixes five small integers once, by a consistency condition
on how anomalies cancel in six dimensions --- a condition that never
mentions the fine-structure constant. Two of them are $J = 7$ and
$N = 3$. The claim is then $\alpha^{-1}(0) = 2^{J} + N^{2} = 128 + 9 =
137$ (\S\ref{sec:zero-parameters}).

Three cautions travel with it. First, $137$ is not the measured value:
the measurement is $137.035999177$, so the formula is low by $0.026\%$,
and the paper says so rather than rounding in its own favor. Second,
$137$ is the value at zero momentum. The coupling changes with energy;
at the mass of the $Z$ boson the same quantity is $127.930$, against
$2^{J} = 128$ alone, high by $0.055\%$. Third --- and this is the one
that should worry you --- with a handful of small integers and a few
permitted operations one can reach a great many targets. Nobody has
computed how many targets this ledger was allowed to aim at. Until
somebody does, the right posture toward the arithmetic is suspended, not
triumphant.

What would make it more than numerology is not the formula. It is that
the same two integers are spent elsewhere with no further freedom ---
$N = 3$ is the number of colors, $J = 7$ sets a marker in the early
universe eleven orders of magnitude away in scale
(\S\ref{ssec:same-integers}) --- and that the same ledger stakes
not-yet-measured quantities on named instruments with dates
(\S\ref{sec:falsifiers}). If those bets lose, $137$ will not rescue
anything.
\end{qabody}

\bigskip
\noindent\textbf{Question 8 (Q-W1-8) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why are there exactly three generations of matter --- three copies of the electron and its quarks?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The framework's answer is a pincer: two bounds arriving
from unrelated directions and meeting at three.

The lower bound is old and belongs to everyone. A physical
matter--antimatter asymmetry in the quark mixing pattern is impossible
with fewer than three families \citep{KobayashiMaskawa1973}; that
asymmetry is observed; so there are at least three. The upper bound is
the framework's own, and it is where the structural point lives. In a
closed theory you can never bound a count from above --- add a fourth
copy of everything and nothing objects, because nothing is paying for it.
The move this paper makes is to treat what we observe as a projection
from a larger space, which makes physics an \emph{open-system} problem;
the copies are then drawn against a finite budget, the budget can be
computed, and it runs out at three \citep{ArisakaP2N3}.

Two honesty notes. The number of families is a different integer from the
number of colors, even though both come out three; the paper is
deliberate about not fusing them, since the two derivations share no
premise. And the lower leg consumes an empirical input --- that
matter--antimatter asymmetry is a measured fact, not a derived one --- so
the result is graded conditional, and it is one row of forty-nine on the
program's sheets, not the headline a casual reader takes it for
(\S\ref{ssec:n3-proportion}). What kills it is simple: one fourth
sequential family (\S\ref{ssec:three-gen}).
\end{qabody}

\bigskip
\noindent\textbf{Question 9 (Q-W1-9) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why has nobody ever seen a single quark on its own?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Start with the picture that is not in dispute. Pull two
quarks apart and the force between them does not fade with distance the
way electric attraction does. The field collapses into a narrow tube of
roughly constant tension --- about sixteen tonnes weight, which is not a
figure of speech --- so the energy stored grows without limit as you
pull. Long before you have separated them it becomes cheaper for the
universe to spend that energy making a fresh quark and antiquark from
nothing. The tube snaps, and you are holding two ordinary particles
instead of one bare quark. This is what Wilson's criterion measures on a
computer: the cost of a loop grows with the area it spans, not its
perimeter \citep{Wilson1974}.

What survives that process is always a color-neutral bundle --- a quark
with an antiquark, or three quarks together. In the standard account
that census is an output of the dynamics, believed because the
simulations show it.

The framework reads it as a grammar instead. Each quark carries one unit
of a three-valued label; the labels add; and a state can be registered by
the boundary at all only if the total closes to a multiple of three
(\S\ref{ssec:singlet-rule}). A lone quark is not forbidden by a force ---
it is not a state of the theory, in the way that half a sentence is not a
sentence. What that reading does not supply, and the paper says so, is the
dynamical half: the value of the tension and the mass gap remain exactly
as open here as in the standard account (\S\ref{ssec:exhibit}).
\end{qabody}

\bigskip
\noindent\textbf{Question 10 (Q-W1-10) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Where did all the antimatter go?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Almost all of it annihilated, and we are the rounding
error --- about six parts in ten billion of the original stock, the part
that had no partner left to annihilate with. The real question is why
there was a remainder at all.

Sakharov wrote down what any answer must contain
\citep{Sakharov1967}. Something must be able to change the number of
baryons. Matter and antimatter must be treated slightly differently by
that something. And it must happen while the universe is out of
equilibrium, or the reverse reaction undoes the result. The Standard
Model has all three ingredients in principle and fails on the arithmetic
by many orders of magnitude.

The framework's chain is specific. Heavy Majorana partners of the
neutrinos sit at about $3.85$~TeV with percent-level splittings; they
fall out of equilibrium and decay with a small bias; washout is strong,
which conveniently erases any memory of initial conditions; the
electroweak vacuum then converts the surviving lepton excess into
baryons, and expansion dilutes the result. The chain lands on
$7.34\times10^{-10}$ against a measured $(6.12\pm0.04)\times10^{-10}$
--- high by $19.9\%$, with nothing fitted (\S\ref{ssec:lepto-chain}).

Now the grade, because that number is easy to over-read. One exponent in
the compression is \emph{postulated}, not derived. And the whole result
carries a declared factor-of-about-three uncertainty from quantities
nobody has computed, so $19.9\%$ is not a precision --- it is a value that
landed comfortably inside a wide band. The genuine claim is that standard
transport physics reconstructs the ledger's number rather than being
tuned to it, and the chain is mortal four separate ways within the decade
(\S\ref{ssec:lepto-grade}).
\end{qabody}

\bigskip
\noindent\textbf{Question 11 (Q-W1-11) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is the universe fine-tuned for life? And doesn't that force a choice between a multiverse and a designer?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Take the observation at full strength first, because it is not manufactured by enthusiasts. Shift the strength of the nuclear force by a few percent and the carbon you are made of does not form in stars. Shift the electron's mass against the proton's and chemistry closes down. Give the vacuum its natural energy density and no galaxy ever assembles. Asked as ``how much room was there to get this wrong,'' the answer keeps coming back: very little.

The usual framing then offers two exits --- a vast ensemble in which ours is the rare workable draw, or a chooser. This framework declines both by refusing their shared premise. Its claim is that the constants are not dials at all. If $\alpha^{-1}(0) = 2^{J} + N^2 = 128 + 9 = 137$ and the weak mixing angle is $L^2/(4N^3) = 25/108 = 0.23148$, read off one small integer ledger, then there is no setting that could have been mis-set: a world ``with slightly different constants'' is not an unlucky alternative but a description that fails a closure condition, the way a triangle with four sides is not a rare triangle.

That is a claim under test, not an established fact, and its honest form is printed in \S\ref{ssec:not-zero}: zero free continuous parameters, plus one discrete completion class whose necessity is \emph{asserted, not proved}. So hold it by its exits. Table~\ref{tab:falsifiers} names them with instruments and dates: a neutrino mass sum bounded below about $58$ meV, a matter share outside $[0.315, 0.322]$, no $3.85$~TeV resonance where one is predicted, a tensor ratio far from $0.0043$. Any one of those and the third option is gone. The specialist treatment of the selection argument is \S\ref{ssec:anthropic}.
\end{qabody}

\bigskip
\noindent\textbf{Question 12 (Q-W1-12) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't our universe just one bubble among infinitely many, and we happen to live in one that works?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That picture deserves stating the way its architects state it, because it is serious work and not a shrug. String compactification has an enormous number of stable solutions once the extra-dimensional shape is frozen in place \citep{KKLT2003}; each gives a different low-energy world, and eternal inflation supplies a way of realizing them; estimates of how many run from $10^{120}$ to $10^{272{,}000}$ depending on the class counted \citep{Douglas2003,TaylorWang2015}. Nothing selects ours, and that literature says so itself --- the swampland program exists to constrain the proliferation from outside \citep{Vafa2005}. Given all that, ``we live in one that works'' is not evasion; it is the remark left to make.

The disagreement here is structural rather than empirical, and narrower than the version usually advertised. It is not a contest over adjustable couplings: string theory has none either, so on that axis the two programs agree, and \S\ref{sec:comparison} says so rather than claiming a victory it has not won. What differs is the size of the remaining discrete set --- five integers with two closure relations, against a discretuum nobody can bound to within hundreds of thousands of orders of magnitude --- and, against this program, neither side has proved its set forced. The landscape does not claim to have; this one claims it and has not.

The sky then adds a constraint both must meet. Regions that never exchanged a signal share one temperature, one kind of electron, and one sign of the matter excess; that third sameness --- of \emph{choices} --- rules out any account in which the choosing happens late \citep{ArisakaA10Cosmos}. Table~\ref{tab:comparison} sets the four postures side by side, and later entries in this Part push the argument further.
\end{qabody}

\bigskip
\noindent\textbf{Question 13 (Q-W1-13) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If every constant is forced by geometry, is my future already written? What happens to free will?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Three different things are compressed into that question, and separating them dissolves most of the alarm.

The first is the claim the framework actually makes. It concerns the descent --- the cascade from a high-energy geometry down to the world of atoms --- and it says that at each stage the outcome is selected rather than drawn: one basin, not a menu with a coin flip. That is what the sky seems to require, since every region chose matter over antimatter with the same sign and the same yield, and no arrangement made after those regions lost contact could have managed it \citep{ArisakaA11Origin}. Call it determinism up to gauge. It is a statement about which \emph{laws} the world ended up with.

The second is whether those laws, once fixed, make the world a wound clock. Here the framework is not on the side people expect. Half of its structure is a fiber whose maps have no inverses --- things are genuinely forgotten --- and the quantum statistics are carried as a measure the dynamics relaxes to rather than as a certainty, with a named analytic step still open \citep{ArisakaA8QM}. Fixed rules and fixed outcomes are different claims, and only the first is asserted.

The third is your deliberation over lunch, and no equation in this paper reaches it. A theory of forced constants stands to a biography roughly as the rules of chess stand to a game: they permit no exceptions and settle no move. Whether that leaves room for what people mean by freedom is a question about persons, and the physics here neither supplies it nor takes it away. The specialist cousin of this question appears later in this Part, beside the entry on where minds fit; the constitutional sense of determinism belongs to the origin companion, not to this paper.
\end{qabody}

\bigskip
\noindent\textbf{Question 14 (Q-W1-14) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why does time run only one way, when the equations don't care which way it runs?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} You have put your finger on the sorest point in classical physics, and the standard repairs are weaker than textbooks imply. Boltzmann's proof that entropy increases smuggles the asymmetry in through an assumption about which collisions are uncorrelated. Loschmidt pointed out in 1876 that reversing every velocity turns an entropy-decreasing history into a lawful entropy-increasing one; Zermelo, in 1896, that a bounded reversible system must return arbitrarily close to where it began. Both are correct. Neither was ever refuted --- they were absorbed, by conceding that the theorem is about a model and not about the mechanics. Coarse-graining does real work and then retrodicts a higher-entropy past, so it needs an assumption that the universe began in an extraordinarily special state. Table~\ref{tab:N-arrows} lists four routes and where each inserts the arrow.

This framework's move is to stop looking for the arrow in the \emph{states} and look at the \emph{maps}. Observation is read as a projection out of a six-dimensional bulk, and a projection that loses anything has no inverse in the algebra of physical maps --- no statistics, no thermodynamic limit required. Loschmidt's construction then has no premise, since it needs a backwards map not in the set; Zermelo's proof needs an invertible measure-preserving map on a finite measure and gets none of the three. A clock, meanwhile, does two jobs on two different fibers: a circle repeats and cannot point, an interval points and cannot repeat (\S\ref{sec:what-is-time}).

One concession, and it matters. That transports run one way follows from the structure; that observation \emph{is} such a projection is a premise of the program, not a theorem of it. Appendix~\ref{app:time} works the whole argument properly.
\end{qabody}

\bigskip
\noindent\textbf{Question 15 (Q-W1-15) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Did time have a beginning? What came before it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Here is what the framework can say, and then, carefully, where it stops.

It can draw the whole history of the world as one line. Plot the logarithm of elapsed time against the descent's own coordinate and you get a line of slope exactly $2$ through the radiation era, kinking to $3/2$ when matter takes over, with eight epochs strung along it (\S\ref{ssec:time-chronology-w1}). Each unit of descent multiplies the clock by $e^2 \simeq 7.4$. The total audits: $\ln(t_0/t_{\rm Pl}) = 140.24 = 36.11 + 91.60 + 12.53$, the terms being launch, radiation era, and everything since. So there is a first entry in that ledger, and it sits at the launch rather than at a point of infinite density.

What the framework does not have is a reading from before it. Its time is a boundary quantity --- the shadow the descent casts on the world we observe --- so asking what came before asks for a reading from a clock the descent had not yet produced. That is not a dodge; standard physics meets the same difficulty from the other side, since in the deepest formulation of gravity we have, the fundamental equation contains no time parameter at all and time must be recovered rather than assumed (Appendix~\ref{app:time}).

So: a beginning to the \emph{descent}, audited, in numbers you can check; no account of a ``before''; and no account of why there is a constitution to be satisfied in the first place. That last is where the regress halts, and it is marked rather than papered over. One caution, since the number is pretty enough to seduce: $140.24$ is close to $4K = 140$, but the closeness carries the flagged grade, not a derivation --- the clock runs longer than $e^{4K}$ Planck times by a factor $e^{0.243} = 1.275$, and no mechanism explains either the agreement or the gap.
\end{qabody}

\bigskip
\noindent\textbf{Question 16 (Q-W1-16) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why is there something rather than nothing?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Physics can move this question one step and not two, and the step is more interesting than it sounds. It comes from asking what ``nothing'' would have to be.

The tempting picture of nothing is a perfectly symmetric state --- no direction preferred, no handedness, matter and antimatter on exactly equal terms. But that is not a description of an alternative world sitting quietly beside ours; it is a description that picks out nothing at all. Grant matter and antimatter exactly equal terms and they annihilate, and there are no atoms. Grant space exact homogeneity and no galaxy forms, since there is no denser place to fall toward. Grant reflection exactness and both mirror forms of every molecule are equally present, and the handed chemistry your cells run on cannot work. Perfect symmetry does not describe an empty room; it describes no particular thing (\S\ref{ssec:empty}).

That reframes the question into one physics can chew on. It is not ``why was the empty option not taken'' but ``how does a symmetric start produce a definite something without a coin being flipped anywhere'' --- a puzzle old enough to have a name: Curie argued in 1894 that the symmetries of the causes reappear in the effects. The program's answer, every clause of it graded, is that the dial the world turns is quantized rather than continuous and the descent that turns it runs one way, so the outcome is definite without being chosen (\S\ref{ssec:symmetry-reading}).

Then it stops. Why there is a structure to be satisfied at all is not a question this or any physics has a method for, and the specialist entry later in this Part stops in the same place --- not from a failure to try harder, but because that is the boundary of what a theory of \emph{this} world can be about.
\end{qabody}

\bigskip
\noindent\textbf{Question 17 (Q-W1-17) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is mathematics discovered or invented? Your framework seems to need an answer.''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It would like one and is not entitled to one, and saying so is more useful than picking a side.

The two positions, fairly: mathematics is a language built for describing patterns, so its describing them is no more remarkable than a hammer hitting things; or the structures are what the world is made of, and finding them is discovery. \citet{Wigner1960} asked why the first answer feels inadequate, and the discomfort is empirical rather than mystical. What sits badly with the invention story is the order of events. The identity governing curvature was written down in the 1880s for reasons internal to differential geometry, mislaid twice, then turned out to force the shape of Einstein's field equations (\S\ref{sec:bianchi}). The refinement at the top of this paper's ladder was built by topologists with no interest in physics and turns out to be the function a physical cascade needs (\S\ref{sec:hs}). Nobody was fitting anything to anything.

Now the part that concerns this framework: suppose the ledger holds up, the integers survive their falsifiers, and the constants really are outputs. What would that show? Something narrower than either camp wants --- that the world's exactly known quantities are of a kind admitting no continuous adjustment. That is a claim about what sort of structure the laws have, not about whether numbers exist independently of minds; a thoroughgoing nominalist can hold it. Failure would settle no more: a wrong ledger is just a wrong ledger.

Where the framework is committed is smaller: it is realist about there being a definite structure the description is \emph{about}, and takes no position on Platonism. The author's own reckoning, held as a discomfort rather than a doctrine, is in the concluding chapter (\S\ref{sec:concluding}).
\end{qabody}

\bigskip
\noindent\textbf{Question 18 (Q-W1-18) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``A handful of small integers explaining this much makes me suspicious. Shouldn't it make you suspicious too?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Yes. It should, and it does. The precedent you half-remember is real: Eddington spent years assembling integer combinations for the fine structure constant and was wrong, and the paper prints the episode.

So here are the instruments, instead of a reassurance. First, every claim carries a grade from a small vocabulary --- derived, derived-conditional, named input, postulated, flagged. Table~\ref{tab:grades} collects them; Appendix~\ref{app:dictionary} says which claims fall when a premise is withdrawn, so you can pull one thread and see what unravels.

Second, the commitments are dated: Table~\ref{tab:falsifiers} gives each a value, a kill criterion, an instrument and a window, and most are two-sided --- which separates a prediction from a bound.

Third, the accounting you want has not been done, and the paper says so. A fair look-elsewhere calculation would count the targets the ledger could aim at, the operations permitted in each expression, and the tolerance counted as a hit. That trials factor has not been computed, so the right reading of the agreements is suspended, not triumphant (\S\ref{ssec:exposure}).

Fourth, the weakest results are labeled as such by their author. The age ledger is FLAGGED --- arithmetic without a mechanism, and not exact: it misses $4K = 140$ by a factor of $1.275$ in the clock. The exponent $K/2$ behind the size of the matter excess is POSTULATED, with no derivation and a declared factor-of-three band, so its landing within twenty percent of the measured baryon-to-photon ratio must never be read as a precision. And the axiom picking $(L,N) = (5,3)$ over $(10,6)$ is a minimality principle --- the kind of premise this paper's own history chapter warns against, and a reader who declines it is entitled to the whole family.
\end{qabody}

\bigskip
\noindent\textbf{Question 19 (Q-W1-19) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should I take one person's framework seriously when thousands of physicists work on other approaches?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} You should not take it seriously on anybody's authority, the author's included. A field of thousands exploring several routes is how the thing is supposed to work, and the people on those routes are doing careful work for good reasons. Single authorship, absence of peer review and arrival all at once are honest grounds for extra scrutiny --- and that scrutiny is a burden on the program, not on you.

What helps is that the useful test does not require you to evaluate the framework at all. Judge the falsifiers instead of the author. Ask of any program: does it name a quantity, a value, a criterion that would kill it, an instrument, and a decade? If it does, you can wait and watch, and the waiting costs you nothing.

So here is a specific thing to watch, chosen because it may fire first. The framework commits to a total neutrino mass of $58.5$ meV, and that number is a \emph{floor} --- the minimum allowed by the measured mass splittings for normal ordering, with no room underneath it (\S\ref{ssec:the-sheet}). Bounds from galaxy surveys combined with the microwave background have been tightening from above for years. If a robust one settles below about $58$ meV the prediction has nowhere to go, and no adjustment is available, because there is no free parameter to adjust. If you want a second, the primordial tensor-to-scalar ratio is committed at $0.0043$ two-sided, and the instruments built to decide it report in the early 2030s (Table~\ref{tab:falsifiers}).

If those readings go the wrong way, this framework ends and nothing else in physics is damaged. That asymmetry is the whole reason it is worth a non-specialist's attention: it is losable, on a schedule, in public.
\end{qabody}

\bigskip
\noindent\textbf{Question 20 (Q-W1-20) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``I am not a physicist and I have got this far. What is the one thing you want me to take away?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} This: the deepest things in physics are protected not because they are symmetric but because they are integers, and an integer has nowhere to move.

That sentence is the destination of the whole ladder, and worth unpacking once. A symmetry is a delicate protection. It says a quantity cannot change so long as a balance holds --- and balances can be spoiled: by a small term nobody anticipated, by an effect that appears only on quantization, by an experiment in a cold laboratory in 1957 that found the world was not mirror-symmetric. Physics has learned that lesson three times over (\S\ref{ssec:failures}). An integer is protected far more crudely. The electron and the proton share no other property, yet their charges are equal in size to twenty-one decimal places \citep{MarinelliMorpurgo1984}; the families of matter number three, and three-point-one is not a different world but no world at all. Nothing continuous can nudge a whole number: between it and its neighbor there is nowhere to go. That is why the top of this ladder is more robust than anything below it, and the argument is about logic rather than beauty (\S\ref{ssec:top-of-ladder}).

Everything else here is either the climb to that statement or a proposal about what stands at the end of it, marked as a proposal throughout.

And now the handoff. The entries that follow are for readers who want the machinery --- what a gauge symmetry really is, why two of Maxwell's equations are free, and where this program is most likely to break. You do not need them to have got the point. But you have already crossed the part most people are told they cannot cross, and none of what follows is in a secret language. Turn the page if you like: you are not the audience physics usually addresses, and you should be.
\end{qabody}

\section{The question: what a conservation law is}
\label{sec:qa}

Two questions on the ground the review starts from. The first is the
one the whole paper is an answer to --- whether a conservation law is a
law of nature or a piece of bookkeeping --- and the answer is that the
mixture changes as one climbs, which is what makes the climb worth
making.


\bigskip
\noindent\textbf{Question 21 (Q-W1-21) (on Part I)}\par\nobreak\smallskip
\noindent\textbf{``Is a conservation law a law of nature or a bookkeeping convention?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Both, and the paper's whole argument is that the mixture changes as
one climbs. Noether's current is a law in the ordinary sense: it depends
on the action, and a different action gives a different current. The
Bianchi identity is bookkeeping: it follows from what a curvature
\emph{is}, and no experiment could conflict with it
(\S\ref{ssec:not-a-law}). The interesting observation is that the more
bookkeeping-like a conservation statement is, the harder it is to break
--- which is Table~\ref{tab:arc-earned}.
\end{qabody}

\bigskip
\noindent\textbf{Question 22 (Q-W1-22) (on Part I)}\par\nobreak\smallskip
\noindent\textbf{``Why do chemists care whether a symmetry is exact?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because an exact symmetry gives a degeneracy and an inexact one gives
a splitting, and the splitting is what is measured. The five $d$ orbitals
are degenerate in a free ion because rotations relate them; put the ion
in a complex and the degeneracy breaks by an amount in the visible range,
which is why transition-metal compounds are colored
(\S\ref{ssec:chemistry}). A chemist reading a spectrum is reading which
symmetries survive and by how much the rest fail.
\end{qabody}

\section{Noether's two theorems}

Three on the first rung. The hardest is the second, on what
``off-shell'' means, because the whole force of Noether's \emph{second}
theorem lives in that word: an identity that holds without using the
equations of motion is a different kind of statement from a current
that is conserved only when they hold.


\bigskip
\noindent\textbf{Question 23 (Q-W1-23) (on Noether)}\par\nobreak\smallskip
\noindent\textbf{``If energy conservation comes from time-translation symmetry, is energy conserved in an expanding universe?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Not globally, and the reason is exactly Noether's. An expanding
cosmology has no timelike Killing vector, so there is no time-translation
symmetry to supply a conserved energy, and the photon energy density does
in fact decrease as the universe expands. What survives is the local
statement $\nabla_\mu T^{\mu\nu} = 0$, which is not a Noether current at
all but a consequence of the identity Eq.~\eqref{eq:C-divG} --- the
second theorem, not the first.
\end{qabody}

\bigskip
\noindent\textbf{Question 24 (Q-W1-24) (on Noether)}\par\nobreak\smallskip
\noindent\textbf{``What exactly is `off-shell', and why does it matter?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} On shell means: evaluated on configurations that satisfy the field
equations. Off shell means: true for every configuration whatsoever,
solution or not. The first theorem gives a conservation law on shell ---
the derivation of \S\ref{ssec:B-first} shows the violating term is
proportional to the field equations. The second gives an identity off
shell. It matters because an off-shell statement cannot be spoiled by
changing the dynamics, and that immunity is the first appearance of the
rigidity the rest of the paper tracks.
\end{qabody}

\bigskip
\noindent\textbf{Question 25 (Q-W1-25) (on Noether)}\par\nobreak\smallskip
\noindent\textbf{``Did Noether know her second theorem was the more general one?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} She certainly knew it was the one she had been asked for. The problem
Hilbert and Klein posed was about energy in general relativity, and the
answer is the second theorem; the first is in the same paper largely as
the contrasting case (\S\ref{ssec:goettingen}). What she cannot have
anticipated is which of the two the next century would quote.
\end{qabody}

\section{Gauge symmetry: redundancy that predicts}

Four on the second rung, and the first of them is the one every
student asks and few textbooks answer squarely. If gauge symmetry is
only redundancy of description, it should predict nothing --- and it
predicts almost everything. The resolution is that a redundancy
constrains what can be written down, which is not the same thing as a
symmetry relating states.


\bigskip
\noindent\textbf{Question 26 (Q-W1-26) (on gauge)}\par\nobreak\smallskip
\noindent\textbf{``If gauge symmetry is only redundancy, why does it predict anything?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because the redundancy has to be consistent, locally, at every point,
and to all orders --- and very few theories manage it. The demand is not
that nature possess a symmetry but that a description with surplus
variables remain unitary and local, and that demand is severe enough to
determine the interactions (\S\ref{ssec:gauge-intuition}). This is a
stronger statement than the original slogan, because a redundancy cannot
turn out to be false, whereas a symmetry can --- and
Chapter~\ref{sec:honest-record} records two that did.
\end{qabody}

\bigskip
\noindent\textbf{Question 27 (Q-W1-27) (on gauge)}\par\nobreak\smallskip
\noindent\textbf{``Does the Higgs boson `break' a gauge symmetry or not?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Not in the sense the phrase suggests, and Elitzur's theorem says a
local gauge symmetry cannot break spontaneously at all
(\S\ref{ssec:elitzur}). What happens is that a scalar acquires a vacuum
value, that value is not gauge invariant so choosing it fixes a gauge,
and the \emph{global} part of the symmetry --- which is a real symmetry
--- is what breaks. The physical content is a change in the spectrum, and
\S\ref{ssec:higgs} does the state count: $4 - 1 = 3$ vectors become
massive, $4 - 3 = 1$ scalar survives.
\end{qabody}

\bigskip
\noindent\textbf{Question 28 (Q-W1-28) (on gauge)}\par\nobreak\smallskip
\noindent\textbf{``Can I measure a gauge potential?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} No, but you can measure its holonomy, and that is not the same as
measuring nothing. The Aharonov--Bohm experiment detects
$\oint A$ in a region where $F$ vanishes identically
(\S\ref{ssec:C-ab}). So $A$ is not observable, $F$ is not the complete
description, and the honest object is the set of phase factors around
loops \citep{WuYang1975}.
\end{qabody}

\bigskip
\noindent\textbf{Question 29 (Q-W1-29) (on gauge)}\par\nobreak\smallskip
\noindent\textbf{``What is the difference between a global and a local U(1) in one sentence?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A global $U(1)$ gives you a charge you can measure; a local one gives
you an identity among your field equations and no new charge at all
(\S\ref{ssec:noether-ii}).
\end{qabody}

\section{The Bianchi identity and monopoles}

Four on the third rung, where the review's central inversion first
bites: the Bianchi identity is not a law, cannot be tested, and is for
exactly that reason harder to break than anything Noether supplies.
The sharpest question is the last --- whether an experiment could still
find a magnetic monopole --- because an identity cannot be violated,
and a discovery would mean the bundle was never the one assumed.


\bigskip
\noindent\textbf{Question 30 (Q-W1-30) (on Bianchi)}\par\nobreak\smallskip
\noindent\textbf{``Are magnetic monopoles forbidden, or just unobserved?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Unobserved. The identity $\dd F = 0$ forbids them only where
$F = \dd A$ globally, and the honest statement is that magnetic charge is
precisely the obstruction to writing $F$ that way
(\S\ref{ssec:no-monopoles}). If a monopole exists, one uses two
potentials on two patches, and the integer measuring their mismatch is
Dirac's.
\end{qabody}

\bigskip
\noindent\textbf{Question 31 (Q-W1-31) (on Bianchi)}\par\nobreak\smallskip
\noindent\textbf{``Why are two of Maxwell's four equations `free'?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because they are not laws. Writing $F = \dd A$ makes $\dd F = 0$
automatic, and $\dd F = 0$ has $\binom{4}{3} = 4$ components which are
$\nabla\cdot\mathbf{B} = 0$ and the three of Faraday's law
(\S\ref{ssec:A-stokes}). The other two come from varying an action,
involve the sources, and would be different for a different action.
\end{qabody}

\bigskip
\noindent\textbf{Question 32 (Q-W1-32) (on Bianchi)}\par\nobreak\smallskip
\noindent\textbf{``The Bianchi identity has a gravitational twin. Does that mean gravity is a gauge theory like the others, or not?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The identity is the same identity, and the theory is not the same
kind of theory. Take the parallel first, because it is exact. Curvature
is built by differentiating a connection, so the antisymmetrized second
derivative collapses: $DF = 0$ in the gauge case, $\nabla_{[\lambda}
R_{\mu\nu]\rho\sigma} = 0$ in the gravitational one, both the Jacobi
identity for a commutator wearing different clothes
(\S\ref{ssec:H-grav}). Contract the second one twice and
$\nabla_\mu G^{\mu\nu} = 0$ holds for every metric whatsoever, which is
what forces the shape of Einstein's equations (\S\ref{ssec:einstein}).
Now the differences, which are structural rather than cosmetic. First,
the frame bundle is soldered to the tangent bundle: its fiber is not an
internal space bolted onto spacetime but the tangent space itself, so a
gravitational gauge transformation moves points, and an internal one
does not. Second, the metric is both the field and the background --- in
Yang--Mills the connection propagates on a geometry it does not
constitute. Third, and most consequential for conservation, there is no
local gauge-invariant energy density for the gravitational field: the
equivalence principle lets one set the connection to zero at any point,
so a local scalar built from it cannot be invariant, and what one gets
instead are quantities defined on a boundary. So the identity
transfers and the interpretation does not. Appendix~\ref{app:bianchi}
keeps the two columns --- identity and field equation --- separate
across all four sectors in Table~\ref{tab:H-split}, which is the place
to check that nothing is being smuggled between them.
\end{qabody}
\bigskip
\noindent\textbf{Question 33 (Q-W1-33) (on Bianchi)}\par\nobreak\smallskip
\noindent\textbf{``If monopoles are excluded by an identity rather than by a law, could an experiment still find one?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Yes, and nothing in the identity would be refuted if it did. That
is the sharp form of the answer, and it is worth being pedantic about.
An identity cannot be violated by an experimental result, because it
follows from what its symbols mean rather than from what the world does.
$\mathrm{d}F = 0$ says that if $F$ is the exterior derivative of a globally
defined potential then no net flux leaves a closed surface. A monopole
discovery would not falsify that conditional; it would falsify the
antecedent. One would learn that the bundle is not the trivial one
assumed, and the identity would go on holding, patch by patch, exactly
as before (\S\ref{ssec:no-monopoles}).
What replaces the global potential is the Wu--Yang construction
\citep{WuYang1975}: one potential on a northern cap, another on a
southern, related by a gauge transformation on the overlapping equator.
The winding number of that transition function around the equator is an
integer, and it is Dirac's --- $eg = n/2$ in units with $\hbar = c = 1$
\citep{Dirac1931}, derived in \S\ref{ssec:dirac} from nothing stronger
than single-valuedness of a wavefunction.
So what a discovery would overturn is a topological assumption about the
gauge bundle of electromagnetism, not a conservation law and not a
theorem. It would also do something more useful than damage: one
monopole anywhere in the universe quantizes every electric charge
everywhere --- and the neutrality of bulk matter is measured at the
level of $10^{-21}$ of the elementary unit
\citep{MarinelliMorpurgo1984}, a number nothing in the Standard Model
explains.
\end{qabody}

\section{Anomalies and Green--Schwarz}

Four on the sixth rung. Anomaly cancellation is the moment
conservation stops being bought with a symmetry and starts being bought
with an integer condition. The question that decides how much of this
review's program survives is the third: whether the six-dimensional
coefficients are derived from the geometry or fitted to the answer.


\bigskip
\noindent\textbf{Question 34 (Q-W1-34) (on Green--Schwarz)}\par\nobreak\smallskip
\noindent\textbf{``What is an anomaly, in one paragraph, without Feynman diagrams?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A symmetry of the classical theory that quantization does not
preserve. The cleanest picture: massless fermions in parallel electric
and magnetic fields have a lowest Landau level that is one-dimensional
and moves one way only; the electric field pushes states through zero
energy at a computable rate; and because the Dirac sea has no bottom,
the supply never runs out. A quantity conserved in every finite
approximation is not conserved in the theory, and the rate of failure is
$e^2 \mathbf{E}\cdot\mathbf{B}/4\pi^2$ (\S\ref{ssec:F-landau}).
\end{qabody}

\bigskip
\noindent\textbf{Question 35 (Q-W1-35) (on Green--Schwarz)}\par\nobreak\smallskip
\noindent\textbf{``How can violating a conservation law help?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because two failures can cancel. Green and Schwarz let the tensor
field's identity fail by a prescribed amount, $\dd H_3 = X_4$, and
arranged for that failure to cancel the fermion loop's
(\S\ref{ssec:let-it-fail}). What is protected afterwards is not any
individual entry but the total --- which is the shift from ``nothing is
lost'' to ``the books balance'' that gives this paper its title. Strictly,
no identity was violated: the object was redefined so that the identity's
premise no longer applied to it.
\end{qabody}

\bigskip
\noindent\textbf{Question 36 (Q-W1-36) (on Green--Schwarz)}\par\nobreak\smallskip
\noindent\textbf{``Are the BI-6 coefficients fitted or derived?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Derived, and the derivation is four lines of arithmetic you can check
(\S\ref{ssec:F-bi6}). The two abelian coefficients are both
$-(L-N)/L = -2/5$ from the ledger integers, with no independent second
parameter. But the honest qualification belongs here too: the ratio is
satisfied by the whole family $(L,N) = (5k,3k)$, and the selection
$k = 1$ is made by an axiom, not by the ratio
(\S\ref{ssec:exposure}).
\end{qabody}

\bigskip
\noindent\textbf{Question 37 (Q-W1-37) (on Green--Schwarz)}\par\nobreak\smallskip
\noindent\textbf{``Anomaly cancellation is already a consistency condition inside the Standard Model. What does BI-6 add that hypercharge assignment does not?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Half the objection is conceded. The
Standard Model already balances its books: per generation, in the
integer normalization, $\sum Y = 0$,
$\sum Y^3 = 0$, and the doublet and triplet sums vanish separately. The
cubic sum is the striking one --- $48 - 1536 + 192 - 432 + 1728 = 0$,
five unrelated numbers canceling exactly (Table~\ref{tab:sm-anomaly}).
Note the direction. Those are conditions \emph{satisfied by}
hypercharges measured one multiplet at a time: they constrain a
spectrum, they do not produce one, and a different anomaly-free
assignment would do as well.
What inflow adds is where the cancellation is paid from. In the
Green--Schwarz arrangement the boundary's Ward identity fails by exactly
what the bulk's modified identity supplies, $\mathrm{d}H_3 = X_4$
\citep{CallanHarvey1985}, so the coefficients are not free parameters of
the four-dimensional theory but are read off a geometry. In this program's six-dimensional instance the four-form
carries $(1, -1, -\tfrac25, -\tfrac25)$, both abelian entries equal to
the trace ratio $(L-N)/L = 2/5$ on the ledger integers $(M,N,L) =
(1,3,5)$, with no independent second parameter (\S\ref{ssec:bi6}).
The grade. Closure of the identity and the ratio's value are theorems
of \citet{ArisakaA3BI6}, imported here.
Selection of $k = 1$ from the family $(5k, 3k)$ is
\emph{derived-conditional} on a minimality axiom (\S\ref{ssec:exposure}).
That this identity is the inflow partner of the observed world is the
program's \emph{proposal} (\S\ref{ssec:inflow}). The
falsifier is not decorative: the abelian factor whose anomaly the
four-form pays is baryon number, left as a rigid charge on a massive
boson --- so the proton is absolutely stable, and one decay event in any
channel ends it (\S\ref{ssec:proton}).
\end{qabody}

\section{Hopkins--Singer: quantization and holonomy}

Five on the seventh rung, the destination. The hardest is the last,
and it is a fair charge: differential cohomology is very heavy
machinery for a result Dirac obtained in 1931 with a wavefunction. The
response concedes the abelian case outright and then says exactly what
the refinement buys that Dirac's argument does not.


\bigskip
\noindent\textbf{Question 38 (Q-W1-38) (on Hopkins--Singer)}\par\nobreak\smallskip
\noindent\textbf{``Why must electric charge be quantized?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} On Dirac's argument, because a wavefunction transported around a
monopole must return to itself, and the two ways of contracting the loop
must agree modulo $2\pi$. The integer is the winding number of the
transition function between the two coordinate patches
(\S\ref{ssec:dirac}). Note the shape of the result: one monopole
anywhere would quantize every charge everywhere.
\end{qabody}

\bigskip
\noindent\textbf{Question 39 (Q-W1-39) (on Hopkins--Singer)}\par\nobreak\smallskip
\noindent\textbf{``What is holonomy, and why is it not in my E\&M textbook?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It is the phase acquired on transport around a closed loop. It is
absent from most textbooks because in a simply connected region every
loop bounds a surface, and then Stokes' theorem makes the holonomy a
consequence of the curvature (\S\ref{ssec:C-holonomy}). It becomes
independent information exactly when loops fail to bound --- around a
solenoid, around a hole, around a compact dimension.
\end{qabody}

\bigskip
\noindent\textbf{Question 40 (Q-W1-40) (on Hopkins--Singer)}\par\nobreak\smallskip
\noindent\textbf{``Why does adding fermions change how the boson field is quantized?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because the fermion determinant contributes a phase, and consistency
of the total requires the boson's flux lattice to sit half a unit off
where one would naively put it. The precise statement is
Eq.~\eqref{eq:G-shift}: the shift is $\lambda/2 = p_1/4$. The elementary
signature needs no topology at all --- $x(x-1)$ steps by an even amount
where $x^2$ steps by an odd one (\S\ref{ssec:qx}).
\end{qabody}

\bigskip
\noindent\textbf{Question 41 (Q-W1-41) (on Hopkins--Singer)}\par\nobreak\smallskip
\noindent\textbf{``Is a cohomology class something I can measure?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} You can measure its consequences, which is the same thing one means
by measuring a charge. The flux through a closed surface is $2\pi n$; the
interference pattern shifts with period $h/q$; the number of families is
three. Each is an integer read off a physical experiment, and the class
is what the integer labels.
\end{qabody}

\bigskip
\noindent\textbf{Question 42 (Q-W1-42) (on Hopkins--Singer)}\par\nobreak\smallskip
\noindent\textbf{``Differential cohomology is very heavy machinery for charge quantization, which Dirac got in 1931 with a wavefunction. Is the top rung necessary or ornamental?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} For the job as stated, ornamental, and the concession should be
unhedged. A single abelian gauge field, one monopole, one
integer: Dirac's argument settles it in a paragraph, using nothing but
single-valuedness of a wavefunction, and $eg = n/2$ needs no
cohomology \citep{Dirac1931}. Anyone who wants only that should
stop at \S\ref{ssec:dirac}.
Three things the refinement buys that Dirac's does not.
First, it handles the field, its potential and the quantization
condition as one object rather than three. The differential cocycle
$(c, h, \omega)$ carries an integer class, a curvature form and a
holonomy phase with a single compatibility condition
(\S\ref{ssec:cocycle}), and it is a genuine refinement: two fields with
identical curvature everywhere can differ physically by a flat
holonomy --- the Aharonov--Bohm situation --- and the cocycle
distinguishes them while the curvature cannot.
Second, the quantization law depends on the fermion content: with
fermions present the flux lattice is displaced by half a characteristic
class, $[G/2\pi] - \lambda/2$ integral rather than $[G/2\pi]$
\citep{Witten1997}. That is not a correction to Dirac's condition; it
changes which configurations exist, and can exclude zero flux
altogether (\S\ref{ssec:witten-shift}). Dirac's argument has no place to
put such a dependence.
Third --- the reason the paper climbs to it --- the class $c$ takes
values in a discrete group, so no continuous deformation moves it. That
is protection without symmetry stated as a theorem about an object
rather than as an observation about a number, and it is the level at
which the statement can be made at all. Appendix~\ref{app:diffcohom} is
the hardest in the paper, and a reader is entitled to judge whether the
three purchases justify it.
\end{qabody}

\section{The GG turn: projection, the Born rule, and the bulk}

Eight on the turn from closed systems to open ones, which is where
this review stops reporting the discipline's consensus and starts
reporting a program. The status question --- whether the Born rule is
assumed or derived --- is answered with its grade printed rather than
rounded up, and the last two ask what a semigroup actually is and what
this adds to forty years of decoherence.


\bigskip
\noindent\textbf{Question 43 (Q-W1-43) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``What does `observation is a projection' mean operationally?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That the theory is not specified until one gives the map from bulk
structure to what is registered: a boundary Hilbert space, a state, the
instruments, and the channels between interventions
(\S\ref{ssec:oggep}). Operationally it means measurement is not an extra
topic bolted on after the dynamics; it is part of stating the dynamics.
The proposition is an axiom, not a derived result.
\end{qabody}

\bigskip
\noindent\textbf{Question 44 (Q-W1-44) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``Why can't a closed theory ever produce an upper bound on a count?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because its tools are admissibility conditions. Anomaly cancellation
says a theory is consistent; integrality says a class lies in the right
lattice. Neither prefers one consistent option to another, and a
four-family world that canceled its anomalies would pass both
(\S\ref{ssec:exclusion}). Excluding a consistent alternative needs a
\emph{cost}, and a cost is a boundary notion --- something must be spent,
and spending is exchange.
\end{qabody}

\bigskip
\noindent\textbf{Question 45 (Q-W1-45) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``Is the Born rule assumed or derived in this framework?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Derived, at a declared grade, and the grade matters. The Born
\emph{form} is settled by a Gleason-type argument on the operational
dictionary. The \emph{dynamical realization} is closed at the relaxation
level and conditional on four named premises, with one analytic residue
outstanding \citep{ArisakaA8QM}. The direction of dependence runs from
the axioms through the boundary law to the Born result and onward, never
backwards, and nothing in the present paper founds quantum mechanics
(\S\ref{ssec:born}).
\end{qabody}

\bigskip
\noindent\textbf{Question 46 (Q-W1-46) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``Is symmetry breaking a loss of information?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} In the cascade picture it is a commitment rather than a loss: the
descent passes a saddle and settles into one basin rather than another,
and what is registered afterwards is which basin
(\S\ref{ssec:gssb}). Information about the alternatives is not destroyed
so much as no longer available at the boundary, which is what the
irreversibility of \S\ref{ssec:gdos} amounts to. Whether to call that a
loss is a matter of where one draws the system.
\end{qabody}

\bigskip
\noindent\textbf{Question 47 (Q-W1-47) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``What would falsify the three-generation claim?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A fourth light family would do it directly. Short of that, the
result is conditional in three declared ways, and each is a place to
push: the empirical input ($CP$ violation), the coupling clause, and one
topological residue (\S\ref{ssec:three-gen}). A reader who rejects the
coupling clause knows exactly which clause they are rejecting, which is
the point of printing it.
\end{qabody}

\bigskip
\noindent\textbf{Question 48 (Q-W1-48) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``Is the six-dimensional bulk `real' or a calculational device?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The program asserts it is real, and this review does not adjudicate
that. What can be said without metaphysics: the question ``where are the
extra dimensions?'' presupposes that they are a larger version of the
room one walks around in, and under \GGEP{} they are not
(\S\ref{ssec:bulk}). The better question is what they do, and the answer
is that they carry the structure the four-dimensional description records
as internal.
\end{qabody}

\bigskip
\noindent\textbf{Question 49 (Q-W1-49) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``What is a semigroup, and why does losing the inverse matter so much?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A group is a set of transformations that compose, contain the
identity, and --- the load-bearing clause --- contain an inverse for
every element. Drop that last clause and what remains is a semigroup:
things still compose associatively, $\Lambda_t \circ \Lambda_s =
\Lambda_{t+s}$, but the parameter runs over $t \ge 0$ only, and there is
no $\Lambda_{-t}$ (\S\ref{ssec:I-step2}). The restriction is not an
omission awaiting repair.
The worked example that makes it concrete is Kadanoff blocking. Average
a spin lattice over $2 \times 2$ cells and you get a new lattice with a
new coupling; the map forward is perfectly well defined, and there is no
map back, because many microscopic configurations gave the same block
average. That is why the renormalization group is called a group and is
not one. In the quantum case the statement is a theorem rather than an
observation: every completely positive trace-preserving map contracts
trace distance, so if a channel and its inverse were both channels,
both inequalities would be equalities and the channel would be unitary.
The only invertible channels are the unitary ones
(Appendix~\ref{app:opensys}). ``No inverse'' therefore means exactly
``the past is not recoverable from the present'' --- not that recovering
it is hard, but that the information determining it is not present in
the current state to be recovered from.
The payoff is a claim about what each kind of theory can produce. A
reversible dynamics admits every consistent history equally and cannot
prefer one; a map that has spent something can be asked what the budget
allows. That is the whole reason Part~III bothers with open systems, and
\S\ref{ssec:exclusion} states it as the difference between an
admissibility condition and a cost.
\end{qabody}
\bigskip
\noindent\textbf{Question 50 (Q-W1-50) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``Decoherence programs have claimed to explain the classical world for forty years. What does this add?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} State the achievement at full strength first, because it is large. Coupling a system to an environment selects a preferred
basis --- einselection --- without anyone choosing it by hand; the
pointer states are the ones the interaction leaves alone; interference
between them is suppressed on timescales so short that the
classical appearance of the everyday world stops being mysterious. That
is a genuine explanation, from ordinary Hamiltonian dynamics.
Now what its own practitioners agree it does not deliver, and this is
not a criticism from outside. It does not say which outcome occurs: a
dephased density matrix is exactly as agnostic about the system's actual
state as the original was, and no equation linear in $\rho$ can produce
a single result. And it does not derive the measure: the weights enter
as the probabilities one wrote down. Gleason's theorem constrains the
\emph{form} of any consistent assignment on Hilbert spaces of dimension
at least three, and says nothing about dynamics --- and fails in
dimension two \citep{Gleason1957}. Both are set out in \S\ref{ssec:I-step2}.
The framework's addition is aimed at the second gap, and its grade must
be stated exactly. The Born \emph{form} is settled by the Gleason-type
argument once the operational dictionary is in place; the
\emph{dynamical realization} --- that the descent into a basin actually
produces those weights --- is closed at the \emph{relaxation} level, and
the Constitutional Born Theorem that assembles it is \emph{conditional}
on four named premises --- the
forward semigroup, a Haar anchor with its recursion, genericity of the
descent, and the basin partition --- with a single analytic residue
outstanding, the field-grade representation step \citep{ArisakaA8QM}.
That is not ``derived'' and should never be written as such
(\S\ref{ssec:born}). It is a conditional theorem with its conditions
printed.
\end{qabody}

\section{The ledger: parameters, mirrors,
confinement, and the excess}
\label{sec:qa-ledger}

Part~\ref{part:ledger} is where the review stops describing a structure
and starts spending it, and it is the Part a referee will press
hardest. The ten questions here follow its chapters in order: what
``zero free parameters'' can honestly mean and what the Standard
Model's own count is; how an integer ratio survives the running of a
coupling; whether $CPT$ is derived or imported; whether $\theta$ is
zero by mechanism or by hand; where the dark-matter line sits against
the instruments built to find it; and what to make of a baryon
asymmetry that lands twenty per cent high. The hardest is the seventh,
because the honest answer is not the flattering one: the compressed
formula carries a postulated exponent, and the silent form it replaced
absorbed the species count outright, so the twenty per cent must be read
as a central value inside a declared band and not as a precision. The tenth asks why a
result the paper itself grades \textsc{flagged} is kept at all.

\bigskip
\noindent\textbf{Question 51 (Q-W1-51) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``What does `zero free parameters' actually mean? Every calculation needs numbers somewhere.''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It does, and this Part never writes the unqualified
phrase for exactly that reason. Two different things get called an
input, and the whole content of the claim is that only one of them is
present. There are inputs of \emph{kind} --- what species of quantity
the construction is allowed to admit at all --- and inputs of
\emph{value} --- a number set by hand and movable afterwards to improve
an agreement. The distinction is not invented for the occasion; it is
the structure of a Wilson line, whose kind is fixed by topology and
whose value is fixed by dynamics, and \S\ref{ssec:kind-value} makes the
point there before the ledger ever uses it.

What is genuinely admitted is a list of twelve items
(\S\ref{ssec:twelve}): five conversion factors $c$, $G_N$, $\hbar$,
$k_B$, $Q_B$; two mathematical constants $e$ and $\pi$; and five
integers $(M,N,L;J,K) = (1,3,5;7,35)$ tied by $K = LJ = M^2+N^2+L^2$ and
$J = 2N+1$. The conversion factors are unit conventions, which every
dimensionful theory performs and none avoids. So the claim, stated in
the form that survives audit (\S\ref{ssec:not-zero}), is: zero free
\emph{continuous} parameters --- no dial adjusted after the theory is
fixed --- plus one frozen discrete completion class whose necessity the
owning papers assert rather than prove, plus named open problems, of
which the charged-fermion mass sheet is the largest.

And note what the claim is really about. It is not that the list is
short; short lists are a matter of taste. It is that the list is
\emph{closed}: that there is nowhere in the construction to bring a
number in from outside. That is a strong claim, and it is a
\emph{reported} one --- the closure theorem carries its owning paper's
grade \citep{ArisakaA4Euclid}, not this review's.
\end{qabody}

\bigskip
\noindent\textbf{Question 52 (Q-W1-52) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``How many parameters does the Standard Model have --- nineteen, twenty-six, or twenty-eight? The literature does not agree, and the count decides how impressive the claim is.''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} All three are correct, for three different theories;
the disagreement is not sloppiness. Nineteen is the pre-1998 theory with
massless neutrinos: three gauge couplings, two Higgs parameters, nine
charged-fermion masses, four CKM entries and $\bar\theta$. Oscillations
add three neutrino masses and four PMNS entries, giving twenty-six. If
the neutrinos are their own antiparticles, two further phases survive
rephasing, and the count reaches twenty-eight. This Part uses the
fullest, $5 + 12 + 4 + 4 + 2 + 1 = 28$ (\S\ref{ssec:count-28}): the gap
between the two is precisely the Dirac-or-Majorana question, and
counting the smaller number assumes an answer nobody has.

None of this is opinion; it is a theorem. Let the couplings live in a
real space of dimension $n_V$, let the flavor group $G$ act by field
redefinitions, and let $H$ be the stabilizer of a generic point. Then
$n_{\rm phys} = n_V - (\dim G - \dim H)$. For the quark sector $Y_u$ and
$Y_d$ carry $2 \times 2 \times 3^2 = 36$ real entries, $G_F = U(3)^3$
has dimension $27$, and the stabilizer is one-dimensional --- it is
$\mathrm{U}(1)_B$, baryon number arriving as the answer to a
linear-algebra question rather than an insight bolted on. Hence $36 -
(27-1) = 10$, decomposing as three up masses, three down masses and four
mixing parameters. Appendix~\ref{app:params} works it in full and gives
$n^2+1$ for $n$ generations.

The count does not decide how impressive the claim is;
\S\ref{ssec:what-derived} does. Of the twenty-eight rows, six are
derived, fourteen derived subject to one named lemma each, eight
asserted; one --- the weak mixing angle --- is a theorem end to end. The
honest formula is the owning paper's: twenty-eight reduced to five
integers, plus named lemmas, plus one measurement.
\end{qabody}

\bigskip
\noindent\textbf{Question 53 (Q-W1-53) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``Couplings run with energy. How can an integer ratio be the value of a running quantity?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It cannot, and the framework does not claim it. An
integer ratio is asserted as the value \emph{at a declared reference
point}; the curve through that point is ordinary renormalization-group
physics, inherited from the boundary theory and predicted by nobody's
ledger.

Take the two cases a reader will check. The fine-structure constant is
quoted twice: $\alpha^{-1}(0) = 2^J + N^2 = 137$ is the zero-momentum
value, against a measured $137.035999177$; $\alpha^{-1}(M_Z) = 2^J =
128$ is the value at the matching scale, against $127.930(8)$. The
$9.106$ that separates them is not a ledger statement at all --- it is a
sum over every charged particle light enough to circulate, and the
electron alone accounts for $-2.566$ of it, twenty-eight per cent, by
one-loop QED (Appendix~\ref{app:params}, \S\ref{ssec:J-running}). The
integers sit at the ends; the flow between them is standard and was
never up for negotiation.

The weak mixing angle is the sharper case, and the sharpest thing to say
about it is against the program's interest. $\sin^2\theta_W = L^2/4N^3
= 25/108 = 0.231481$ agrees with the effective leptonic angle
$0.23155(4)$ to $0.03\%$ --- and sits $+6.5\sigma$ from the
$\overline{\rm MS}$ angle $0.23122(4)$. Both measured numbers are
correct; they are different quantities. A closed form quoted without its
scheme and scale has not yet made a testable statement.

The structural reason the framework expects fixed values anywhere is
that its integers live on the compact circle's holonomy data, which is
quantized and has nowhere to flow, while what runs is the boundary's
readout. That is the same distinction Part~II drew between universality
and quantization: dynamics can protect a ratio from change, but only
topology can select which ratio.
\end{qabody}

\bigskip
\noindent\textbf{Question 54 (Q-W1-54) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``Is $CPT$ derived in this framework, or imported from quantum field theory?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Imported, then given an anatomy --- two different
services, which the paper does not confuse.

The classic theorem is Jost's, on three hypotheses: Lorentz invariance
of the vacuum and $S$-matrix; locality --- fields commuting or
anticommuting at spacelike separation, interactions polynomial at a
point --- and a hermitian Hamiltonian with spectrum bounded below, with
the normal spin-statistics connection. Drop any one and the theorem
goes. The proof runs through analyticity, not algebra: positivity of
energy buys analyticity on the forward tube, Lorentz invariance extends
it to the extended tube where real Jost points live, the total inversion
$x \mapsto -x$ lies in the identity component of the complex Lorentz
group \emph{because spacetime has an even number of dimensions}, and
locality closes the loop (Appendix~\ref{app:discrete}).

What the framework adds is what the three mirrors \emph{are}. The bulk
carries three orientable factors --- the fiber circle, the spatial
slice, the causal clock --- and $C$, $P$, $T$ are their reversals, one
each. The boundary builds its ``multiply by $i$'' out of the clock's
orientation alone, so a mirror is antiunitary if and only if it contains
$T$: the table becomes a one-line computation about which reversal
touches the source of $i$, not an assignment. Kramers' degeneracy and
the absence of a $T$ quantum number fall out.

The grade travels with it: the reading is complementary, not a
replacement. Jost reaches interacting fields and says nothing about what
the mirrors are; the orientation account names the anatomy and proves
the interacting case only conditionally (\S\ref{sec:discrete}). Either
way the stakes are identical: by Greenberg's converse a $CPT$-violating
signal takes Lorentz invariance down with it.
\end{qabody}

\bigskip
\noindent\textbf{Question 55 (Q-W1-55) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``The strong $CP$ problem: is $\bar\theta$ zero here by mechanism, or set to zero by hand?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} By mechanism --- the same closure that confines quarks.
Write the angle as $\bar\theta = \theta_{\rm top} + \arg\det(M_u M_d)$
and read both pieces through the color triangle
(\S\ref{ssec:theta-zero}).

The first piece: the four-dimensional seam is a projected image of the
bulk, whose ledger licenses no free continuous $\theta$-source, and the
color holonomy on the compact cycle is a quantized center element --- a
cube root of unity, not a dial. There is no continuous angle available
to be nonzero. That step is derived. The second piece: the quark
determinant phases are relative holonomies carried around the same
closed loop, and a loop returning to the identity contributes no net
phase. That step is \emph{derived-conditional} --- it rests on the
codebook happening to be real at first order, and is not proved end to
end.

So the standing commitment is not a hard zero but a quantum-defect
leakage $\bar\theta \lesssim 5\times10^{-18}$, seven orders below the
neutron electric-dipole bound of $10^{-10}$; a measured dipole above
that envelope is the falsifier.

On Peccei--Quinn, stated as its advocates state it: a new global
symmetry, spontaneously broken, relaxes $\bar\theta$ dynamically to zero
and leaves a light pseudoscalar as its Goldstone boson --- economical,
predictive of a particle, and a mechanism rather than a fudge. The
disagreement is structural, not empirical. There, $\bar\theta$ remains a
genuine parameter that a field is introduced to relax; here the claim is
that it does not exist, because a closed loop has no phase to fine. The
problem changes shape rather than being solved by an added device. And
the framework's own light line, the Kaxion, does no work here: it lives
on the dark sector's information circle, not the color one.
\end{qabody}

\bigskip
\noindent\textbf{Question 56 (Q-W1-56) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``The Kaxion is a specific mass and frequency. How does it sit against the standard QCD-axion window, and what experiment settles it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It sits inside the region resonant-cavity haloscopes
were built for, which is why it is the cohort's cleanest live falsifier
rather than a promissory note. The commitment is a mass of $8.1 \pm
0.5\,\mu$eV and a laboratory frequency of $1.95$~GHz, with a searched
band of $1.59$--$2.13$~GHz.

Read the two intervals carefully, because they are not the same kind of
interval and they will not shrink for the same reasons. The mass error
is continuous, and it is geometric. The frequency band is the image of
something discrete: the integer cap loading $\Xi_{\rm cap} \in [35,47]$
around its benchmark $\Xi_{\rm cap} = K + N + L = 43$. A null result
excludes different things depending on which of the two it exhausts,
which is why \S\ref{sec:cosmos-sheet} keeps them apart. The band also carries
predicted internal structure --- a doublet split of one part in $K$,
about $56$~MHz --- which is what makes a scan decisive rather than
merely thorough.

Against the standard picture: the QCD axion's mass is tied to its decay
constant by the QCD relation and the classic allowed window spans
decades, so a $\mu$eV line is compatible with that hypothesis but does
not select it. The Kaxion is a different object --- it is not the
Peccei--Quinn field, and it does not carry the strong-$CP$ job, which
\S\ref{ssec:theta-zero} assigns to the closure of the color loop
instead. What distinguishes the two experimentally is the specificity:
four independent handles on one object --- a mass, a bounded frequency
window, a coupling on a known line, and the $1/K$ splitting.

The kill criterion is stated in the falsifier table
(\S\ref{sec:falsifiers}): the window scanned to depth with no line, by
haloscopes, in the window the owning paper declares as 2026--2035. No
renegotiation is available afterwards.
\end{qabody}

\bigskip
\noindent\textbf{Question 57 (Q-W1-57) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``The baryon asymmetry comes out twenty per cent high. Is that a success or a failure?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Neither, and reading it as either is the mistake this
answer exists to prevent. The pipeline gives $\eta_B \simeq 7.04 \times
(28/79) \times (1.25/g_*) \times e^{-K/2} \simeq 7.34\times10^{-10}$ against a
measured $(6.12 \pm 0.04)\times10^{-10}$: a central value $19.9\%$ high.
That number must not be quoted as a precision, and the parent paper says
so before this review does.

Two reasons, both structural. First, the exponent $K/2$ in the
throughput $\epsilon^{\rm res}\kappa = e^{-K/2}$ is \textbf{postulated}.
The chain lands on it --- an explicit seesaw-plus-washout transport
reproduces $2.5\times10^{-8}$ to displayed precision --- but the
compression is geometrically suggestive rather than derived, and its
derivation is an open work order. Second, the compressed formula
took the heavy-state abundance as $n_N/s \sim 1/g_*$, where a
relativistic species in equilibrium gives $0.416/g_*$ and the triplet
gives $1.25/g_*$. That silent form absorbed the species count outright,
comfortably inside the declared factor-$\sim3$ band of uncomputed $O(1)$
coefficients; the displayed form carries it.

So the honest statement is: a central value sitting twenty per cent from
the measurement, inside a band three times wider in each direction. Read
that way it is a consistency check that could have failed by ten orders
of magnitude and did not --- the Standard Model's own CKM route feeds
the inventory at $\sim10^{-20}$, and any account that lands within a
factor of a few has done something. What it is not is a measurement of
anything, and the genuine claims are elsewhere: the reconstruction of
$e^{-K/2}$ by an independent transport calculation, and the dial count
of zero. The assembled result carries grade D$^*$ with the named
on-resonance input attached (\S\ref{ssec:lepto-grade}).
\end{qabody}

\bigskip
\noindent\textbf{Question 58 (Q-W1-58) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``Sakharov's third condition needs a departure from equilibrium. What supplies it here?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The descent itself, at a specific address. The lock
crossing sits at $M_{\rm lock} = \tfrac12 M_{\rm Pl} e^{-K} \simeq
3.85$~TeV, where the Majorana seesaw partners --- parked at the lock
with percent-level splittings of order $1/K$ --- fall out of equilibrium
and decay (\S\ref{ssec:sakharov}). Two features matter more than the
address. The crossing is \emph{prompt}: no supercooling, no secondary
inflation, no late entropy release, so an asymmetry made there survives
undiluted and the microwave background counts the same coin an age
later. And $3.85$~TeV is collider-adjacent, which converts the account
from a story into a schedule.

Underneath the address is the structural answer, and it is why a closed
theory cannot supply the third condition and this one can. Transport
along the open scale fiber is a \emph{semigroup} --- no inverses, no
return --- while transport on the compact circle is a group, all
holonomy and all memory (\S\ref{ssec:symmetry-reading}). A group has no
arrow. A semigroup has one intrinsically, and each gate is crossed at a
definite radial coordinate and cannot be recrossed. So no first-order
phase transition is needed to manufacture irreversibility --- fortunate,
since at the measured Higgs mass the Standard Model's electroweak
transition is a smooth crossover and supplies none.

What remains conditional, plainly. The semigroup supplies
\emph{directedness} without recourse to typicality; it does not by
itself supply the \emph{gradient} --- the observed rate of entropy
production and the smallness of the early-universe entropy still ride on
the descent's boundary data (Appendix~\ref{app:time}). And the transport
block that turns this epoch into a number carries the cohort's sharpest
named input, the on-resonance condition.
\end{qabody}

\bigskip
\noindent\textbf{Question 59 (Q-W1-59) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``Is the proton stable in this framework, and if not, at what lifetime?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Stable --- not long-lived, stable --- at the owning
paper's statement, conditional on one named descent step. Baryon number
is the winding of the compact cycle: the center identity gives $B = t/3$
on every fragment of the spectrum, exactly and not merely modulo one, so
a quark is a third of a wrap and a baryon is one full wrap
(\S\ref{ssec:lock-winding}). A closed loop cannot be continuously
unwound.

The guard has to be stated so that it forbids something, and this is
where loose versions fail. Take the channel that matters, $p \to
e^{+}\pi^{0}$. Both sides are color singlets of total triality zero, so
the singlet rule of \S\ref{ssec:singlet-rule} permits it; and $B-L$ is
conserved across it, so a $B-L$ guard permits it too. That channel is
why grand unification expects proton decay. What forbids it here is the
exact additive charge: the rigid $\mathrm{U}(1)_B$ survives the
Stueckelberg mass of its boson, no local operator moves the winding
integer, and $\Delta B = 1$ has no carrier. The second guard closes the
one door that does change $B$: the sphaleron, which moves $\Delta B =
\Delta L = 3n$ --- three units at a time, one per generation --- and so
cannot deliver a single unit either. Two guards, and the interesting
channel fails both while passing the color test.

Where a number is wanted against detector reach, the benchmark the
geometry's own suppression returns is about $3\times10^{35}$~yr, and the
cohort's falsifier ledger carries the more conservative $\tau_p >
10^{35}$~yr --- both above the Super-Kamiokande floor and inside
Hyper-Kamiokande and DUNE's reach in the 2030s. But the row is not a
value with a tolerance. It is a prohibition, and one event in any
channel ends the program with nothing to renegotiate
(\S\ref{ssec:sharpest}).
\end{qabody}

\bigskip
\noindent\textbf{Question 60 (Q-W1-60) (on the ledger)}\par\nobreak\smallskip
\noindent\textbf{``The age-of-the-universe ledger is graded FLAGGED. Why keep a result you do not believe?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Separate two claims sitting in the same equation; only
one is flagged. The chronology closes: $\ln(t_0/t_{\rm Pl}) = 140.24 =
36.11 + 91.60 + 12.53$, launch plus radiation era plus
matter-and-$\Lambda$, the segments computed by integrating
$\mathrm{d}\ln t/\mathrm{d}r = 2$ through the epochs. Against the direct
evaluation from $t_0 = 13.787$~Gyr and $t_{\rm Pl} =
5.3912\times10^{-44}$~s, which gives $140.243$, that is a real check ---
standard kinematics plus a derived reheating temperature --- and it
passes (\S\ref{ssec:time-chronology-w1}).

What is flagged is the coincidence beside it: $4K = 140$, and
$140.24/140 = 1.0017$. Here is the deflation, worth arithmetic rather
than admiration. Agreement to $0.17\%$ in a logarithm of order $140$
means the underlying ratio $t_0/t_{\rm Pl}$ differs from $e^{4K}$ by a
factor $e^{0.243} = 1.275$. To make the identity exact the universe
would have to be $10.8$~Gyr old rather than $13.8$ --- excluded many
times over. So a mechanism that predicted $4K$ exactly would be in
\emph{tension} with the measured age, not in support of it. Two parts in
a thousand in the log is twenty-eight per cent in the quantity.

Which answers the question: it is kept \emph{because} it is flagged. The
same small alphabet of integers that produced this Part's hits also
produces near-misses, and a framework that printed only the hits would
make its own trials factor unauditable --- which is to say,
unfalsifiable. Writing the near-miss down, labeling it arithmetic
without a mechanism, and refusing to lean on it is what keeps the
skeptic's question answerable. What would change the grade is a
derivation of the $O(1)$ factor; until one exists, the honest thing to
do with a suggestive integer is exactly this (Appendix~\ref{app:time}).
\end{qabody}

\section{The program as a whole}

Six that step back from the rungs and ask about the thing they are
rungs of. The two that matter most are adjacent and deliberately so:
the single strongest objection to the program, and the single
strongest piece of evidence for it. A reader who reads only those two
will have the shape of the disagreement.


\bigskip
\noindent\textbf{Question 61 (Q-W1-61) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``If gauge symmetry isn't a symmetry and Bianchi isn't a law, what is fundamental?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} On the reading of this paper: consistency requirements. Not
``nature possesses this invariance'' but ``no description can be built
otherwise''. The top of the ladder is the clearest case --- a phase must
close when carried around a loop, and an integer follows with no symmetry
anywhere in the argument (\S\ref{ssec:top-of-ladder}).
\end{qabody}

\bigskip
\noindent\textbf{Question 62 (Q-W1-62) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Why should constants of nature be integers?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Most are not. The observation is narrower and stranger: the ones that
are known \emph{exactly} are integers --- the charge ratio to better than
one part in $10^{21}$, the three colors, the three families
(\S\ref{ssec:integers}). The paper's answer is that a quantity taking
values in $\mathbb{Z}$ cannot degrade under a small symmetry breaking,
whereas a Noether charge can; so in a thoroughly broken world the
exactly-known quantities are precisely the counts
(\S\ref{ssec:counting-survives}).
\end{qabody}

\bigskip
\noindent\textbf{Question 63 (Q-W1-63) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``What is the single strongest objection to this program?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That a minimality axiom does load-bearing work. \S\ref{ssec:exposure}
names it: the trace ratio admits the family $(5k,3k)$, and only
\AXkw{}-4 selects $k = 1$. That is exactly the shape of the reasoning
Chapter~\ref{sec:honest-record} convicts in the naturalness case --- a
principle imposed because the alternative is unforced --- and the
program is on the wrong side of its own line there. The paper's answer
is not that the premise is safe but that it is named.
\end{qabody}

\bigskip
\noindent\textbf{Question 64 (Q-W1-64) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``What is the single strongest piece of evidence for it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The $\kappa$-bridge (\S\ref{ssec:kappa}), because it is agreement
rather than fit. Two descriptions with no shared working parts --- a
topological pushforward from six dimensions and the Berry bundle of
ordinary boundary quantum mechanics --- compute the same object and
return the same answer, with no adjustable quantity anywhere. Whether one
finds that persuasive depends on how one weighs unforced agreement, but
it is a different kind of evidence from a successful parameter fit.
\end{qabody}

\bigskip
\noindent\textbf{Question 65 (Q-W1-65) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Where has symmetry-first reasoning misled physics?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Three times, at least. Parity was assumed until 1956 and is violated
maximally. $CP$ was the repair and broke in 1964. Naturalness predicted
superpartners near a TeV and the preferred region is now excluded
(\S\ref{ssec:failures}). The pattern is sharper than ``symmetry reasoning
is unreliable'': symmetry read \emph{off the data} has an excellent
record, and symmetry imposed \emph{for beauty} has a poor one.
\end{qabody}

\bigskip
\noindent\textbf{Question 66 (Q-W1-66) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``What should I read next, and in what order?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} For the standard physics: \citet{Nakahara2003} for the geometry, then
\citet{PeskinSchroeder1995} Ch.~19 for anomalies, then
\citet{HopkinsSinger2005} last and slowly. For the program: the
constitution \citep{ArisakaA1Constitution} for the axioms, then the
boundary paper \citep{ArisakaA5GDOS} for the open-system move, then the
family-number paper \citep{ArisakaP2N3} for the sharpest claim and its
conditioning. If you read one member paper, read the last of those, and
read its conditioning section first.
\end{qabody}


\section{Wilson and Ward: the two steps the short list skipped}

Five on the fourth and fifth rungs, which a short telling of this
story tends to jump over. The most useful is the second --- how an
object that depends on the choice of gauge at every point along its
length can close into something physical --- because the answer is the
mechanism the whole nonlocal half of gauge theory runs on.


\bigskip
\noindent\textbf{Question 67 (Q-W1-67) (on Wilson)}\par\nobreak\smallskip
\noindent\textbf{``What is a Wilson loop, and what does it measure that a local field cannot?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A Wilson loop is the phase a charged probe accumulates when carried once around a closed path, $W=\exp\!\big(i\oint A\big)$. Its value is gauge-invariant even though the potential $A$ inside the integral is not, so it is a genuine observable built from an unobservable. What it measures is what no local field at a point can report: the \emph{holonomy} --- how much the internal phase has been twisted around the loop as a whole. Two configurations can agree on the field strength everywhere a probe travels and still disagree on the loop, which is why the loop, not the local field, is the honest carrier of gauge content. Step~S4 of this paper is the rung on which that fact becomes structural.
\end{qabody}

\bigskip
\noindent\textbf{Question 68 (Q-W1-68) (on Wilson)}\par\nobreak\smallskip
\noindent\textbf{``If a Wilson line depends on the choice of gauge, how can the loop be physical?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because closing the path is exactly what cancels the dependence. An open Wilson line from one point to another rotates under a gauge change by factors attached to its two endpoints; when the endpoints coincide --- a loop --- those factors are inverses and annihilate, leaving a number every observer agrees on. The lesson generalizes past electromagnetism and is one this paper leans on repeatedly: the physical content of a redundant description lives in its \emph{closed} objects, the ones on which the redundancy has nothing left to act.
\end{qabody}

\bigskip
\noindent\textbf{Question 69 (Q-W1-69) (on Ward)}\par\nobreak\smallskip
\noindent\textbf{``What is the Ward identity, in one paragraph and without diagrams?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A Ward identity is the quantum survival certificate of a classical conservation law. Classically, a continuous symmetry gives a conserved current (Step~S1); quantizing the theory could in principle spoil that conservation, and the Ward identity is the exact statement that, for the symmetries the measure respects, it does not --- the current stays conserved inside every correlation function, order by order, not just on the classical solution. It is the reason a symmetry argument can be trusted after quantization rather than merely before it, and Step~S5 is where the paper cashes that trust.
\end{qabody}

\bigskip
\noindent\textbf{Question 70 (Q-W1-70) (on Ward)}\par\nobreak\smallskip
\noindent\textbf{``Why does the Ward identity survive quantization when a naive symmetry argument would not?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because it is a statement about the \emph{measure}, not about a single field configuration. A naive symmetry argument asks whether the action is invariant; the Ward identity asks whether the action \emph{and} the integration measure are, together. When both are, the identity holds exactly and the current is conserved in the full quantum theory; when the measure is not invariant, the failure is an anomaly (Steps~S6 and the entry on anomalies above). So the Ward identity and the anomaly are two outcomes of the same question --- does the measure share the action's symmetry? --- which is why this paper treats them as neighboring rungs rather than distant topics.
\end{qabody}

\bigskip
\noindent\textbf{Question 71 (Q-W1-71) (on Wilson)}\par\nobreak\smallskip
\noindent\textbf{``The Aharonov--Bohm effect keeps being cited. What exactly does it settle about the reality of the potential?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It settles that the potential carries physical information the field strength does not. An electron kept entirely in a region where the magnetic field is zero still has its interference pattern shifted by a solenoid it never touches, and the shift is precisely the Wilson loop enclosing the flux. So the local field cannot be the whole story --- something non-local, the holonomy of the potential around the inaccessible region, is doing measurable work. The effect does \emph{not} promote the gauge-dependent potential at a point to an observable; it promotes the gauge-invariant loop. That distinction --- redundant locally, physical globally --- is the whole content of Step~S4.
\end{qabody}

\section{Symmetry breaking, the arrow of time, and why anything exists}

Six on the discussion Part. The hardest is the last: whether the
second law comes out of this framework or goes into it. The response
does not claim more than the structure supplies, and it concedes what
Loschmidt and Zermelo established rather than treating them as
historical curiosities.


\bigskip
\noindent\textbf{Question 72 (Q-W1-72) (on Part V)}\par\nobreak\smallskip
\noindent\textbf{``If perfect symmetry is `a description of nothing at all', why did physicists ever expect the world to be symmetric?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because symmetry has been an extraordinarily good guide, and its failures are rarer than its successes and easier to forget. Every conserved quantity a chemist trusts, every selection rule, every multiplet in the particle tables came from taking a symmetry seriously, so the habit of expecting more symmetry is well earned. The paper's claim is not that the habit is foolish but that it has a characteristic failure mode --- it misleads exactly when a symmetry is \emph{imposed} for its beauty rather than \emph{read} off a measurement, and the honest record (Chapter~\ref{sec:honest-record}) lists three occasions when it did. A world in \emph{exact} possession of all its symmetries would contain no matter, no structure, and no handed chemistry; what we have is symmetry plus exactly the breaking that permits existence, which is a more delicate object than the symmetry alone.
\end{qabody}

\bigskip
\noindent\textbf{Question 73 (Q-W1-73) (on Part V)}\par\nobreak\smallskip
\noindent\textbf{``Is the arrow of time derived in this framework, or assumed?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Assumed, and the paper says so at the one place it matters most --- the reference apparatus flags, as the single row to read first, that the arrow of time is an axiom of the program, not a theorem of it. What \emph{is} derived, given that arrow, is a great deal: that the descent runs monotonically downhill in a Lyapunov functional, that its conservation laws survive as identities of the whole descent rather than symmetries of its endpoint, and that the breaking accumulates in a definite order. But the directedness itself is inherited as a constitutional input from the open-system boundary law \citep{ArisakaA5GDOS}, and calling it derived would be exactly the kind of overclaim this paper is built to avoid.
\end{qabody}

\bigskip
\noindent\textbf{Question 74 (Q-W1-74) (on Part V)}\par\nobreak\smallskip
\noindent\textbf{``You call breaking `the productive step'. Isn't that just relabeling a defect as a virtue?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It would be, if the relabeling bought nothing, so the test is whether it buys anything. It does: placing the breaking in the \emph{middle} of a pipeline, with stages before and after, makes the count of the world's families a count of breakings, ties the generations to the critical points of a descent rather than to three copies of a thing, and lets exact integer statements survive a broken world instead of degrading with it. A relabeling that reorganizes those facts is a reading, not a slogan. The paper also states plainly that the word ``productive'' is its own emphasis rather than the cohort's, and that a reader who dislikes it may drop it and lose no result --- which is the opposite of hiding a defect behind a virtue.
\end{qabody}

\bigskip
\noindent\textbf{Question 75 (Q-W1-75) (on Part V)}\par\nobreak\smallskip
\noindent\textbf{``The Concluding remark retired the word `design' for `forced'. If both describe the same world, what is the difference?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The difference is everything that matters to a scientist: one word can be taken away by a measurement and the other cannot. ``The world is designed'' names no step, forbids no observation, and survives whatever a telescope finds --- which is precisely why it explains nothing about \emph{this} world in particular. ``What is forced is symmetry plus exactly the breaking that permits existence'' names a mechanism, and a mechanism can fail: a fourth light family, cosmic shares off the $\pi$-locked values, a tensor ratio far from the predicted one would each end it. The Concluding remark stops using the older word not because the conviction weakened but because it finally became specific enough to lose, and the cohort states the same conclusion in four words --- not contingent, not lucky, not designed; forced \citep{ArisakaA11Origin}.
\end{qabody}

\bigskip
\noindent\textbf{Question 76 (Q-W1-76) (on Part V)}\par\nobreak\smallskip
\noindent\textbf{``What are the three horizon problems, and why does the third need more than inflation?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} They are a ladder of samenesses across regions that never exchanged a signal. The first, from 1981, is sameness of \emph{conditions} --- one temperature to a part in $10^5$ --- and inflation answers it. The second is sameness of \emph{things} --- an electron here identical to one across the sky --- answered by the identity of the underlying geometry. The third is sameness of \emph{choices} --- every region chose matter over antimatter with one sign, one yield, one mechanism --- and inflation alone cannot supply it, because inflation stretches a choice but does not make the choosing deterministic. That last rung needs the breaking itself to select a unique outcome rather than sample a menu, which is the cohort's constitutional commitment; the ladder is owned and answered in the cosmology paper \citep{ArisakaA10Cosmos}.
\end{qabody}

\bigskip
\noindent\textbf{Question 77 (Q-W1-77) (on Part V)}\par\nobreak\smallskip
\noindent\textbf{``Does the second law come out of this framework, or go into it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A little of both, stated carefully. The \emph{direction} goes in: it is the arrow-of-time axiom of the previous answer. The \emph{monotonicity} comes out: given that direction, the open-system boundary law makes a definite functional decrease along every admissible trajectory, so that ``entropy does not decrease'' becomes a theorem of the descent rather than a separate postulate \citep{ArisakaA5GDOS}. So the framework does not claim to conjure the second law from nothing --- no honest framework can, since the direction has to enter somewhere --- but it does claim to reduce two laws to one axiom plus a mechanism, which is the kind of economy the whole program is after.
\end{qabody}

\section{The multiverse, the landscape, and what can lose}

Six on the position this review most directly opposes, stated at
full strength before it is answered. The decisive one is the fifth ---
what single measurement could kill this program and leave the
landscape untouched --- because that asymmetry, and not any argument
about elegance, is the disagreement. The sixth turns to many-worlds,
which is the same structural complaint in a quantum register.


\bigskip
\noindent\textbf{Question 78 (Q-W1-78) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Isn't our universe just one bubble in a vast multiverse, and we happen to live in the one that works?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That is one influential story, and it is a hypothesis rather than a finding --- the one this program is built to make unnecessary. The multiverse explains life-friendly numbers by giving up on deriving them: they become an accident of our address in an unobservable ensemble. This paper takes the opposite route, treating the numbers as outputs to be computed from one geometry with no dial to turn. If that succeeds there is nothing for a multiverse to do, because there was never a menu of alternatives to select from. The honest status is that several target values are still conditional on named, stated conjectures; but the wager is clear and it is testable this decade, which no multiverse claim is.
\end{qabody}

\bigskip
\noindent\textbf{Question 79 (Q-W1-79) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``String theory has no adjustable coupling either --- the freedom is in the vacuum. Why is this different from the landscape?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The location of the freedom is the whole difference. In the landscape picture the theory admits an astronomical number of self-consistent vacua and nothing selects ours, so the constants become a draw. This program's central bet is a single word against that: uniqueness --- that the constitutional axioms plus the anomaly and integer ledgers leave \emph{one} admissible geometry, not a vast family. Whether that uniqueness is fully proved is exactly the boundary the paper marks honestly; the internal consistency is shown, the no-other-geometry theorem is a stated conjecture. What is not conjectural is the exposure: a theory with no continuous free parameters cannot retreat to a landscape when the data press, because it has no knobs left to re-tune.
\end{qabody}

\bigskip
\noindent\textbf{Question 80 (Q-W1-80) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Do the three horizon problems falsify the multiverse?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It is tempting to say yes, and the truth is sharper and worth more. The landscape does not contradict the observed samenesses; it accommodates them, by the same early-choice-then-inflate mechanism this program uses, and it stores its variety outside the horizon where no measurement reaches. So the samenesses cannot execute it. What the three problems \emph{do} kill is every account in which the choosing happens \emph{late}, after regions fall out of contact: such a sky would carry matter--antimatter annihilation seams, and it carries none \citep{ArisakaA10Cosmos}. That drives every framework, this one included, into the early-choice corner --- and there the landscape can no longer explain why \emph{this} vacuum, \emph{this} sign; it can only report that each happens somewhere. The right verdict is not falsification but exile: the sameness of the sky sends the multiverse to a place where it can neither explain nor be tested.
\end{qabody}

\bigskip
\noindent\textbf{Question 81 (Q-W1-81) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{```Everything possible happens somewhere' does cover the facts. Why refuse to count it as an explanation?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Because it has the same logical shape as a claim this paper also declines from the other side of the aisle: \emph{the world was designed}. Both statements are compatible with anything a telescope could ever find; neither names a step that could have failed or risks anything on tomorrow's data. That is exactly why neither functions as an explanation of the particular world in front of us. An explanation, in the sense this paper tries to practice, is a chain with exposed links --- name the mechanism, compute the number, say what reading breaks it. The same razor cuts both comfortable alternatives, the theological and the statistical, and for the same reason; the paper accepts the discipline of being losable instead.
\end{qabody}

\bigskip
\noindent\textbf{Question 82 (Q-W1-82) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``What single measurement could kill this program and leave the landscape untouched?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The cleanest is the primordial tensor-to-scalar ratio. The cohort's launch is a derived plateau predicting a specific small value that CMB-S4 and LiteBIRD are built to decide this decade \citep{ArisakaA10Cosmos}; a result far from it, in either direction, ends the derivation outright. The landscape is inconvenienced not at all by the same measurement, since there is always another vacuum to accommodate whatever is seen. That asymmetry is the point of the whole exercise: one program has placed a two-sided bet a funded instrument settles, the other retains the right to reinterpret the outcome, and the reader may decide which posture better deserves the word \emph{prediction}.
\end{qabody}

\bigskip
\noindent\textbf{Question 83 (Q-W1-83) (on the GG turn)}\par\nobreak\smallskip
\noindent\textbf{``What about many-worlds --- every quantum possibility realized in a branching reality?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That is a different proposal from the cosmological multiverse and the paper's stance on it is narrower. The branching picture is one reading of the measurement problem; this framework's relevant claim is about the Born rule, which it treats not as a postulate but as an output of the boundary law's relaxation to a fixed measure \citep{ArisakaA8QM}, closed at the relaxation level with a single analytic step still open. If that program succeeds, the statistics of outcomes are forced by the dynamics rather than distributed over branches, and the explanatory work the many-worlds picture is invoked to do is done instead by a computed measure. The paper does not claim to have refuted the branching interpretation; it claims to make it unnecessary in the same sense it makes the landscape unnecessary --- by deriving what the alternative would leave to be posited.
\end{qabody}

\section{Integers, $\pi$, and the numerology worry}

Five on the objection a careful reader should raise first and press
hardest. The strongest is the look-elsewhere question: how many targets
the ledger was allowed to aim at, and where that accounting is. The
honest answer is that nobody has computed it, which is why the response
directs the reader to the falsifiers instead of to the arithmetic.


\bigskip
\noindent\textbf{Question 84 (Q-W1-84) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``With $\pi$, $e$, and a few integers you can approximate almost any number. How is the ledger not curve-fitting?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The worry is the right one, and the answer is that the ledger is not allowed to shop. Curve-fitting has one free choice per target; the program's claim is that a \emph{single} small set of integers $(1,3,5;7,35)$, fixed once by an anomaly-closure condition, is spent across many unrelated targets at once, so that the same integer must work in a collider coupling and in a recombination redshift or the whole scheme fails together. The discipline that separates the two is stated as a rule the paper holds itself to: every count is to be derived and verified before it is typeset, and the look-elsewhere question (next entry) must be answered rather than waved away. Where that discipline is not yet met, the entry is graded conditional rather than claimed.
\end{qabody}

\bigskip
\noindent\textbf{Question 85 (Q-W1-85) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``$2^7+3^2=137$ is Eddington redux. With a small alphabet of integers and constants one reaches most targets. How is this different?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Eddington's failure is the correct cautionary tale, and the honest difference is not that this program is immune but that it is \emph{exposed} where he was not. Eddington adjusted his combinatorics to hit a single measured number; the claim here is that one ledger, fixed by a prior anomaly condition, is forced to hit \emph{many} numbers simultaneously and stakes falsifiers on the ones not yet measured. Whether the small-integer coincidences survive a proper look-elsewhere accounting is a question the paper says must be settled quantitatively, not asserted; until it is, the pattern is offered as an observation to be earned, not a proof. The reader is right to keep the Eddington precedent in hand --- so does the paper.
\end{qabody}

\bigskip
\noindent\textbf{Question 86 (Q-W1-86) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Were the ledger integers chosen before or after the data were known?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} After, unavoidably --- the couplings and redshifts they are matched to were measured long ago, so no one can honestly claim a blind prediction of them. What can be claimed is structural priority rather than temporal: the integers are fixed by an anomaly-closure condition stated without reference to those data, and the same integers are then reused elsewhere with no further freedom, so a later measurement they were not tuned to can still break them. The program's answer to the ``chosen after'' worry is therefore not a claim of innocence but a set of standing bets on quantities not yet measured, which is the only form of before-the-fact a mature theory of already-measured constants can offer.
\end{qabody}

\bigskip
\noindent\textbf{Question 87 (Q-W1-87) (on Hopkins--Singer)}\par\nobreak\smallskip
\noindent\textbf{``Why should the same integer $J=7$ fix both a collider coupling measured at $91$~GeV and a redshift in the recombination era?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That coincidence, if it holds, is the strongest single thing the paper can put in front of a skeptic, and it is also where the paper is most careful. One integer, $J=7$, fixes a coupling measured at a collider and, through $1+z_{\rm sw}=e^{J}$, a topological marker eleven orders of magnitude away in scale and a century away in the laboratory. The care is in what the marker is: it is the corridor switch of the descent and \emph{not} the epoch of atom formation, and the two are distinguished exactly where they are derived, so the agreement is not quietly borrowed from the more familiar number. A single integer doing two unrelated jobs is a stronger fact than one number being fit twice --- provided the jobs really are unrelated, which is why the paper insists on separating them in print.
\end{qabody}

\bigskip
\noindent\textbf{Question 88 (Q-W1-88) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``The look-elsewhere effect: how many targets did the ledger have to hit, and where is that accounting?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} This is the objection the paper takes most seriously and does not yet claim to have closed. A fair look-elsewhere accounting must count the targets the ledger was allowed to aim at, the operations permitted in forming each monomial, and the tolerance counted as a hit, and only then ask how surprising the collection of matches is. That global trials factor has not been computed, and until it is, the correct reading of the numerical agreements is suspended rather than triumphant. The paper's position is that the burden is real and lands on the program, not the reader --- which is why the strongest claims are staked on \emph{forced} relations and on not-yet-measured quantities, where the look-elsewhere worry has no purchase.
\end{qabody}

\section{Method, honesty, and the cohort}

Five on the standing of the work rather than its content: single
authorship, absent refereeing, a review that imports its own central
results. The sharpest is the fourth --- what survives if the
conditional results are never closed --- and the answer is a short
list, given explicitly, rather than a reassurance.


\bigskip
\noindent\textbf{Question 89 (Q-W1-89) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``The cohort is single-author, unrefereed preprints. Why should anyone accept it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} No one should accept it on authority, and the paper does not ask them to --- it asks to be checked. Single authorship and the absence of peer review are reasons for scrutiny, not dismissal, and the program's answer to both is the same: every quantitative claim is meant to be regenerable from stated inputs, carries an explicit grade, and names what would break it. That is a heavier exposure than most refereed work accepts, and it is deliberately so, because a program that arrives all at once and outside the usual channels has to hand its critics the tools to attack it. Acceptance is the wrong frame; the right one is whether the falsifiers survive contact with the data, and that is a question for the sky and the community, not for me.
\end{qabody}

\bigskip
\noindent\textbf{Question 90 (Q-W1-90) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``The coordinated papers of the cohort arriving at once is unusual. How should the community judge a program that does not arrive incrementally?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} By its exposed seams rather than its arrival. An incremental literature lets the community test each claim as it lands; a program delivered whole denies that convenience and owes something in return --- an explicit map of what rests on what, so a reader can pull any single thread and see what unravels. That is what the grades and the cross-paper dependencies are for: they turn ``take it or leave it whole'' into ``here is exactly which result to attack first.'' The unusual delivery is a fair reason for caution and an unfair reason for dismissal; the honest test is the same one applied to any theory, whether a named measurement can end it, and the program has tried to make that test easy rather than hard.
\end{qabody}

\bigskip
\noindent\textbf{Question 91 (Q-W1-91) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``This paper is a review that imports its central results. What does it itself prove?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Very little in the way of new theorems, and it says so --- its contribution is not a fresh derivation but a reading. What is its own is the arc: the claim that seven results from Noether to Hopkins--Singer form a single ladder in which conservation laws grow harder to break as they say less about motion, and that the same ladder, read one way, is a review of established physics and, read another, the frame for a proposal at declared grades. The Part~III proposal leans on the cohort for its content and is graded accordingly; the paper is careful that a reader may accept the review and reject the proposal without inconsistency. Calling a synthesis a proof would be the overclaim the paper exists to avoid; its value, if it has one, is organisational and interpretive, and it stakes only what a review can honestly stake.
\end{qabody}

\bigskip
\noindent\textbf{Question 92 (Q-W1-92) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``If several of the `conditional' results are never closed, what survives?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The review survives entirely, because it rests on established physics that no conjecture of this program can touch. Above that, the structural observations survive: that the breaking sits in the middle of a pipeline, that families count breakings, that some statements fail only in whole numbers and so survive a broken world. What would be lost is the strong form of the proposal --- the specific derived values that depend on the named open lemmas --- and the paper is arranged so that loss is visible rather than hidden, each dependent claim carrying its grade and its exposure. A reader who bets against every conjecture still keeps the ladder; that is the sense in which the paper tries not to make its whole worth hostage to its riskiest step.
\end{qabody}

\bigskip
\noindent\textbf{Question 93 (Q-W1-93) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Name the single result whose collapse would do the most damage.''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The uniqueness of the geometry. Almost everything the proposal claims beyond the review --- that the numbers are outputs rather than choices, that there is no landscape to be lost in, that the samenesses of the sky are inherited rather than arranged --- rests on there being \emph{one} admissible geometry, and that no-other-geometry theorem is a stated conjecture rather than a closed result. If a second admissible geometry were exhibited, the program would not merely lose a prediction; it would lose the reason its predictions are forced at all, and would fall back to being one model among many. The paper names this as the load-bearing exposure rather than burying it, on the principle that a theory should announce where it is most likely to break.
\end{qabody}

\section{Necessity, mind, and why there is something}

Nine that a reader carries home after the physics is done. Several
are the technical cousins of the curious reader's questions at the head
of this Part and are answered here at depth. The hardest is the one
about circularity: whether physics written by humans concluding that
humans were inevitable can be anything but a loop. It is answered by
showing that the loop is traversed once, in one direction, with no step
reaching back.


\bigskip
\noindent\textbf{Question 94 (Q-W1-94) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Isn't `necessity' just a mathematical designer replacing a divine one --- moving the mystery rather than solving it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The two are different in the one way that matters: a designer explains by intention and forbids nothing, while necessity explains by mechanism and forbids a great deal. If a constant is the forced output of one geometry, it could not have been otherwise, and there is no chooser left with anything to choose --- the value stands whether or not anyone is present to admire it. The mystery is not moved so much as narrowed: it retreats from ``why these values'' to ``why this constitution,'' which the paper marks openly as the place the regress currently stops. That is a smaller and sharper mystery than a designer, and, unlike a designer, it is one a measurement can still argue with.
\end{qabody}

\bigskip
\noindent\textbf{Question 95 (Q-W1-95) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Is the anthropic principle the real explanation, and `necessity' just anthropics in disguise?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} They are opposite kinds of explanation, and a measurement separates them. Anthropic reasoning presupposes a range of possible values and prunes it by our existence, so it predicts constants scattered anywhere in the life-permitting window, typically near its edge. A necessity claim denies the range: the value is forced, so it should sit at a specific place fixed before any observer existed, at the center of nothing in particular. This program predicts specific values and stakes them; if they were merely anthropically allowed they would not need to be exactly those. The paper's own section against the anthropic move makes the further point that its derived launch places the relevant sameness \emph{before} any matter existed, so it is inherited from one causal patch rather than selected region by region --- an argument about ordering that a selection account cannot make.
\end{qabody}

\bigskip
\noindent\textbf{Question 96 (Q-W1-96) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``If everything is forced by geometry, is the universe deterministic --- is my future already written?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} The cohort's determinism is a statement about the descent, not a claim about your biography, and the distinction is worth keeping. What the framework asserts is that the cascade selects a unique attractor at each layer rather than sampling a menu --- determinism up to gauge, in the constitutional sense \citep{ArisakaA11Origin} --- which is what the third horizon problem needs and what makes the sky's uniform choices intelligible. It does \emph{not} follow that a mind is a filmstrip already exposed: the same framework carries the Born rule as a relaxed measure rather than a certainty \citep{ArisakaA8QM}, and the question of freedom lives in the cognitive layer treated by the companion papers, not in the cosmological one. The safe short answer is that the \emph{laws} are forced; how much that constrains a person is a different question the physics here does not settle.
\end{qabody}

\bigskip
\noindent\textbf{Question 97 (Q-W1-97) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Where do observers and minds fit? Is consciousness an afterthought bolted onto the physics?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} Not an afterthought in the cohort's telling, though this paper is not where that case is made. The origin companion treats mind as the far end of a single cascade --- the same descent that yields particles and chemistry continuing, layer by layer, to cells, to nervous systems, to a world-model that closes a loop with the universe it models \citep{ArisakaA11Origin}. Whether that continuity succeeds as an explanation of consciousness is well outside what a review of symmetry and conservation can adjudicate, and it would be a category error to claim it here. What this paper contributes is only the bottom of that ladder: the sense in which the world's exact statements survive its breaking, which is the precondition for there being stable things to be an observer made of at all.
\end{qabody}

\bigskip
\noindent\textbf{Question 98 (Q-W1-98) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Why is there something rather than nothing? Does this framework even touch it?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It touches it and then stops, deliberately, at the line it cannot cross. The framework can argue why, \emph{given} that there is a world, the world has the particular structure it does --- forced integers, a handed chemistry, an arrow of time --- and even that a symmetric ``nothing-like'' state is empty of matter and structure, so that existence and breaking come together. What it does not claim is to explain why there is a constitution to be satisfied at all, rather than no constitution and no world; that is the place the regress halts, and the paper marks it as such rather than dressing a physical mechanism in metaphysical clothes. Honesty about where the explanation ends is part of the explanation's value.
\end{qabody}

\bigskip
\noindent\textbf{Question 99 (Q-W1-99) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Isn't it circular for physics written by humans to conclude that humans were inevitable?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It would be circular only if human existence were used as a \emph{premise} in the derivation, and it is not. The constants are argued from geometry and anomaly conditions that make no reference to observers; that complex systems, and eventually observers, then appear is offered as a \emph{consequence} read off the descent, in one direction, with no step reaching back to justify an earlier one by appeal to us. The loop closes at the end --- observers articulating the constitution that produced them --- but a loop traversed once in a fixed direction is not a circular argument; it is a self-consistency the paper claims, not a premise it assumes. Whether that consistency is a genuine explanation or an elegant restatement is a fair question, and one the paper leaves open rather than resolving by fiat.
\end{qabody}

\bigskip
\noindent\textbf{Question 100 (Q-W1-100) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Every generation's physicists think they are near a `theory of everything'. Why is this any different?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} It may not be, and the paper's only honest defense is the shape of its exposure rather than the confidence of its claims. Past confident frameworks failed, and this one may join them; what it offers against that fate is not certainty but losability --- a short list of measurements, several arriving this decade, any one of which ends it. A theory that names its own executioners in advance and hands the community the axe is behaving differently from one that explains every outcome after the fact, whatever its eventual verdict. The reader should keep the graveyard of theories-of-everything firmly in view; the paper asks only to be judged by whether its stated falsifiers survive, which is the one standard that has ever distinguished the survivors from the rest.
\end{qabody}

\bigskip
\noindent\textbf{Question 101 (Q-W1-101) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``Suppose you are wrong. What would it actually take to convince you?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} A short, concrete list, and the paper commits to it rather than reserving the right to reinterpret. A fourth light family would do it immediately. So would a demonstration that the anomaly coefficients are not locked --- that the ratio fixing them can be shifted without cost --- since that would turn a determination back into a convention. So would the cosmic shares settling stably away from the $\pi$-locked values, or a primordial tensor ratio far from the predicted one, or a second admissible geometry (the entry on the most damaging collapse above). None of these is decades away or forever inaccessible; the point of stating them plainly is that a conviction which cannot be moved is not worth holding, and the paper would rather print its exit conditions than have a referee find them.
\end{qabody}

\bigskip
\noindent\textbf{Question 102 (Q-W1-102) (on the whole)}\par\nobreak\smallskip
\noindent\textbf{``After all of it --- the seven steps, the turn, the grades and the open lemmas --- what is the one thing you would want a skeptical reader to carry away?''}\par\nobreak\smallskip
\begin{qabody}
\emph{Response.} That the program has tried to be the kind of thing that can be wrong. The history of grand unifying pictures is mostly a history of frameworks that explained everything and forbade nothing, and the single discipline this paper holds to --- across the review and the graded proposal alike --- is that a claim worth making is a claim a measurement could end. The seven steps are established physics and need no defense; the turn is a reading, marked as one; the derived numbers are staked, several of them on instruments now being built. A reader who remembers only one sentence should remember that the paper would rather be judged by its falsifiers than by its elegance --- because elegance has misled physics before, and a falsifier, unlike a conviction, can be taken away. If that stance is kept and the predictions survive, the picture earns its place; if they do not, it should be discarded, and the paper has tried to make discarding it easy. That is the whole of what is asked.
\end{qabody}


\bigskip
\noindent
That is the last question. What remains is the apparatus for checking the
answers. Part~\ref{part:references} lists every work this review draws on
--- the cohort papers first, the external literature second --- and each
entry carries a paragraph rather than a line: what the work established,
what it cost the field to establish it, how this review reads it including
where the reading departs from it, and where in these pages it is first
cited, as a link a reader can follow. A reader who wants to attack a claim
should begin there, because that is where each claim's owner is named.

%% ===================================================================
%%  PART IX --- REFERENCES
%% ===================================================================
\GGPart[References]{References}{Cited works in two groups --- the
\GGTheory{} cohort and the external literature --- each entry carrying
a relevance note and the location of its first citation}
\label{part:references}
\renewcommand{\bibsection}{\vspace{1.2em}}

\noindent Unlike a conventional reference list, every entry below
carries a full \emph{Relevance} paragraph rather than a line of
bibliographic courtesy. Each one says four things: what the cited work
established, what it cost the field to establish it, how this review
reads it --- including, where it applies, where this reading departs
from the cited work rather than building on it --- and, ending every
note, exactly where in this paper the work is first cited, as a live
cross-reference a reader can follow.

\noindent The list is in two sections, in line with the \GGTheory{}
house style. \textbf{Section A} lists the cohort papers this review
draws on, in the program's canonical order; \textbf{Section B} the
external literature, alphabetically by first author. Earlier drafts
grouped Section~B by the rung of the ladder each work belongs to; that
ordering now lives in the reading list at the end of the section, where
it is more useful, and the alphabet governs the entries themselves so
that a reader who wants one work can find it. There are no orphan
entries: every work listed is cited in the text, and every citation in
the text resolves here. All shipped cohort preprints are available at \url{https://preprints.arisaka-gg.org/}.


\section*{A.\ Papers by Katsushi Arisaka (the \GGTheory{} cohort)}

\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K. (2026).} The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems. Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.}
The charter of the program, and the paper this review is most exposed
to. It states four axioms and seven postulates, of which two are used by
name throughout: \GGEP{}, that geometry and gauge structure are one
composite object rather than two structures laid side by side, and
\OGGEP{}, that observation is a projection from bulk to boundary rather
than a passive reading. The third is the one the review flags as the
program's most exposed premise --- the minimality clause \MSMF{},
which forbids continuous free parameters outright and therefore cannot
be weakened without weakening every closed form downstream. A reader who
wants to reject the ladder at its foundation should reject it here,
because everything the later rungs claim about exactness inherits from
this document rather than from any argument made in these pages. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K. (2026).} \GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{}). Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.}
The bulk paper: it proves that the axioms force the six-dimensional
arena rather than permitting it, and derives the \OGNfive{} integer
ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ with its closure identities
$K = LJ = M^2 + N^2 + L^2$ and $J = 2N + 1$. Those identities are the
arithmetic quoted in \S\ref{ssec:exposure}, and they matter to this
review for a reason that has nothing to do with their values: an integer
reached by two routes with no shared premise is a different kind of
object from an integer reached by one, and the ladder's last rung is
about exactly that difference. Where this paper departs from the parent
is in register rather than content --- \ATwo{} proves the forcing, and
the review only asks what a forced arena is worth as an explanation. First cited in \S\ref{ssec:bi6}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K. (2026).} Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{}). Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.}
The home paper of the six-dimensional, non-supersymmetric instance of
the \GS{} mechanism, and the source of most of the technical content of
Part~IV: the four-component anomaly polynomial $X_4$, the locked
coefficient vector $(1, -1, -2/5, -2/5)$, the trace-ratio lock
$(L - N)/L = 2/5$, the primitive integral lift, and the $a = 1$
Wu--Chern--Simons refinement whose quadratic function is the $q(x)$ of
\S\ref{ssec:qx}. It is also the source of the review's sharpest honest
qualification. \AThree{}'s own minimality theorem shows the trace ratio
alone admits the whole family $(L, N) = (5k, 3k)$; the selection of
$k = 1$ comes from elsewhere, and \S\ref{ssec:exposure} says so rather
than letting the ratio carry more weight than it can bear. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K. (2026).} From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics. Arisaka-GG-A4-Euclid.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.}
The input-discipline paper: the five dimensional constants, the
topological constants $\pi$ and $e$, and the proved integer ledger,
assembled into a declaration that the program spends nothing else.
Its role in this review is negative and therefore easy to miss --- it is
the paper that makes ``zero free continuous parameters'' a checkable
claim rather than a slogan, by listing exhaustively what \emph{is}
spent. The ladder's closing question, whether exactness can be had
without a symmetry to guarantee it, only has force if the alternative
(tuning) has been closed off first, and \AFour{} is where that closing
is done. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A5-\GDOS{})]{ArisakaA5GDOS}
\textbf{Arisaka, K. (2026).} \GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT. Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.}
The boundary paper, and the technical foundation of the review's
open-system turn (Part~V). It constructs the projection from the bulk to
the four-dimensional boundary as a completely positive map, derives the
quantum postulates as theorems of that construction rather than
assuming them, and supplies the master equation whose dissipator does
the irreversible work. The review uses it for one structural point above
all: a projection that composes forward and does not invert is the
reason a symmetry can be exact in the bulk and broken in what a
laboratory reads, which is the sentence the whole ladder has been
climbing toward. Readers approaching the program for the first time
are directed here second, after the constitution. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K. (2026).} The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind. Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.}
The runtime paper: the four-pillar structure and the cascade
$\DGI \to \GFF \to \GSSB \to \GA$ that carries a bulk configuration down
to a selected boundary branch. Its Morse analysis --- a unique deepest
minimum on each stratum --- is what the review's account of
deterministic breaking leans on in \S\ref{ssec:breaking-source},
because a landscape with a unique floor breaks symmetry
\emph{deterministically}, and that is a different
claim from the textbook's ``the system falls into one of the equivalent
minima.'' The distinction is the review's central departure from the
standard account of spontaneous symmetry breaking, and it is
\ASix{}'s theorem rather than this paper's argument that carries it. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A7-\QiFour{})]{ArisakaA7Qi4}
\textbf{Arisaka, K. (2026).} The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}. Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.}
The quantum-information backbone: four pillars metering what crosses
the seam, with the Arisaka equation relating the energy quantum, the
information quantum, and the geometry. The review assumes it rather than
develops it --- \S\ref{ssec:qi4-assumed} says so explicitly --- because
the ladder's argument runs through conservation and protection, not
through the information accounting. It is cited so that a reader
following the program's internal structure can see where the metering
lives, and so that the assumption is visible rather than silent. First cited in \S\ref{ssec:qi4-assumed}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K. (2026).} Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary. Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.}
The quantum-mechanics member of the applications trio: the postulates
read as theorems of the boundary projection, with decoherence and
einselection inherited whole and the Born rule closed at the relaxation
level with its one analytic residue named. This review cites it for the
Born discussion of \S\ref{ssec:born} and, more importantly, for a
methodological example it repeats: \AEight{} concedes exactly what
decoherence supplies before claiming what it does not, and the ladder's
seventh rung is written in the same register. Where the two papers
differ is scope --- \AEight{} argues the physics, and the review only
asks what kind of explanatory move has been made. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K. (2026).} The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry. Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.}
The Standard-Model paper: the twenty-eight parameters derived from the
ledger with no continuous freedom, graded and audited entry by entry.
The review draws on it for the parameter sheet of
\S\ref{ssec:the-sheet} and for the closed forms quoted there, but its
deeper use is as the empirical face of the ladder's last rung. If
protection without symmetry is real, it should show up as integers and
rational partitions where the field expected dials --- and
\ANine{}'s codebook is the place to check whether it does. The review
takes the numbers at their parent's grades and does not re-derive them. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K. (2026).} The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry. Arisaka-GG-A10-Cosmos.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.}
The cosmology paper: the $\Lambda$CDM inventory, the descent, and the
spectral observables derived from the same ledger the particle sector
spends. The review cites it for the cosmological table of
\S\ref{ssec:cosmos-table} and for the two-quantity discipline that
governs the corridor switch and the atomic decoupling redshift --- two
numbers that agree to a fraction of a per cent and must never be quoted
as one. That discipline is a small thing to insist on and a good example
of the habit the ladder is arguing for: an exact quantity and a measured
one can sit beside each other, but they are not the same object, and a
framework that fuses them has lost the distinction its whole case rests
on. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K. (2026).} The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program. Arisaka-GG-A11-Origin.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
No result of the present paper depends on it.
\par\noindent\textbf{Relevance.}
The capstone of the applications: the descent traced from the
unification edge to chemistry, life, and mind. This review cites it in
\S\ref{ssec:not-zero} for the one claim it needs from that territory ---
that the chain from a protected integer to a persisting molecule is
continuous, with no step at which the exactness is handed off to
something else. The review does not endorse the capstone's reach, and
says so; what it takes is the structural point that a conservation law
which fails at the fourteenth decimal place would not merely be
inelegant but would break a chain with a biological end. First cited in
\S\ref{ssec:not-zero}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{Arisaka, K. (2026).} Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability. Arisaka-GG-P2-N3.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.}
The family-number paper, and the sharpest single claim in the cohort:
$N_g = 3$ bounded from below and from above inside one declared axiom
system, by two arguments that share no premise. The review cites it as
the cleanest available instance of the thing the ladder ends on --- an
exact integer that is not the output of a symmetry --- and recommends it
to readers as the third paper to read, with one instruction attached:
read the conditioning section before the result. A pincer argument is only as strong as the axioms both jaws are hung from, and \textsf{GG-P2-N3} states them in the open. Its Appendix~H carries the cohort's two-sided audit --- the bulk's anomaly ledger and the boundary's adiabatic theorem computing the same line bundle independently, in different languages, with no shared parts --- which is the six-dimensional instance of the identity this review's fourth and fifth rungs circle, and the reason the anomaly bookkeeping of Part~IV is an audit rather than a convention chosen to make the books balance. First cited in the Executive Summary.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{Arisaka, K. (2026).} The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter. Arisaka-GG-P5-CPT.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.}
The cohort's treatment of $C$, $P$ and $T$ as orientation reversals of
the three real bulk factors, and the parent of two chapters of this
review: the mirrors of \S\ref{sec:discrete} and the nine-step
leptogenesis chain of \S\ref{sec:leptogenesis} are its results, read
here at the grades it assigns them. Where this review compresses a
derivation to a paragraph, that paper is the authority. First cited in
\S\ref{sec:discrete}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{Arisaka, K. (2026).} The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}. Arisaka-GG-P6-Proton.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.}
The proton paper, and the source of the color-triangle material of
\S\ref{sec:triangle}: baryon number as a winding of the compact
geometry, the $\mathbb{Z}_3$ registration rule, and the singlet
condition that forbids the decay the tanks were built to see. It is
also the cohort's sharpest instance of the seventh rung --- a
conservation law protected by an integer rather than by a symmetry, and
therefore with nothing to break. First cited in \S\ref{sec:triangle}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K. (2026).} The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}.
Arisaka-GG-P1-KK.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.}
The first member of the Predictions Branch, and the cohort's most
immediate exposure: a resonance whose mass is fixed by the same lock
scale the parameter sheet of \S\ref{ssec:the-sheet} spends, in a range
a running machine can reach. The review cites it in
\S\ref{ssec:cohort-map-w1} as one of the seven papers by which the
program can be retired in public, and takes no position on the
prediction itself. Its interest for the argument of these pages is
structural rather than numerical: a framework whose exact quantities are
integers still has to say something continuous somewhere, and this is
where the cohort says it and prints the falsifier. First cited in
\S\ref{ssec:cohort-map-w1}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K. (2026).} The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}. Arisaka-GG-P3-Electron.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.}
The lightest charged particle read as the unique ground state of its
stratum, importing the family count of \citet{ArisakaP2N3} with its
conditioning attached. The review cites it in
\S\ref{ssec:cohort-map-w1} as the Predictions Branch's second entry.
What is worth a reader's attention here is the method rather than the
result: the paper does not fit the electron, it asks which
configuration a descent must land in and checks what lands, which is
the same style of argument the exclusion claim of
\S\ref{ssec:exclusion} rests on. First cited in
\S\ref{ssec:cohort-map-w1}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K. (2026).} The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}. Arisaka-GG-P4-Real.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.}
The origin-of-$i$ paper: the single imaginary unit of boundary physics
derived from two real structures of the bulk --- the compact phase
circle and the causal clock --- rather than assumed with the formalism.
It stands on the Foundations Branch flank of the map
(\S\ref{ssec:cohort-map-w1}) and feeds the quantum readout. The review
does not use its result, and cites it for a reason that belongs to
Part~II rather than Part~III: the fourth rung of the ladder is a phase
on a loop, and a program that takes phases seriously owes an account
of where the phase's $i$ came from. This is the cohort's attempt to pay
that debt. First cited in \S\ref{ssec:cohort-map-w1}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K. (2026).} The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}. Arisaka-GG-C1-Kaxion.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.}
The dark-matter candidate of the program, and the cleanest example of
the shape this review keeps pointing at: a particle whose existence is
not postulated but follows from a compact direction already required by
the bulk, with a mass fixed rather than fitted and a haloscope band an
experiment can close. Cited in \S\ref{ssec:cohort-map-w1} as the first
entry of the Cosmology \& Astrophysics Branch. The review reports the
band and does not check the calculation behind it. First cited in
\S\ref{ssec:cohort-map-w1}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{Arisaka, K. (2026).} The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$. Arisaka-GG-C2-Hubble.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.}
The paper that reads the two discordant expansion rates as the two ends
of one corridor rather than as a measurement to be split. Cited in
\S\ref{ssec:cohort-map-w1}. It is also the cohort's own working example
of the discipline this review argues for at length in
\S\ref{ssec:not-zero}: two numbers that nearly agree are not thereby the
same object, and the paper keeps the topological corridor switch and the
atomic decoupling redshift apart on principle even where the arithmetic
would tempt one to fuse them. First cited in
\S\ref{ssec:cohort-map-w1}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K. (2026).} Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity.
Arisaka-GG-C3-BH.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.}
The horizon read as a boundary port of the kind AX-3 postulates, with
the Bekenstein--Hawking entropy recovered exactly from induced gravity.
Cited in \S\ref{ssec:cohort-map-w1}. Of the seven exposed papers this is
the one closest to the argument of Part~III, because it is the case
where taking the open-system reading literally is not an
interpretive choice: a horizon is a boundary whether or not a formalism
has room for one, and the paper's claim is that the entropy is what the
reading returns. First cited in \S\ref{ssec:cohort-map-w1}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{Arisaka, K. (2026).} The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology. Arisaka-GG-W2-Holonomy.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.}
The companion white paper, and this review's sibling: where these pages
climb the century of symmetry, that one takes the single fork the climb
arrives at --- holonomy requires closed loops, and an interval has none
--- and carries it across sixteen standing problems. It proves the group
law for transport on the compact fiber and the semigroup law for
transport along the open one, which is the distinction
\S\ref{ssec:breaking-source} and the \GRIP{} chapter of
Part~\ref{part:turn} both lean on. First cited in \S\ref{sec:grip}.

\end{thebibliography}

%% Section B, added v15.0.  The prose above this bibliography has
%% always promised it -- "Section B the external literature,
%% alphabetically by first author" -- and there was no heading and no
%% second environment: 92 external works printed under a heading
%% reading "Papers by Katsushi Arisaka".  Section 12 item 10 requires
%% the two environments in every arXiv package.  Nothing is reordered
%% and no entry or key is touched; the boundary is where the entries
%% themselves put it, after ArisakaW2Holonomy and before Adler1969.
\section*{B.\ Other references}

\begin{thebibliography}{99}

\bibitem[Adler(1969)]{Adler1969}
\textbf{Adler, S. L.} Axial-vector vertex in spinor electrodynamics.
\emph{Physical Review} \textbf{177} (1969) 2426--2438.
\par\noindent\textbf{Relevance.}
The paper that found the axial anomaly in spinor electrodynamics: a
current conserved by the classical equations of motion, and not
conserved once the theory is quantized. It is the first place the field
met a conservation law that the symmetry alone could not deliver, and
that is why the review's fourth rung begins here. The anomaly is
usually taught as a pathology to be canceled; the ladder reads it the
other way, as the first evidence that conservation and symmetry are
separable ideas which had been carried under one name since 1918. First cited in \S\ref{ssec:anomalies}.

\bibitem[Adler \& Bardeen(1969)]{AdlerBardeen1969}
\textbf{Adler, S. L., \& Bardeen, W. A.} Absence of higher-order corrections in the anomalous axial-vector divergence equation.
\emph{Physical Review} \textbf{182} (1969) 1517--1536.
\par\noindent\textbf{Relevance.}
The non-renormalization theorem: the anomaly coefficient computed at one
loop receives no corrections at any higher order. This is a stronger and
stranger result than the anomaly itself, and it is the technical reason
the review can treat anomaly coefficients as \emph{integers} rather than
as quantities that happen to be small. A number immune to perturbative
correction is not being protected by a symmetry --- there is no symmetry
in sight --- and \S\ref{ssec:E-measure} uses this to make the ladder's
central distinction concrete rather than programmatic. First cited in \S\ref{ssec:E-measure}.

\bibitem[Aharonov \& Bohm(1959)]{AharonovBohm1959}
\textbf{Aharonov, Y., \& Bohm, D.} Significance of electromagnetic potentials in the quantum theory.
\emph{Physical Review} \textbf{115} (1959) 485--491.
\par\noindent\textbf{Relevance.}
The demonstration that the electromagnetic potential carries physical
content through the holonomy of a loop, in a region where the field
strength vanishes identically. It is the experiment that made
``redundancy'' an unsatisfying description of gauge structure: something
gauge-dependent is doing observable work, and what is observable is the
phase around a closed circuit. \S\ref{ssec:measurable} uses it as the
hinge of the second rung --- the point at which the answer to ``what is
physically real in a gauge theory?'' stopped being ``the field
strength'' and became ``the holonomy.''. First cited in \S\ref{ssec:measurable}.

\bibitem[Alvarez-Gaum\'e \& Witten(1984)]{AlvarezGaumeWitten1984}
\textbf{Alvarez-Gaum\'e, L., \& Witten, E.} Gravitational anomalies.
\emph{Nuclear Physics B} \textbf{234} (1984) 269--330.
\par\noindent\textbf{Relevance.}
The gravitational anomalies paper: the index-theoretic computation that
established which spacetime dimensions and which matter content are
consistent, and made anomaly cancellation a design constraint rather
than a check performed after the fact. The review cites it in
\S\ref{ssec:F-wz} for the descent structure it shares with the
Wess--Zumino consistency conditions, and for a point of intellectual
history the fifth rung depends on: by 1984 the field had learned to read
an anomaly as an obstruction with a topological home, which is the
reading the program inherits. First cited in \S\ref{ssec:F-wz}.

\bibitem[Atiyah \& Singer(1963)]{AtiyahSinger1963}
\textbf{Atiyah, M. F., \& Singer, I. M.} The index of elliptic operators on compact manifolds.
\emph{Bulletin of the American Mathematical Society} \textbf{69} (1963) 422--433.
\par\noindent\textbf{Relevance.}
The theorem that ties an analytic count --- the difference between the
numbers of left- and right-handed zero modes of a Dirac operator --- to a
topological integral over the manifold the operator lives on. It is the
single mathematical fact the ladder's top rung rests on: it is why the
amount by which a Ward identity fails is a whole number rather than a
small quantity, and why nothing continuous can move it.
\S\ref{ssec:E-measure} uses the index in exactly that role, and a reader
who wants the review's central claim in one line can take it from here.
First cited in \S\ref{ssec:E-measure}.

\bibitem[ATLAS Collaboration(2026)]{ATLASsusy2026}
\textbf{ATLAS Collaboration.} Searches for strong production of supersymmetric particles with the ATLAS detector.
\emph{arXiv:2601.19695} (2026).
\par\noindent\textbf{Relevance.}
The current exclusion limits on supersymmetric partners from the full
LHC dataset. The review cites this in the Executive Summary and in
\S\ref{ssec:failures} for one purpose, and it is not polemical: the
naturalness argument was the single most confident prediction the
symmetry-first tradition made about the electroweak scale, it was made
by serious people for good reasons, and the partners are not there at
the masses the argument required. A review of a century of
symmetry-first reasoning that omitted its sharpest recent
disappointment would be a poor review, and the ladder's seventh rung is
in part an attempt to say what that disappointment was evidence
\emph{of}. First cited in the Executive Summary.

\bibitem[Baez \& Muniain(1994)]{Baez1994}
\textbf{Baez, J. C., \& Muniain, J. P.} \emph{Gauge Fields, Knots and Gravity.}
World Scientific (1994).
\par\noindent\textbf{Relevance.}
\emph{Gauge Fields, Knots and Gravity}: the friendliest route into the
bundle language the appendices assume, written for readers who know
physics and are meeting differential geometry properly for the first
time. It is recommended as the first book on the reading ladder of
Part~\ref{part:appendices} because it does the one thing most
introductions skip --- it explains \emph{why} a connection is the right
object before writing its components --- and a reader who has that
motivation will find Appendix~\ref{app:connections} short rather than
dense. First cited in Part~\ref{part:appendices}.

\bibitem[Bell \& Jackiw(1969)]{BellJackiw1969}
\textbf{Bell, J. S., \& Jackiw, R.} A PCAC puzzle: $\pi^0 \to \gamma\gamma$ in the $\sigma$-model.
\emph{Il Nuovo Cimento A} \textbf{60} (1969) 47--61.
\par\noindent\textbf{Relevance.}
The independent discovery of the axial anomaly, arrived at through the
$\pi^0 \to \gamma\gamma$ puzzle: the decay rate demanded by current
algebra was wrong, and the anomaly was what made the measured rate
correct. Cited alongside \citet{Adler1969} throughout the fourth rung,
and worth reading first of the two by anyone who wants to see the
anomaly arrive as an experimental necessity rather than as a
regularization subtlety. The lesson the review draws is that the first
anomaly was found because a symmetry-based prediction \emph{failed
against data}, which is not how the story is usually told. First cited in \S\ref{ssec:anomalies}.

\bibitem[Bennett(1982)]{Bennett1982}
\textbf{Bennett, C. H.} The thermodynamics of computation --- a review.
\emph{International Journal of Theoretical Physics} \textbf{21} (1982) 905--940.
\par\noindent\textbf{Relevance.}
The paper that closed Maxwell's demon by putting the cost in the right
place. Measurement can be made free; forgetting cannot, so the demon's
books balance only when the memory is cleared and \citet{Landauer1961}
is charged for the clearing. \S\ref{ssec:N-landauer} uses the argument
for a structural point rather than a thermodynamic one: the
irreversibility sits in a map that has no inverse, not in anybody's
ignorance, which is the reading the boundary paper's arrow needs.
First cited in \S\ref{ssec:N-landauer}.

\bibitem[Berry(1984)]{Berry1984}
\textbf{Berry, M. V.} Quantal phase factors accompanying adiabatic changes.
\emph{Proceedings of the Royal Society of London A} \textbf{392} (1984) 45--57.
\par\noindent\textbf{Relevance.}
The geometric phase: a quantum system carried adiabatically around a
closed circuit in parameter space returns with a phase that depends only
on the circuit's geometry, not on the rate of traversal. Berry's paper
turned holonomy from a formal structure into something a laboratory
measures routinely, and it is cited in \S\ref{ssec:kappa} for exactly
that --- the review's sixth rung needs holonomy to be an ordinary
observable rather than an exotic one, and after 1984 it is. The
historical curiosity that the effect had been found and forgotten
several times before is itself part of the ladder's argument about what
the field was and was not equipped to notice. First cited in \S\ref{ssec:kappa}.

\bibitem[Bianchi(1902)]{Bianchi1902}
\textbf{Bianchi, L.} Sui simboli a quattro indici e sulla curvatura di Riemann.
\emph{Rendiconti della Reale Accademia Nazionale dei Lincei} (5) \textbf{11} (1902) 3--7.
\par\noindent\textbf{Relevance.}
The identity that carries Bianchi's name, in the paper where he stated
it --- twenty-two years after \citet{Voss1880}, which is the point
\S\ref{ssec:bianchi-history} makes. The review dwells on the attribution
not out of antiquarian interest but because the identity's status is the
third rung's whole subject: $dF = 0$ is not a conservation law derived
from a symmetry, it is a consequence of $F = dA$, true of any curvature
of any connection, and owing nothing to Noether. A structure that
predates and outranks the symmetry argument is exactly what the ladder
is looking for. First cited in \S\ref{ssec:bianchi-history}.

\bibitem[Blackmond(2010)]{Blackmond2010}
\textbf{Blackmond, D. G.} The origin of biological homochirality.
\emph{Cold Spring Harbor Perspectives in Biology} \textbf{2} (2010) a002147.
\par\noindent\textbf{Relevance.}
The review of biological homochirality: why life uses one enantiomer and
not the other, what mechanisms can amplify a small initial imbalance,
and how much of the story remains open. The paper is cited in
\S\ref{ssec:chirality} for a single structural point the ladder needs at
its far end --- that a chiral preference, once written, is propagated
and amplified by ordinary chemistry rather than re-decided at every
step. The review makes no claim about the origin of terrestrial
homochirality and says so; what it takes is the demonstration that
one-way selection is a real and studied phenomenon at the molecular
scale. First cited in \S\ref{ssec:chirality}.

\bibitem[Bogomolny(1976)]{Bogomolny1976}
\textbf{Bogomolny, E. B.} Stability of classical solutions.
\emph{Soviet Journal of Nuclear Physics} \textbf{24} (1976) 449.
\par\noindent\textbf{Relevance.}
The bound that carries Bogomolny's name: a topological lower limit on
the energy of a field configuration, saturated by solutions of a
first-order equation. It is cited in \S\ref{ssec:C-bundles} as an early
and clean instance of a physical quantity bounded by a topological
class rather than by a symmetry --- the energy cannot go below the
bound, and the reason has nothing to do with any conserved current. The
review reads the Bogomolny bound as a rung the field climbed without
noticing what it was standing on. First cited in \S\ref{ssec:C-bundles}.

\bibitem[Bott \& Tu(1982)]{BottTu1982}
\textbf{Bott, R., \& Tu, L. W.} \emph{Differential Forms in Algebraic Topology.}
Springer (1982).
\par\noindent\textbf{Relevance.}
\emph{Differential Forms in Algebraic Topology}: the book to go to when
the physics texts stop being enough. It is recommended in
Part~\ref{part:appendices} as the second stage of the mathematical
reading, for readers who want the machinery behind characteristic
classes and the de Rham theory properly rather than operationally. It
is a mathematician's book and makes no concessions, which is precisely
its value here --- a reader who works through it will be able to check
the review's claims about differential cohomology instead of taking
them. First cited in Part~\ref{part:appendices}.

\bibitem[Bousso \& Polchinski(2000)]{BoussoPolchinski2000}
\textbf{Bousso, R., \& Polchinski, J.} Quantization of four-form fluxes
and dynamical neutralization of the cosmological constant.
\emph{Journal of High Energy Physics} \textbf{06} (2000) 006.
\par\noindent\textbf{Relevance.}
The mechanism that turned anthropic reasoning into a construction:
quantized four-form fluxes whose combinations produce a discretuum of
vacuum energies dense enough that some member sits near the observed
value. It is cited in \S\ref{ssec:where-freedom} as the clearest
statement of the position this review argues against --- and the
disagreement is structural rather than empirical. A framework whose
census permits one arena has no room to accommodate a different answer
and can be killed by a single measurement; a discretuum absorbs any
result. Which of those is a virtue is the question the seventh rung
puts to the reader, and it is put honestly. First cited in \S\ref{ssec:where-freedom}.

\bibitem[Brading(2002)]{Brading2002}
\textbf{Brading, K. A.} Which symmetry? Noether, Weyl, and conservation of electric charge.
\emph{Studies in History and Philosophy of Modern Physics} \textbf{33} (2002) 3--22.
\par\noindent\textbf{Relevance.}
The philosophical reconstruction of Noether's second theorem and what it
actually asserts: that a local symmetry yields identities among the
equations of motion rather than new conserved currents. The review
leans on it in \S\ref{ssec:goettingen} because the distinction between
the two theorems is the ladder's first rung and is routinely blurred in
physics teaching, where ``Noether's theorem'' means the first one and
the second is either omitted or misdescribed. Brading's paper is the
cleanest short statement of what was proved and what was not. First cited in \S\ref{ssec:goettingen}.

\bibitem[Byers(1998)]{Byers1998}
\textbf{Byers, N.} E. Noether's discovery of the deep connection between symmetries and conservation laws.
In \emph{The Heritage of Emmy Noether}, Israel Mathematical Conference Proceedings \textbf{12} (1999); arXiv:physics/9807044 (preprint 1998).
\par\noindent\textbf{Relevance.}
The historical account of how Noether's theorems came to be written:
what Hilbert and Klein were actually asking about energy conservation in
general relativity, why the question was urgent in G\"ottingen in 1918,
and how the second theorem answered it. \S\ref{ssec:goettingen} uses
this to make a point the review returns to repeatedly --- the second
theorem was the one the physicists of the day needed, and it is the one
the discipline subsequently forgot. Reading the history back makes the
ladder's first step look less like a reinterpretation and more like a
recovery. First cited in \S\ref{ssec:goettingen}.

\bibitem[Callan \& Harvey(1985)]{CallanHarvey1985}
\textbf{Callan, C. G., \& Harvey, J. A.} Anomalies and fermion zero modes on strings and domain walls.
\emph{Nuclear Physics B} \textbf{250} (1985) 427--436.
\par\noindent\textbf{Relevance.}
Anomaly inflow: a boundary anomaly canceled not within the boundary
theory but by a current flowing in from the bulk, so that the two sides
of an interface compute the same object in different languages. This is
the ancestor of the fifth rung's central move and is cited as such. The
review's use of it is structural: inflow shows that an anomaly's
cancellation can be a fact about a \emph{pair} of theories rather than
about either alone, and once that is admitted, the question ``what
guarantees this exactness?'' has an answer that is not a symmetry. The six-dimensional instance the program uses is carried out in \citet[Appendix~H]{ArisakaP2N3}. First cited in \S\ref{ssec:inflow}.

\bibitem[Casas \& Ibarra(2001)]{CasasIbarra2001}
\textbf{Casas, J. A., \& Ibarra, A.} Oscillating neutrinos and $\mu \to e,\gamma$.
\emph{Nuclear Physics B} \textbf{618} (2001) 171--204.
\par\noindent\textbf{Relevance.}
The parametrization that separates what neutrino experiments can measure
from what leptogenesis actually uses: the seesaw's Yukawa matrix written
with a complex orthogonal matrix $R$ carrying the parameters no
low-energy measurement reaches. The review cites it for one consequence,
in \S\ref{ssec:alphabet-w1} and again in \S\ref{ssec:M-step2} --- the
asymmetry is blind to the PMNS phase, so the phase DUNE will measure is
not the reason there is matter. That is a disappointment worth stating
plainly, and it is this paper's algebra that makes it precise.
First cited in \S\ref{ssec:alphabet-w1}.

\bibitem[Cheeger \& Simons(1985)]{CheegerSimons1985}
\textbf{Cheeger, J., \& Simons, J.} Differential characters and geometric invariants.
In \emph{Geometry and Topology}, Lecture Notes in Mathematics \textbf{1167}, Springer (1985) 50--80. (Notes circulated from 1973.)
\par\noindent\textbf{Relevance.}
Differential characters: the construction that refines a characteristic
class to an object carrying both its integral topological content and
its differential-form representative, so that a quantity can be exactly
integral and continuously varying at once. It is the mathematical spine
of the review's sixth rung, and \S\ref{ssec:cocycle} is written directly
against it. This is primary literature and assumes a good deal; a reader
meeting it for the first time should come to it after
\citet{Nakahara2003} and \citet{BottTu1982}, and should expect the
payoff to be conceptual rather than computational. First cited in \S\ref{ssec:cocycle}.

\bibitem[Cheng \& Li(1984)]{ChengLi1984}
\textbf{Cheng, T.-P., \& Li, L.-F.} \emph{Gauge Theory of Elementary Particle Physics.}
Oxford University Press (1984).
\par\noindent\textbf{Relevance.}
\emph{Gauge Theory of Elementary Particle Physics}: the text recommended
for the anomaly material of Appendix~\ref{app:anomalies}, because it
works the triangle diagram, the consistency conditions, and the
cancellation arithmetic at a level that a reader can reproduce with
paper. It is recommended alongside \citet{PeskinSchroeder1995} rather
than instead of it --- Cheng and Li are better on the group theory of
anomaly cancellation, Peskin and Schroeder on the field-theoretic
machinery, and the review's fourth rung needs both. First cited in Part~\ref{part:appendices}.

\bibitem[Christenson \emph{et al.}(1964)]{Christenson1964}
\textbf{Christenson, J. H., Cronin, J. W., Fitch, V. L., \& Turlay, R.} Evidence for the $2\pi$ decay of the $K^0_2$ meson.
\emph{Physical Review Letters} \textbf{13} (1964) 138--140.
\par\noindent\textbf{Relevance.}
The discovery of CP violation in neutral kaon decay: a two-pion final
state at the level of $2 \times 10^{-3}$, where CP conservation forbids
it entirely. The result was unwanted, small, and correct. The review
cites it in the Executive Summary and in \S\ref{ssec:failures} as the
second of the three occasions on which a symmetry everyone had assumed
turned out to be approximate --- and as the one that mattered most,
because it converted the matter--antimatter asymmetry of the universe
from a brute fact into a question physics could address. First cited in the Executive Summary.

\bibitem[Coleman \& Weinberg(1973)]{ColemanWeinberg1973}
\textbf{Coleman, S., \& Weinberg, E.} Radiative corrections as the origin of spontaneous symmetry breaking.
\emph{Physical Review D} \textbf{7} (1973) 1888--1910.
\par\noindent\textbf{Relevance.}
The demonstration that a potential flat at the classical level need not
stay flat: the one-loop vacuum energy can pick out a minimum where the
classical theory had none, so the direction the world takes is a quantum
effect rather than an input. Appendix~\ref{app:wilson} runs the same
computation for the Wilson-line angle, which is where the review's
second rung stops being kinematics --- the phase is not merely allowed
to be physical, it is driven to a definite value by the tower whose
masses move with it. First cited in \S\ref{ssec:D-potential}.

\bibitem[Curie(1894)]{Curie1894}
\textbf{Curie, P.} Sur la sym\'etrie dans les ph\'enom\`enes physiques, sym\'etrie d'un champ \'electrique et d'un champ magn\'etique.
\emph{Journal de Physique Th\'eorique et Appliqu\'ee} \textbf{3} (1894) 393--415.
\par\noindent\textbf{Relevance.}
The oldest statement of the problem the review's capstone answers: the
symmetries of the causes are found in the effects, so an asymmetric
world should not come from a symmetric start. Taken at face value it
forbids our universe, and the three standard escapes each pay for the
escape in a currency \S\ref{ssec:symmetry-reading} itemizes. The reading
offered there is that the principle is not violated but applied at the
wrong level: what the effect inherits from the cause is the arithmetic
of a groove and a descent, not a coin flip.
First cited in \S\ref{ssec:symmetry-reading}.

\bibitem[Davidson \& Ibarra(2002)]{DavidsonIbarra2002}
\textbf{Davidson, S., \& Ibarra, A.} A lower bound on the right-handed neutrino mass from leptogenesis.
\emph{Physics Letters B} \textbf{535} (2002) 25--32.
\par\noindent\textbf{Relevance.}
The ceiling on the decay asymmetry of the lightest heavy neutrino, and
with it a floor of about $10^{9}$~GeV on its mass if leptogenesis is to
be hierarchical. \S\ref{ssec:M-chain} quotes the bound and evaluates it,
because it is the reason the chain of Part~IV cannot be hierarchical and
must be resonant instead: the bound is what forces the near-degenerate
regime rather than a taste for it.
First cited in \S\ref{ssec:M-chain}.

\bibitem[Dirac(1931)]{Dirac1931}
\textbf{Dirac, P. A. M.} Quantised singularities in the electromagnetic field.
\emph{Proceedings of the Royal Society of London A} \textbf{133} (1931) 60--72.
\par\noindent\textbf{Relevance.}
The magnetic monopole paper, and the origin of every topologically
protected integer in physics: the consistency of a charged quantum
particle in a monopole field forces $eg = n\hbar c/2$ with $n$ an
integer. The review cites it in the Executive Summary as the ladder's
earliest sighting of the destination --- an exact quantization that
follows from topology rather than from a dynamical equation or a
symmetry group. That the argument works ``if a single monopole exists
anywhere'' is also the historical model for paying for exactness with a
topological hypothesis instead of a fitted parameter. First cited in the Executive Summary.

\bibitem[Douglas(2003)]{Douglas2003}
\textbf{Douglas, M.~R.} The statistics of string/M theory vacua.
\emph{Journal of High Energy Physics} \textbf{05} (2003) 046.
\par\noindent\textbf{Relevance.}
The statistics of string vacua: the counting program that made the
landscape a quantitative subject rather than a rhetorical one. Cited in
\S\ref{ssec:steelman}, where the review states the opposing position at
its strongest before disagreeing with it. Douglas's work is the reason
the disagreement has to be structural: if the vacuum count really is
astronomical and the distribution really is as computed, then a
program that computes the constants is attempting something that
cannot be done, and the review says so plainly rather than arguing
around it. First cited in \S\ref{ssec:steelman}.

\bibitem[Elitzur(1975)]{Elitzur1975}
\textbf{Elitzur, S.} Impossibility of spontaneously breaking local symmetries.
\emph{Physical Review D} \textbf{12} (1975) 3978--3982.
\par\noindent\textbf{Relevance.}
The theorem that a local gauge symmetry cannot be spontaneously broken:
gauge-non-invariant order parameters have vanishing expectation values,
and what the Higgs mechanism does is remove a redundancy in a chosen
gauge rather than break a symmetry. The result is standard,
uncontroversial, and routinely ignored in the language physicists
actually use --- which is why it opens the review's second rung and is
cited from the Executive Summary onward. If Elitzur is right, then the
most famous instance of ``spontaneous symmetry breaking'' in the
Standard Model is not one, and the ladder's later rungs are an attempt
to say what it is instead. First cited in the Executive Summary.

\bibitem[Englert \& Brout(1964)]{Englert1964}
\textbf{Englert, F., \& Brout, R.} Broken symmetry and the mass of gauge vector mesons.
\emph{Physical Review Letters} \textbf{13} (1964) 321--323.
\par\noindent\textbf{Relevance.}
The first of the three 1964 papers on mass generation through a
spontaneously broken gauge theory, submitted before the others and
often cited last. The review treats the three as one event in
\S\ref{ssec:higgs} and cites them together, partly for fairness and
partly because the differences between them matter to the second rung:
Englert and Brout work with the vacuum polarization and never write the
scalar boson explicitly, which is a reminder that what the mechanism
\emph{is} was not obvious even to its authors. First cited in \S\ref{ssec:higgs}.

\bibitem[Fradkin \& Shenker(1979)]{FradkinShenker1979}
\textbf{Fradkin, E., \& Shenker, S. H.} Phase diagrams of lattice gauge theories with Higgs fields.
\emph{Physical Review D} \textbf{19} (1979) 3682--3697.
\par\noindent\textbf{Relevance.}
The lattice result that the confining and Higgs regimes of a gauge
theory with fundamental scalars are analytically connected --- no phase
boundary separates them, so ``the symmetry is broken'' and ``the
symmetry is confined'' do not describe distinct phases. Together with
\citet{Elitzur1975} this is the technical core of the review's second
rung. It is cited in \S\ref{ssec:elitzur} because it makes the point
quantitatively rather than as a theorem about order parameters: the
distinction physicists draw is not merely unmeasurable in principle, it
is absent from the phase diagram. First cited in \S\ref{ssec:elitzur}.

\bibitem[Freed(2002)]{Freed2002}
\textbf{Freed, D. S.} Dirac charge quantization and generalized differential cohomology.
\emph{Surveys in Differential Geometry} \textbf{7} (2002) 129--194; arXiv:hep-th/0011220.
\par\noindent\textbf{Relevance.}
The formulation of Dirac charge quantization in differential cohomology,
and the paper that makes precise what a ``quantized'' field
configuration is when the underlying object is not simply an integer.
Cited in \S\ref{ssec:G-characters} for the character-group language the
sixth rung uses. Like \citet{CheegerSimons1985} and \citet{Witten1997},
this is primary mathematical literature; the review quotes its results
and does not reproduce its arguments, and a reader who wants to check
them should budget accordingly. First cited in \S\ref{ssec:G-characters}.

\bibitem[Fujikawa(1979)]{Fujikawa1979}
\textbf{Fujikawa, K.} Path integral measure for gauge invariant fermion theories.
\emph{Physical Review Letters} \textbf{42} (1979) 1195--1198.
\par\noindent\textbf{Relevance.}
The path-integral derivation of the anomaly: it lives in the
non-invariance of the fermionic measure, and its coefficient is an index
--- a number counting zero modes. This is the single most important
external result for the review's fourth rung, because it relocates the
anomaly from a property of a particular diagram to a property of the
space the path integral is taken over. \S\ref{ssec:E-measure} follows
Fujikawa's argument in outline for exactly that reason: an integer that
counts something cannot drift, and no symmetry was required to hold it
still. First cited in \S\ref{ssec:E-measure}.

\bibitem[Garwin, Lederman \& Weinrich(1957)]{Garwin1957}
\textbf{Garwin, R. L., Lederman, L. M., \& Weinrich, M.} Observations of the failure of conservation of parity and charge conjugation in meson decays: the magnetic moment of the free muon.
\emph{Physical Review} \textbf{105} (1957) 1415--1417.
\par\noindent\textbf{Relevance.}
The muon-decay experiment that confirmed parity violation within days of
the cobalt-60 result, using entirely different apparatus and a different
decay. Cited in \S\ref{ssec:failures} alongside \citet{Wu1957} because
the review's point about the fall of parity is not that one experiment
overturned an assumption but that two independent measurements did so
within a week --- which is what made the overthrow immediate rather than
contested, and which is the standard the review holds the program's
own claims to. First cited in \S\ref{ssec:failures}.

\bibitem[Gavela \emph{et al.}(1994)]{Gavela1994}
\textbf{Gavela, M. B., Hern\'andez, P., Orloff, J., \& P\`ene, O.} Standard model $CP$ violation and baryon asymmetry.
\emph{Modern Physics Letters A} \textbf{9} (1994) 795--810.
\par\noindent\textbf{Relevance.}
The calculation that closed Standard-Model baryogenesis on the numbers
rather than on taste: run the CKM phase honestly through the hot
electroweak era and what it can deliver falls some ten orders of
magnitude short, because the Jarlskog invariant enters through quark
masses that are negligible at those temperatures. It is cited in
\S\ref{sec:leptogenesis} as one of the two independent halves of the
failure that sends the question to the neutrino sector, and it is a
model of how to shut a possibility --- by computing it, not by
disliking it. First cited in \S\ref{sec:leptogenesis}.

\bibitem[Georgi \& Glashow(1974)]{GeorgiGlashow1974}
\textbf{Georgi, H., \& Glashow, S. L.} Unity of all elementary particle forces.
\emph{Physical Review Letters} \textbf{32} (1974) 438--441.
\par\noindent\textbf{Relevance.}
Minimal $SU(5)$: the proposal that the three Standard-Model factors are
remnants of one simple group, explaining charge quantization and
predicting proton decay. The review cites it in \S\ref{ssec:sun} as the
high-water mark of symmetry-first reasoning --- the moment at which
choosing the right group looked like it would deliver the world --- and
returns to it in the seventh rung, because the minimal version's
prediction was excluded by measurement. The idea that the gauge
structure is the shadow of something simpler has never been abandoned,
and the program this review surveys is a version of it that answers
to geometry rather than to group theory. First cited in \S\ref{ssec:sun}.

\bibitem[Gleason(1957)]{Gleason1957}
\textbf{Gleason, A. M.} Measures on the closed subspaces of a Hilbert space.
\emph{Journal of Mathematics and Mechanics} \textbf{6} (1957) 885--893.
\par\noindent\textbf{Relevance.}
Gleason's theorem: on a Hilbert space of dimension three or more, the
only probability measure on the lattice of projections is the trace
rule. It is the strongest available statement that the Born measure is
not an independent postulate but a consequence of the structure it sits
in, given non-contextuality --- and the qualifier is where the interest
lies. \S\ref{ssec:born} cites it as the boundary of what can be proved
without additional physical input, which is precisely the boundary the
program's open-system derivation claims to cross. First cited in \S\ref{ssec:born}.

\bibitem[Goldstone(1961)]{Goldstone1961}
\textbf{Goldstone, J.} Field theories with ``superconductor'' solutions.
\emph{Il Nuovo Cimento} \textbf{19} (1961) 154--164.
\par\noindent\textbf{Relevance.}
The theorem that made spontaneous breaking predictive: break a
continuous symmetry and a massless mode appears for every broken
direction, which is what turns a statement about a vacuum into something
a detector can look for. \S\ref{ssec:symmetry-reading} cites it with
\citet{Nambu1960} for what it does and does not do --- it explains why
the symmetric state is unstable, and correctly, and by construction
cannot say which of the degenerate directions the world took. That gap
is the one the chapter's quantized dial is meant to close.
First cited in \S\ref{ssec:symmetry-reading}.

\bibitem[Gorini, Kossakowski \& Sudarshan(1976)]{GoriniKossakowskiSudarshan1976}
\textbf{Gorini, V., Kossakowski, A., \& Sudarshan, E. C. G.} Completely positive dynamical semigroups of $N$-level systems.
\emph{Journal of Mathematical Physics} \textbf{17} (1976) 821--825.
\par\noindent\textbf{Relevance.}
The classification of generators of completely positive dynamical
semigroups in finite dimensions, published independently of
\citet{Lindblad1976} and jointly canonical with it. The review cites the
pair in \S\ref{ssec:gdos} for the structure the program's boundary
dynamics inherits, and for one word in the title that the ladder cares
about more than the theorem: \emph{semigroup}. A generator with no
inverse is the formal reason a bulk symmetry can survive a projection
that the boundary reading does not, and the two 1976 papers are where
that structure was pinned down. First cited in \S\ref{ssec:gdos}.

\bibitem[Green \& Schwarz(1984)]{GreenSchwarz1984}
\textbf{Green, M. B., \& Schwarz, J. H.} Anomaly cancellations in supersymmetric $D = 10$ gauge theory and superstring theory.
\emph{Physics Letters B} \textbf{149} (1984) 117--122.
\par\noindent\textbf{Relevance.}
The anomaly-cancellation mechanism that rescued ten-dimensional string
theory: a two-form whose modified Bianchi identity absorbs an otherwise
fatal anomaly, at the price of a rigid relation among the theory's
group-theoretic coefficients. The review cites it from the Executive
Summary onward for the mechanism rather than the setting, since the
program's six-dimensional, non-supersymmetric instance is the subject
of \citet{ArisakaA3BI6}. The general moral belongs to the fifth rung: a
cancellation that fixes coefficients is a constraint on what can exist,
not a condition to be maintained order by order. First cited in the Executive Summary.

\bibitem[Greenberg(2002)]{Greenberg2002}
\textbf{Greenberg, O. W.} $CPT$ violation implies violation of Lorentz invariance.
\emph{Physical Review Letters} \textbf{89} (2002) 231602.
\par\noindent\textbf{Relevance.}
The converse to \citet{Jost1957}: in an interacting local field theory a
$CPT$-violating signal cannot be a small local repair, because it takes
Lorentz invariance down with it. The review leans on this in
\S\ref{sec:discrete} to fix the experimental stakes --- the orientation
reading of the three mirrors and the classic theorem differ about
\emph{what} the mirrors are, but they stand or fall on the same
measurement, and that measurement is about spacetime rather than about
any particular model. First cited in \S\ref{sec:discrete}.

\bibitem[Guralnik, Hagen \& Kibble(1964)]{GuralnikHagenKibble1964}
\textbf{Guralnik, G. S., Hagen, C. R., \& Kibble, T. W. B.} Global conservation laws and massless particles.
\emph{Physical Review Letters} \textbf{13} (1964) 585--587.
\par\noindent\textbf{Relevance.}
The third of the 1964 papers, and the one that states the mechanism most
carefully --- it is explicit that no symmetry is broken in the physical
Hilbert space, and that the Goldstone theorem is evaded rather than
violated. The review gives it particular weight in \S\ref{ssec:higgs}
for that reason: the objection the second rung raises against the
standard telling was already answered correctly, in print, in 1964, by
authors whose formulation the textbooks then largely set aside in favor
of a more picturesque one. First cited in \S\ref{ssec:higgs}.

\bibitem[Harvey \& Turner(1990)]{HarveyTurner1990}
\textbf{Harvey, J. A., \& Turner, M. S.} Cosmological baryon and lepton number in the presence of electroweak fermion number violation.
\emph{Physical Review D} \textbf{42} (1990) 3344--3349.
\par\noindent\textbf{Relevance.}
The equilibrium bookkeeping that fixes how much of a lepton excess the
sphalerons hand on to baryons: $28/79$ for one Higgs doublet, a number
that follows from the conserved charges and the chemical potentials
rather than from any dynamical detail. \S\ref{ssec:lepto-chain} uses it
as one of the fixed factors of the leptogenesis pipeline, and it is
worth saying that the ratio is a counting result and not a fitted one
--- the integers come from the spectrum.
First cited in \S\ref{ssec:lepto-chain}.

\bibitem[Hatanaka, Inami \& Lim(1998)]{HatanakaInamiLim1998}
\textbf{Hatanaka, H., Inami, T., \& Lim, C. S.} The gauge hierarchy problem and higher-dimensional gauge theories.
\emph{Modern Physics Letters A} \textbf{13} (1998) 2601--2612.
\par\noindent\textbf{Relevance.}
The paper that showed the Wilson-line vacuum expectation value is finite
and calculable --- free of the ultraviolet divergence that afflicts an
ordinary scalar mass, because the quantity is non-local and no local
counterterm can touch it. \S\ref{ssec:finite} uses this as the sharpest
technical illustration the second rung has: a physical vacuum parameter
protected not by a symmetry but by \emph{where it lives}, which is the
distinction the whole ladder is climbing toward. It is also the clearest
answer available to the naturalness objection, and it was in print
twenty years before the objection became urgent. First cited in \S\ref{ssec:finite}.

\bibitem[Higgs(1964)]{Higgs1964}
\textbf{Higgs, P. W.} Broken symmetries and the masses of gauge bosons.
\emph{Physical Review Letters} \textbf{13} (1964) 508--509.
\par\noindent\textbf{Relevance.}
The paper that named the boson: the second of the 1964 trio, and the one
that states explicitly that the mechanism leaves a massive scalar
excitation behind. Cited with its companions in \S\ref{ssec:higgs}. The
review's interest is in what the three papers together did \emph{not}
settle --- whether a local symmetry is being broken, or a redundancy
removed --- a question \citet{Elitzur1975} answered eleven years later
and which the vocabulary of the field never absorbed. First cited in \S\ref{ssec:higgs}.

\bibitem[Hopkins \& Singer(2005)]{HopkinsSinger2005}
\textbf{Hopkins, M. J., \& Singer, I. M.} Quadratic functions in geometry, topology, and M-theory.
\emph{Journal of Differential Geometry} \textbf{70} (2005) 329--452.
\par\noindent\textbf{Relevance.}
The terminus of the ladder. Differential cohomology, developed here in
the form the physics literature now uses, is the mathematical setting in
which a quantity can be protected by an integer invariant with no
symmetry group anywhere in the argument. The review cites it from the
Executive Summary onward because it is the existence proof that the
category the seventh rung needs is not a hopeful invention: exactness
that is not Noetherian has a home, it has been built, and it is where
the program's integers live. Whether the physics is right is a
separate question; whether the mathematics permits the claim is settled,
and this is where it was settled. First cited in the Executive Summary.

\bibitem[Hosotani(1983)]{Hosotani1983}
\textbf{Hosotani, Y.} Dynamical mass generation by compact extra dimensions.
\emph{Physics Letters B} \textbf{126} (1983) 309--313.
\par\noindent\textbf{Relevance.}
The mechanism by which a gauge field's Wilson line around a compact
extra dimension acquires a vacuum expectation value and breaks the gauge
symmetry --- symmetry breaking with no elementary scalar, driven purely
by holonomy. It is cited from the Executive Summary onward as the most
direct precedent in the literature for the reading the program
generalizes: a physical, measurable vacuum label that is a holonomy and
nothing else. The review's second rung is in large part an argument that
this mechanism deserved more attention than it received, and got less
because the field already had a scalar it liked. First cited in the Executive Summary.

\bibitem[Hosotani(1989)]{Hosotani1989}
\textbf{Hosotani, Y.} Dynamics of non-integrable phases and gauge symmetry breaking.
\emph{Annals of Physics} \textbf{190} (1989) 233--253.
\par\noindent\textbf{Relevance.}
The systematic development of the mechanism six years on: the dynamics
of the Wilson-line phase, the effective potential, and the pattern of
breaking it produces. Cited in \S\ref{ssec:hosotani} where the review
needs the mechanism worked rather than announced. Reading the 1983 and
1989 papers in sequence also makes a historical point the ladder cares
about --- the idea was complete and public long before the community had
a reason to want it, which is the ordinary way structural ideas arrive. First cited in \S\ref{ssec:hosotani}.

\bibitem[Jarlskog(1985)]{Jarlskog1985}
\textbf{Jarlskog, C.} Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal $CP$ nonconservation.
\emph{Physical Review Letters} \textbf{55} (1985) 1039--1042.
\par\noindent\textbf{Relevance.}
The invariant that says how much $CP$ violation a three-generation model
contains without reference to any choice of phase convention --- one
number, rephasing-invariant, vanishing exactly when the mixing can be
made real. It earns its place here twice: it is the cleanest example in
the flavor sector of a physical quantity extracted from a redundant
description, which is the second rung's lesson, and it is the number
\S\ref{ssec:cp-pentagon} recomputes from the pentagon's geometry.
First cited in \S\ref{ssec:cp-pentagon}.

\bibitem[Jost(1957)]{Jost1957}
\textbf{Jost, R.} Eine Bemerkung zum $CPT$-Theorem.
\emph{Helvetica Physica Acta} \textbf{30} (1957) 409--416.
\par\noindent\textbf{Relevance.}
The $CPT$ theorem in the form the field still uses: three hypotheses ---
Lorentz invariance, locality, and a Hamiltonian bounded below with the
normal spin--statistics connection --- and a proof that runs through
analyticity rather than algebra. It reaches interacting fields, which is
its strength, and says nothing about \emph{what} the three mirrors are,
which is where \S\ref{sec:discrete} and Appendix~\ref{app:discrete} pick
the question up. The review is explicit that its orientation reading is
complementary to this result and not a replacement for it.
First cited in \S\ref{sec:discrete}.

\bibitem[Kachru \emph{et al.}(2003)]{KKLT2003}
\textbf{Kachru, S., Kallosh, R., Linde, A., \& Trivedi, S.~P.}
de Sitter vacua in string theory.
\emph{Physical Review D} \textbf{68} (2003) 046005.
\par\noindent\textbf{Relevance.}
The construction that made metastable de Sitter vacua with stabilized
moduli concrete, and with them the landscape as a working framework
rather than a possibility. Cited in \S\ref{ssec:where-freedom}. The
review's disagreement with this line of work is stated as a difference
in what the two programs are willing to leave undetermined, not as a
criticism of the construction, which is an achievement of a kind the
program surveyed here has not matched. First cited in \S\ref{ssec:where-freedom}.

\bibitem[Kobayashi \& Maskawa(1973)]{KobayashiMaskawa1973}
\textbf{Kobayashi, M., \& Maskawa, T.} $CP$ violation in the renormalizable theory of weak interaction.
\emph{Progress of Theoretical Physics} \textbf{49} (1973) 652--657.
\par\noindent\textbf{Relevance.}
The observation that CP violation in the quark sector requires at least
three generations, because only then does the mixing matrix retain a
physical phase after all rephasings are exhausted. It is a counting
argument, and \S\ref{ssec:three-gen} cites it as the field's most
successful one --- a structural prediction from representation theory
alone, confirmed when the third family appeared. The review holds it up
as the standard the program's own counting claims should be judged
against, and notes what the argument did \emph{not} do: it required
three, and did not explain why there are exactly three. First cited in \S\ref{ssec:three-gen}.

\bibitem[Kosmann-Schwarzbach(2011)]{KosmannSchwarzbach2011}
\textbf{Kosmann-Schwarzbach, Y.} \emph{The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century.}
Springer (2011).
\par\noindent\textbf{Relevance.}
\emph{The Noether Theorems}: the scholarly translation and commentary,
and the reference of record for what the 1918 paper says. The review
cites it in Appendix~\ref{app:noether} where the two theorems are stated
carefully, because much of the physics literature quotes Noether at
second hand and the second theorem in particular is frequently
paraphrased into something she did not prove. A reader who suspects the
review of over-reading the first rung should check it here. First cited in Appendix~\ref{app:noether}.

\bibitem[Landauer(1961)]{Landauer1961}
\textbf{Landauer, R.} Irreversibility and heat generation in the computing process.
\emph{IBM Journal of Research and Development} \textbf{5} (1961) 183--191.
\par\noindent\textbf{Relevance.}
The bound that put a price on forgetting: erasing one bit at temperature
$T$ costs at least $k_B T \ln 2$, because the erasing map compresses
phase space and a compression has to be paid for somewhere.
\S\ref{ssec:N-landauer} derives it twice and evaluates it, and reads it
for the structural point the review needs --- the irreversibility is in
a map with no inverse, which is a statement about the dynamics rather
than about anybody's ignorance.
First cited in \S\ref{ssec:N-landauer}.

\bibitem[Lee \& Yang(1956)]{LeeYang1956}
\textbf{Lee, T. D., \& Yang, C. N.} Question of parity conservation in weak interactions.
\emph{Physical Review} \textbf{104} (1956) 254--258.
\par\noindent\textbf{Relevance.}
The paper that noticed nobody had ever tested parity in the weak
interaction, surveyed the existing data to confirm the gap, and proposed
the experiments that would close it. The review cites it from the
Executive Summary as the discipline's best example of a symmetry assumed
for so long that the assumption became invisible --- the reason the
seventh rung asks not ``which symmetries hold?'' but ``why did we expect
them to?'' The episode is the ladder's cautionary anchor: the mirrors
were taken to be exact because no one had looked. First cited in the Executive Summary.

\bibitem[Lindblad(1976)]{Lindblad1976}
\textbf{Lindblad, G.} On the generators of quantum dynamical semigroups.
\emph{Communications in Mathematical Physics} \textbf{48} (1976) 119--130.
\par\noindent\textbf{Relevance.}
The generator of a quantum dynamical semigroup, in the form that carries
Lindblad's name. Cited with \citet{GoriniKossakowskiSudarshan1976} in
\S\ref{ssec:gdos}. The review uses it for a structural claim rather than
a computational one: the boundary dynamics of an open system is
generated by something that composes forward and does not invert, and
that irreversibility is not an approximation introduced for
convenience --- it is what the completely positive structure of the map
requires. The ladder's fifth and sixth rungs both stand on this. First cited in \S\ref{ssec:gdos}.

\bibitem[Manton(1979)]{Manton1979}
\textbf{Manton, N. S.} A new six-dimensional approach to the Weinberg--Salam model.
\emph{Nuclear Physics B} \textbf{158} (1979) 141--153.
\par\noindent\textbf{Relevance.}
The early paper on symmetry breaking by topology in a gauge theory on a
space with non-trivial first homotopy, predating Hosotani's treatment
and reaching the same structural conclusion. \S\ref{ssec:hosotani}
cites it as the beginning of the line, and as another instance of the
pattern the review keeps noticing: the mechanism was available, was
correct, and was not adopted, because the field had no shortage of
scalars and no pressure to look for an alternative. First cited in \S\ref{ssec:hosotani}.

\bibitem[Marinelli \& Morpurgo(1984)]{MarinelliMorpurgo1984}
\textbf{Marinelli, M., \& Morpurgo, G.} The electric neutrality of matter: a summary.
\emph{Physics Letters B} \textbf{137} (1984) 439--442.
\par\noindent\textbf{Relevance.}
The Stanford-style search for fractional residual charge on bulk matter,
and the bound it set: no fractional charge at the level of $10^{-21}$ of
an electron charge per nucleon. The review cites it in
\S\ref{ssec:integers} as the most direct experimental statement
available that charge quantization is exact rather than merely
accurate --- and the exactness is the point. A quantity measured to
twenty-one decimal places and found integral is either protected by
something or is the most remarkable coincidence in the record, and the
ladder's sixth rung is about which. First cited in \S\ref{ssec:integers}.

\bibitem[Milnor \& Stasheff(1974)]{MilnorStasheff1974}
\textbf{Milnor, J. W., \& Stasheff, J. D.} \emph{Characteristic Classes.}
Annals of Mathematics Studies \textbf{76}, Princeton University Press (1974).
\par\noindent\textbf{Relevance.}
\emph{Characteristic Classes}: the standard reference, and the one the
review cites in \S\ref{ssec:G-wu} for the Wu formula and the Stiefel--Whitney
machinery behind it. It is a mathematician's text and assumes a first
course in algebraic topology; the appendices quote its results rather
than developing them, and a reader who wants the derivations should come
here after \citet{BottTu1982}. First cited in \S\ref{ssec:G-wu}.

\bibitem[Misner, Thorne \& Wheeler(1973)]{MisnerThorneWheeler1973}
\textbf{Misner, C. W., Thorne, K. S., \& Wheeler, J. A.} \emph{Gravitation.}
W. H. Freeman (1973).
\par\noindent\textbf{Relevance.}
\emph{Gravitation}: recommended in Part~\ref{part:appendices} for the
gravitational half of Appendix~\ref{app:connections}, and for a
pedagogical habit the review tries to imitate --- say what the object is
before writing its components. Its pictures of a vector carried around a
closed circuit on a curved surface remain the clearest illustration in
print of the fact that holonomy is a property of a loop and of nothing
else, which is the image the entire ladder rests on. First cited in Part~\ref{part:appendices}.

\bibitem[Nakahara(2003)]{Nakahara2003}
\textbf{Nakahara, M.} \emph{Geometry, Topology and Physics.}
2nd ed., Institute of Physics (2003).
\par\noindent\textbf{Relevance.}
\emph{Geometry, Topology and Physics}: the bridge text, and the single
most useful reference for the technical content of the appendices. Fiber
bundles, connections, characteristic classes, homotopy and homology,
index theorems and anomalies, all developed for physicists and all used
here. It is recommended in Part~\ref{part:appendices} as the second book
on the ladder, after \citet{Baez1994} and before \citet{BottTu1982}, and
a reader who works its bundle chapters will find that the review's
claims about holonomy are the first consequences of the definitions
rather than exotic mathematics. First cited in Part~\ref{part:appendices}.

\bibitem[Nambu(1960)]{Nambu1960}
\textbf{Nambu, Y.} Quasi-particles and gauge invariance in the theory of superconductivity.
\emph{Physical Review} \textbf{117} (1960) 648--663.
\par\noindent\textbf{Relevance.}
Where spontaneous symmetry breaking entered particle physics, carried
across from superconductivity: the ground state need not have the
symmetry of the laws, and the spectrum is where you find out that it
does not. Cited with \citet{Goldstone1961} in
\S\ref{ssec:symmetry-reading} as the tradition the review draws a
structural contrast with rather than argues against --- the mechanism is
right about the instability and silent about the outcome, and it is the
silence, not the mechanism, that the chapter is trying to remove.
First cited in \S\ref{ssec:symmetry-reading}.

\bibitem[Nielsen \& Ninomiya(1983)]{NielsenNinomiya1983}
\textbf{Nielsen, H. B., \& Ninomiya, M.} The Adler--Bell--Jackiw anomaly and Weyl fermions in a crystal.
\emph{Physics Letters B} \textbf{130} (1983) 389--396.
\par\noindent\textbf{Relevance.}
The no-go theorem for chiral fermions on a lattice: a local, hermitian,
translation-invariant lattice action with the right continuum limit
necessarily produces fermions in pairs of opposite chirality. It is
cited in \S\ref{ssec:anomalies} because it makes the anomaly a
statement about what can be \emph{constructed}, not merely about what
diagrams evaluate to --- the obstruction survives regularization
because it is topological. The review reads this as the fourth rung's
strongest single result: a symmetry that cannot be realized on any
lattice is being told something by the geometry. First cited in \S\ref{ssec:anomalies}.

\bibitem[Noether(1918)]{Noether1918}
\textbf{Noether, E.} Invariante Variationsprobleme.
\emph{Nachrichten von der Gesellschaft der Wissenschaften zu G\"ottingen, Mathematisch-Physikalische Klasse} (1918) 235--257.
\par\noindent\textbf{Relevance.}
The source of both theorems and of the distinction between them that the
review's first rung turns on. The first theorem converts a continuous
global symmetry into a conserved current; the second converts a local
symmetry into identities among the equations of motion --- constraints
rather than new conservation laws. The review's whole argument depends
on that difference being real and on the discipline having largely lost
it, so the paper is cited from the Executive Summary and again wherever
the ladder needs to say which theorem is being invoked. Read in the
original, the second theorem is unmistakably the one Hilbert and Klein
had asked for. First cited in \S\ref{sec:zero-parameters}.

\bibitem[Particle Data Group(2024)]{PDG2024}
\textbf{Navas, S. et al. (Particle Data Group).} Review of Particle
Physics. \emph{Physical Review D} \textbf{110} (2024) 030001.
\par\noindent\textbf{Relevance.}
The Review of Particle Physics: the reference against which every
particle-sector number in this paper is checked, and the source of the
parameter sheet discussed in \S\ref{ssec:the-sheet}. It is cited also
for a discipline the review tries to keep --- every quoted value carries
a scheme, a scale, and an uncertainty, and a comparison that drops any
of the three is not a comparison. The sheet's own structure is the
seventh rung's evidence: the numbers the Standard Model cannot compute
are listed there, in public, with their errors, which is why the claim
that a framework computes them is checkable rather than rhetorical.
First cited in \S\ref{ssec:fatal}.

\bibitem[Peccei \& Quinn(1977)]{PecceiQuinn1977}
\textbf{Peccei, R. D., \& Quinn, H. R.} $CP$ conservation in the presence of pseudoparticles.
\emph{Physical Review Letters} \textbf{38} (1977) 1440--1443.
\par\noindent\textbf{Relevance.}
The standard answer to the strong-$CP$ problem: promote $\bar\theta$ to
a field, let a new global symmetry relax it to zero, and accept a light
pseudoscalar as the price. It is a good solution, and the review says so
before saying where it differs --- \S\ref{ssec:theta-zero} reaches
$\bar\theta = 0$ by closure rather than by relaxation, so on that
account there is no new light particle to look for, and
\S\ref{sec:qa-ledger} states the contrast as its advocates would state
it. First cited in \S\ref{ssec:theta-zero}.

\bibitem[Peskin \& Schroeder(1995)]{PeskinSchroeder1995}
\textbf{Peskin, M. E., \& Schroeder, D. V.} \emph{An Introduction to Quantum Field Theory.}
Addison-Wesley (1995).
\par\noindent\textbf{Relevance.}
\emph{An Introduction to Quantum Field Theory}: the text most readers of
this review were trained on, recommended in Part~\ref{part:appendices}
for the field-theoretic machinery the anomaly appendix assumes --- the
triangle diagram, the regularization, the renormalization of gauge
theories. It is the calculational spine; the conceptual framing of the
anomaly is better found in \citet{Fujikawa1979}, and the group theory of
its cancellation in \citet{ChengLi1984}. First cited in Part~\ref{part:appendices}.

\bibitem[Pilaftsis \& Underwood(2004)]{PilaftsisUnderwood2004}
\textbf{Pilaftsis, A., \& Underwood, T. E. J.} Resonant leptogenesis.
\emph{Nuclear Physics B} \textbf{692} (2004) 303--345.
\par\noindent\textbf{Relevance.}
The self-energy resonance: when two heavy Majorana states are nearly
degenerate, the $CP$ asymmetry in their decays is enhanced by the ratio
of the splitting to the width, and saturates when the two are
comparable. \S\ref{ssec:lepto-chain} needs this regime rather than
preferring it --- the framework's heavy states are split at the percent
level, and the on-resonance condition is the chain's one named input,
carried in public as a conjecture with a candidate mechanism attached.
First cited in \S\ref{ssec:lepto-chain}.

\bibitem[Planck Collaboration(2020)]{Planck2018}
\textbf{Aghanim, N. et al. (Planck Collaboration).} Planck 2018
results. VI. Cosmological parameters.
\emph{Astronomy \& Astrophysics} \textbf{641} (2020) A6.
\par\noindent\textbf{Relevance.}
The cosmological parameters from the full Planck mission, and the source
of the measured column in the cosmological table of
\S\ref{ssec:cosmos-table}. The review quotes it rather than analyzing
it, and observes one discipline in the quoting: the framework's corridor
switch and the atomic decoupling redshift Planck reports are two
distinct quantities that agree to a fraction of a per cent, and the
cohort's rule is to name which is which at every site. That is a small
piece of hygiene, and the ladder's argument about exact versus measured
quantities is exactly the reason it matters. First cited in
\S\ref{sec:leptogenesis}.

\bibitem[Sakharov(1967)]{Sakharov1967}
\textbf{Sakharov, A. D.} Violation of $CP$ invariance, $C$ asymmetry, and baryon asymmetry of the universe.
\emph{JETP Letters} \textbf{5} (1967) 24--27.
\par\noindent\textbf{Relevance.}
The three conditions --- baryon-number violation, C and CP violation,
and departure from thermal equilibrium --- without which no dynamical
mechanism can produce a matter excess from a symmetric start. The review
cites them in \S\ref{ssec:empty} for a structural observation rather
than a cosmological one: two of the three are statements about which
symmetries fail, and the third is a statement about irreversibility.
That the recipe for a universe with matter in it requires both kinds of
ingredient is, on the ladder's reading, not a coincidence. First cited in
\S\ref{sec:leptogenesis}.

\bibitem[Slavnov(1972)]{Slavnov1972}
\textbf{Slavnov, A. A.} Ward identities in gauge theories.
\emph{Theoretical and Mathematical Physics} \textbf{10} (1972) 99--107.
\par\noindent\textbf{Relevance.}
The non-abelian generalization of the Ward identity, derived
independently of \citet{Taylor1971} and jointly canonical with it. The
review cites the pair in \S\ref{ssec:slavnov} for the second rung's
technical core: the identities are consequences of the gauge
\emph{redundancy}, not of a symmetry of the physical states, and they
constrain Green's functions rather than supplying conserved charges.
That is Noether's second theorem doing its work in a modern setting, and
recognizing it as such is one of the review's small recoveries. First cited in \S\ref{ssec:slavnov}.

\bibitem[Susskind(2003)]{Susskind2003}
\textbf{Susskind, L.} The anthropic landscape of string theory.
\emph{arXiv:hep-th/0302219} (2003).
\par\noindent\textbf{Relevance.}
The essay that named the anthropic landscape and argued that a vast
vacuum count combined with eternal inflation makes environmental
selection the correct explanation for the constants rather than a
failure of explanation. \S\ref{ssec:steelman} states this position at
full strength before disagreeing with it, because a review that
caricatured it would be worthless. The two positions make different
bets and both are exposed: if the constants are environmental, a
program that computes them is attempting the impossible and will
fail --- publicly, and soon. First cited in \S\ref{ssec:steelman}.

\bibitem['t Hooft(1971)]{tHooft1971}
\textbf{'t Hooft, G.} Renormalizable Lagrangians for massive Yang--Mills fields.
\emph{Nuclear Physics B} \textbf{35} (1971) 167--188.
\par\noindent\textbf{Relevance.}
The proof that spontaneously broken non-abelian gauge theories are
renormalizable --- the result that turned the electroweak model from a
curiosity into the Standard Model, and made the three-page electroweak
paper of 1967 the most cited in the field. The review cites it in
\S\ref{ssec:higgs} for the historical weight it carries: the field's
confidence in the symmetry-breaking picture rests to a remarkable degree
on this calculation, and the calculation is about the gauge-fixing
machinery rather than about any symmetry of the physical states. First cited in \S\ref{ssec:higgs}.

\bibitem['t Hooft(1980)]{tHooft1980}
\textbf{'t Hooft, G.} Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking.
In \emph{Recent Developments in Gauge Theories}, NATO Advanced Study Institutes Series B \textbf{59}, Plenum (1980) 135--157.
\par\noindent\textbf{Relevance.}
The naturalness criterion: a small dimensionless parameter is natural if
setting it to zero enhances the symmetry, because then the smallness is
protected against radiative corrections. It is the most influential
single idea in model-building since the Standard Model was completed,
and \S\ref{ssec:failures} treats it with care rather than dismissal. The
review's claim is not that the criterion is wrong but that its symmetry
clause is too narrow: protection can also come from topology, with no
symmetry anywhere \citep{HopkinsSinger2005}, and a generation of
model-building looked for the first kind because the second had not been
named. First cited in \S\ref{ssec:failures}.

\bibitem[Takahashi(1957)]{Takahashi1957}
\textbf{Takahashi, Y.} On the generalized Ward identity.
\emph{Nuovo Cimento} \textbf{6} (1957) 371--375.
\par\noindent\textbf{Relevance.}
The generalization of Ward's identity to arbitrary momenta, and the
half of the Ward--Takahashi identity that makes it a tool rather than a
special case. Cited with \citet{Ward1950} from the Executive Summary
onward. The review's interest is in what the identity \emph{is}: a
relation among Green's functions forced by the redundancy of the
description, exactly of the type Noether's second theorem predicts, and
routinely described in textbooks as a consequence of ``gauge
symmetry'' without the distinction being drawn. First cited in the Executive Summary.

\bibitem[Taylor(1971)]{Taylor1971}
\textbf{Taylor, J. C.} Ward identities and charge renormalization of the Yang--Mills field.
\emph{Nuclear Physics B} \textbf{33} (1971) 436--444.
\par\noindent\textbf{Relevance.}
The Ward identity extended to non-abelian gauge theories, obtained
independently of \citet{Slavnov1972}. Cited in \S\ref{ssec:slavnov} as
the first half of the canonical pair. The review reads the
Slavnov--Taylor identities as the second rung's most consequential
technical fact: the machinery that makes non-abelian gauge theory
renormalizable is machinery of redundancy, and once that is seen, the
question of what a gauge symmetry \emph{is} stops being philosophical. First cited in \S\ref{ssec:slavnov}.

\bibitem[Taylor \& Wang(2015)]{TaylorWang2015}
\textbf{Taylor, W., \& Wang, Y.-N.} The F-theory geometry with most
flux vacua.
\emph{Journal of High Energy Physics} \textbf{12} (2015) 164.
\par\noindent\textbf{Relevance.}
The result that a large fraction of the flux-vacuum count in F-theory is
concentrated on a small number of geometries, which sharpens the
landscape question from ``how many vacua?'' to ``how are they
distributed?'' Cited in \S\ref{ssec:steelman}. The review includes it
because the steelman of the opposing position is stronger with this
result than without it: a landscape with structure is a more serious
scientific object than a landscape that is merely large. First cited in \S\ref{ssec:steelman}.

\bibitem[Vafa(2005)]{Vafa2005}
\textbf{Vafa, C.} The string landscape and the swampland.
\emph{arXiv:hep-th/0509212} (2005).
\par\noindent\textbf{Relevance.}
The swampland program: the proposal that most effective field theories
which look consistent cannot be completed into quantum gravity, so that
the landscape is bounded by criteria of a new kind. Cited in
\S\ref{ssec:where-freedom} as the closest thing in the string literature
to the move this review is interested in --- an attempt to say what
\emph{cannot} exist, rather than to catalog what can. The two
programs differ on where the constraints come from, and the review
notes the convergence of instinct without overstating it. First cited in \S\ref{ssec:where-freedom}.

\bibitem[Voss(1880)]{Voss1880}
\textbf{Voss, A.} Zur Theorie der Transformation quadratischer Differentialausdr\"ucke und der Kr\"ummung h\"oherer Mannigfaltigkeiten.
\emph{Mathematische Annalen} \textbf{16} (1880) 129--178.
\par\noindent\textbf{Relevance.}
The paper in which the identity later attributed to Bianchi was first
stated, twenty-two years before \citet{Bianchi1902}. The review dwells
on this in \S\ref{ssec:bianchi-history} for a reason that is not
antiquarian: the misattribution is a small instance of the larger
pattern the ladder documents, in which structural facts that owe nothing
to symmetry are recorded, forgotten, and rediscovered, because the
discipline's index is organized around the symmetry-first story and
these results do not file neatly under it. First cited in \S\ref{ssec:bianchi-history}.

\bibitem[Ward(1950)]{Ward1950}
\textbf{Ward, J. C.} An identity in quantum electrodynamics.
\emph{Physical Review} \textbf{78} (1950) 182.
\par\noindent\textbf{Relevance.}
The identity relating the electron self-energy and the vertex function
in quantum electrodynamics --- the first of the relations that make the
theory renormalizable. Cited from the Executive Summary onward as the
second rung's opening exhibit. Ward derived it as a technical fact about
Feynman diagrams; the review's point is that it is Noether's second
theorem in disguise, and that a century of teaching has presented it as
a consequence of gauge \emph{symmetry} when it is a consequence of gauge
\emph{redundancy}. The two produce the same equation and mean different
things. First cited in the Executive Summary.

\bibitem[Weinberg(1987)]{Weinberg1987}
\textbf{Weinberg, S.} Anthropic bound on the cosmological constant.
\emph{Physical Review Letters} \textbf{59}, 2607 (1987).
\par\noindent\textbf{Relevance.}
The one anthropic prediction that preceded its measurement: an upper
bound on $\Lambda$ of roughly the observed order, a decade before the
supernova result, and closer than any first-principles estimate then
available. It is cited here because a review that asks to be judged on
dated forward commitments is obliged to credit the opposing
program's, and \S\ref{ssec:steelman} does. Nothing in this paper's
own reading of $\Lambda$ depends on it. First cited in
\S\ref{ssec:steelman}.

\bibitem[Weinberg(1996)]{Weinberg1996}
\textbf{Weinberg, S.} \emph{The Quantum Theory of Fields, Vol.~II.}
Cambridge University Press (1996).
\par\noindent\textbf{Relevance.}
\emph{The Quantum Theory of Fields}, Volume II: the reference the review
recommends last on the field-theory ladder of
Part~\ref{part:appendices}, for readers who want the non-abelian gauge
structure, the anomalies, and the effective-action formalism derived
rather than presented. Weinberg is unusually careful about which
features of the standard framework are forced and which are
conventional, which is the question this review asks of every structure
it inherits, and Volume II is where that care is most useful to it. First cited in Part~\ref{part:appendices}.

\bibitem[Wess \& Zumino(1971)]{WessZumino1971}
\textbf{Wess, J., \& Zumino, B.} Consequences of anomalous Ward identities.
\emph{Physics Letters B} \textbf{37} (1971) 95--97.
\par\noindent\textbf{Relevance.}
The consistency conditions: the anomaly cannot be an arbitrary
functional, because the gauge transformations it obstructs form a group,
and the group's composition law constrains it. \S\ref{ssec:E-brst}
cites this as the point at which the anomaly stopped being a nuisance
and became an object with a classification --- the descent structure
follows, and with it the modern reading in which anomalies are
cohomological rather than accidental. It is the fourth rung's turning
point. First cited in \S\ref{ssec:E-brst}.

\bibitem[Weyl(1929)]{Weyl1929}
\textbf{Weyl, H.} Elektron und Gravitation. I.
\emph{Zeitschrift f\"ur Physik} \textbf{56} (1929) 330--352.
\par\noindent\textbf{Relevance.}
The paper that gave the gauge principle its modern form: invariance of
the Dirac field under a local change of \emph{phase} requires the
electromagnetic potential, whose transformation law is fixed by the
requirement. Weyl had tried the same idea a decade earlier with a local
change of \emph{scale}, which failed; the survival of the name
``gauge'' from the failed version is an accident the field never
corrected. \S\ref{ssec:phase} cites it as the second rung's foundation
and notes the irony the ladder eventually resolves --- Weyl's instinct
that the local structure was geometric was right, and the geometry was
not the one he tried. First cited in \S\ref{ssec:phase}.

\bibitem[Wigner(1960)]{Wigner1960}
\textbf{Wigner, E. P.} The unreasonable effectiveness of mathematics in the natural sciences.
\emph{Communications on Pure and Applied Mathematics} \textbf{13} (1960) 1--14.
\par\noindent\textbf{Relevance.}
The essay on the unreasonable effectiveness of mathematics in the
natural sciences. The review cites it in the Executive Summary because
the ladder's terminus is a specific instance of Wigner's puzzle: if
exactness in physics is guaranteed by integers that label bundles, then
the mathematics is not being applied to the world from outside --- it is
what the world's structure is being read back as. Whether that dissolves
the puzzle or relocates it is a question the review raises and does not
claim to settle. First cited in the Executive Summary.

\bibitem[Wilson(1974)]{Wilson1974}
\textbf{Wilson, K. G.} Confinement of quarks.
\emph{Physical Review D} \textbf{10} (1974) 2445--2459.
\par\noindent\textbf{Relevance.}
The paper that put gauge theory on a lattice and identified the
gauge-invariant content of the theory as the set of holonomies around
closed loops. This is the technical fact the review's second rung is
built on: Wilson's variables are what a gauge theory \emph{is}, and they
are loop functionals, so the question of what is physically real in such
a theory has a concrete answer that is neither the potential nor the
field strength. \S\ref{ssec:elitzur} uses it together with
\citet{Elitzur1975} to make the redundancy reading unavoidable rather
than optional. First cited in \S\ref{ssec:elitzur}.

\bibitem[Witten(1985)]{Witten1985}
\textbf{Witten, E.} Symmetry breaking patterns in superstring models.
\emph{Nuclear Physics B} \textbf{258} (1985) 75--100.
\par\noindent\textbf{Relevance.}
The global anomaly: an obstruction invisible to the perturbative
triangle, detected instead by a homotopy class --- $\pi_4(SU(2)) =
\mathbb{Z}_2$ --- so that a theory can be free of every local anomaly
and still be inconsistent. \S\ref{ssec:sun} cites it because it is the
cleanest demonstration in the fourth rung's territory that the
constraint being enforced is topological. No amount of diagram
computation finds it; counting does. First cited in \S\ref{ssec:sun}.

\bibitem[Witten(1997)]{Witten1997}
\textbf{Witten, E.} On flux quantization in M-theory and the effective action.
\emph{Journal of Geometry and Physics} \textbf{22} (1997) 1--13.
\par\noindent\textbf{Relevance.}
The paper that identified the shift in the quantization condition for
the M-theory four-form, and with it made differential cohomology a
working tool in physics rather than a formal option. Cited in
\S\ref{ssec:witten-shift} for the sixth rung's central technical move:
a quantity that is neither an integer nor a free real number, but an
integer shifted by a computable half-integral characteristic class.
This is primary literature; the review states the result and points a
reader to \citet{Freed2002} and \citet{HopkinsSinger2005} for the
framework it lives in. First cited in \S\ref{ssec:witten-shift}.

\bibitem[Wu \emph{et al.}(1957)]{Wu1957}
\textbf{Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D., \& Hudson, R. P.} Experimental test of parity conservation in beta decay.
\emph{Physical Review} \textbf{105} (1957) 1413--1415.
\par\noindent\textbf{Relevance.}
The cobalt-60 experiment: polarized nuclei emitting beta electrons
preferentially against their spin direction, in maximal violation of
parity. The result arrived within months of the \citet{LeeYang1956}
proposal and overturned an assumption that had organized physics for
decades. The review cites it from the Executive Summary as the first of
the three great failures of symmetry-first expectation, and notes what
made it decisive: the violation was not small, and \citet{Garwin1957}
confirmed it independently within a week. First cited in the Executive Summary.

\bibitem[Wu \& Yang(1975)]{WuYang1975}
\textbf{Wu, T. T., \& Yang, C. N.} Concept of nonintegrable phase factors and global formulation of gauge fields.
\emph{Physical Review D} \textbf{12} (1975) 3845--3857.
\par\noindent\textbf{Relevance.}
The paper that identified the physically meaningful content of a gauge
field as the non-integrable phase factor --- the holonomy --- and gave
the global bundle formulation in which the Aharonov--Bohm effect and the
Dirac monopole become two aspects of one statement. It is the historical
hinge between gauge theory as written by physicists and gauge theory as
understood by geometers, and \S\ref{ssec:hosotani-history} cites it as
the moment the second rung's vocabulary became available. Everything the
review says afterwards about holonomy is written in the dictionary this
paper supplies. First cited in \S\ref{ssec:hosotani-history}.

\bibitem[Yang(1980)]{Yang1980}
\textbf{Yang, C. N.} Einstein's impact on theoretical physics.
\emph{Physics Today} \textbf{33} (1980), no.~6, 42--49.
\par\noindent\textbf{Relevance.}
Yang's own retrospective on the fiber-bundle formulation and its
relation to gauge theory, written a quarter century after
\citet{YangMills1954}. The review cites it in the Executive Summary
because it is the clearest first-hand statement of the surprise --- that
physicists had been rediscovering, in their own notation, structures
mathematicians had built for other reasons. That convergence is the
ladder's subject in miniature, and Yang's account of encountering it is
worth reading before any of the technical material. First cited in the Executive Summary.

\bibitem[Yang \& Mills(1954)]{YangMills1954}
\textbf{Yang, C. N., \& Mills, R. L.} Conservation of isotopic spin and isotopic gauge invariance.
\emph{Physical Review} \textbf{96} (1954) 191--195.
\par\noindent\textbf{Relevance.}
The generalization of the gauge principle to a non-abelian group,
written for isotopic spin and rejected at the time for predicting
massless charged vector bosons nobody had seen. It is now the
mathematical spine of three of the four interactions. \S\ref{ssec:sun}
cites it as the moment force ceased to be a list and became a
representation-theoretic question --- and as the beginning of the
symmetry-first tradition whose triumphs and disappointments the review's
seventh rung tries to weigh fairly. First cited in \S\ref{ssec:sun}.

\bibitem[Zee(2010)]{Zee2010}
\textbf{Zee, A.} \emph{Quantum Field Theory in a Nutshell.}
2nd ed., Princeton University Press (2010).
\par\noindent\textbf{Relevance.}
\emph{Quantum Field Theory in a Nutshell}: recommended in
Part~\ref{part:appendices} for readers who want the ideas of the anomaly
and effective-field-theory appendices before the technology. Zee is
better than any of the alternatives at saying \emph{why} a structure is
there, which is what this review's appendices assume a reader already
has. It is the book to read alongside the ladder's fourth rung, not
after it. First cited in Part~\ref{part:appendices}.

\end{thebibliography}

\par\medskip\noindent{\bfseries Further reading, by level}\par\medskip

\noindent
A reader coming to this material for the first time will do best in this
order. \emph{To begin:} \citet{Baez1994} for the geometry and
\citet{Zee2010} for the field theory, both of which keep the physics in
view. \emph{To work through Part~\ref{part:appendices}:} \citet{Nakahara2003} throughout,
with \citet{PeskinSchroeder1995} for Appendix~\ref{app:anomalies} and
\citet{MisnerThorneWheeler1973} for the gravitational half of
Appendix~\ref{app:connections}. \emph{To go past it:}
\citet{BottTu1982} and \citet{Weinberg1996}. \emph{Last, and slowly:}
\citet{CheegerSimons1985}, \citet{Witten1997}, \citet{Freed2002} and
\citet{HopkinsSinger2005}, which are primary literature and assume the
rest.

\noindent
For the program itself, the order that costs a reader least is: the
constitution \citep{ArisakaA1Constitution} for the axioms, then the
boundary paper \citep{ArisakaA5GDOS} for the open-system move, then the
family-number paper \citep{ArisakaP2N3} for the sharpest claim --- and in
that last one, read the conditioning section before the result.


\end{document}
