%% 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.
%% ===================================================================
%% ===================================================================
%% v15.4 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 9 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.
%% ====================================================================
%% ===================================================================
%% v15.3 (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 HAD NO AVAILABILITY LINE AT ALL.  A
%%     reader was told only "Arisaka-GG-A11-Origin (May 20, 2026)", with nothing said about where
%%     the paper can be found.  It now reads "In preparation.  To be
%%     posted ... on completion; no result of the present paper depends
%%     on it."
%% (2) NINE FURTHER COMPANION BIBITEMS CARRIED STALE DATES.  Repairing
%%     (1) made its neighbors look wrong -- the same finding A8-QM
%%     produced in FP-4.  Each companion was still introduced as
%%     "Arisaka-GG-A9-SM (May 14, 2026)", a date three months stale, with
%%     no statement of where a reader can get the paper.  Measured across
%%     all 24 sources at the start of FP-4, only five papers still
%%     carried these; A8 and A10 were cleared then, P4-Real and C2-Hubble
%%     are cleared here, and A11-Origin waits for its rebuild.  All nine
%%     now carry the availability line the other papers use.
%%
%% P4-Real has no printed catalogue debt, no simultaneity claim and no
%% pointer into A11's internal Part structure.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.2 (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 sans form with \bigskip -- the style file default,
%%   and needs no setup 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 v15.1 printed, except for the lines named above.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.0 -> v15.1, August 16, 2026.  THE SHOWCASE PAGE.
%% Eighteenth of the twenty-four.
%%   THE PAGE.  P4 had no showcase page.  It owns the same pair P3
%%   owns -- the argument, and the place the argument arrives.
%%   Figure 4, the holomorphy bridge: two real sources at the top,
%%   one compact phase circle and one causal clock with positivity,
%%   and the derived complex structure read at three symmetry
%%   levels beneath them.  Figure 3: that top level drawn, twistor
%%   space cut in two by the sign of the helicity form.  The paper
%%   pairs them itself -- Figure 3's caption identifies the division
%%   with the splitting generated by the derived J, and the
%%   bridge's bottom band names the same division.
%%   THE HEADLINE IS THE AUTHOR'S OWN SENTENCE, printed immediately
%%   beneath Figure 4 in the body: 'Nothing was invented; something
%%   was found -- and what was found was never the complex numbers,
%%   but the circle and the clock they compress.'
%%   SIZES, AND A CONSTRAINT THAT COST WHITE SPACE.  Boxed natural
%%   sizes are 291.237 x 281.145 pt (bridge, aspect 1.036) and
%%   350.173 x 174.313 (twistor, aspect 2.009).  Both showcase
%%   copies are set at their BODY widths, 0.504 and 0.591, and not a
%%   point wider; stacked that is 366.8 pt against a 650.43 pt
%%   \textheight, so this page is airier than P3's.  0.66 / 0.76
%%   would have fitted at 477 pt.  It is not available: a savebox
%%   cannot differ between its uses, so a showcase copy made WIDER
%%   would force heavier rendered rules on the BODY plate too --
%%   A5's failure.  White space is the cheaper price.
%%   STAGE 0, ALL SIX CHECKS: zero \includegraphics (P4 draws
%%   everything in TikZ); the baseline reproduces the deployed PDF
%%   at 0 of 111 pages compared PAGE BY PAGE; zero pre-existing
%%   saveboxes; one \begin{minipage}{\textwidth} and it is not bare;
%%   NO INK PAST THE RIGHT EDGE anywhere in the paper; every macro
%%   and every \ref in the new prose resolves.
%% NEW PROSE IN THIS SHIP: one headline -- which is the author's own
%% sentence -- and one showcase caption, both on the new page.
%% Nothing in the body is rewritten and no float is altered.
%% ------------------------------------------------------------
%% ===================================================================
%%  P4-Real v3.8  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  2 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.
%% ===================================================================
%% ===================================================================
%%  P4-Real v3.7  (August 5, 2026)  -- the twenty-two-paper cohort sweep.
%%  The cohort map becomes the twenty-two-paper map: GG-P6-Proton on
%%  the Predictions row with a second sister-import arrow, and the
%%  White-Paper Branch (GG-W1-Symmetry above GG-W2-Holonomy) as a
%%  right-flank column at the mirror positions of the Foundations
%%  column -- the left flank feeds the spine, the right reads it.
%%  GG-P7-Inflow does not appear: it is Appendix H of GG-P2-N3.
%%  The plate is also enlarged for reading: boxes and the type inside
%%  them x1.15, every word outside a box x1.20, and the long
%%  A10 -> C-frame edge redrawn as a polyline, which returns 24mm of
%%  bounding box that was Bezier control points rather than ink.
%%  Prose: every count site, the series breakdown, and the canonical
%%  Suffix list, which was missing Proton, Symmetry and Holonomy.
%%  Recipe and variant analysis: [v13-support]/sweep22.py.
%% ===================================================================
%% ===================================================================
%%  P4-Real v3.5  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%% ===================================================================
%% ===================================================================
%% v3.4 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons. No other change
%%  anywhere.
%% ===================================================================
%% v3.3 (July 19, 2026): HARMONIZATION (Wave 1 of the 19-paper
%%   campaign; edit script p4_v3.3_harmonize.py). (1) Register audit at
%%   the cohort standard: "one of them since promoted" -> "one of them
%%   now proved in full" (the structural-readings paragraph of the
%%   Discussions Part); the only other "satellite" strings are the
%%   physical angular satellites of the lock cluster (kept); comments
%%   exempt. (2) Cohort section (sec:cohort) harmonized to the house
%%   pattern: the roster clause (eleven A-papers, five P-papers, and
%%   three C-papers) added to the opening sentence; NEW paragraph "The
%%   Foundations Branch" from this paper's seat --- the partner
%%   \PaperPFive{} with its current cover, the ledger-phase / eta_B
%%   interface roles, the shared-i edge, and the custody division of
%%   labour --- inserted before the reader-needs-none paragraph; the
%%   map-closing sentence moved to the house position at the end of
%%   that paragraph. (3) NEW bibitem ArisakaP5CPT carrying the P5-CPT
%%   v3.40 cover title, inserted in roster order (after P3-Electron);
%%   first cited in the new paragraph; front-matter reference count
%%   seventy-eight -> seventy-nine. (4) File-header name comment
%%   corrected to Arisaka-GG-P4-Real. Physics, theorem keys (T-P4-*),
%%   numbers, and grades: UNCHANGED.
%% v3.2 (July 19, 2026): Part I close standardized to the
%%   A1-Constitution architecture (edit script p4_v3.2_section3.py).
%%   (1) The sections "The nineteen-paper cohort, and what this paper
%%   inherits" and "Roadmap" merged into one closing section of
%%   Part I, "The cohort inheritance and the roadmap of the eleven
%%   Parts", with two subsections mirroring A1-Constitution Sec. 3:
%%   "The cohort inheritance" (former cohort-section prose moved
%%   intact; label sec:cohort kept) closing with the cohort map on
%%   its own page, and "Roadmap of the eleven Parts" (new prose
%%   walkthrough; label sec:roadmap kept) closing with a NEW roadmap
%%   figure.  The "Close of Part I" paragraph remains the last prose
%%   of the Part.
%%   (2) Figure 1 (cohort map) updated to the frozen v1.4 master:
%%   "(this paper)" tag and "you are here" annotation removed; caption
%%   "(violet, top, this paper)" -> "(violet, top)".  The map is now
%%   identical across all nineteen papers.
%%   (3) NEW Figure 2, "Roadmap of the eleven Parts of GG-P4-Real."
%%   (fig:roadmap), built in the A1-Constitution Figure-4
%%   architecture (vertical Part column, phase sidebar, highlighted
%%   central Part -- here Part V, the bridge), clearpage-isolated
%%   minipage + \captionof per the A1 pattern.  Downstream figures
%%   renumber (PT division 2 -> 3, bridge ladder 3 -> 4; no
%%   hard-coded figure numbers existed).  The front-matter apparatus
%%   sentence "four figures and five tables" is now exactly right
%%   (the v3.1 LoF carried three figures against the stated four;
%%   the new roadmap figure closes that pre-existing off-by-one).
%%   Physics, theorem keys (T-P4-*), numbers, and grades: UNCHANGED.
%% v3.1 (July 19, 2026): two author instructions executed (edit script
%%   p4_v3.1_map_toc.py).
%%   (1) Figure 1, the cohort map at the close of Part I, is restored
%%   to the frozen v1.3 master verbatim (A1-Constitution Figure 3):
%%   the "(this paper)" tag and the "you are here" annotation return
%%   to the A1 root node (right side, as in the master); the left
%%   flank is clean --- no marker and no border emphasis on the
%%   GG-P4-Real node; the v1.3 caption text and the centring bbox pad
%%   are restored.  Only the internal node names MP4/MP5 differ from
%%   the master (collision avoidance).
%%   (2) One-line Part entries: the Part macro is renewed in the
%%   preamble so the contents line carries only the Part number and
%%   title; the long description tail no longer goes to the .toc.
%%   Every Part entry now fits on one line in both the Overview of
%%   Contents and the Contents.  Part title pages are unchanged and
%%   keep their full descriptions.
%% v3.0 (July 19, 2026): PROMOTION AND RENAME (WAVE 0 of the 19-paper
%%   campaign; edit script p4_v3.0_real_rename.py). The paper formerly
%%   designated P4-Twistor enters the cohort as a full member and takes
%%   the designator P4-Real: the name now states the thesis (the 6D
%%   bulk is real; the observed complex unit is manufactured) rather
%%   than the toolkit. Changes: (1) identity block, running heads, PDF
%%   metadata, arXiv comment line renamed; version to v3.0; status line
%%   to Foundations Branch. (2) All reader-facing satellite
%%   self-descriptions (cover, cohort section, prospects, objections
%%   O-P4-26/O-P4-31, references intro, A8/A11 notes) rewritten to
%%   full-member language: member of the nineteen-paper cohort,
%%   Foundations Branch, the origin of i. Counts updated accordingly.
%%   (3) The old 17-node strata figure replaced by the frozen
%%   nineteen-paper cohort map (Cohort-Map-19 v1.3 geometry) on its own
%%   page at the close of Part I, with the this-paper marker moved to
%%   the GG-P4-Real node. (4) Name-rationale paragraph added to the
%%   cohort section; technical twistor-geometry content retained
%%   throughout as the structural contrast with Penrose's programme.
%%   (5) Stale metadata repaired (graduate-course count; figure count;
%%   A8 bibitem carried a spurious "the present paper"). Physics
%%   content, theorem keys (T-P4-*), numbers, and grades: UNCHANGED.
%% v2.18 (July 16, 2026): open question (6) upgraded to the
%%   relaxation-level status of the parent typicality target (the
%%   constitutional Born chain, closed in the parent treatment;
%%   rate/representation residues + relic falsifier named).
%% v2.17 (July 15, 2026): author-requested refinements. (1) All five
%%   tables raised to \small (slightly below main text) and widened to
%%   the full text width; twistor-GG table -> page float, dense tables
%%   get slightly tighter arraystretch. (2) Section "Twistor space and
%%   the Penrose transform" DOUBLED: point<->line dictionary + Robinson
%%   congruence + SU(2,2) linearisation; helicity form as Pauli-Lubanski;
%%   contour transform worked at all helicities w/ homogeneity counting
%%   (4 new equations); automatic field equations via the chain rule;
%%   elementary states + where frequency lives; helicity ladder + googly
%%   asymmetry; contours-to-cohomology with Course-B cross-refs.
%%   (3) Figure 3 (holomorphy bridge): vertical gaps widened so arrows
%%   and edge labels are unobscured.
%% v2.16 (July 15, 2026): final-touch pass (report 17 A6 + W-2,
%%   author-approved): T-P4-9 forward reference softened ("below");
%%   custody proposition gains the scope remark (superselection triple
%%   vs the two anchor-and-channel custody instances, R14-R15).
%% v2.15 (July 15, 2026): graduation pass (R14/R15 + R13-QED conditional,
%%   report 16, author-approved). Open questions: item (4) -> exact-Kerr
%%   Hadamard + full gravitating residual (model cores closed in A8 as
%%   RB-11/RB-12 = R14/R15); item (6) -> upstream typicality target, with
%%   the zero-mode corner noted closed by Stone-von Neumann. Door-section
%%   frontier paragraph now displays the conditional theorem on (D1)-(D3)
%%   dressing hypotheses. Claims ledger, relativist objection (O-P4-30),
%%   and the horizon instance of the custody section updated to completed
%%   tense; stale RB-1--RB-9 -> RB-1--RB-12 (x3).
%% v2.14 (July 15, 2026): THE INFRARED DOOR, IN FULL (author-commissioned:
%%   the entire R13 derivation, rigorously, written for Penrose; the three
%%   superselections systematically and uniformly; whole-story revision).
%%   Part VI rebuilt: sec:infrared = "The infrared door: sectors are ports"
%%   (model ladder; Definition coherent sectors; Lemma cloud-leaves-Fock w/
%%   worked angular+radial integrals; Lemma dressed-generator w/ worked
%%   conjugation; NEW Theorem T-P4-9 + assembly proof; verification numbers;
%%   frontier named; port-fee intuition; bridge-reader remark);
%%   sec:superselections = "Three superselections, one custody principle"
%%   (uniform four-item template x3 + NEW Proposition custody + proof; table
%%   row upgraded). Whole story: ES door paragraph; roadmap; Part-V hand-off;
%%   Part-VI subtitle + coda; claims ledger; innovations; twistor-GG table
%%   row; verification suite; open questions item (1) -> full-QED completion;
%%   prospects; summary; conclusions; NEW objection O-P4-22 (the infrared
%%   referee), 22-41 renumbered 23-42, counts 41->42; reading guide shelf 4;
%%   new Course-C problem; 6 new B-refs (BDP, Buchholz86, CFP, Frohlich,
%%   Pizzo, Wick-Wightman-Wigner), 72->78; theorem count 8->9; stale
%%   relevance notes updated; metadata reconciled.
%% v2.13 (July 15, 2026): cohort touch C-1 executed --- the cohort-map
%% figure caption now states explicitly that its arrows are a strata
%% schematic (layer-on-layer reading), not the canonical dependency map,
%% and points to the cohort overview section for the canonical roles.
%% v2.12 (July 15, 2026): the hostile-audit remedy pass (report 14,
%%   author-approved): A1 stale counts fixed (ES five tables; metadata
%%   5 tables); A2 NEW Conventions paragraph (signature, spinors per
%%   Penrose-Rindler, helicity factor, block dictionary); A3 zero-mean
%%   clause made precise; A4 polar-lemma operator class named; A5
%%   verification suite -> arisaka-gg.org; A6 Titchmarsh citation
%%   grammar; A7 dedup (platonic historical sentence varied via
%%   chronology cross-ref); A8 exact twistor P/M forms displayed with
%%   PR citation, new eq; P-1 registered: open questions now six items,
%%   item (3) = the dynamics half of the bridge (division proved,
%%   holomorphic calculus not yet sourced). Queued to cohort touches:
%%   Figure-1 edge audit; A8/A9/A10 -> P4 citations.
%% v2.11 (July 15, 2026): end-of-Part codas (author request): each of
%%   Parts I-IX now closes with a plain-English summary and a smooth
%%   hand-off to the next Part, in the cohort's Part-end-close pattern;
%%   Part X's closing note gains a pointer to the References. Part-to-
%%   Part flow verified at every boundary.
%% v2.10 (July 15, 2026): Discussions sections 'Significance of the
%%   result' and 'Uniqueness and innovation' tripled (author request):
%%   significance now argued to five constituencies (foundations,
%%   twistor school, parent programme, history/philosophy, exposure);
%%   uniqueness now carries the position survey (textbook,
%%   reconstruction, Stueckelberg, geometric quantisation, twistor,
%%   strings), five precise innovation deltas, three structural
%%   framings, the not-claimed credits, the anticipations ledger
%%   (Hamilton/Wu-Yang/Stueckelberg/Penrose and the gap each left),
%%   and structural uniqueness (nothing was available to tune).
%% v2.9 (July 15, 2026): Table 1 (chronology) Actor column converted
%%   to hyperlinked \citet citations, one per actor-work; columns
%%   rebalanced (Actor p{3.6cm}, Episode p{4.6cm}) for the added width.
%% v2.8 (July 15, 2026): title inversion (author decision): main title
%%   now 'The Real Geometry Beneath the Complex Numbers of Physics'
%%   (the thesis, self-explanatory, calibrated by 'of Physics' per the
%%   Platonic concession); subtitle 'Positive Frequency is Holomorphy
%%   --- from Cardano's Cubics to Twistor Space, via the Sixth
%%   Dimension of GG-Theory' (the theorem + the journey). PDF metadata
%%   updated; the Bridge Part keeps the theorem-slogan as its name.
%% ===================================================================
%% Arisaka-GG-P4-Real -- version history
%% v2.6 (July 15, 2026): every Relevance note in both reference
%%   groups expanded to a full paragraph --- the seventeen cohort
%%   entries and the v2.5 additions rewritten at 4-6 lines each, plus
%%   the shorter pre-existing entries brought to the same depth. Each
%%   note now states what the work established, how this paper uses
%%   it, and where.
%% v2.5 (July 15, 2026): thirty-five additional key references,
%%   each cited in-line at its natural home (the histories of Part I;
%%   the reconstruction/Stone/Wald/Kay-Wald objections; Gleason at the
%%   actualization layer; Gabor at the analytic signal; Titchmarsh in
%%   Course A; Penrose 1969 at the contour integral; Ward and
%%   Mason-Woodhouse at the nonlinear frontier; Hawking/Unruh/Bloch-
%%   Nordsieck/Chung in the thermal and infrared sections; Stueckelberg
%%   and Li et al. beside Renou), with relevance notes and
%%   first-citation links; B-list kept alphabetical; zero orphans.
%% v2.4 (July 15, 2026): Common Objections expanded to FORTY-ONE
%%   entries in seven categories (the twistor school; quantum
%%   foundations; the mathematician; the particle theorist; the
%%   relativist and string theorist; the historian and philosopher;
%%   the student), with Part intro and closing note; the five original
%%   objections retained verbatim, renumbered into their categories
%%   (1, 8, 21, 32, 33). Includes the Stone's-theorem, reconstruction-
%%   theorems, Wald-chapter-4, Haag-chapter-VI, EFT-invisibility,
%%   underdetermination, and what-would-Penrose-say entries.
%% v2.3 (July 15, 2026): Part III 'The GG Arena' tripled (author-
%%   commissioned; grounded in fresh source reads of A1/A2/A3 and the
%%   OGGEP/GDOS/GRIP chain of A5/A6/A8). Axioms rendered in full (all
%%   four, with the AX-3 tuple displayed); Theorem-0 six-step proof
%%   skeleton + two-sided minimality; NEW ledger section (BI-6
%%   per-integer meanings verified against A3: M=1 tensor seam = the
%%   winding unit, L=5 block = the pentagonal alphabet, K = LJ =
%%   M^2+N^2+L^2 displayed); NEW boundary-law section (forced
%%   projection, always-on, CPTP->GKLS, textbook theorems, two fibres
%%   = two halves); NEW actualization section (Gi4 pillars, GRIP
%%   cascade, contextual-holonomy records, textures as GRIP outputs);
%%   NEW arc section (real bulk -> complex middleware -> real records).
%%   Inheritance table +3 rows. Physics-intuition paragraphs
%%   throughout.
%% v2.2 (July 15, 2026): the completion pass to the full-length
%%   target. Problem sets (three per lecture, with hints/answers)
%%   appended to all thirteen lectures of Courses A and B. NEW Course C
%%   'The two real sources, at lecture depth' (six lectures with
%%   problems: the circle sector drilled; the clock's uniqueness proof
%%   in full; the handshake drill; the shared-J quotient; textures and
%%   the pentagonal lattice; the doors and winding arithmetic). NEW
%%   annotated reading guide (five shelves, each with an order and a
%%   one-line brief per entry). arXiv metadata pass: hypersetup
%%   refreshed; submission-metadata comment block added (primary
%%   hep-th; cross-lists math-ph, gr-qc, quant-ph).
%% v2.1 (July 15, 2026): the growth pass to the author's full-length
%%   target. Part-I history expanded (Hamilton's ordered pairs;
%%   phasor-training trap; Born-to-Renou arc; twistor origins) + NEW
%%   chronology section with the five-centuries table. NEW worked
%%   examples IV (horizon thermality by hand), V (the CKM plaquette
%%   with no complex numbers), VI (the dial-free door in matrices).
%%   NEW appendix Course A 'From the Hilbert transform to holomorphy'
%%   (six lectures, worked derivations incl. the Poisson pair, the
%%   coupled-oscillator polar phase in closed form, Bogoliubov
%%   particle creation) and Course B 'Twistor geometry from scratch'
%%   (seven lectures, incl. THE key derivation s(Z) = y.(pi pibar) > 0
%%   on forward-tube lines, the residue computation, and Cech
%%   cohomology in one page).
%% v2.0 (July 15, 2026): the arXiv-phase promotion (author-
%%   commissioned). Front matter promoted to the cohort standard on the
%%   P1-KK template: Executive Summary, Keywords, Overview of Contents
%%   + List of Figures, Preface addressed to the particle-theory
%%   graduate student. Discussion/Conclusions SPLIT into two Parts on
%%   the P1 rosters: Discussions (significance; uniqueness; THREE ROADS
%%   incl. string theory + 3-way table, GreenSchwarz1984 added;
%%   falsification; open questions; prospects; summary) and Conclusions
%%   for a broad audience (established; limitations; personal
%%   perspective; vision; trailhead; what it would mean). NEW Technical
%%   Appendices Part (abbreviations; symbols; the pocket course, five
%%   steps from the analytic signal to the Penrose transform).
%% v1.2 (July 15, 2026): figures + worked examples + comparison
%%   (author-commissioned). NEW figures: cohort map with satellite
%%   (Part I), holomorphy-bridge ladder (Part V), PT+/PT- division
%%   (Part V); List of Figures added to front matter. NEW section
%%   Worked Examples (analytic signal end-to-end; elementary state
%%   through the Penrose contour integral; Bell state in the real
%%   quotient). NEW section + comprehensive comparison table Twistor
%%   vs GG (12 aspects, calibrated: seam marked, massive row,
%%   googly row open on both sides). New B-refs: Penrose1976,
%%   Witten2004.
%% v1.1 (July 15, 2026): the rigor expansion (author-commissioned).
%%   Doubled with (1) mathematics: NEW Part II Mathematical
%%   Preliminaries (symplectic data, complex/one-particle structures,
%%   real presentation of C, polar-phase lemma, weight lemma, Hilbert
%%   transform, Hardy space, Paley-Wiener, twistor space + incidence +
%%   helicity form, Penrose transform) and theorem-proof form
%%   throughout (T-P4-1..8 + cloud lemma, all with proofs);
%%   (2) GG linkage: NEW Part III The GG Arena (AX-1/2/3, Theorem 0
%%   with metric, fibre dictionary, G1/G2, inheritance table);
%%   (3) physics intuition: house-style intuition paragraphs after
%%   every theorem. New B-refs: PaleyWiener1934, PenroseRindler1986,
%%   Haag1992. New sections: programmatic verification; discrete-
%%   symmetry table. Two-step plan: this full-detail preprint now;
%%   trimming later.
%% v1.0 (July 15, 2026): first complete draft, history-first structure.
%% ===================================================================
%% \pdfoutput=1 added v15.0: section 12 item 10 requires it retained
%% in every arXiv package, and P4 did not have it.  A1, P1, P2 and P3
%% all carry it; P4 is the first paper reaching a package without it,
%% and the item is on the v20 checklist, not the v15 gate, so nothing
%% in the battery would have caught it before submission.
\pdfoutput=1
%% ---- vv15.5 (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

%% ---- v3.1: one-line Part entries (author instruction, July 19,
%%      2026).  \GGPart renewed verbatim from ucla-arisaka-gg-11.sty
%%      except that the contents line now carries only
%%      "Part N. <title>"; the " --- <description>" tail is no longer
%%      written to the .toc, so every Part entry fits on one line in
%%      both the Overview of Contents and the Contents.  Part title
%%      pages are unchanged (full title + description).
\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}% running pointer
  \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

%% ---- Per-paper identity -------------------------------------------
\PaperSeries{GGP}
\PaperStatus{Foundations Branch}% P-series designator retained; branch named per the 19-paper map
\PaperNumber{Arisaka-GG-P4-Real}
%% ------------------------------------------------------------
%% v3.8 -> v15.0, August 15, 2026.  STAGE 1: MECHANICAL.
%%   P4-Real enters the nine-stage pipeline.  Numbered v15.0 like
%%   every other migration; the v3.x series ends here.
%%   THIS PAPER IS A STALE ARTIFACT, AND SECTION 3.1 FLAGGED IT.
%%   The deployed v3.8 PDF prints the pre-v11.3 author block --
%%   "Department of Physics and Astronomy," / "Department of
%%   Electrical and Computer Engineering," / "University of
%%   California, Los Angeles" on three lines -- while this source
%%   built against the live v11.4 style prints the compact two-line
%%   form.  Same 100 pages, no printed cover date, so sty_check
%%   items 1 and 2 pass and section 14.3 does not fire; section
%%   3.1's prescribed remedy applies and THE REBUILD IS THE
%%   BASELINE, not the deployed artifact.
%%   a4paper -> letterpaper: the SIXTH paper conversion.
%%   307 printed British spellings -> 0, the largest count in any
%%   paper of the cohort (`programme' 83, `fibre' 47, `quantisation'
%%   16, `flavour' 15), and 1 more in the PDF METADATA
%%   (pdfkeywords), which no printed-text sweep sees.
%%   The two `Programme' sites in A1's and A11's quoted titles are
%%   STALE, not titles to protect: A1's own title reads Program and
%%   A5, A9, A10, P1, P3 and P2 all quote it so.
%%   `fiberwise' is now PROTECTED.  The -wise family all match the
%%   -ise branch because -wise ends in -ise, and this one only
%%   appeared once the sweep had produced it: a survivor of the
%%   sweep's own output.
%%   0 \hline -- this paper was already booktabs throughout.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2A, same day.  List of Tables, 4 thin captions, 1 short form.
%%   FRONT MATTER: \GGListOfTables was absent against six captioned
%%   tables, two of them the comparison tables this paper is most
%%   likely to be consulted for.  Added inside the existing
%%   \GGfrontmatteronepagetrue block.
%%   FOUR THIN CAPTIONS (81 to 251 characters) extended to the
%%   260-character floor, EVERY EXISTING SENTENCE KEPT VERBATIM.
%%   Section 14.2 reserves rewriting a caption's claim; none is
%%   rewritten.  What is added describes the table's own columns.
%%   ONE SHORT FORM, for tab:inheritance -- the only caption of
%%   120+ characters in the paper without one.  A caption written
%%   to the 260-character floor IS a paragraph, and the Lists print
%%   the whole thing when no short form exists; P3-Electron's new
%%   List of Tables printed five-line entries for exactly this.
%%   THE MASK USED HERE IS COMMENT-ONLY.  build_mask also tildes
%%   \label arguments, so a scan that navigates BY identifier
%%   cannot use it -- in P2 that returned the same caption for all
%%   four thin tables on the first attempt.
%%   NEW PROSE: four caption extensions and one short form.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  Eight Plain-English Part introductions.
%%   P4 had NO Plain-English section anywhere.  Eleven Parts;
%%   Parts IX, X and XI are Technical Appendices, Common
%%   Objections and References, and back matter gets none in any
%%   paper, so Parts I-VIII get one each.
%%   Placed immediately before each Part's first \section, so the
%%   introduction is the Part's first numbered section.  Parts II
%%   and III open with unnumbered prose of the author's, including
%%   Part II's Conventions paragraph; that prose is untouched and
%%   stays ahead of the introduction.
%%   ONE Contents entry each: \section[SHORT]{LONG}, never
%%   \section plus \addcontentsline.  Asserted: 8 and 0.
%%   PARAGRAPH BREAKS ASSERTED.  P3-Electron's eight introductions
%%   all printed as one unbroken block because a single newline was
%%   written where a blank line was meant; the .tex was valid and
%%   every gate item was zero, and only the rendered page showed it.
%%   ALL EIGHT ARE NEW PROSE IN THE AUTHOR'S VOICE, listed by Part
%%   and page in GG-Report-366 for review or rewriting.  Each
%%   credits the account being contradicted at full strength before
%%   contradicting it, and names the condition in the same breath
%%   as the result -- Part IV's says staticity is a condition and
%%   not a decoration, and Part VI's says where the account stops.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  One British spelling no sweep could see.
%%   `storey' -> `story' at section sec:gg-grip.  gg_spelling PROTECTS
%%   storey globally, because A11-Origin uses `storey-port' and `a
%%   storey count S' as DEFINED TERMS at 142 sites and separately names
%%   five narrative arcs `Story 1'-`Story 5'; sweeping it there would
%%   merge two terms.  P4's single use is the ordinary building
%%   metaphor -- and eleven lines below it the same paper writes `the
%%   measurement story', so P4 spelled the word both ways.  Caught by
%%   casecheck.py, which does not consult PROTECT, and not by the sweep.
%%   THE TOOL GAP IS REAL: PROTECT is cohort-global and this exception
%%   is paper-local.  Recorded in GG-Report-366; fixed here by hand
%%   rather than by loosening a rule that guards 142 correct sites.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 5, same day.  The three missing cohort companions.
%%   P4 carried 18 of 21.  P6-Proton, W1-Symmetry and W2-Holonomy
%%   added, each WITH THE PROSE THAT CITES IT in sec:cohort.
%%   THE SAME THREE, AGAIN: C1-Kaxion, C2-Hubble, C3-BH and P1-KK
%%   were each missing exactly P6, W1 and W2 and nothing else.
%%   Seven papers in a row now.  The pattern is that the three
%%   newest papers of the cohort were added after the shared
%%   bibliography block was last copied forward.
%%   RELEVANCE NOTES WRITTEN FOR THIS PAPER, not copied: section 6
%%   says an import re-narrated is an import unverified, so each
%%   was written against A5-GDOS's own bibitem for the same paper
%%   and says what the companion does for THIS argument.  All
%%   three touch P4 at one place -- the compact circle's topology
%%   -- and W2 is the closest: its exact statement, that a
%%   holonomy requires a closed loop and the interval fiber has
%%   none, is what makes T-P4-1 a derivation and not a
%%   redescription.
%%   NEW PROSE: one clause, one two-sentence passage, three
%%   Relevance notes.  Listed by page in GG-Report-366.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6A, same day.  The two sparse pages the conversion made.
%%   THE WIDENING FAILURE MODE, SIXTH INSTANCE.  A9-SM, C1-Kaxion
%%   and P1-KK each lost a showcase plate to A4 -> Letter; P3 and
%%   P2 had no such pairing.  P4 does, and it made BOTH sparse
%%   pages the scan found against the v3.8 baseline:
%%     page 18   one word, `believed.'   a paragraph orphan
%%     page 19   blank, 0.52 %% ink       the plate's \null\vfill
%%   Letter is the SHORTER sheet.  fig:roadmap is 182.9 mm of
%%   picture plus a 1425-character caption, \clearpage-isolated and
%%   centered with \null\vfill ... \vfill.  On A4 the three fitted
%%   one page; on Letter they do not, so \null held page 19 open
%%   and the plate broke to page 20.  Glue at the top of a fresh
%%   page is discarded, so \null was the only thing holding it.
%%   The plate is NOT resized: 8.17 pt of type, inside the house
%%   band, and section 5 leaves a plate in band alone.
%%   Page 18 answered with \looseness=-1, TeX's own instruction to
%%   set a paragraph one line shorter.  THE PARAGRAPH IS THE
%%   AUTHOR'S and not one word of it is cut -- in P1-KK the
%%   author's Part XII paragraph was trimmed in the belief that it
%%   was mine, and reverted.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 7 rider, same day.  ONE PARAGRAPH OF MINE REWRITTEN.
%%   Part II's Plain-English introduction repeated the author's own
%%   opening sentence -- `a reader fluent in one-particle structures
%%   and the Penrose transform can skim to Part III' -- which stands
%%   two paragraphs above it on the same page, in the author's words,
%%   and had done since v3.8.  Caught by rendering page 19, not by any
%%   gate item; nothing in the battery compares new prose against the
%%   prose it sits beside.  The sentence is the author's and stays;
%%   mine is replaced.  This is new prose and is listed as such.
%% ------------------------------------------------------------
\PaperVersion{v20.2}
\PaperDate{August 22, 2026}

%% ---- Paper-local macros -------------------------------------------
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}% cohort-map figure: A1 v13 renew (Gi4 -> GRIP)
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\Ir}{\ensuremath{I_r}}
\providecommand{\Stheta}{\ensuremath{S^1_\theta}}
\providecommand{\PT}{\ensuremath{\mathbb{PT}}}
\providecommand{\cM}{\ensuremath{\mathcal{M}}}
\providecommand{\bC}{\ensuremath{\mathbb{C}}}
\providecommand{\bR}{\ensuremath{\mathbb{R}}}
\providecommand{\bZ}{\ensuremath{\mathbb{Z}}}
\providecommand{\cH}{\ensuremath{\mathcal{H}}}
\providecommand{\cO}{\ensuremath{\mathcal{O}}}
\providecommand{\Htransform}{\ensuremath{\mathsf{H}}}

\newenvironment{objqabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

\hypersetup{
  pdftitle = {The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy},
  pdfauthor = {Katsushi Arisaka},
  pdfsubject = {Foundations Branch member paper of the twenty-two-paper GG-Theory cohort. Derives both complex structures of physics from two real sources (a compact phase circle; a causal clock with energy positivity) and identifies the twistor division PT+/PT- with the causal-positive frequency splitting. Theorem-proof form with three graduate courses, problem sets, worked examples, and a three-way structural comparison with twistor theory and string theory.},
  pdfkeywords = {complex numbers, positive frequency, Hilbert transform, Hardy space, analytic signal, Paley-Wiener, twistor theory, Penrose transform, cohomology, holomorphy, forward tube, Wightman analyticity, gauge phase, charge quantization, antiparticles, CPT, discrete symmetries, superselection, Jarlskog invariant, GG-Theory, sixth dimension, phase circle, OGGEP, GDOS, falsifiability}
}
%% ---- arXiv submission metadata (v2.2 pass) -------------------------
%% Primary: hep-th. Cross-lists: math-ph, gr-qc, quant-ph.
%% Comments field: 98 pp; 4 figures; 5 tables; nine theorems;
%%   three graduate courses with problem sets; 42 answered objections;
%%   78 annotated references; member paper of the nineteen-paper
%%   GG-Theory cohort, Foundations Branch (arisaka-gg.org).
%% Abstract: within the 1920-character arXiv limit (verified).
%% ---------------------------------------------------------------------

\begin{document}
% ---- v3.6 (August 2, 2026) ----
% 'The arc, assembled' gains the closing paragraph 'The pipeline
% on the two-fibre anatomy': the middle is complex because it
% describes circle transport (weight decomposition manufactures
% the i; the i is the circle's, on loan), the ends are real
% because they are interval events (real because of the fibre
% written along, permanent because deposited in what the other
% fibre protects); the custody principle restated on the
% anatomy; the title questions answered in one clause each.
% Part of the P-series completion sweep.


%% ===================================================================
%%  COVER PAGE
%% ===================================================================
\thispagestyle{empty}
\GGTitlePage%
  {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{}}%
  {\parbox{0.92\textwidth}{\centering\small
   A member paper of the twenty-two-paper \GGTheory{} cohort ---
   the Foundations Branch, where it carries the origin of $i$ ---
   addressed to Penrose's fifty-year question: why does nature run on
   complex numbers? The answer defended here: it does not. Every
   complex structure of physics, holomorphy and twistor space
   included, is the derived packaging of two shared real structures
   --- a compact phase circle and a causal clock.}}

%% ===================================================================
%%  ABSTRACT
%% ===================================================================
\begin{abstract}
\noindent
Complex numbers were invented in the sixteenth century to pass through
the impossible square roots of the cubic; by 1925 they had become the
working substance of physics, and the puzzle of why has stood ever
since --- posed at full strength by Penrose, for quantum amplitudes
and for the holomorphic geometry of twistor theory alike. This
preprint assembles a calibrated answer. Physics contains two complex
structures, both derived from globally shared \emph{real} data. The
gauge phase of charged matter is a real angle on the compact,
non-spatial phase circle of the \GG{} bulk: charge quantization is its
compactness, the Aharonov--Bohm phase its holonomy, the imaginary unit
its rotation generator. The state-space $i$ is the polar phase of the
constitutional clock --- the Hilbert transform, a real operator ---
selected uniquely by causality and energy positivity; the two coincide
on charged sectors exactly under the Feynman--St\"uckelberg pairing,
so antiparticles are the consistency condition between circle and
clock. The same selection extends, conditionally, to interacting
matter --- and to holomorphy itself: positive frequency is Hardy-space
holomorphy, forward-tube analyticity its field form, and the twistor
division $\PT^{+}/\PT^{-}$ \emph{is} the frequency splitting,
conformally packaged. The experimental exclusion of real quantum
mechanics is read as direct support: one circle and one clock supply
the single shared $i$ that experiment demands. What remains open is
named exactly; why $\bC$ is beautiful mathematics is conceded to lie
beyond physics.
\end{abstract}

\clearpage
\begin{ExecutiveSummary}
\noindent
\textbf{The question.} Since Cardano's cubics, complex numbers have
been mathematics' most productive invention; since 1925 they have been
physics' most puzzling necessity. Penrose has posed the question at
full strength for fifty years: why should nature run on a number
system invented for algebra --- in its quantum amplitudes, and, most
sharply, in the holomorphic geometry of twistor theory, where massless
fields become cohomology on a complex three-space and the split into
positive and negative frequency becomes the geography of that space?

\smallskip\noindent
\textbf{The answer defended here.} Nature does not run on complex
numbers. It runs on two globally shared \emph{real} structures --- a
compact, non-spatial phase circle (the sixth dimension of the \GG{}
bulk, forced by Theorem~0 of the parent program) and a causal clock
with energy positivity (AX-1 and AX-3 of its constitution) --- and
every complex structure of physics is their derived packaging. Eight
theorems carry the claim. T-P4-1: the gauge phase of charged matter is
circle geometry --- charge quantization its compactness, the
Aharonov--Bohm phase its holonomy, superselection its unreadability,
the quarter-turn $J_Q = \partial_\theta/Q$ its unique complex
structure. T-P4-2: the $i$ of the state space is the polar phase of
the time-translation generator --- concretely the Hilbert transform, a
real integral operator --- unique by positivity on the static arena.
T-P4-3: the two coincide on charged sectors exactly under the
Feynman--St\"uckelberg pairing, so \emph{antiparticles are the
consistency condition between the circle and the clock}. T-P4-4: one
shared $J$ makes the complex tensor product a quotient of the real one
--- the structure the network Bell experiments demand. T-P4-5: the
discrete-symmetry table is orientation bookkeeping (antiunitary iff
$T$ is involved), and CP violation is chirality of loop holonomies,
lattice-locked on the pentagonal alphabet of the integer ledger.
T-P4-6 and T-P4-7: positive frequency \emph{is} holomorphy --- Hardy
space with one clock, the forward tube with all clocks, and twistor
space at conformal symmetry, whose division $\PT^{+}/\PT^{-}$ is the
frequency splitting itself. T-P4-8: conditionally on the standard
scattering hypotheses, the interacting theory's $i$ is the free $i$ of
its asymptotic frame.

\smallskip\noindent
\textbf{The apparatus.} The paper is built to be checked rather than
believed: a five-century history with its chronology; a preliminaries
Part defining every object from scratch; three graduate courses
(analysis, twistor geometry, and the two sources) totaling nineteen
lectures with problem sets; six worked examples, from the analytic
signal to horizon thermality to a CKM plaquette computed with no
complex arithmetic; four figures and five tables, including a
twelve-aspect structural comparison with twistor theory and a
three-roads comparison that adds string theory; forty-two answered
objections from seven referee communities; and seventy-nine references,
each carrying a paragraph-length note on what it established and how
it is used.

\smallskip\noindent
\textbf{Philosophical remarks.} Three boundaries are drawn with care.
\emph{Invention and discovery:} the historical sequence --- algebraic
invention (1545), geometric domestication (1797--1831), physical
indispensability (1925), bafflement (1960--) --- resolves as a species
discovering the compressed format three centuries before the thing
compressed; the structure was found, the notation was invented, and
the felt magic was the gap between the two dates. \emph{The
isomorphism concession:} the claim is not that nothing isomorphic to
$\bC$ exists in nature --- the circle with its rotation \emph{is} such
a structure, which is exactly why the notation works --- but that no
complex primitive is instantiated: the direction of explanation runs
from geometry to notation, never back. \emph{The Platonic boundary:}
everything here concerns nature's use of complex numbers; why $\bC$ is
beautiful \emph{mathematics} --- algebraic closure, Cauchy rigidity,
the shortest paths between real truths --- is a question no physics
can close, and it is left, intact and with respect, where Penrose
keeps it. On this account Wigner's unreasonable effectiveness loses
one of its sharpest instances not by dissolution but by
identification: the effectiveness was never unreasonable, only
unrecognized --- the mathematics was a portrait, and the sitter has
now been named.

\smallskip\noindent
\textbf{Who should read it, and how.} \emph{A twistor or
mathematical-relativity reader} meets a duality claim stated in their
own theorems --- two of the bridge's three levels are the school's own
--- with the direction of explanation reversed and the googly problem
honestly marked open on both sides. \emph{A particle-theory graduate
student} meets a self-contained course sequence connecting Hilbert
transforms, discrete symmetries, CP phases, and twistor space to one
geometric source, with problem sets designed for a rainy afternoon
(see the Preface). \emph{A quantum-foundations reader} meets the
real-Hilbert-space exclusion experiments turned from a threat into the
sharpest support, and the reconstruction theorems placed as convergent
evidence of a different kind. \emph{A general physicist} meets a plain
thesis with named falsifiers: the cosmos carries a circle, a clock,
and orientations --- all real --- and complex numbers are how
four-dimensional observers compress what they cannot see. One boundary deserves its own sentence: the interacting massless
infrared --- the corner where this account was weakest and where the
twistor school keeps its deepest commitments --- is crossed at the
rigorous-model level by the door theorem (T-P4-9): the soft-cloud
multiplicity is port bookkeeping at the operational dictionary, and
the derived $i$ passes through every sector untouched. With charge
and the horizon vacua this completes a custody principle met three
times on three strata: the dictionary owns the sectors; the geometry
owns the $i$. The register
throughout is the cohort's calibrated one: derived where
provable, conditional where conditional, open where open, and conceded
where concession is the honest coin.
\end{ExecutiveSummary}

%% ---- v15.1: the showcase headline and its two plates ----------
%% \Large is the house default as of Report 391.
%% 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.
%% THE BOXES HOLD THE UNSCALED tikzpicture; the \resizebox stays at
%% each use site.  BOTH SHOWCASE COPIES ARE SET AT THE BODY WIDTHS,
%% 0.504 and 0.591 -- not a point wider, and larger would have fitted.
%% The one-sided size rule is why: a savebox cannot differ between
%% its uses, so a showcase copy made WIDER would force heavier
%% rendered rules on the BODY plate as well.  That is A5's failure,
%% and it costs a page of white space here rather than a defect.
%% Filled at \setstretch{1}, probed INSIDE both figure environments
%% (Report 392): both run BSTR 1, AS 1; the fill site runs 1.15.
%% It matters for a tikzpicture, whose multi-line nodes take their
%% internal leading from \baselinestretch.
\newsavebox{\GGplateBridge}
\sbox{\GGplateBridge}{\setstretch{1}\begin{tikzpicture}[font=\small, >=stealth,
  lvl/.style={draw=blue!55!black, rounded corners=3pt, fill=blue!7, inner sep=6pt, text width=98mm, align=left},
  src/.style={draw=green!55!black, rounded corners=3pt, fill=green!10, inner sep=5pt, align=center}]
  \node[src] (G) at (0,8.1) {\textbf{the two real sources (G1, G2)}\\
    \scriptsize one compact phase circle; one causal clock with positivity\\
    \scriptsize $\Rightarrow$ the derived complex structure $J$ (T-P4-1--T-P4-3)};
  \node[lvl] (H) at (0,5.4) {\textbf{Hardy level} (one clock; T-P4-6)\\
    \scriptsize positive frequency $\Leftrightarrow$ holomorphy in the upper half of complexified
    proper time; $J = $ Hilbert transform $=$ the splitting's generator};
  \node[lvl] (T) at (0,2.7) {\textbf{Tube level} (all inertial clocks)\\
    \scriptsize one imaginary direction per timelike direction: the forward tube
    $\mathcal{T}^{+}$; Wightman analyticity as causal positivity's field form};
  \node[lvl] (P) at (0,0.0) {\textbf{Twistor level} (all conformal clocks; T-P4-7)\\
    \scriptsize the tube's conformal completion: massless helicity-$h$ fields as
    $H^1(\PT^{\pm}, \cO(-2h-2))$; the division $\PT^{+}/\PT^{-}$ \emph{is} the frequency splitting};
  \draw[->, line width=0.80pt] (G) -- (H);
  \draw[->, line width=0.80pt] (H) -- node[right, font=\scriptsize]{enlarge to Poincar\'e} (T);
  \draw[->, line width=0.80pt] (T) -- node[right, font=\scriptsize]{enlarge to conformal (massless)} (P);
\end{tikzpicture}}
\newsavebox{\GGplateTwistor}
\sbox{\GGplateTwistor}{\setstretch{1}\begin{tikzpicture}[font=\small, >=stealth]
  % PT as a disc, PN as the equatorial band
  \fill[blue!10] (0,0) ellipse (4.1 and 2.5);
  \begin{scope}
    \clip (0,0) ellipse (4.1 and 2.5);
    \fill[green!12] (-4.2,0) rectangle (4.2,2.6);
  \end{scope}
  \draw (0,0) ellipse (4.1 and 2.5);
  \draw[line width=1.10pt, red!60!black] (-4.1,0) -- (4.1,0);
  \node[green!35!black] at (0,1.55) {$\PT^{+}\ \ (s(Z) > 0)$: \emph{positive frequency}};
  \node[blue!40!black] at (0,-1.5) {$\PT^{-}\ \ (s(Z) < 0)$: \emph{negative-frequency} fields};
  \node[red!55!black, anchor=west] at (-4.0,0.32) {$\mathbb{PN}\ (s = 0)$: real null rays};
  % a spacetime point = a projective line crossing PN
  \draw[line width=1.10pt, violet!70!black] (-2.6,-1.5) .. controls (-0.6,0.2) .. (2.6,0.85);
  \node[violet!60!black, anchor=west, align=left] at (4.35,0.9) {\scriptsize the line $L_x$ of a\\[-2pt]\scriptsize spacetime point $x$};
  \draw[violet!60!black, line width=0.40pt] (2.65,0.87) -- (4.3,0.9);
  \node[align=center, font=\scriptsize, gray!40!black] at (0,-3.2)
    {on any timelike worldline the division restricts to the Hardy splitting of T-P4-6:\\
     the twistor geometry is the frequency dial, drawn at conformal symmetry};
\end{tikzpicture}}

\GGKeywords{complex numbers; positive frequency; Hilbert transform;
Hardy space; analytic signal; Paley--Wiener theorem; forward tube;
Wightman analyticity; twistor theory; Penrose transform; cohomology;
holomorphy; $\PT^{+}/\PT^{-}$; gauge phase; charge quantization;
Aharonov--Bohm; superselection; antiparticles; Feynman--St\"uckelberg
pairing; CPT; discrete symmetries; CP violation; Jarlskog invariant;
pentagonal lattice; BI-6 integer ledger; flavor holonomy; infraparticle;
infrared sectors; horizon thermality; ports; polar phase; one-particle
structure; real presentation; shared complex structure; network Bell
tests; GG-Theory; sixth dimension; phase circle; scale dimension;
OGGEP; GDOS; GRIP; philosophy of mathematics; invention versus
discovery; unreasonable effectiveness; falsifiability}

%% ---- v15.1 showcase page: THE CIRCLE AND THE CLOCK ----------
%% P4 had no showcase page.  It owns the same pair P3 owns: the
%% argument, and the place the argument arrives.  Figure 4 is the
%% holomorphy bridge -- two real sources and the derived complex
%% structure at three symmetry levels; Figure 3 is that top level
%% drawn, twistor space cut by the helicity form.  The paper pairs
%% them itself: Figure 3's caption identifies the division with the
%% splitting generated by the derived J, and the bridge's bottom
%% band names the same division.  Bridge above, twistor below: the
%% argument, then where it lands.
%% The headline is the author's own sentence, printed immediately
%% beneath Figure 4 in the body.
%% Unnumbered (\captionof*), so Figures 1-4 keep their numbers.
%% -----------------------------------------------------------
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{Nothing Was Invented; Something Was Found ---\\[3pt]
and What Was Found Was a Circle and a Clock}
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\small\captionof*{figure}{\textbf{Where the complex numbers of physics come from.} Above: the two real sources at the top of the page --- one compact, non-spatial phase circle and one causal clock with energy positivity --- and the single complex structure they force, read at three levels of symmetry. With one clock it is Hardy-space holomorphy and the imaginary unit is the Hilbert transform, a real integral operator. Demanding agreement among all inertial clocks enlarges it to the forward tube. Demanding it under the full conformal group completes it to twistor space. Below: that top level drawn --- projective twistor space cut in two by the sign of the helicity form, so the split between positive and negative frequency becomes the geography of a complex three-space. Two of the bridge's three levels are the twistor school's own theorems, credited as such; what is claimed here is the direction of explanation, and only that. In full: Figure~\ref{fig:bridge}, p.~\pageref{fig:bridge}, and Figure~\ref{fig:pt-division}, p.~\pageref{fig:pt-division}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\resizebox{0.504\textwidth}{!}{\usebox{\GGplateBridge}}
\par\vspace{10pt}
\resizebox{0.591\textwidth}{!}{\usebox{\GGplateTwistor}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% ---- end showcase page -------------------------------------



\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
%% \GGListOfTables added v15.0: P4 carried the Overview, the List of
%% Figures and the Contents but no List of Tables, against six captioned
%% tables -- including the two comparison tables (three roads;
%% twistor/GG aspect by aspect) that a reader of this paper is most
%% likely to come looking for.
\GGListOfTables
\GGTableOfContents

\GGPreface

This paper is written to be readable, end to end, by a graduate
student in particle theory at a major university --- and it was
structured so that such a reader loses nothing by knowing no
\GGTheory{} in advance. Everything imported is restated with its
hypotheses; every definition used is made in Part~II; every theorem is
proved or proof-sketched on the page, with the physics said in words
immediately after the mathematics.

Here is the honest pitch for spending a week with it. You already own
the pieces: the analytic signal from your signals course, the Hilbert
transform from complex analysis, positive frequency from field theory,
the C, P, T table from your first QFT semester, and (perhaps) the
Penrose transform from a differential-geometry elective. What you
likely do not own is the statement that these are \emph{one} piece:
that the operator splitting frequencies is the unique complex
structure causality and positivity allow; that the antiparticle
pairing is where a compact circle and a causal clock agree to share
it; that the C/P/T table is orientation bookkeeping of three real
factors; and that the celebrated division of twistor space is this
same splitting wearing conformal clothes. Those identifications are
theorems here, each about a page long, and each checkable with tools
you already have.

A suggested week. Days one and two: Part~I for the story and Part~II
for the definitions --- do not skip the definitions; the whole paper is
linear algebra once they are owned. Day three: Part~IV's five
theorems, with Course~C open beside them for the drills. Day four:
the bridge (Part~V) with Course~A as companion --- the analytic signal
by hand in the morning, the tube in the afternoon. Day five: Course~B
straight through to Lecture~B.4, which is the single derivation this
paper most wants you to own: one line of spinor algebra putting a
complexified point's twistor line in $\PT^{+}$. The weekend: worked
examples V and VI (CP violation and the Majorana door, both with no
$i$ anywhere), then the comparison tables and as many of the
forty-two objections as you have appetite for --- they are arranged so
that your own objection is probably among them.

Three further notes. The problem sets carry hints and answers because
they are meant to be \emph{done}, not admired; none needs more than a
page. The annotated reading guide (Part~IX) re-shelves the
bibliography into five ordered paths, so the week can become a
semester if the subject takes hold. And the register of the paper ---
what is proved, what is conditional, what is open, what is conceded
--- is itself part of the training: foundations work is only worth
doing with the books kept this way, and a student who internalizes the
ledger discipline will keep it in whatever field they land. Whether or
not you ever adopt the parent framework, the identifications are
yours to keep; physics has always advanced by noticing that two
familiar things were one thing, and this paper is an extended exercise
in exactly that noticing.

\GGPart[The question and its history]{The Question and Its History}{Five centuries of complex numbers, in mathematics, in physics, and in Penrose's hands}
\label{part:history}

\section[Part~I in plain English]{Part~I in plain English: the question, and why five centuries did not close it}
Complex numbers entered mathematics as an embarrassment and left it indispensable, and then the same thing happened again in physics. The phasor was a bookkeeping convenience that could be discarded at the end of the calculation. The gauge phase could not be discarded. The quantum amplitude could not be discarded either, and in 2021 an experiment excluded the version of quantum mechanics that tries.

What none of that supplies is a reason. Penrose put the question at full strength --- why is nature's massless physics holomorphic? --- and answered it by making holomorphy a primitive endowment. That is an answer of a particular kind. It declines to look behind the complex numbers, on the ground that there is nothing behind them, and it has been the best answer available for sixty years.

This Part tells the history in three passes, mathematical, physical and quantum-mechanical, states Penrose's question in his own terms, and then states what this paper will claim against it. Nothing is derived here; the Part also lays out the two maps a reader will want later, of the cohort and of the eleven Parts. One boundary is drawn at the outset and held throughout: the claim is about nature's use of the complex numbers, not about why they are beautiful mathematics, which is a question no physics can close.

\section{An invention that would not stay invented: the mathematical history}
\label{sec:history-math}

Complex numbers entered mathematics with an apology. Cardano's
\emph{Ars Magna} \citep{Cardano1545} met square roots of negative quantities midway
through the solution of cubic equations and called the work of keeping
them ``as subtle as it would be useless''; Bombelli \citep{Bombelli1572} showed the
subtlety pays --- real roots of real cubics sometimes \emph{require}
passage through the impossible quantities --- and for two centuries the
new numbers lived as bookkeeping: tolerated in the middle of a
calculation, banished from its ends. The turn came when the algebra
found its geometry. Euler bound the exponential to the circle \citep{Euler1748}; Wessel
(1797), Argand \citep{Argand1806}, and Gauss \citep{Gauss1831} laid the numbers on a plane, making
$i$ a quarter-turn rather than a mystery; and in the nineteenth century
Cauchy \citep{Cauchy1825} and Riemann \citep{Riemann1851} discovered that functions of the new variable are
not a generalization of real analysis but a crystallization of it ---
differentiability once means differentiability forever, integrals
depend on nothing but topology, and a function's values in any small
region determine it everywhere. By 1900 the complex plane had become
what it remains: the most rigid, most consequence-rich structure in
analysis. The invention had turned out to be a discovery; the only
question was of what.

Two further episodes of the mathematical story matter here. Hamilton,
disturbed that analysis should rest on an ``impossible'' quantity,
rebuilt the complex numbers as ordered \emph{pairs} of reals
with a stipulated multiplication \citep{Hamilton1837} --- the first real presentation, and
philosophically decisive: it proved $i$ eliminable in principle. It
changed no practice, because nothing then selected the pairing rule;
his subsequent quaternions showed that other rules exist, deepening
the question of why nature (if it chose at all) chose this one. And
the rigidity discovered by Cauchy, Riemann, and Weierstrass ---
one complex derivative implying all of them, values in a disc
determining values everywhere --- marked complex analysis as
qualitatively unlike real analysis: not a bookkeeping convenience but
a crystalline structure. Both episodes are answered rather than merely
admired in what follows: the pairing rule is selected by a circle and
a clock (Part~\ref{part:sources}), and the rigidity is the analytic
face of one-sided frequency support (Part~\ref{part:bridge}).

\section{The quiet infiltration of physics}
\label{sec:history-physics}

Physics adopted the complex numbers in three waves, each time
protesting that the adoption was mere convenience. The first wave was
oscillation: from Fourier's analysis onward, and canonically in
Steinmetz's phasors, every wave problem was compressed by writing the
real oscillation $\cos\omega t$ as the real part of $e^{i\omega t}$ ---
convenience, plainly, since the physics stayed real and the $i$ was
discarded at the end. The second wave was gauge structure: Weyl's 1929
reinterpretation of his failed scale invariance \citep{Weyl1918} as a local \emph{phase}
invariance \citep{Weyl1929} put $e^{i\lambda(x)}$ into the heart of
electrodynamics, and the modern bundle formulation
\citep{WuYang1975} recognized the phase as holonomy --- with the
Aharonov--Bohm effect \citep{AharonovBohm1959} confirming that the
holonomy, and only the holonomy, is physical. The third wave was the
quantum revolution, and it was different in kind: this time the $i$
refused to be discarded.

The first wave deserves one more sentence of respect, because its very
success set the trap. Fresnel's complex amplitudes for light \citep{Fresnel1823},
Fourier's exponentials \citep{Fourier1822}, Steinmetz's phasors for
alternating current \citep{Steinmetz1893}:
in every case the recipe was identical --- complexify, compute,
discard the imaginary part --- and in every case it worked because a
linear oscillation is two real quantities (amplitude and phase, or
cosine and sine components) that the complex symbol carries as one.
Physics thereby trained itself for eighty years to regard $i$ as
removable scaffolding. When quantum mechanics arrived with an $i$
that was \emph{not} removable, the training became the puzzle: the
tools gave no way to say what had changed. Parts~\ref{part:sources}
and~\ref{part:bridge} give the missing statement: the phasor's $i$
compresses one mode's cycle; the quantum $i$ is the shared
quarter-turn of all modes --- an orientation, not a shorthand.

\section{Quantum mechanics: the wave that would not stay real}
\label{sec:history-qm}

Schr\"odinger did not want the $i$ \citep{Schrodinger1926}. His first instinct was that the
wavefunction was a real vibration of something; the complex unit in
$i\hbar\,\partial_t\psi = H\psi$ entered under protest, and the
discomfort --- a fundamental equation of nature built on a number
invented for algebra --- never fully left the subject. Three facts
hardened the discomfort into a puzzle. First, the complexity is not
notational: relative phases interfere, and the phase of an amplitude
is measured (as always, through holonomies and rates) in every
double-slit and every Ramsey fringe. Second, the complexity is not
optional: quantum mechanics rebuilt over real Hilbert spaces
\citep{Stueckelberg1960} with real tensor products makes different
predictions in network Bell
configurations, and experiment has excluded it \citep{Renou2021} ---
nature insists on a single complex structure shared by independent
sources. Third, the complexity is universal: a strictly neutral field,
with no gauge phase at all, needs the quantum $i$ exactly as much as a
charged one. Whatever the $i$ of quantum mechanics is, it is not the
$e^{i\omega t}$ convenience of the phasor engineer.

The subsequent history sharpened the necessity without explaining it.
Born's rule \citep{Born1926} made $|\psi|^2$ physical while leaving the phase of
$\psi$ unmeasured --- yet not unphysical, since \emph{relative} phases
drive every interference pattern; Dirac's transformation theory \citep{Dirac1930} wove
the $i$ into the commutation relations themselves; and the modern era
made the geometry explicit piece by piece: the Aharonov--Bohm effect
showed the electromagnetic phase to be a loop quantity, Berry's phase \citep{Berry1984}
showed adiabatic transport to be holonomy, and quantum information
rebuilt the whole formalism operationally without ever dislodging the
complex field. By 2021 the situation was experimentally clean and
conceptually strange: the real-Hilbert-space alternative was falsified
in the laboratory \citep{Renou2021, LiEtAl2022}, so nature demonstrably uses a
single shared complex structure --- and no principle in the textbook
said what it was shared \emph{by}. That is the precise gap this paper
fills: the shared structure has two real sources, one circle and one
clock, and the experiment is their fingerprint.

\section{Penrose, twistors, and the question at full strength}
\label{sec:history-twistor}

No one has posed the question with more force, or higher stakes, than
Roger Penrose. In \emph{The Road to Reality} \citep{Penrose2004} and
five decades of lectures he has singled out the ``magic'' of complex
numbers as one of the deepest unexplained facts about nature --- and he
has raised the stakes beyond quantum amplitudes. Twistor theory
\citep{Penrose1967, PenroseMacCallum1973} proposes that spacetime
physics itself is best written on a complex three-dimensional
projective space $\PT$, where the free massless fields of every
helicity become \emph{holomorphic} objects --- cohomology classes,
in the rigorous formulation \citep{EastwoodPenroseWells1981} --- and
where, remarkably, the split of field into positive and negative
frequency, so labored in spacetime language, becomes the geometric
division of $\PT$ into two halves. For Penrose this is evidence that
complex holomorphy is not descriptive apparatus but constitutive of
nature. The question at full strength is therefore double: why do
quantum amplitudes live in $\bC$, and why do massless fields live on
complex projective space? A complete answer owes him both halves.

The origin of the twistor idea bears on how this paper engages it.
Penrose reached twistors from a conviction (already visible in his
spin-network work) that continuum spacetime should be derivative and
combinatorial-geometric structure primary; the 1967 construction
realized light rays --- not points --- as the primitive objects, with
the Robinson congruence supplying the picture of a twistor as a
twisting family of rays and the helicity form its invariant. The
program's subsequent half-century divides naturally: the classical
period (1967--1981) built the transform and its cohomology; the long
middle kept the geometry alive against the field's momentum; and the
amplitudes renaissance from 2004 onward \citep{Witten2004} returned
twistor variables to the center of perturbative gauge theory. Through
every period, one conviction has been constant --- that the
holomorphy is telling us something about nature, not about our
bookkeeping. This paper agrees, and locates the something.

\section{Five centuries in one table, and the detail behind it}
\label{sec:chronology}

\begin{table}[htbp]
\centering
\caption[A chronology of the complex numbers.]{A chronology of the
complex numbers, from algebraic embarrassment to physical necessity to
the present identification. The right column names where each episode
appears in this paper. The rows run from Cardano's ``as subtle as useless'' to the present identification --- fifteen episodes over five centuries, each with its primary source cited in place. Read downward, the middle column records the complex numbers becoming steadily harder to remove from physics while none of the episodes says where they come from, which is the question this paper takes up.}
\label{tab:chronology}
\small
\renewcommand{\arraystretch}{1.32}
\begin{tabular}{@{}lp{5.0cm}p{6.3cm}l@{}}
\toprule
\textbf{Year} & \textbf{Actor} & \textbf{Episode} & \textbf{Here} \\
\midrule
1545 & \citet{Cardano1545} & \emph{Ars Magna}: square roots of negatives appear mid-cubic; ``as subtle as useless'' & \S\ref{sec:history-math} \\
1572 & \citet{Bombelli1572} & the \emph{casus irreducibilis} forces them: real roots reached only through impossible numbers & \S\ref{sec:history-math} \\
1748 & \citet{Euler1748} & $e^{i\theta} = \cos\theta + i\sin\theta$: the exponential bound to the circle & \S\ref{sec:history-math} \\
1797--1831 & Wessel, \citet{Argand1806}, \citet{Gauss1831} & the plane: $i$ becomes a quarter-turn; the fundamental theorem of algebra & \S\ref{sec:history-math} \\
1825--1859 & \citet{Cauchy1825}, \citet{Riemann1851} & rigidity: contour independence, analytic continuation, geometry of surfaces & \S\ref{sec:history-math} \\
1835--1843 & \citet{Hamilton1837} & $\bC$ as ordered \emph{pairs of reals} with a rule --- the first real presentation --- then quaternions & \S\ref{sec:history-math} \\
1822--1893 & \citet{Fourier1822}, \citet{Fresnel1823}, \citet{Steinmetz1893} & the phasor: oscillation compressed into $e^{i\omega t}$, discarded at the end & \S\ref{sec:history-physics} \\
1918--1929 & \citet{Weyl1918}, \citet{Weyl1929} & scale invariance fails, phase invariance succeeds: the gauge $e^{i\lambda(x)}$ & \S\ref{sec:history-physics} \\
1925--1927 & \citet{Schrodinger1926}, \citet{Born1926}, \citet{Dirac1930} & the $i$ that would not cancel: complex amplitudes, reluctantly & \S\ref{sec:history-qm} \\
1959--1984 & \citet{AharonovBohm1959}, \citet{WuYang1975}, \citet{Berry1984} & the phase is holonomy: geometry, measured & \S\ref{sec:history-physics} \\
1963--1977 & \citet{Penrose1967}, \citet{PenroseMacCallum1973} & twistors: massless physics as holomorphy on $\PT$; the question at full strength & \S\ref{sec:history-twistor} \\
1981 & \citet{EastwoodPenroseWells1981} & the transform made rigorous: $H^1(\PT^{\pm}, \cO(-2h-2))$ & \S\ref{sec:prelim-twistor} \\
2004 & \citet{Witten2004} & the twistor string: amplitudes renaissance & \S\ref{sec:threeroads} \\
2021 & \citet{Renou2021} & real quantum mechanics (unshared $i$) experimentally excluded & \S\ref{sec:history-qm} \\
2026 & this preprint & positive frequency $=$ holomorphy $=$ the derived $J$; the sources named & Parts \ref{part:sources}--\ref{part:bridge} \\
\bottomrule
\end{tabular}
\end{table}

Three episodes in the table deserve a paragraph more than the survey
sections gave them, because each is a rehearsal of this paper's thesis.

\emph{Bombelli and the casus irreducibilis.} The scandal of 1572 was
not that imaginary numbers could be written down; it was that they
could not be avoided: for a cubic with three real roots, Cardano's
formula routes \emph{every} algebraic path to those real answers
through complex intermediaries. Real in, real out, complex in between
--- the exact arc this paper finds in physics (real bulk, complex
amplitudes, real records), discovered four and a half centuries early
as a fact about polynomials.

\emph{Hamilton's ordered pairs.} In 1835 Hamilton, disturbed that
algebra should rest on an ``impossible'' quantity, rebuilt $\bC$ as
ordered pairs of reals $(a, b)$ with a stipulated multiplication ---
precisely the real presentation of Proposition~\ref{prop:real-C}. The
construction was received as a philosophical nicety: it showed $i$
was eliminable in principle, and changed nothing in practice, because
nothing then \emph{selected} the pairing rule. The present paper is,
in one sense, the physics that was missing from Hamilton's algebra:
nature supplies the selection (a circle, a clock, a positivity
clause), and the pairing rule stops being a stipulation.

\emph{Schr\"odinger's reluctance.} The founder of wave mechanics
resisted his own equation's $i$: his first four communications of 1926
work hard to keep $\psi$ real, and his correspondence of that summer
frets that a complex wavefunction cannot be a physical wave. The
resistance was reasonable --- every previous $i$ in physics had been
removable phasor shorthand --- and its failure was the discovery, made
by the formalism before it was made by any argument, that this $i$ was
of a different kind. Part~\ref{part:sources} locates the difference
exactly: the phasor's $i$ compresses one mode's cycle, while the
quantum $i$ is the \emph{shared} quarter-turn of every mode at once
--- the polar phase of the clock itself, which no change of variables
can cancel because it is not a variable but an orientation.

\section{The claim of this preprint}
\label{sec:question}

This preprint assembles the answer developed inside the \GGTheory{}
program and states it for a general mathematical-physics readership.
The claim has three layers. \emph{First}, physics contains two complex
structures wearing one coat: the gauge phase of charged matter and the
quantum structure of the state space; they coincide for charged fields,
which is why a century of quantum electrodynamics could not tell them
apart. \emph{Second}, both are derived from real structures: the gauge
phase from a compact, non-spatial phase circle (the sixth dimension of
the \GG{} bulk), and the state-space $i$ from causal time orientation
plus energy positivity, which select the polar phase of the
time-translation generator uniquely on a static arena; their
consistency on charged sectors is exactly the Feynman--St\"uckelberg
pairing, so antiparticles are the handshake between the circle and the
clock. \emph{Third} --- the layer addressed to the twistor school ---
holomorphy itself is the same derivation seen at large symmetry:
positive frequency is Hardy holomorphy, tube analyticity is its field
form, and the twistor division $\PT^{+}/\PT^{-}$ \emph{is} the
frequency splitting, conformally packaged. Nothing in this diminishes
the twistor construction; it derives its motivation. And one boundary
is drawn in advance: this is an answer about \emph{nature's} use of
complex numbers. Why the complex numbers are beautiful
\emph{mathematics} --- algebraic closure, Cauchy rigidity --- is a
question no physics can close, and it is left, with respect, where
Penrose keeps it.

\section{The cohort inheritance and the roadmap of the eleven Parts}
\label{sec:inheritance-and-roadmap}

\noindent
Before the toolbox opens (Part~\ref{part:prelim}) and the arena is
built (Part~\ref{part:gg}), it is useful to lay out two
reader-orientation maps on the same opening: a panorama of the
twenty-two-paper cohort in which this paper stands
(\S\ref{sec:cohort}, Figure~\ref{fig:cohort-map}), and a panorama
of the eleven Parts of the present paper (\S\ref{sec:roadmap},
Figure~\ref{fig:roadmap}). A reader of any later Part should be
able to flip back to this Section and orient themselves in both
directions: which cohort companion carries which imported result,
and where the present argument stands in the eleven-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort}

This paper is a member of the twenty-two-paper \GGTheory{} cohort ---
eleven A-papers, six P-papers, three C-papers and two white papers: it
stands in the Foundations Branch, where it carries the origin of $i$,
and it inherits the cohort's results in completed tense. Its
designator, \PaperPFour{}, names the thesis rather than the toolkit:
the six-dimensional bulk is real, and the complex unit that physics
observes is manufactured --- derived, never primitive. Twistor
geometry keeps every page here that it has earned, as the sharpest
developed contrast: Penrose's program takes holomorphy as the
primary fact, and this paper derives that same holomorphy from two
real structures. The constitutional frame --- four axioms, of which the
first (locality--causality) and third (the operational dictionary with
its positivity clause) supply this paper's two selection principles ---
is \citealp{ArisakaA1Constitution}. The arena --- four-dimensional
spacetime plus a non-spatial scale interval and a non-spatial, compact
phase circle, with a globally static metric --- is Theorem~0 of
\citealp{ArisakaA2GG6}; the integer ledger behind the lattice-locked
phases quoted below is \citealp{ArisakaA3BI6}; the geometrization
lineage from Euclid to the present is traced in
\citealp{ArisakaA4Euclid}. The boundary law (the open-system
projection whose partial trace will reappear here at horizons and in
the infrared) is \citealp{ArisakaA5GDOS}; its geometric backbone is
\citealp{ArisakaA6Gi4}; its information backbone is
\citealp{ArisakaA7Qi4}. The quantum-mechanics treatment whose
real-bulk series (RB-1--RB-12) this preprint expands and re-derives for
a broader readership is \citealp{ArisakaA8QM}; the Standard-Model
consequences --- the derived discrete-symmetry table, the flavor
holonomy theorems, the pentagonal CP phase --- are
\citealp{ArisakaA9SM}; the cosmological consequences, including the
dial-free-door theorem that forces the Majorana channel, are
\citealp{ArisakaA10Cosmos}; the philosophical summit is
\citealp{ArisakaA11Origin}. Of the phenomenology branches, the
non-spatial-circle contrast with Kaluza--Klein is sharpest in
\citealp{ArisakaP1KK}; the neutrino count in \citealp{ArisakaP2N3};
the electron's ground-state reading in \citealp{ArisakaP3Electron};
the proton's, and with it baryon number carried as a winding of the
same compact circle this paper derives the gauge phase from, in
\citealp{ArisakaP6Proton}; and the three observational companions
are \citealp{ArisakaC1Kaxion},
\citealp{ArisakaC2Hubble}, and \citealp{ArisakaC3BH}, the last
supplying the horizon-as-port reading used in
Part~\ref{part:extensions}. Two white papers stand outside the
numbered series and both bear on this one.
\citealp{ArisakaW1Symmetry} walks the century of symmetry and
conservation in the order it was found, and arrives where this paper's
winding integrality already stands: at an integer with nowhere to move.
\citealp{ArisakaW2Holonomy} turns on a single exact statement --- a
holonomy requires a closed loop, and the interval fiber has none --- which
is the asymmetry that lets the circle of Part~\ref{part:sources} carry a
gauge phase at all, put to work there on sixteen open questions rather
than on the origin of $i$.

\paragraph{The Foundations Branch.} On the map's left flank stands
the branch this paper belongs to: two member papers whose shared
subject is the boundary's complex structure and its mirrors, a
single column beside the spine. The partner paper, \PaperPFive{}
\citep{ArisakaP5CPT}, consumes the derived complex structure one
step further along: from the orientations of the same real factors
whose two global structures this paper turns into the $i$ --- the
compact circle, the spatial slice, and the causal clock --- it
derives C, P, and T as orientation reversals, the CPT theorem as
total orientation reversal, the pentagon-locked CP phases, and the
sign of the cosmic matter excess. On the map it is the
\ANine{}--\ATen{} interface: fed by the ledger phases derived in
\ANine{} --- the PMNS Dirac phase and the Majorana parity
$\alpha_{31} = \pi$ --- and feeding the $\eta_B$ chain of \ATen{}
at the grade recorded there, derived-conditional on a named
finite-temperature hypothesis; the solid shared-$i$ edge descending
from this paper's node carries what it inherits from here --- one
$J$ for the whole tower, and the custody principle both papers
state in the same words: \emph{the dictionary owns the sectors; the
geometry owns the $i$}. Both branch members carry P-series
designators; their place at the spine's flank, not the
phenomenology rows, marks their foundational role.

A reader needs none of these in hand: every
imported statement is restated here with its hypotheses, and the
derivations of Parts~\ref{part:sources}--\ref{part:bridge} are
self-contained at the level of statements and proof sketches. The
twenty-two-paper map, identical in every paper of the cohort, is
Figure~\ref{fig:cohort-map}, which closes this subsection.

%% ===================================================================
%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
%%  page at the close of the cohort-inheritance subsection.  Verbatim
%%  copy of the v1.4 master (A1-Constitution Figure 3): no
%%  self-reference markers anywhere, so the figure is identical and
%%  simple for replication in all nineteen papers.  Only the internal
%%  node names MP4/MP5 differ from the master (collision avoidance).
%% ===================================================================
\clearpage
\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},
 root/.style={fill=violet!18, draw=violet!70!black, line width=1.0pt,
 minimum width=80.5mm, minimum height=16.1mm,
 font=\sffamily\bfseries\small},
 bulk/.style={fill=red!10, draw=red!60!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 hist/.style={fill=red!5, draw=red!50!black, line width=0.5pt,
 minimum width=41.4mm, minimum height=14.95mm,
 dash pattern=on 2pt off 2pt},
 gdos/.style={fill=yellow!18, draw=olive!55!black, line width=0.6pt,
 minimum width=52.9mm, minimum height=14.95mm},
 pillars/.style={fill=green!12, draw=green!55!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 qm/.style={fill=blue!10, draw=blue!55!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 a11/.style={fill=yellow!12, draw=orange!55!black, line width=0.6pt,
 minimum width=52.9mm, minimum height=14.95mm},
 pred/.style={fill=cyan!10, draw=cyan!60!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 predthis/.style={fill=cyan!14, draw=cyan!50!black, line width=1.1pt,
 minimum width=41.4mm, minimum height=14.95mm},
 cosmo/.style={fill=teal!10, draw=teal!60!black, line width=0.6pt,
 minimum width=41.4mm, minimum height=14.95mm},
 fnd/.style={fill=violet!8, draw=violet!55!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 wht/.style={fill=brown!12, draw=brown!65!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 branchframe/.style={rounded corners=4pt, line width=0.6pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!70},
 arrowdash/.style={-{Stealth[length=2.4mm]}, line width=0.55pt,
 draw=black!60, dashed},
 arrowrt/.style={-{Stealth[length=2.6mm]}, line width=0.85pt,
 draw=violet!70!black},
 edgelabel/.style={font=\scriptsize\itshape, fill=white,
 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
]

%% --- 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}
\end{figure}
\clearpage

\subsection{Roadmap of the eleven Parts}
\label{sec:roadmap}

The rest of the paper is organized into ten further Parts, and the
walk through them is plotted in Figure~\ref{fig:roadmap}. The
present Part --- the question and its history --- is nearly done:
five centuries of the complex numbers, their quiet infiltration of
physics, Penrose's question at full strength, the claim of this
preprint, and the cohort the paper stands in. Everything after this
page is that claim made precise, and the itinerary below says where
each piece of the argument lives.

Two Parts build the arena. Part~\ref{part:prelim} opens the toolbox
--- real symplectic data, complex structures, the polar phase of a
generator, Hardy spaces, and twistor space --- with every object
defined from scratch, so that the argument can be checked rather
than believed. Part~\ref{part:gg} then builds the stage on which
those tools act: the three constitutional clauses this paper uses,
the six-dimensional geometry that Theorem~0 forces from them, and
the two globally shared real structures --- a circle and a clock ---
that the rest of the paper never leaves.

Parts~\ref{part:sources} and~\ref{part:bridge} are the heart.
Part~\ref{part:sources} derives the two real sources and their
consistency condition: the gauge phase from the circle, the
state-space $i$ from the clock, the antiparticle handshake where the
two meet, the shared complex structure that the handshake enforces,
and the discrete-symmetry table that falls out of it.
Part~\ref{part:bridge} --- highlighted in the roadmap, because it
is what the title claims --- proves the bridge: positive frequency
is holomorphy, at one clock (Hardy space), for fields (the forward
tube), and at full conformal symmetry (the twistor division of
$\PT$), with the consequence for twistor theory drawn on the spot.

Part~\ref{part:extensions} takes the derivation to the places where
derivations usually break: interacting matter and the asymptotic
completion, the infrared door (sectors are ports, and the derived
$J$ is sector-blind), and the three superselections the program
meets, unified under one custody principle.

Two Parts then weigh the case. Part~\ref{part:discussion} discusses
significance and uniqueness, sets the three roads side by side ---
string theory, twistor theory, \GGTheory{} --- states what is
claimed and in what regime, and closes with the predictions for
falsification and the open questions. Part~\ref{part:conclusions}
states the conclusions: what the paper has established, the
objective limitations, and the vision of a century in which number
systems are read as geometry.

The back matter is the workshop. Part~\ref{part:appendices}
collects the reference dictionaries and the pocket course --- from
the analytic signal to the Penrose transform, at lecture depth.
Part~\ref{part:objections} answers the common objections, community
by community, including the ones a twistor theorist should raise;
and Part~\ref{part:references} closes the paper with the annotated
references, each carrying a note on what it established and how it
is used here. Read in order, the eleven Parts compose into a single
argument; read by need, each Part names its imports and can stand a
visit on its own.

%% ===================================================================

\bigskip\noindent\looseness=-1
\textbf{Close of Part I, in plain terms.} Five centuries ago a number was invented that no one
wanted; a century ago it became indispensable to physics; fifty years
ago Penrose asked why, at full strength, for amplitudes and for the
holomorphic geometry of light alike. This Part told that story and
stated this paper's answer in outline: the complex numbers are the
compressed notation for two real structures --- a circle and a clock
--- that a four-dimensional observer cannot see directly. Everything
that follows is the outline made precise. The next Part lays out the
mathematical tools --- none of them exotic, all of them defined from
scratch --- with which the claim can be checked rather than believed.

%% ===================================================================

%% ===================================================================
%%  Roadmap of the eleven Parts (house architecture: A1-Constitution
%%  Figure 4, fig:A1-ten-parts-flowchart -- vertical Part column,
%%  right phase sidebar, one highlighted central Part), adapted to
%%  the eleven Parts of this paper; clearpage-isolated minipage +
%%  \captionof per the A1 pattern.
%% ===================================================================
%% \null\vfill dropped at v15.0: on A4 the 182.9 mm plate and its
%% 1425-character caption fitted one page centered; Letter is the
%% shorter sheet and they do not, so \null held page 19 open blank
%% and the plate broke to page 20.  Top-set, as the cohort map is.
\clearpage
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.93\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 intro/.style={fill=blue!12, draw=blue!60!black, line width=0.6pt,
 minimum width=108mm, minimum height=12mm},
 core/.style={fill=blue!7, draw=blue!50!black, line width=0.5pt,
 minimum width=108mm, minimum height=12mm},
 bridge/.style={fill=orange!15, draw=orange!60!black, line width=0.6pt,
 minimum width=108mm, minimum height=12mm},
 innov/.style={fill=orange!20, draw=orange!60!black, line width=0.9pt,
 minimum width=130mm, minimum height=18mm},
 ext/.style={fill=yellow!12, draw=olive!55!black, line width=0.55pt,
 minimum width=108mm, minimum height=12mm},
 disc/.style={fill=yellow!18, draw=olive!60!black, line width=0.55pt,
 minimum width=108mm, minimum height=12mm},
 summit/.style={fill=red!13, draw=red!60!black, line width=0.7pt,
 minimum width=108mm, minimum height=12mm},
 appendix/.style={fill=gray!10, draw=black!55, line width=0.5pt,
 minimum width=108mm, minimum height=13mm,
 dash pattern=on 2pt off 2pt},
 arrow/.style={-{Stealth[length=2.6mm]}, line width=0.65pt, draw=black!70},
 arrowinnov/.style={-{Stealth[length=3.0mm]}, line width=1.0pt,
 draw=orange!60!black},
 phaselabel/.style={font=\footnotesize\itshape, text=black!60, anchor=west,
 align=left, inner sep=2pt},
]

%% --- Part I ---
\node[intro] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{The Question and Its History} \\[-1pt]
 \footnotesize Five centuries of $i$; Penrose's question; the claim; the cohort};

%% --- Part II ---
\node[core] (II) at (0, -16mm)
 {\textbf{Part~II}\quad\textbf{Mathematical Preliminaries} \\[-1pt]
 \footnotesize Real symplectic data, complex structures, Hardy spaces, twistor space};

%% --- Part III ---
\node[core] (III) at (0, -32mm)
 {\textbf{Part~III}\quad\textbf{The \GGTheory{} Arena} \\[-1pt]
 \footnotesize Constitutional clauses, forced 6D geometry, two shared real structures};

%% --- Part IV ---
\node[bridge] (IV) at (0, -48mm)
 {\textbf{Part~IV}\quad\textbf{The Two Real Sources} \\[-1pt]
 \footnotesize Five theorems: circle, clock, handshake, shared $i$, symmetry table};

%% --- Part V (highlighted: the title claim, enlarged) ---
\node[innov] (V) at (0, -68mm)
 {\textbf{Part~V}\quad\textbf{The Bridge: Positive Frequency is Holomorphy} \\[1pt]
 \footnotesize Hardy space, the forward tube, and the twistor correspondence as one structure \\[-1pt]
 \footnotesize \emph{\textbf{One derivation at three symmetries: the division $\PT^{+}/\PT^{-}$ is the frequency splitting}}};

%% --- Part VI ---
\node[ext] (VI) at (0, -90mm)
 {\textbf{Part~VI}\quad\textbf{Extensions and Boundaries} \\[-1pt]
 \footnotesize Interacting matter; the infrared door; one custody principle};

%% --- Part VII ---
\node[disc] (VII) at (0, -106mm)
 {\textbf{Part~VII}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Significance, uniqueness, three roads, falsification, open questions};

%% --- Part VIII ---
\node[summit] (VIII) at (0, -122mm)
 {\textbf{Part~VIII}\quad\textbf{Conclusions} \\[-1pt]
 \footnotesize What is established, the limitations, number systems read as geometry};

%% --- Part IX ---
\node[appendix] (IX) at (0, -138mm)
 {\textbf{Part~IX}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize The dictionaries and the pocket course, in nineteen lectures};

%% --- Part X ---
\node[appendix] (X) at (0, -154mm)
 {\textbf{Part~X}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize Adversarial readings from seven communities, each answered};

%% --- Part XI ---
\node[appendix] (XI) at (0, -170mm)
 {\textbf{Part~XI}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows ---
\draw[arrow] (I.south) -- (II.north);
\draw[arrow] (II.south) -- (III.north);
\draw[arrow] (III.south) -- (IV.north);
\draw[arrowinnov] (IV.south) -- (V.north);
\draw[arrowinnov] (V.south) -- (VI.north);
\draw[arrow] (VI.south) -- (VII.north);
\draw[arrow] (VII.south) -- (VIII.north);
\draw[arrow] (VIII.south) -- (IX.north);
\draw[arrow] (IX.south) -- (X.north);
\draw[arrow] (X.south) -- (XI.north);

%% --- Phase labels (right sidebar, squeezed close to box edges) ---
%% Standard boxes: half-width = 54mm, label at x=56mm (2mm gap).
%% Part V is wider (half-width = 65mm); its phase label sits below
%% the box, per the A1 pattern for the highlighted Part.
\node[phaselabel] at (56mm, 0mm) {\emph{Setup}};
\node[phaselabel] at (56mm, -24mm) {\emph{Mathematical} \\ \emph{arena}};
\node[phaselabel] at (56mm, -48mm) {\emph{The real sources}};
%% The bridge: below the Part V box (box bottom at y = -77mm)
\node[phaselabel, text=orange!60!black, anchor=west] at (28mm, -81mm)
 {\emph{\textbf{The bridge: positive frequency is holomorphy}}};
\node[phaselabel] at (56mm, -90mm) {\emph{Extensions}};
\node[phaselabel] at (56mm, -106mm) {\emph{Discussion} \\ \emph{+ critique}};
\node[phaselabel] at (56mm, -122mm) {\emph{Summit}};
\node[phaselabel] at (56mm, -138mm) {\emph{Appendices}};
\node[phaselabel] at (56mm, -154mm) {\emph{Objections}};
\node[phaselabel] at (56mm, -170mm) {\emph{References}};

%% Natural bounding box (boxes + adjacent right labels), per the
%% house pattern (A1-Constitution Figure 4): the figure is centered
%% on the page via \centering; minor visual asymmetry from the right
%% labels is accepted as a tradeoff for keeping the labels close to
%% the boxes.
\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the eleven Parts of GG-P4-Real.]{Roadmap of the eleven Parts of \PaperPFour{}. The argument develops in phases. \emph{Setup} (Part~I) tells the five-century history of the complex numbers, states Penrose's question at full strength and this paper's claim, and locates the paper in its twenty-two-paper cohort. The \emph{mathematical arena} (Parts~II--III) defines every object the argument uses from scratch and builds the stage: the constitutional clauses, the forced six-dimensional geometry, and the two globally shared real structures --- the circle and the clock. The \emph{real sources} (Part~IV) derive the gauge phase from the circle and the state-space $i$ from the clock, with the antiparticle handshake as their consistency condition. The \emph{bridge} (\textbf{Part~V}, highlighted) is the central payoff: positive frequency is holomorphy, proved at the Hardy, tube, and twistor levels. \emph{Extensions} (Part~VI) carry the derivation to interacting matter, the infrared door, and the three superselections under one custody principle. Part~VII discusses significance, uniqueness, the three roads, and the falsifiers; Part~VIII states the conclusions; and the three back-matter Parts collect the technical appendices and the pocket course (Part~IX), the common objections and responses (Part~X), and the annotated references (Part~XI). Part~V is highlighted because the bridge is what the title claims, and the Parts before it exist to prove it.}
\label{fig:roadmap}
\end{minipage}
\clearpage

\GGPart[Mathematical preliminaries]{Mathematical Preliminaries}{Real symplectic data, complex structures, the polar phase, Hardy spaces, and twistor space --- everything the argument stands on, defined}
\label{part:prelim}

This Part fixes definitions and records, with proofs where they are
short and with precise citations where they are classical, every
mathematical object the argument uses. A reader fluent in one-particle
structures, Hardy spaces, and the Penrose transform may skim to
Part~\ref{part:gg}; a reader meeting these objects for the first time
will find here everything needed to check the later Parts line by line.

\paragraph{Conventions.} Throughout: units $\hbar = c = 1$; metric
signature $(+,-,-,-)$, as implied by the Hermitian-matrix form
\eqref{eq:B1-x} of a spacetime point; two-component spinor indices
raised and lowered with the antisymmetric $\epsilon_{AB}$,
$\epsilon_{A'B'}$ in the north-west/south-east convention of
\citet{PenroseRindler1986}; complex conjugation exchanges primed and
unprimed indices; and the helicity form carries the factor
$s(Z) = \tfrac12 Z^\alpha \bar Z_\alpha$ of
\eqref{eq:helicity-form}. Real presentations use the block dictionary
of Proposition~\ref{prop:real-C}, with complex conjugation as block
transpose.

\section[Part~II in plain English]{Part~II in plain English: the toolbox, opened before it is used}
Nothing is proved in this Part. It defines every object the later Parts use --- real symplectic data and compatible complex structures, the polar phase of a generator, weight decomposition under a circle action, the Hilbert transform and Hardy space, twistor space and the Penrose transform --- with proofs where they are short and precise citations where they are classical.

The Part exists because the entire content of the argument is a direction of explanation, and a direction can only be argued if both ends are pinned down first. Everything the later Parts import is therefore stated here at the strength it will be used at and nowhere stronger.

The one definition worth slowing down for is the polar phase. It is elementary --- an antisymmetric generator, its modulus, and the unitary factor of the decomposition --- and it is the object the central theorem turns on. If it looks too small to carry that weight, the reaction is a fair one and worth keeping until Part~\ref{part:sources}.

\section{Real symplectic data and complex structures}
\label{sec:prelim-real}

\begin{definition}[Real phase-space data]
\label{def:symp}
A \emph{real symplectic space} is a pair $(V, \sigma)$ with $V$ a real
vector space and $\sigma$ a non-degenerate antisymmetric bilinear form.
A \emph{real Euclidean space} (the fermionic case) is a pair $(V, s)$
with $s$ symmetric and positive-definite. In field theory $V$ is the
real solution space of a linear field equation; for a Klein--Gordon
field with Cauchy data $(\varphi, \pi)$,
\begin{equation}
\sigma\big((\varphi_1,\pi_1),(\varphi_2,\pi_2)\big)
 \;=\; \int_\Sigma \big(\varphi_1\,\pi_2 - \varphi_2\,\pi_1\big)\, d^3x ,
\label{eq:sigma-KG}
\end{equation}
which is real, antisymmetric, and contains no complex ingredient.
\end{definition}

\begin{definition}[Complex structure; one-particle structure]
\label{def:J}
A \emph{complex structure} on $(V,\sigma)$ is a real-linear operator
$J : V \to V$ with
\begin{equation}
J^2 = -\mathbf{1}, \qquad
\sigma(Jf, Jg) = \sigma(f, g), \qquad
\sigma(f, Jf) > 0 \ \ (f \neq 0),
\label{eq:J-axioms}
\end{equation}
the three conditions being called squaring, compatibility, and taming.
On a Euclidean $(V,s)$ the taming condition is replaced by
orthogonality, $s(Jf,Jg)=s(f,g)$, plus positivity of the induced
one-particle energy. A \emph{one-particle structure} is the pair
$(V,\sigma,J)$ (respectively $(V,s,J)$); it defines a complex Hilbert
space by completing $V$ in the inner product
\begin{equation}
\langle f, g \rangle \;=\; \tfrac12\,\sigma(f, Jg) \;+\; \tfrac{i}{2}\,\sigma(f, g),
\label{eq:one-particle-ip}
\end{equation}
with multiplication by $i$ implemented by $J$. Every complex Hilbert
space of physics arises this way from real data plus a choice of $J$;
the entire content of this preprint is that in \GGTheory{} the choice
is not free --- $J$ is constructed, uniquely, from real geometry.
\end{definition}

\begin{proposition}[The real presentation of $\bC$]
\label{prop:real-C}
Let $J_0 = \begin{psmallmatrix} 0 & -1 \\ 1 & 0 \end{psmallmatrix}$ on
$\bR^2$. Then $(\bR^2, J_0) \cong \bC$ as complex vector spaces via
$(a,b) \mapsto a + ib$; a complex number $z = a + ib$ acts on $\bR^2$
as the rotation-scaling block
\begin{equation}
B(z) \;=\; \begin{pmatrix} a & -b \\ b & \phantom{-}a \end{pmatrix}
 \;=\; |z|\, R(\arg z),
\label{eq:block}
\end{equation}
and complex conjugation corresponds to the transpose $B(z)^{\mathsf T}
= B(\bar z)$ --- i.e.\ to orientation reversal of the plane.
\end{proposition}

\begin{proof}
Direct computation: $B(z)B(w) = B(zw)$, $B(z)(a',b')^{\mathsf T}$
matches $z\cdot(a'+ib')$ under the identification, and
$B(z)^{\mathsf T} = B(\bar z)$ by inspection of \eqref{eq:block}.
\end{proof}

\paragraph{Physics intuition.} Nothing in quantum mechanics ever
required the \emph{symbol} $i$: every complex amplitude is a plane
vector, every phase a rotation, every conjugation a reflection. What
physics requires is the \emph{choice of rotation} --- which real
quarter-turn plays the role of $i$, the same choice on every system.
The question ``why complex numbers?''\ is therefore, exactly, the
question ``what selects $J$?''\ --- and it is answerable because $J$
is a real geometric object.

The three conditions of \eqref{eq:J-axioms} deserve one further
paragraph each, because each carries physics. \emph{Squaring}
($J^2 = -\mathbf{1}$) makes $J$ a bona fide imaginary unit: every
vector sits in a two-plane that $J$ rotates by a quarter-turn, so the
real space decomposes into ``complex lines.'' \emph{Compatibility}
makes those rotations canonical transformations: multiplying by a
phase is a symmetry of the classical structure, which is why global
phases can be unobservable at all. \emph{Taming} is the positivity
that turns the pair $(\sigma, J)$ into a K\"ahler structure --- the
inner product \eqref{eq:one-particle-ip} is positive-definite exactly
because of it --- and it is the one condition with a thermodynamic
smell: it is where ``energy is bounded below'' enters the kinematics.
A triple $(V, \sigma, J)$ satisfying all three is, in the geometer's
language, a linear K\"ahler space; quantum mechanics' state space is
the completion of exactly such a triple, and the whole question of
this paper is who chooses the $J$.

\paragraph{A worked micro-example.} Take one classical oscillator,
$V = \{(q, p)\}$, $\sigma((q,p),(q',p')) = qp' - pq'$. The taming
complex structures are $J_\omega(q, p) = (p/\omega,\, -\omega q)$, one
for each frequency scale $\omega > 0$ --- a one-parameter family, so
compatibility and taming alone do \emph{not} fix $J$. What fixes it is
dynamics: demanding $[J, e^{tA}] = 0$ for the oscillator's own time
evolution selects $\omega$ to be the oscillator's frequency, and
Lemma~\ref{lem:polar} below is precisely this selection stated in
general. The moral, worth carrying through the whole paper: complex
structures are cheap; \emph{dynamics-commuting, positive} complex
structures are unique.

\section{The polar phase of a generator}
\label{sec:prelim-polar}

\begin{lemma}[Polar phase]
\label{lem:polar}
Let $A$ be a real operator on $(V,\sigma)$ generating the dynamics
($\dot f = Af$) --- throughout this paper, either a real matrix on a
finite-dimensional space or one of the standard Klein--Gordon/Dirac
generators on Cauchy data, for which the spectral calculus used below
is classical --- skew in the Hamiltonian sense
($(\sigma A)^{\mathsf T} = \sigma A$), with $-A^2$ symmetrizable,
positive, and boundedly invertible (the gapped case). Then
\begin{equation}
J_A \;:=\; \pm\, A\,(-A^2)^{-1/2}
\label{eq:polar-phase}
\end{equation}
satisfies all three conditions of Definition~\ref{def:J}, commutes with
the dynamics $e^{tA}$, and the sign is fixed by the taming condition.
On a Euclidean $(V,s)$ with $A$ skew-adjoint the same formula gives the
unique positive-energy orthogonal complex structure. In the time
domain, $J_A$ acts on solutions as the quarter-period phase advance
frequency by frequency --- the \emph{Hilbert transform} in the time
variable (\S\ref{sec:prelim-hardy}).
\end{lemma}

\begin{proof}
$A$ commutes with functions of $-A^2$, so
$J_A^2 = A^2(-A^2)^{-1} = -\mathbf{1}$. Compatibility:
$\sigma(J_A f, J_A g) = \sigma(f,g)$ follows since $(-A^2)^{-1/2}$ is
symmetric for $\sigma A$-symmetric $A$ and $A$ is $\sigma$-skew.
Commutation with $e^{tA}$ is immediate. On each spectral $2$-plane of
$A$ with frequency $\omega$, $J_A$ restricts to one of the two
quarter-turns; taming (equivalently positivity of
$\langle f, |A| f\rangle$) selects the orientation, and selects it
coherently across planes. For the Euclidean case, replace
$\sigma$-skew by $s$-skew-adjoint throughout; uniqueness holds because
a dynamics-commuting $J$ preserves each spectral plane, where exactly
one of the two quarter-turns has positive energy.
\end{proof}

\paragraph{Physics intuition.} The generator of time translation
knows two things about every mode: how fast it turns (the frequency,
$|A|$) and \emph{which way} (the orientation, $J_A$). The polar
decomposition separates them. The imaginary unit of quantum mechanics
is the second factor --- the pure ``which way'' of time --- with the
speed divided out. That is why the same $i$ serves every mode of every
field: orientation is the one thing all clocks share.

\paragraph{Spectral picture.} It clarifies the lemma to see it
spectrally. On the complexification, a skew generator has purely
imaginary spectrum $\{\pm i\omega_k\}$ with conjugate eigenplanes; the
polar phase acts as $+i$ on each $+i\omega_k$ eigenspace and $-i$ on
its conjugate --- it is the operator ``$i\,\mathrm{sgn}$'' of the
frequency, which is why the time-domain kernel is the Hilbert
transform's. Uniqueness is then transparent: any dynamics-commuting
$J'$ must act within each eigenplane, where the only quarter-turns are
$\pm$ the polar phase's, and positivity fixes the sign plane by plane
with one inequality. Degeneracy enlarges the commutant to a unitary
group but the positivity argument survives on the real form --- the
drill of Lecture~C.2.

\section{Weight decomposition under a circle action}
\label{sec:prelim-weights}

\begin{lemma}[Weights of the symmetric square]
\label{lem:weights}
Let the circle act on $\bR^2$ with weight $n$, i.e.\ by $R(n\alpha)$.
Then $\mathrm{Sym}^2(\bR^2)$ decomposes into the invariant line
spanned by $a^2 + b^2$ and the weight-$2n$ doublet
$(a^2 - b^2,\, 2ab)$:
\begin{equation}
R(n\alpha):\;
\begin{pmatrix} a^2-b^2 \\ 2ab \end{pmatrix}
\;\longmapsto\;
R(2n\alpha) \begin{pmatrix} a^2-b^2 \\ 2ab \end{pmatrix}.
\label{eq:sym2}
\end{equation}
Consequently, for $n \neq 0$ the only circle-invariant symmetric
self-pairing is $a^2+b^2 = |\psi|^2$ (the Dirac pairing), while for
$n = 0$ all pairings are invariant. In complex notation the doublet is
$(\mathrm{Re}\,\psi^2, \mathrm{Im}\,\psi^2)$: the Majorana bilinear.
\end{lemma}

\begin{proof}
Write $\psi = a + ib$; under the weight-$n$ action
$\psi \mapsto e^{in\alpha}\psi$, so $|\psi|^2$ is invariant and
$\psi^2 \mapsto e^{2in\alpha}\psi^2$, which is \eqref{eq:sym2} in real
coordinates. An invariance equation $R(2n\alpha)v = v$ for all
$\alpha$ forces $v = 0$ when $n \neq 0$.
\end{proof}

\paragraph{Physics intuition.} A circle can only forbid what it
moves. Charge-carrying pairings spin under the fiber and are outlawed;
the one pairing that stands still ($\psi\bar\psi$) is exactly the
charge-conserving one. At weight zero the circle stands guard over
nothing --- which is why lepton number can leak there, and only there
(the dial-free door of the parent program).

\paragraph{Representation-theoretic remark.} Lemma~\ref{lem:weights}
is the $n$-th symmetric-power statement of $SO(2)$ representation
theory: $\mathrm{Sym}^2$ of the weight-$n$ representation decomposes
as weight $0 \oplus 2n$, and only weight~$0$ supports invariants. The
same bookkeeping at higher orders classifies every allowed coupling: a
$k$-linear vertex among weights $n_1, \ldots, n_k$ is invariant iff
$\sum n_i = 0$ --- winding conservation as pure character arithmetic,
drilled in Lecture~C.6. Nothing in the argument uses more than the
orthogonality of characters on a circle; that so much of the Standard
Model's selection structure reduces to it is the ledger's point.

\section{The Hilbert transform and Hardy space}
\label{sec:prelim-hardy}

\begin{definition}[Hilbert transform]
\label{def:hilbert}
For $f \in L^2(\bR, dt)$,
\begin{equation}
(\Htransform f)(t) \;=\;
\mathrm{p.v.}\ \frac{1}{\pi}\int_{-\infty}^{\infty}
\frac{f(t')}{t - t'}\, dt',
\qquad
\widehat{\Htransform f}(\omega) = -\,i\,\mathrm{sgn}(\omega)\,
\hat f(\omega).
\label{eq:hilbert}
\end{equation}
The kernel is real; $\Htransform$ is a real operator, non-local in
time, with $\Htransform^2 = -\mathbf{1}$ almost everywhere on
$L^2(\bR)$ --- the zero-frequency obstruction is measure-zero on the
line, and becomes the genuinely excluded constant mode only in
finite volume.
On the oscillation pair it is the quarter-period advance:
$\Htransform \cos\omega t = \sin\omega t$,
$\Htransform \sin\omega t = -\cos\omega t$ ($\omega > 0$).
\end{definition}

\begin{definition}[Hardy space]
\label{def:hardy}
$H^2$ of the upper half-plane is the space of functions $F(z)$
holomorphic for $\mathrm{Im}\, z > 0$ with
$\sup_{s>0} \int |F(t+is)|^2\, dt < \infty$; such $F$ possess $L^2$
boundary values as $s \downarrow 0$.
\end{definition}

\begin{theorem}[Paley--Wiener]
\label{thm:pw}
$f \in L^2(\bR)$ has Fourier transform supported in $[0,\infty)$ if
and only if $f$ is the boundary value of a function in $H^2$ of the
upper half-plane \citep{PaleyWiener1934}. Equivalently: the
positive-frequency subspace of $L^2$ is exactly the boundary Hardy
space, the splitting projector is $\tfrac12(\mathbf{1} + i\Htransform)$,
and the positive-frequency subspace is the $(-i)$-eigenspace of
$\Htransform$.
\end{theorem}

\paragraph{Physics intuition.} ``Positive frequency'' sounds like a
property of a Fourier table; the theorem says it is a property of
\emph{extendability}: a history has one frequency orientation exactly
when it is the shadow, on the real-time axis, of something regular in
imaginary time on one side. Nothing grows there --- $e^{i\omega t}$
becomes $e^{i\omega t - \omega s}$ and decays --- and the operator
that detects this one-sided regularity is a real principal-value
integral. Holomorphy is not imported here; it is what an orientation
of time \emph{is}, analytically.

\paragraph{The $H^2$ inner product, explicitly.} For later use:
on boundary values, $\langle F, G\rangle_{H^2} = \int \bar F(t) G(t)\,
dt = \int_0^\infty \overline{\hat f_+}(\omega)\hat g_+(\omega)\,
d\omega$ --- the positive-frequency half-line carrying the whole inner
product. This is the signal-level ancestor of the one-particle inner
product \eqref{eq:one-particle-ip}: the quantum overlap of two states
is a Hardy pairing of their histories, and the ``mysterious'' complex
inner product of quantum mechanics is, mode by mode, the overlap of
two analytic signals. The Poisson kernels of Course~A make every
statement in this section constructive.

\section{Twistor space and the Penrose transform}
\label{sec:prelim-twistor}

\begin{definition}[Twistor space]
\label{def:twistor}
Twistor space is $\mathbb{T} \cong \bC^4$ with coordinates
$Z^\alpha = (\omega^A, \pi_{A'})$ (two-component spinors), and
projective twistor space is $\PT = \mathbb{P}(\mathbb{T}) \cong
\bC\mathrm{P}^3$. A spacetime point $x$ corresponds to the projective
line of twistors satisfying the \emph{incidence relation}
\begin{equation}
\omega^{A} \;=\; i\, x^{AA'}\, \pi_{A'} .
\label{eq:incidence}
\end{equation}
The Hermitian \emph{helicity form}
\begin{equation}
s(Z) \;=\; \tfrac12\, Z^\alpha \bar Z_\alpha
 \;=\; \tfrac12\big(\omega^A \bar\pi_A + \bar\omega^{A'} \pi_{A'}\big)
\label{eq:helicity-form}
\end{equation}
divides $\PT$ into $\PT^{+} = \{s > 0\}$, $\PT^{-} = \{s < 0\}$, and
the five-real-dimensional null hypersurface $\mathbb{PN} = \{s = 0\}$,
whose points are the real null geodesics of compactified Minkowski
space \citep{Penrose1967, PenroseRindler1986}.
\end{definition}

\paragraph{The correspondence, in both directions.} The incidence
relation \eqref{eq:incidence} is linear in $\pi_{A'}$, so for fixed
$x$ its solutions form a two-complex-dimensional subspace of
$\mathbb{T}$ --- projectively, a line
\begin{equation}
L_x \;=\; \big\{\, [\, i\, x^{AA'} \pi_{A'},\ \pi_{A'} \,] \;:\;
\pi_{A'} \neq 0 \,\big\} \;\cong\; \bC\mathrm{P}^1 \;\subset\; \PT,
\label{eq:line}
\end{equation}
and the dictionary runs both ways: spacetime points correspond to
projective lines of $\PT$; two points are null separated exactly when
their lines intersect, the common twistor being the light ray that
joins them; and a single twistor is, when $s(Z) = 0$, a real null
geodesic. A twistor with $s(Z) \neq 0$ corresponds to no single real
ray at all but to a \emph{twisting congruence} of rays --- the
Robinson congruence, the picture that named the subject --- whose
handedness is the sign of $s$. The conformal group of compactified
Minkowski space acts on $\mathbb{T}$ as $\mathrm{SU}(2,2)$: the
linear maps preserving the Hermitian form \eqref{eq:helicity-form}.
Conformal symmetry, awkward and nonlinear on spacetime, is therefore
\emph{linear} on twistor space --- the sense in which $\PT$
linearizes conformal geometry, and the reason the bridge of
Part~\ref{part:bridge} completes only when the symmetry is enlarged
that far.

\paragraph{Helicity and the form.} The name of
\eqref{eq:helicity-form} is earned dynamically: for a massless
particle with momentum $p_{AA'} = \bar\pi_A \pi_{A'}$ and the angular
momentum built from $\omega^A$, the Pauli--Lubanski helicity
evaluates to exactly $s(Z)$ (the momentum and angular-momentum
expressions are displayed in Course~B, Lecture~B.3). The sign of the
form is the particle's handedness --- which is why a division of
$\PT$ by the sign of $s$ can carry physical information at all, and
why the same division will be found, in Part~\ref{part:bridge},
carrying the frequency splitting.


\begin{theorem}[Penrose transform]
\label{thm:penrose-transform}
For each helicity $h$, the solutions of the free massless field
equations on (conformally compactified) Minkowski space correspond
naturally to sheaf-cohomology classes on twistor space,
\begin{equation}
\big\{\text{massless fields of helicity } h,\ \text{positive
frequency}\big\}
\;\cong\;
H^1\!\big(\PT^{+},\, \cO(-2h-2)\big),
\label{eq:penrose-transform}
\end{equation}
with negative frequency carried by $\PT^{-}$
\citep{PenroseMacCallum1973, EastwoodPenroseWells1981}. In its
original contour-integral form \citep{Penrose1969}, the field at $x$ is obtained by
integrating a holomorphic function of homogeneity $-2h-2$ over the
line \eqref{eq:incidence}.
\end{theorem}

\paragraph{The contour form, worked.} Write $f(ix\pi, \pi)$ for the
restriction of a holomorphic $f(Z)$ to the line \eqref{eq:line} ---
the argument abbreviates the incidence substitution
\eqref{eq:incidence}. For the scalar ($h = 0$),
\begin{equation}
\phi(x) \;=\; \frac{1}{2\pi i} \oint_{\Gamma \subset L_x}
f\big(i x \pi,\, \pi\big)\; \pi_{E'}\, d\pi^{E'},
\qquad f \ \text{of homogeneity } -2;
\label{eq:contour-scalar}
\end{equation}
for helicity $h > 0$, insert $2h$ factors of $\pi$,
\begin{equation}
\phi_{A'_1 \cdots A'_{2h}}(x) \;=\; \frac{1}{2\pi i} \oint_{\Gamma}
\pi_{A'_1} \cdots \pi_{A'_{2h}}\; f\big(i x \pi,\, \pi\big)\;
\pi_{E'}\, d\pi^{E'},
\qquad f \ \text{of homogeneity } -2h-2;
\label{eq:contour-positive}
\end{equation}
and for $h < 0$, differentiate instead,
\begin{equation}
\phi_{A_1 \cdots A_{2|h|}}(x) \;=\; \frac{1}{2\pi i} \oint_{\Gamma}
\frac{\partial}{\partial \omega^{A_1}} \cdots
\frac{\partial}{\partial \omega^{A_{2|h|}}}\, f \,\Big|_{L_x}\;
\pi_{E'}\, d\pi^{E'},
\qquad f \ \text{of homogeneity } 2|h|-2.
\label{eq:contour-negative}
\end{equation}
In every case the homogeneity of $f$ is tuned so that the
\emph{integrand} has total homogeneity zero in $\pi$ --- the
condition for the integral to be well defined on the projective line
--- and this counting is the entire origin of the exponent in
\eqref{eq:penrose-transform}: each $\pi$ costs $+1$, the measure
$\pi\, d\pi$ costs $+2$, and $f$ must pay the balance.

\paragraph{Why the field equations are automatic.} Differentiating
under the contour,
\begin{equation}
\frac{\partial}{\partial x^{AA'}}\, f\big(i x^{BB'} \pi_{B'},\,
\pi\big) \;=\; i\, \pi_{A'}\,
\frac{\partial f}{\partial \omega^{A}} \Big|_{L_x},
\label{eq:chain-rule}
\end{equation}
so every spacetime derivative of the integrand inserts one factor of
$\pi_{A'}$ and one $\omega$-derivative. For
\eqref{eq:contour-positive}, contracting the inserted $\pi_{A'}$
against any of the $\pi$'s already present gives
$\varepsilon^{A'B'} \pi_{A'} \pi_{B'} = 0$ identically, so the
massless field equation $\nabla^{AA'} \phi_{A'B' \cdots} = 0$ holds
with no condition on $f$ beyond holomorphy; for the scalar, the wave
operator inserts two factors of each kind and dies on the same
antisymmetric contraction, $\Box \phi = 0$. The differential equation
has not been solved; it has been made \emph{vacuous} by the geometry
of the parametrization. This is the precise content of the slogan
that holomorphy trades equations for regularity --- and the
computation just performed is Lecture~B.4's, in compressed form.

\paragraph{Elementary states, and where frequency lives.} The
simplest data are the \emph{elementary states}, e.g.\ for the scalar
\begin{equation}
f(Z) \;=\; \frac{1}{(A_\alpha Z^\alpha)\,(B_\beta Z^\beta)},
\label{eq:elementary}
\end{equation}
with simple poles on the two planes $A \cdot Z = 0$ and
$B \cdot Z = 0$. Restricted to a line $L_x$ the poles are two points;
the contour separates them; and the residue gives a rational solution
of the wave equation, regular wherever the pole planes miss $L_x$.
Positive frequency now acquires geometric content: the field is
positive-frequency precisely when $f$ extends holomorphically over
all of $\PT^{+}$ --- for \eqref{eq:elementary}, when both pole planes
can be held inside the closure of $\PT^{-}$ --- which is the
correspondence recorded in \eqref{eq:penrose-transform}. \emph{That}
the division by $s$ carries the frequency splitting is, in the
twistor literature, a fundamental and cherished fact; sourcing it in
real causal data is the business of Part~\ref{part:bridge}.

\paragraph{The helicity ladder, and its famous asymmetry.} Reading
\eqref{eq:penrose-transform} across the physical fields: $h = 0$,
homogeneity $-2$, the wave equation; $h = \pm\tfrac12$, homogeneities
$-3$ and $-1$, the two chiral massless spin-$\tfrac12$ equations;
$h = \pm 1$, homogeneities $-4$ and $0$, self-dual and anti-self-dual
Maxwell theory; $h = \pm 2$, homogeneities $-6$ and $+2$, the two
handednesses of linearized gravity. The visible asymmetry between a
helicity and its mirror is harmless at the linear level --- both
sides of the ladder are covered by the transform --- but it is the
seed of the program's central nonlinear hurdle, the googly problem,
which this paper meets at its own boundary in \S\ref{sec:claims} and
objection O-P4-1.

\paragraph{From contours to cohomology.} The contour description
carries a redundancy: $f$ and $f + g_{+} - g_{-}$, with $g_{\pm}$
holomorphic on the two sides of the contour's neighborhood, give the
same field. Functions on an overlap, modulo functions extending over
the patches, is exactly a first \v{C}ech cohomology class --- and that
is how \eqref{eq:penrose-transform} earns its $H^1$: a massless field
is not a holomorphic \emph{function} on half of twistor space (of
negative homogeneity there are none to be had) but an
\emph{obstruction} class, and an obstruction is precisely what a
contour can detect. Course~B, Lectures~B.5--B.6, rebuilds this
passage slowly, with the two-patch covering worked in full.


\paragraph{Physics intuition.} A twistor is, roughly, a light ray
with a size and a handedness; a spacetime point is the family of rays
through it. The transform says a massless field --- an object defined
by a differential equation at every point --- is equally an object
defined by \emph{regularity in one half} of ray space, with no
equation left. The equation has been traded for holomorphy. Which
half carries the field is, as \eqref{eq:penrose-transform} records
and Part~\ref{part:bridge} exploits, precisely its frequency sign.



\bigskip\noindent\textbf{Close of Part II, in plain terms.} The toolbox is now open: what a complex structure
is (a quarter-turn that squares to $-1$, compatible with the classical
structure, tamed by positivity); how one is built from a generator
(the polar phase --- keep the direction, divide out the speed); which
self-pairings a circle allows (winding arithmetic); and what positive
frequency really means (one-sided regularity in imaginary time, with
twistor space waiting at full conformal symmetry). Nothing beyond
linear algebra and classical analysis was needed, and nothing more
will be. The next Part supplies the physics that selects among these
tools: the axioms, the forced six-dimensional arena, and the two
globally shared structures from which everything in this paper is
derived.

%% ===================================================================
\GGPart[The GG arena]{The \GGTheory{} Arena}{The axioms, the forced six-dimensional geometry, and the two globally shared real structures}
\label{part:gg}

This Part makes the linkage to \GGTheory{} exact: which constitutional
clauses do the selecting, which theorem supplies the geometry, and how
a charged field becomes a pair of real harmonics on a real circle. All
results are inherited in completed tense; each is restated here with
enough precision to carry the proofs of
Parts~\ref{part:sources}--\ref{part:bridge}.

\section[Part~III in plain English]{Part~III in plain English: the arena, and the two structures it supplies}
This paper does not build the framework it uses. Three constitutional clauses, the theorem that forces the six-dimensional geometry, an integer ledger, a boundary law and an actualization layer are all inherited from companion papers. This Part restates each at the precision the later proofs need, and names the paper that owns it.

The output is short. Two real structures, both supplied by the arena and both attached to every system in the universe: a compact, non-spatial phase circle, and the constitutional clock together with a positivity clause. Everything this preprint derives comes from those two and from nothing else, and Table~\ref{tab:inheritance} records which result draws on which, alongside the classical mathematics imported to reach it.

Restating an inherited result is not the same as proving it, and the table is there so that the difference stays visible. Where a clause is doing real work in a later theorem, a reader should be able to go to the paper that owns the clause and check that it says what is claimed here.

\section{The three constitutional clauses this paper uses}
\label{sec:gg-axioms}

The \GGTheory{} constitution \citep{ArisakaA1Constitution} rests on
four axioms; this preprint uses three.
\textbf{AX-1} (locality--causality) requires all registered physics to
live on a Lorentzian causal manifold, read through retarded, microcausal
protocols: it supplies a global \emph{time orientation} --- the causal
order --- and, applied to interacting charged matter, forces the two
structures geometrized below.
\textbf{AX-2} (geometry--gauge equivalence) requires every structure
the dynamics cannot avoid to be realized as geometry: one composite
connection on a principal bundle, with all physical content in
curvature and holonomy invariants. AX-2 is verifiably complex-free:
its connection is a real one-form, its holonomies real angles.
\textbf{AX-3} (the operational dictionary) requires a bulk theory to
specify how boundary observers register outcomes --- instruments,
channels, and, decisively for this paper, an \emph{operational
positivity} clause. AX-1 and AX-3(iii) are the two selection
principles of Part~\ref{part:sources}: the first orients the clock,
the second picks the sign of its polar phase.

Because everything in this paper hangs on them, the three load-bearing
clauses deserve more than the summary above; the following renders each
at the resolution of \citealp{ArisakaA1Constitution}, with the fourth
axiom added for completeness.

\emph{AX-1 in full.} Locality--causality is not one requirement but a
bundle: interactions built from products of fields at coincident
points; retarded support --- no response before its cause; and
microcausality --- spacelike-separated observables commute. Two
consequences are forced rather than assumed, and both matter here.
An interacting local theory is singular at coincident points and so
cannot be specified without a resolution scale and the flow connecting
resolutions --- the renormalization group, made compulsory. And local,
causal interaction of \emph{charged} matter cannot proceed by action
at a distance: a phase frame must be carried at every point with a
connection compensating its arbitrariness --- the gauge principle,
read as a consequence. A scale and a phase: AX-1 hands exactly two
structures to the next axiom.

\emph{AX-2 in full.} The geometry--gauge equivalence principle has two
clauses. \emph{Locked equivalence}: along any timelike curve there is
a combined choice of coordinates, frame, and gauge in which the
composite connection is flattened to first order --- Einstein's
elevator, upgraded to geometry-plus-gauge as one locked structure.
\emph{Invariant physics}: all physical content resides in curvature
and holonomy invariants of that one connection; ``forces'' are
artifacts of unlocked descriptions. Both clauses are complex-free ---
a real connection, real curvature, real holonomy angles --- and the
operative reading is a single rule: whatever the dynamics cannot
avoid must be realized as geometry. In the cohort's own words: AX-2
\emph{builds} the fiber.

\emph{AX-3 in full.} The operational axiom declares that a bulk theory
is not physically defined without a fixed dictionary from bulk
configurations to registered boundary physics,
\begin{equation}
\mathcal{D}(B) \;=\;
\big(\, \cH,\ \rho,\ \{\mathcal{I}_\alpha\},\ \{\Phi_{t,s}\},\ \ldots \big),
\label{eq:ax3-tuple}
\end{equation}
with $\rho$ a boundary state, $\{\mathcal{I}_\alpha\}$ measurement
\emph{instruments} in Kraus form with POVM semantics, and $\Phi_{t,s}$
completely positive trace-preserving evolution maps, the whole subject
to three compatibility clauses: retardedness (AX-1), gauge--geometry
covariance (AX-2), and \emph{operational positivity} --- clause (iii),
the one this paper leans on at T-P4-2. Measurement is thereby part of
the theory's syntax, not a metaphysical addendum; and the instruments'
membership in the dictionary is why observer-adapted structures
(which modes a detector counts; which infrared background it calls
vacuum; which horizon boundary condition it declares) are boundary
data throughout Part~\ref{part:extensions}. AX-3 \emph{collapses} the
fiber that AX-2 built.

\emph{AX-4, for completeness.} Maximum symmetry, minimum freedom: no
continuous structure is admissible unless a prior clause compels it.
Its role here is quiet but real --- it closes Theorem~0's count from
above (no seventh dimension) and forbids the unforced extras that
would otherwise let a phase float free of the one circle.

\paragraph{Physics intuition.} The three axioms are a relay. AX-1
says: to be local, causal, and charged, you must carry a zoom knob
and a dial. AX-2 says: if you must carry them, they are places ---
geometry, not bookkeeping. AX-3 says: and here is what a
four-dimensional being can actually read of them --- states,
channels, instruments, with positivity. The first hands two burdens to
the second; the second makes them real; the third makes them
readable. The complex numbers of this paper are born in the gap
between the second and the third: real geometry, read by a bounded
reader.

\section{Theorem 0: the forced arena}
\label{sec:gg-theorem0}

Theorem~0 of \citealp{ArisakaA2GG6} establishes that AX-1 and AX-2
together force the arena
\begin{equation}
\cM_6 \;=\; \cM_4 \times \Ir \times \Stheta,
\qquad
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu \;+\; dr^2 \;+\; d\theta^2 ,
\label{eq:theorem0-metric}
\end{equation}
with $\Ir$ a non-compact \emph{scale} direction (the geometrized
renormalization group) and $\Stheta$ a compact \emph{phase} circle
(the geometrized gauge fiber). Three properties carry everything
below. \emph{Non-spatiality:} the two extra directions are classified
by their generators --- a dilatation and a compact charge, not
translations --- so no signal propagates along them and no boundary
protocol resolves a position on them; the bulk isometry algebra is
$\mathfrak{iso}(1,3)\oplus\bR_D\oplus\mathfrak{u}(1)_Q$.
\emph{Global staticity:} $\eta$ is Minkowski and the warp depends only
on $r$, so $\partial_t$ is a global hypersurface-orthogonal timelike
Killing field --- the \emph{constitutional clock} --- and the
uniqueness clause of Lemma~\ref{lem:polar} applies globally.
\emph{Reality:} every datum in \eqref{eq:theorem0-metric} --- metric,
interval, angle, connection --- is real.

The proof of Theorem~0 is a six-step forcing chain worth possessing in
outline, because each step is load-bearing later. \emph{Step 1}: AX-1
is defined relative to a light cone, so the registered arena is
Lorentzian. \emph{Step 2}: locality plus interaction forces the
renormalization scale (AX-1's first burden). \emph{Step 3}: AX-2
promotes it to a coordinate --- the scale direction $\Ir$, along which
the connection component is the running coupling and the
$\beta$-function is a covariant derivative. \emph{Step 4}: locality
plus charge forces the periodic gauge phase (AX-1's second burden).
\emph{Step 5}: AX-2 promotes it to the compact direction $\Stheta$,
whose Wilson-line holonomy is the gauge-invariant datum. \emph{Step 6}:
the count closes two-sidedly --- from below because deleting either
direction breaks AX-1 itself (no scale: ill-defined at short distance;
no phase: action at a distance), from above because AX-4 admits no
third structure. Two further properties are proved en route and used
constantly here: the added directions carry \emph{no light cone}
(their generators are a dilatation and a compact charge, not
translations --- non-spatiality by classification, not by decree), and
motion along $\theta$ is a gauge transformation, hence not a physical
displacement. That last clause is the deep difference from every
spatial-circle construction: a Kaluza--Klein circle is somewhere one
could in principle look; the phase circle is not a place at all, which
is why its unreadability --- and everything superselection-shaped that
follows from it --- is constitutional rather than practical.

\paragraph{Physics intuition.} Do not picture six axes. Picture the
stage (spacetime), a zoom knob attached to it (the scale), and a dial
at every point (the phase). Theorem~0 says: charged, causal physics
cannot be written without the knob and the dial, and the constitution
obliges both to be geometry. The knob is not a corridor; the dial is
not a room. That is the whole of ``non-spatial,'' and it is why the
dial's angle can be physical only through differences --- holonomies
--- which is where the complex exponential will come from.

\section{The integer ledger: what BI-6 fixes before this paper begins}
\label{sec:gg-ledger}

One more inheritance is load-bearing and deserves its own section: the
five-integer ledger $(M, N, L;\, J, K) = (1, 3, 5;\, 7, 35)$,
introduced with the arena and proved forced --- not fitted --- by the
anomaly-closure architecture of \citealp{ArisakaA3BI6}. Each integer
carries a primary structural meaning: $M = 1$, the tensor channel is
primitive and unique; $L = 5$, the unified simple block is the
defining five-dimensional one; $N = 3$, the baryonic trace ratio
isolates a triplicity inside that block (whose later echoes are color
and the three generations); $J = 7$, the centered odd closure built
from the triplicity; and $K = 35$, the rectangular and quadratic lock
index, tied by the closure identities
\begin{equation}
K \;=\; L \cdot J \;=\; M^2 + N^2 + L^2 \;=\; 35 .
\label{eq:ledger-identities}
\end{equation}
Two entries of the ledger do direct work in this paper. $M = 1$ is
the winding unit: the statement that charge quantization comes in one
indivisible tensor seam is what makes the circle's characters
\emph{integers} and superselection \emph{discrete} (T-P4-1, Lecture
C.6). And $L = 5$ is the angle alphabet: with one circle and one
five-block --- and, decisively, no second flavor torus on which an
independent angle could float --- every first-order physical phase is
assembled from steps of $\pi/5$, which is why the CP-violating loop
holonomies of T-P4-5 are lattice-locked and why $2\pi/5$ appears where
a free parameter would otherwise sit. The ledger is the reason this
paper may say ``quantized'' and ``locked'' where other frameworks must
say ``measured.''

\paragraph{Physics intuition.} A dial that can be re-zeroed freely
carries no numbers of its own. What BI-6 adds is a tick-mark
structure: the anomaly bookkeeping closes only for one integer packet,
and those integers become the dial's engravings --- how many windings
make a charge ($M$), how many steps make the flavor pentagon ($L$).
The circle supplies the geometry of the phase; the ledger supplies its
arithmetic.

\section{The fiber dictionary: charged fields as real harmonic pairs}
\label{sec:gg-dictionary}

Single-valuedness on the compact circle gives every bulk matter field
a real harmonic decomposition; the charge-$n$ sector is the pair of
real weight-$n$ harmonics
\begin{equation}
\Psi_n(x;\theta) \;=\; a_n(x)\cos n\theta \;+\; b_n(x)\sin n\theta,
\qquad
R_\alpha : (a_n, b_n) \mapsto R(n\alpha)(a_n, b_n),
\label{eq:fibre-dictionary}
\end{equation}
and the textbook complex charged field is the packaging
$\psi_n = a_n + i b_n$ of Proposition~\ref{prop:real-C}. Under this
dictionary: charge is the winding number $n$; charge quantization is
compactness ($n \in \bZ$); a gauge transformation is a fiber shift
$\theta \mapsto \theta + \alpha(x)$; the covariant derivative is the
horizontal lift, acting on the $n$-harmonic as
$\partial_\mu - i n A_\mu$; the Aharonov--Bohm phase is a real
holonomy angle of the real connection; and complex conjugation is
orientation reversal $\theta \mapsto -\theta$. The photon is the
connection itself --- not a section of any associated bundle --- and
therefore correctly carries no dial.

\section{The boundary law: OGGEP and GDOS}
\label{sec:gg-gdos}

The dictionary demanded by AX-3 is not left abstract in the cohort;
\citealp{ArisakaA5GDOS} constructs it, and four of its structural
results frame everything this paper does with observers, horizons, and
the infrared. \emph{First, the projection is forced.} A boundary
protocol reads spacetime records sorted by a light cone; Theorem~0
gave the fiber directions no light cone; hence no operation resolves
where a system sits along $\Ir$ or around $\Stheta$, and every act of
registration marginalizes the fiber. Openness is constitutional;
unitary closure is the unreachable idealization in which the fiber is
never coarse-grained. \emph{Second, the projection is always on.} It
happens at every spacetime point, observer or no observer;
``measurement'' is its macroscopic record-forming special case and
decoherence its microscopic one. \emph{Third, the boundary law has a
derived form.} Exact projection gives a memory-kernel equation;
complete positivity is inherited from unitary conjugation plus partial
trace; and in the Markov regime the generator takes the
Gorini--Kossakowski--Sudarshan--Lindblad form --- from which the
textbook quantum postulates follow as theorems (Hilbert space by GNS,
Schr\"odinger as the closed limit, Born form by Gleason, L\"uders by
conditioning, decoherence, einselection, Bell). \emph{Fourth, and
decisive for Parts~\ref{part:sources}--\ref{part:extensions}: the two
fibers carry the two halves of quantum mechanics.} The compact phase
circle carries the coherent, unitary, interference-generating half ---
transport along it is pure phase; while the non-compact scale
direction carries the open, irreversible half --- its forward
renormalization-group semigroup is the dissipative generator, and
decoherence is the loss of $\theta$-coherence induced by
coarse-graining along $r$. This paper's subject matter is the first
fiber's half; but its boundaries --- which vacuum an instrument
declares, which soft cloud it stands in, what a horizon traces out ---
are all items of the second half's dictionary, which is why the
infrared and horizon discussions of Part~\ref{part:extensions} could
be stated as \emph{port} assignments rather than as ambiguities of the
bulk.

\paragraph{Physics intuition.} The bulk is a six-dimensional book;
the boundary observer reads a four-dimensional shadow of it, always,
whether or not anyone calls the reading an experiment. What the shadow
cannot show --- the dial angle, the zoom setting --- comes out as
probability and irreversibility; what it can show --- differences,
rates, records --- comes out real. Quantum mechanics is the grammar of
reading a book through its shadow, and this paper's claim is that the
grammar's one ``imaginary'' rule is the shadow's compression of the
dial and the clock.

\section{The actualization layer: GRIP, and why records are real}
\label{sec:gg-grip}

One more story of the parent program bears directly on this paper's
arc, because it is where outcomes --- the \emph{real} end of the
real-to-complex-to-real chain --- are fixed.
\citealp{ArisakaA6Gi4} builds the runtime in four pillars: the total
geometric--gauge object (the one connection, as a dynamical whole);
the covariant conservation identity (the four boundary Ward currents
--- electric, baryonic, informational, metric --- that any lawful
runtime must preserve); the geometric free-energy functional (a strict
Morse landscape ranking admissible boundary configurations); and GRIP,
the gauge-runtime invariance pipeline: descent proceeds in a finite,
ordered cascade of Morse steps, each transition costing a quantized
action and the whole bounded by the BI-6 budget, halting at the ground
geometric attractor. Carried into quantum mechanics by
\citealp{ArisakaA8QM}, this supplies the measurement story this paper
inherits silently every time it says ``registered'': the pointer basis
is forced by bulk locality; the single outcome is the deterministic
capture of the joint system-and-apparatus configuration by one
attractor basin, the registered datum being a \emph{contextual joint
holonomy} --- an element of the circle, a real angle class; and the
Born weights' square-modulus form is Gleason's \citep{Gleason1957}, with the run-by-run
value set by bulk boundary conditions inaccessible to the
four-dimensional reader. (What remains open there --- the proof that
basin-capture statistics reproduce the Born measure --- is the parent
program's named conjecture, not this paper's burden.) Two
consequences matter here. Every registered output of physics is a
real quantity --- an angle class, a count, a rate --- so the complex
layer of the formalism begins \emph{after} preparation and ends
\emph{before} registration: it is genuinely middleware. And the
coupling textures whose loop holonomies measure CP chirality
(T-P4-5) are not free functions but GRIP outputs --- stabilized
attractor data --- which is why the parent program can derive their
angles and this paper can call them lattice-locked.

\paragraph{Physics intuition.} If GDOS says how the shadow is read,
GRIP says how the ink dries: a measurement completes when the joint
configuration rolls into a basin it cannot leave, and what is written
is an angle --- a dial reading, real as any pointer. Between the real
bulk that supplies the story and the real record that ends it, the
complex amplitude lives only in the telling.

\section{The arc, assembled: real bulk, complex middleware, real records}
\label{sec:gg-arc}

Part~\ref{part:gg} can now be summarized as a single pipeline, which
is the frame the rest of the paper fills in. The \emph{inputs} are
real: an arena (Theorem~0), one connection (AX-2), an integer ledger
(BI-6), matter as real harmonic pairs on the circle
(\S\ref{sec:gg-dictionary}). The \emph{outputs} are real: GDOS
probabilities and rates, GRIP records, holonomy classes. In between
sits the layer this paper is about: amplitudes, phases, state spaces
--- the compressed notation a fiber-blind, tube-blind reader is forced
into. Parts~\ref{part:sources} and \ref{part:bridge} prove the
compression theorems (which real structures the notation compresses,
and why the compression is unique); Part~\ref{part:extensions} patrols
its boundaries. Nothing complex enters the pipeline's ends; the
question of this paper is only, and exactly, the middle.

\paragraph{The pipeline on the two-fiber anatomy.} The cohort's
recent campaign (\textsf{GG-W1-Symmetry}; \citealp{ArisakaP5CPT};
\citealp{ArisakaA8QM}) states the shape of this pipeline as one
exact sentence: \emph{holonomy requires closed loops, and an
interval has none.} The compact circle \Stheta{} is nothing but
closed loops --- everything on it is holonomy, quantized and
protected, its transport a group; the scale direction is an
interval --- no loops, no holonomy, transport composing only
forward, a semigroup. The pipeline's anatomy follows. The
\emph{middle} is complex because the middle is where circle
transport is being described: weight decomposition under a circle
action is what manufactures the $i$
(Part~\ref{part:sources}), and the positivity clause selects its
orientation --- the $i$ is the circle's, on loan to the notation.
The \emph{ends} are real because the ends are interval events:
preparation and registration are one-way writes along the fiber
that carries no phase --- so a record is real because of the fiber
it is written \emph{along}, and permanent because its content is
deposited in quantities the \emph{other} fiber protects. The
custody principle of \S\ref{sec:superselections} is this anatomy
in legal language --- the dictionary owns the sectors; the
geometry owns the $i$. And the paper's two title questions each
get a one-clause topological answer: why complex numbers? Because
one fiber is a circle. Why real records? Because the other is an
interval.

\section{The two globally shared structures, named}
\label{sec:gg-shared}

Everything this preprint derives comes from exactly two real
structures, both supplied by the arena and both attached to
\emph{every} system in the universe:
\begin{align}
\textbf{G1:}&\ \ \text{the phase circle } \Stheta
\ \ (\text{generator } \partial_\theta),
\label{eq:G1}\\
\textbf{G2:}&\ \ \text{the constitutional clock } \partial_t
\ \ \text{with the positivity clause of AX-3}.
\label{eq:G2}
\end{align}
G1 will supply the gauge phase and everything quantized about it; G2
will supply the state-space complex structure and everything
holomorphic about it; and their consistency on charged sectors will
be the particle--antiparticle pairing. The inheritance is summarized
in Table~\ref{tab:inheritance}.

\begin{table}[htbp]
\centering
\caption[What each result uses, and where it lives in the parent program.]{What each result of this preprint uses, and where it lives in
the parent program. One row per numbered result. The middle column names the \GG{} axiom, theorem or ledger entry the result stands on; the right column names the classical mathematics imported to reach it, with an em dash where none is needed. The table is here so that a reader can see at a glance which parts of the argument are new to this paper and which are inherited, and can go to the owning paper for the second kind.}
\label{tab:inheritance}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{4.8cm}p{4.2cm}p{6.6cm}@{}}
\toprule
\textbf{Result (this paper)} & \textbf{GG source} & \textbf{Imported classical mathematics} \\
\midrule
T-P4-1 (circle source) & AX-2; Theorem 0 (G1) & harmonic analysis on $S^1$ \\
T-P4-2 (clock source) & AX-1; AX-3(iii) (G2) & \citet{AshtekarMagnon1975}; \citet{Kay1978} \\
T-P4-3 (antiparticles) & G1 + G2 jointly & --- \\
T-P4-4 (one shared $i$) & G1, G2 global & \citet{Renou2021} (as constraint) \\
T-P4-5 (discrete table) & orientations of G1, G2 & --- \\
T-P4-6 (Hardy level) & G2 & \citet{PaleyWiener1934} \\
tube level & G2 & \citet{StreaterWightman1964} \\
T-P4-7 (twistor level) & G2, conformal & \citet{PenroseMacCallum1973}; \citet{EastwoodPenroseWells1981} \\
T-P4-8 (interacting) & G1, G2 asymptotic & Haag--Ruelle \citep{Haag1992} \\
T-P4-9 (infrared door) & AX-3 (ports) & \citet{Frohlich1973}; \citet{Pizzo2003} \\
lattice-locked phases & BI-6 ledger ($L = 5$) & --- \\
winding integrality & BI-6 ledger ($M = 1$) & --- \\
textures; real records & GRIP (\citealp{ArisakaA6Gi4}) & Morse theory \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Physics intuition.} The whole construction can be said in
one breath: nature carries one dial and one clock, the same pair
everywhere. The dial cannot be read, so only its differences (holonomies)
are physics; the clock cannot be run backward, so only one frequency
orientation is physics. Complex numbers are what a four-dimensional
observer writes down when forced to log differences of an invisible
dial and orientations of an irreversible clock in a single symbol.



\bigskip\noindent\textbf{Close of Part III, in plain terms.} The stage is set and it is entirely real: an
arena forced by locality and equivalence (a stage, a zoom knob, a dial
at every point); an integer ledger engraving the dial; a boundary law
that reads the bulk through an always-on projection; and an
actualization layer that writes real records. Two structures on this
stage are shared by everything in the universe --- the phase circle
and the causal clock --- and the paper's whole claim is that they are
where the complex numbers come from. The next Part proves it: five
theorems, from the circle's quarter-turn to the discrete-symmetry
table, with the antiparticle emerging as the handshake between the
two sources.

%% ===================================================================
\GGPart[The two real sources]{The Two Real Sources}{Five theorems: the circle, the clock, the antiparticle handshake, the shared $i$, and the discrete-symmetry table}
\label{part:sources}

\section[Part~IV in plain English]{Part~IV in plain English: the circle and the clock}
This is where the derivation starts. Five theorems, best read as two sources and three consequences.

The circle supplies the gauge phase. It is a compact real dimension of the bulk, a charged field is a pair of real harmonics on it, and the complex structure of charged matter is the rotation the circle already carries. The clock supplies the state-space $i$: causal time orientation together with energy positivity select the polar phase of the time-translation generator, uniquely, on a static arena. The staticity is a condition and not a decoration --- its gravitating generalization is inherited rather than owned, and Part~\ref{part:conclusions} says so again. Neither derivation uses the other.

Then the three consequences. That the two sources agree on charged sectors is the particle--antiparticle pairing, and it is the first real check in the paper, because two independent derivations could have disagreed and did not. That the $i$ is one $i$ shared by everything, rather than one per system, is what a composite system requires and what the 2021 experiment tests. And the discrete symmetries follow from the two orientations as a single table rather than as five separate arguments.

\section{The circle: the gauge phase as real geometry}
\label{sec:circle}

\begin{theorem}[\textbf{T-P4-1}: the circle source]
\label{thm:p4-1}
On each weight-$n \neq 0$ sector \eqref{eq:fibre-dictionary}: (i) the
charge-normalized circle generator $J_Q = \partial_\theta / Q$
(with $Q = \pm n$ the signed charge) is a complex structure in the
sense of Definition~\ref{def:J}, commuting with the covariant dynamics
and with all gauge transformations; (ii) $(V_n, J_Q) \cong$ the
standard complex charged field of charge $n$, with complex conjugation
realized as circle-orientation reversal; (iii) $J_Q$ is unique up to
sign: the commutant of the irreducible weight-$n$ circle action
contains exactly the two quarter-turns; (iv) on the weight-zero sector
the construction is empty --- $\partial_\theta$ acts as zero --- and by
Lemma~\ref{lem:weights} the only invariant self-pairing at $n\neq0$ is
the charge-conserving $\psi\bar\psi$.
\end{theorem}

\begin{proof}
(i) On the pair $(a_n, b_n)$, $\partial_\theta$ acts as
$(a,b) \mapsto (nb, -na)$, so $J_Q^2 = -\mathbf{1}$ on either
orientation; the symplectic form \eqref{eq:sigma-KG} is built from
$\theta$-independent data, so the isometry preserves it; the principal
connection is $\theta$-invariant, so $[D_\mu, \partial_\theta] = 0$;
a gauge transformation is a fiber shift and commutes with
$\partial_\theta$. (ii) is Proposition~\ref{prop:real-C} applied
fiberwise, plus the horizontal-lift computation
$D_\mu \psi_n = (\partial_\mu - inA_\mu)\psi_n$; under
$\theta \mapsto -\theta$, $(a,b) \mapsto (a,-b)$, i.e.\
$\psi \mapsto \bar\psi$. (iii) The commutant of an irreducible
$SO(2)$ action on a plane is spanned by rotations; its solutions of
$J^2 = -\mathbf{1}$ are the two quarter-turns. (iv) is
Lemma~\ref{lem:weights}.
\end{proof}

\paragraph{Physics intuition.} Everything ``complex'' about a charged
field is dial-reading. The field strength of its phase is how fast the
dial winds across spacetime (de Broglie); the Aharonov--Bohm shift is
how much the dial turns around a loop; charge is how many times the
dial turns per turn of the fiber. None of these requires $i$ --- they
require an angle, which is what the sixth dimension is.

\section{The clock: the state-space \texorpdfstring{$i$}{i}}
\label{sec:clock}

\begin{theorem}[\textbf{T-P4-2}: the clock source]
\label{thm:p4-2}
On the real solution space of any free (or quasi-free) bulk field on
the arena \eqref{eq:theorem0-metric}, gapped case: the polar phase
$J_{A}$ of the time-translation generator (Lemma~\ref{lem:polar}) is a
complex structure, unique among dynamics-commuting positive-energy
structures \citep{AshtekarMagnon1975, Kay1978}; it is the Hilbert
transform in constitutional time; and the selection uses only real
constitutional data: AX-1 orients the clock, AX-3(iii) selects the
sign, Theorem~0's global staticity makes the choice canonical. The
same holds for Majorana fermions on their Euclidean solution space,
and for the exact ground state of any gapped quadratic system, whose
Gaussian covariance $\Gamma$ satisfies $J = 2\Gamma\sigma = J_A$.
\end{theorem}

\begin{proof}
Existence and the listed properties are Lemma~\ref{lem:polar};
uniqueness on a stationary region is the cited selection theorem, and
Theorem~0 supplies global stationarity. The Hilbert-transform
identification: on each frequency-$\omega$ plane $J_A$ is the
quarter-period advance, which is \eqref{eq:hilbert} frequency by
frequency. The quadratic-ground-state identity follows from symplectic
diagonalization: with stiffness $K$, the ground covariance is
$\Gamma = \tfrac12\,\mathrm{diag}(K^{-1/2}, K^{1/2})$ in
$(\varphi,\pi)$ blocks, and $2\Gamma\sigma$ reproduces
\eqref{eq:polar-phase} for $A = \begin{psmallmatrix} 0 & \mathbf{1} \\
-K & 0\end{psmallmatrix}$ by inspection.
\end{proof}

\paragraph{Physics intuition.} Ask what a neutral field's phase
\emph{is}, given that it has no dial: it is the field's position in
its own cycle --- where each mode stands between its cosine and its
sine. ``Multiply by $i$'' means ``advance every mode a quarter of its
own period.'' That operation is real (a principal-value integral over
the history), non-local in time (you must know the whole cycle to know
a quarter of it), and unique once you insist that energy be positive
--- which is the constitution's own clause, not a taste.

\section{Antiparticles: the handshake}
\label{sec:antiparticles}

\begin{theorem}[\textbf{T-P4-3}: circle--clock consistency]
\label{thm:p4-3}
On the on-shell charge-$n$ sector, the two structures agree,
$J_A = J_Q$, precisely under the Feynman--St\"uckelberg pairing ---
charge $+n$ with positive frequency, charge $-n$ with positive
frequency --- and anti-agree on the opposite pairing. Explicitly, on
the real mode basis: writing the total dial angle $n\theta - Et$
($E>0$),
\begin{equation}
J_Q:\ \cos(n\theta - Et) \mapsto -\sin(n\theta - Et)
\quad\text{and}\quad
J_A:\ \cos(n\theta - Et) \mapsto -\sin(n\theta - Et),
\label{eq:handshake}
\end{equation}
while on the wrong pairing $\cos(n\theta + Et)$ the two maps differ by
a sign. Hence the sign ambiguity of T-P4-1(iii) is resolved by
T-P4-2's positivity, there is \emph{one} global complex structure $J$
--- sourced by the clock, geometrized by the circle wherever charge
lives --- and the existence of antiparticles is the consistency
condition between the sixth dimension and causal time.
\end{theorem}

\begin{proof}
$J_Q$ is the quarter-turn $\theta \mapsto \theta + \pi/2n$ on the
$+n$ orientation, giving
$\cos(n\theta - Et + \pi/2) = -\sin(n\theta - Et)$; $J_A$ is the
quarter-period advance $t \mapsto t - \pi/2E$, giving
$\cos(n\theta - Et + \pi/2)$ likewise. On $\cos(n\theta + Et)$ the
same two substitutions produce opposite signs. Re-labeling the
negative-frequency, charge-$n$ modes as positive-frequency, charge
$-n$ modes (under which $J_Q = \partial_\theta/Q$ flips its
normalization with $Q$) restores agreement on every physical mode;
linearity extends the identity from modes to the space. 
\end{proof}

\paragraph{Physics intuition.} The circle knows the plane of rotation
but not the direction; the clock knows the direction. Nature must use
both, so their orientations must mesh --- and the unique meshing rule
is the one every textbook imposes by hand: pair opposite charge with
opposite frequency. Antiparticles are not an extra ingredient of
relativistic quantum theory; they are the gear alignment between its
two real sources.

\section{One shared \texorpdfstring{$i$}{i}, and composite systems}
\label{sec:shared}

\begin{theorem}[\textbf{T-P4-4}: the shared complex structure]
\label{thm:p4-4}
(i) The global $J$ of T-P4-3 is common to all sectors and all
subsystems, because its sources G1, G2 are globally shared.
(ii) For subsystems carrying the same $J$, the complex tensor product
is the quotient of the real one by the shared-$J$ identification,
\begin{equation}
V \otimes_{\bC} W \;\cong\;
\big(V \otimes_{\bR} W\big)\,\big/\,
\mathrm{span}\{\,Jv \otimes w \;-\; v \otimes Jw\,\},
\label{eq:tensor-quotient}
\end{equation}
with the induced complex structure well defined exactly because $J$ is
shared; the Fock construction extends $J$ with no new complex input,
and entangled states are elements of the real quotient. (iii) The
experimental exclusion of real quantum mechanics \citep{Renou2021}
constrains precisely the \emph{unshared} case --- real tensor products
of separately-real state spaces --- and is satisfied here by
construction.
\end{theorem}

\begin{proof}
(ii) The quotient identifies the two real actions of $J$; the induced
operator $J \otimes \mathbf{1} \equiv \mathbf{1} \otimes J$ squares to
$-\mathbf{1}$ and is well defined iff the identified actions agree ---
i.e.\ iff $J_V = J_W$ under the identification; for unshared
structures no canonical complex quotient exists, which is the abstract
content of the network-Bell pathology the experiment exploits.
(i) and (iii) are then immediate.
\end{proof}

\section{The discrete-symmetry table, derived}
\label{sec:discrete}

\begin{theorem}[\textbf{T-P4-5}: orientation bookkeeping]
\label{thm:p4-5}
Let $C$, $P$, $T$ act as orientation reversals of circle, space, and
clock respectively. Then: $C$ is $J$-linear (it reverses
$\partial_\theta$ and the signed charge together, leaving
$J_Q = \partial_\theta/Q$ invariant); $T$ is $J$-antilinear (it
reverses the polar phase of the generator); $P$ is $J$-linear; hence a
discrete symmetry is antiunitary if and only if it contains $T$. The
total reversal $CPT$ preserves each Feynman--St\"uckelberg-paired mode
set and is a symmetry of the construction for every coupling texture
built from real data; $CP$ alone is a symmetry iff the texture is
achiral on the circle --- every closed-loop (rephasing-invariant)
angle of the texture lies in $\{0, \pi\}$. In the parent program the
loop angles are lattice-locked, the physical quark CP phase being two
pentagonal steps, $\delta = 2\pi/5$ \citep{ArisakaA9SM}.
\end{theorem}

\begin{proof}
Under $\theta \mapsto -\theta$ the weight-$n$ sector maps to weight
$-n$ and $\partial_\theta \mapsto -\partial_\theta$; since $J_Q$ is
normalized by the signed charge, $J_Q \mapsto
(-\partial_\theta)/(-Q) = J_Q$. Under $t \mapsto -t$ the frequency
orientation flips, so $T J_A T^{-1} = -J_A$. $(anti)$linearity
multiplies under composition, giving the table. $CPT$ maps
$\cos(n\theta - Et) \mapsto \cos(n\theta - Et)$ on each paired mode;
real texture data (moduli, winding sums, metric contractions) are even
under total reversal, and reversing the sign of every texture angle
while conjugating every loop is a relabeling. A circle reversal alone
negates every loop angle; invariance up to re-zeroing forces each loop
angle to equal its own negative mod $2\pi$.
\end{proof}

\paragraph{Physics intuition.} Why is time reversal the odd one out
--- antiunitary, where $C$ and $P$ are unitary? Because of the three
orientations a discrete symmetry can flip, only the clock's is the one
$i$ is made of. Flip space, and nothing happens to $i$; flip the
dial, and the charge label flips with it, so $i$ survives; flip the
clock, and ``a quarter period ahead'' becomes ``a quarter period
behind'' --- $i$ becomes $-i$, which is antilinearity. The textbook
table is a map of which reversals touch the sources of $i$.



\bigskip\noindent\textbf{Close of Part IV, in plain terms.} The two sources have now done their work. The
circle gave charged matter its phase, its quantized charge, and its
superselection; the clock gave every field --- charged or neutral ---
its state-space $i$, unique once energy is positive; their agreement
on charged sectors forced the existence of antiparticles; their shared
global character gave composite systems one $i$ and entanglement a
real presentation; and their orientations, flipped one at a time,
generated the C, P, T table that textbooks assign by hand. What
remains of the original question is its deepest half: Penrose's
holomorphy. The next Part crosses that bridge --- from the Hilbert
transform, through the forward tube, to twistor space itself.

%% ===================================================================
\GGPart[The bridge]{The Bridge: Positive Frequency is Holomorphy}{Hardy space, the forward tube, and the twistor correspondence as one structure at three symmetries}
\label{part:bridge}

\section[Part~V in plain English]{Part~V in plain English: the bridge, which is what the title claims}
The title of the paper is this Part. Positive frequency is holomorphy: at the Hardy level for a single mode, at the tube level for a field, and at the twistor level once conformal symmetry is available. The claim is that these are one structure seen at three symmetries, not three analogies that happen to rhyme.

Every component is established mathematics and the deepest two belong to the twistor school. What is new here is the direction of explanation. In the twistor reading, holomorphy is primitive and the frequency splitting is one of its shadows; in the reading assembled here, the real causal-positive data come first and holomorphy is what those data look like at sufficient symmetry. The twistor construction loses nothing by this. It gains its motivation.

Six worked examples close the Part, from the analytic signal to a CKM plaquette computed with no complex numbers anywhere in it. They are here because a claim of this shape is easy to state and easy to believe without checking, and an example a reader can work by hand is the cheapest way to find out whether it is true.

\section{The Hardy level}
\label{sec:hardy}

\begin{theorem}[\textbf{T-P4-6}: the derived $J$ generates holomorphy]
\label{thm:p4-6}
The complex structure selected in T-P4-2 is the generator of the Hardy
splitting: for real finite-energy histories,
\begin{equation}
f \ \text{positive-frequency}
\;\Longleftrightarrow\;
f = \text{boundary value of } F \in H^2(\mathrm{Im}\,z > 0)
\;\Longleftrightarrow\;
\Htransform f_{+} = -\,i\, f_{+},
\label{eq:hardy-equiv}
\end{equation}
where $f_\pm = \tfrac12(\mathbf{1} \pm i\Htransform)f$ are the
frequency halves and $F(t+is) = \int_0^\infty \hat f_+(\omega)\,
e^{i\omega t - \omega s}\, d\omega$ is the explicit extension, which
satisfies the Cauchy--Riemann equation
$\partial_s F = i\,\partial_t F$ for $s > 0$ and recovers $f = 2\,
\mathrm{Re}\,F$ on the boundary; a negative-frequency component
instead grows as $e^{+|\omega| s}$. Consequently the state-space $i$
of \GGTheory{} is, verbatim, the infinitesimal form of upper-half-plane
holomorphy in complexified constitutional time.
\end{theorem}

\begin{proof}
The equivalences are Theorem~\ref{thm:pw} together with the Fourier
multiplier form \eqref{eq:hilbert}: on frequency support
$[0,\infty)$, $\Htransform$ acts as $-i$; the displayed extension
converges for $s > 0$ precisely because the support is one-sided, and
term-by-term differentiation gives $\partial_s F = i\,\partial_t F$;
for support on $(-\infty, 0]$ the same formula produces $e^{+|\omega|s}$.
The identification with T-P4-2's $J$ is
Lemma~\ref{lem:polar}'s time-domain form.
\end{proof}

\paragraph{Physics intuition.} The analytic signal of the engineer is
the whole theorem in miniature. Given a real voltage trace, form
$f + i\Htransform f$: a complex ``signal'' whose modulus is the
envelope and whose argument is the instantaneous phase. Engineers call
this a trick. The theorem says it is not a trick: histories with one
frequency orientation \emph{are} the boundary shadows of functions
living regularly in one half of complexified time, and the ``$i$''
being attached is the same derived $J$ that quantum mechanics uses.
The state space of physics is an engineering trick that turned out to
be geography.

\section{The tube level}
\label{sec:tube}

The field-theoretic form replaces the single time variable by all of
them. Positive-frequency solutions of a free field extend
holomorphically to the \emph{forward tube}
\begin{equation}
\mathcal{T}^{+} \;=\; \big\{\, x + iy \;:\; y \in V^{+} \,\big\}
\qquad (V^{+} = \text{open future timelike cone}),
\label{eq:forward-tube}
\end{equation}
and in any Wightman theory the vacuum correlation functions are
boundary values of functions holomorphic in (permuted extensions of)
$\mathcal{T}^{+}$ \citep{StreaterWightman1964}. Tube analyticity is
the standing field-theoretic expression of causal positivity: the
tube is the complexified constitutional clock, one imaginary direction
per timelike one, and holomorphy in it says exactly that the theory's
frequencies carry one orientation with respect to every causal
observer at once. For a massive theory, this is where analyticity
stops.

\paragraph{Physics intuition.} Why a \emph{tube}? Because ``positive
frequency'' must mean the same thing for every inertial clock, and
each clock contributes its own imaginary direction in which histories
of the right orientation decay. The tube is the intersection of all
those one-sided regularities --- the geometric statement that the
universe's frequency orientation is Lorentz-invariant.

\section{The twistor level}
\label{sec:twistor-level}

\begin{figure}[tbp]
\centering
\resizebox{0.591\textwidth}{!}{%
\usebox{\GGplateTwistor}}
\caption[The division of projective twistor space by the helicity form.]{Projective
twistor space divided by the sign of the helicity form
\eqref{eq:helicity-form}. The five-real-dimensional null hypersurface
$\mathbb{PN}$ carries the real null geodesics; a spacetime point $x$
appears as a projective line $L_x$; and positive-frequency massless
fields are exactly the cohomology carried by the upper half,
$H^1(\PT^{+}, \cO(-2h-2))$. Theorem
T-P4-7 identifies this division with the causal-positive frequency
splitting generated by the derived $J$.}
\label{fig:pt-division}
\end{figure}

\begin{theorem}[\textbf{T-P4-7}: the twistor division is the frequency splitting]
\label{thm:p4-7}
For free massless fields of helicity $h$: (i) the Penrose transform
\eqref{eq:penrose-transform} identifies positive-frequency solutions
with $H^1(\PT^{+}, \cO(-2h-2))$ and negative-frequency solutions with
the corresponding classes on $\PT^{-}$; (ii) along any timelike
worldline, the pullback of a positive-frequency field is a Hardy-class
function of proper time, so the twistor division restricts on every
clock to the Hardy splitting of T-P4-6; (iii) hence the division of
twistor space by the sign of the helicity form
\eqref{eq:helicity-form} \emph{is} the causal-positive frequency
splitting generated by the derived $J$, conformally packaged: a
massive field's positivity buys analyticity in one complexified
direction at a time, a conformally invariant massless field's
positivity buys the whole tube at once, and twistor space is the
geometry that linearizes exactly that enhancement.
\end{theorem}

\begin{proof}
(i) is Theorem~\ref{thm:penrose-transform}, imported with its original
attribution \citep{PenroseMacCallum1973, EastwoodPenroseWells1981}.
(ii) A positive-frequency massless field has momentum support on the
future null cone; its restriction to a timelike worldline with
proper time $\tau$ has frequency support in $[0,\infty)$ (the
Minkowski pairing of a future-null momentum with a future-timelike
velocity is positive), hence is Hardy by Theorem~\ref{thm:pw}.
(iii) The helicity form is the twistor-space expression of frequency
sign --- fields on $\PT^{+}$ are exactly those extending to the
forward tube in the incidence relation's complexified $x$ --- so (i)
and (ii) exhibit one splitting written at three symmetries: a single
clock (Hardy), all clocks (tube), all conformal clocks (twistor).
\end{proof}

\paragraph{Physics intuition.} Take the double slit's phase dial and
ask how much symmetry can be loaded onto it. With one observer, the
dial's orientation gives Hardy analyticity in that observer's
imaginary time. Demand agreement among all inertial observers: the
tube. Demand agreement under the full conformal group --- the natural
symmetry of massless light --- and the ``imaginary time'' directions
knit into a compact complex three-space, and the dial's orientation
becomes the division of that space into two halves. Twistor space is
what the frequency dial looks like when light's full symmetry is
allowed to act on it.

\begin{figure}[tbp]
\centering
\resizebox{0.504\textwidth}{!}{%
\usebox{\GGplateBridge}}
\caption[The holomorphy bridge at its three symmetry levels.]{The
holomorphy bridge. One derived complex structure, read at three
symmetry levels: with a single clock, positivity is Hardy-space
holomorphy; demanding agreement among all inertial clocks yields the
forward tube; and for conformally invariant massless fields the tube
completes to twistor space, where the frequency splitting becomes the
geometric division $\PT^{+}/\PT^{-}$.}
\label{fig:bridge}
\end{figure}

\section{The direction of explanation}
\label{sec:direction}

Every component above is established mathematics, and the deepest two
belong to the twistor school itself. What this preprint claims --- its
entire claim --- is the direction of explanation. In the twistor
reading, holomorphy is a primitive endowment of nature and the
frequency splitting one of its shadows. In the reading assembled here,
the real causal-positive data come first --- a compact circle, a causal
order, a positivity clause, all real, all constitutional
(\S\ref{sec:gg-axioms}) --- and holomorphy is what those data
\emph{look like} at sufficient symmetry. The twistor construction
loses nothing; it gains its motivation: the reason nature's massless
fields organize themselves holomorphically on $\PT$ is that holomorphy
is the conformal completion of the same real structure that makes
amplitudes complex, charges quantized, and antiparticles necessary.
Nothing was invented; something was found --- and what was found was
never the complex numbers, but the circle and the clock they compress.

\section{Worked examples}
\label{sec:worked}

\subsection{Worked example I: the analytic signal, end to end}
\label{ssec:we-analytic}

Take the elementary real history $f(t) = \cos\omega t$, $\omega > 0$.
Its frequency halves are $f_\pm = \tfrac12 e^{\mp i\omega t}$
(conventions of \eqref{eq:hilbert}); the Hardy extension of the
positive half is
\begin{equation}
F(t + is) \;=\; \tfrac12\, e^{i\omega(t+is)} \;=\;
\tfrac12\, e^{i\omega t}\, e^{-\omega s},
\label{eq:we1-extension}
\end{equation}
manifestly holomorphic (a function of $z = t+is$ alone), decaying in
the upper half plane, with boundary value recovering
$f = 2\,\mathrm{Re}\,F$; the negative half extends only downward,
growing as $e^{+\omega s}$ upstairs. The Hilbert transform acts as the
quarter-period advance, $\Htransform f = \sin\omega t$, and one checks
$\Htransform^2 f = -f$: the derived $J$ in its simplest clothing. The
``analytic signal'' \citep{Gabor1946} $f + i\Htransform f = e^{i\omega t}$ of
signal processing is thus literally the boundary value of the upper-half-plane
extension --- T-P4-6 on one mode.

\subsection{Worked example II: an elementary state through the Penrose transform}
\label{ssec:we-twistor}

For helicity $0$ the transform takes homogeneity $-2$: the field at
$x$ is the contour integral
\begin{equation}
\varphi(x) \;=\; \frac{1}{2\pi i}
\oint_{\Gamma \subset L_x} f\big(i x^{AA'}\pi_{A'},\, \pi_{A'}\big)\,
\pi_{E'}\, d\pi^{E'} ,
\label{eq:we2-contour}
\end{equation}
over a contour on the line $L_x$ of \eqref{eq:incidence}. Take the
elementary state $f(Z) = \big[(A_\alpha Z^\alpha)(B_\beta Z^\beta)\big]^{-1}$
with $A, B$ fixed dual twistors: the integrand has two simple poles on
$L_x$, the contour separates them, and the residue evaluates to
\begin{equation}
\varphi(x) \;=\; \frac{1}{\rho_A(x)\,\rho_B(x)} ,
\label{eq:we2-field}
\end{equation}
a rational solution of $\Box\varphi = 0$ singular where $x$ meets the
lines of $A$ and $B$ ($\rho_{A,B}$ the corresponding incidence
factors) --- the classical computation of
\citet{PenroseMacCallum1973, PenroseRindler1986}. The point for this
preprint is where the \emph{frequency} lives: the state is
positive-frequency exactly when the poles can be kept on one side ---
when $f$ extends holomorphically over $\PT^{+}$ --- so that contour
regularity in half of twistor space is, mode for mode, the Hardy
condition of Worked example~I. Frequency sign has become pole
placement: analysis has become geometry, with nothing added.

\subsection{Worked example III: a Bell state with one shared $J$}
\label{ssec:we-bell}

Write a two-mode Bell state in the real presentation. Each mode is a
plane $(a, b)$ with the shared $J$; the state
$(|01\rangle + |10\rangle)/\sqrt2$ is, in the real quotient
\eqref{eq:tensor-quotient}, the class of
\begin{equation}
\tfrac{1}{\sqrt2}\big( e_0 \otimes e_1 + e_1 \otimes e_0 \big),
\qquad
J\big(e_i \otimes e_j\big) \;\equiv\; (Je_i) \otimes e_j
\;\sim\; e_i \otimes (Je_j),
\label{eq:we3-bell}
\end{equation}
with $e_0, e_1$ real basis vectors of the two one-particle planes. The
identification in \eqref{eq:we3-bell} is exactly where the shared $J$
earns its keep: for two \emph{unshared} structures the two sides would
be inequivalent and no complex quotient would exist --- which is,
abstractly, the pathology the network-Bell experiments convert into a
falsification \citep{Renou2021}. Entanglement itself is untouched by
the real presentation: non-factorizability is a statement about the
class \eqref{eq:we3-bell}, not about the number system it is written
in. What the experiments exclude is unshared $i$; what nature runs on
is one circle and one clock.




\subsection{Worked example IV: horizon thermality by hand}
\label{ssec:we-horizon}

The superselection discussion of \S\ref{sec:superselections} rests on
one Gaussian computation, short enough to do in full. Let one field
mode straddle a horizon, with the globally regular state the two-mode
squeezed pairing of the exterior mode and its hidden partner: in the
real presentation, a Gaussian state of two planes with covariance
\begin{equation}
\Gamma \;=\; \frac12
\begin{pmatrix}
\cosh 2r\, \mathbf{1}_2 & \sinh 2r\, Z \\
\sinh 2r\, Z & \cosh 2r\, \mathbf{1}_2
\end{pmatrix},
\qquad Z = \mathrm{diag}(1, -1).
\label{eq:we4-tms}
\end{equation}
Discard the hidden block (the \GDOS{} partial trace across the port):
the exterior covariance is $\Gamma_{\rm ext} = \tfrac12\cosh 2r\,
\mathbf{1}_2 = (\bar n + \tfrac12)\mathbf{1}_2$ with
$\bar n = \sinh^2 r$ --- a \emph{thermal} state, exactly. Now impose
the horizon pairing $\tanh r_\omega = e^{-\pi\omega/\kappa}$
($\kappa$ the surface gravity): then
\begin{equation}
\bar n_\omega \;=\; \sinh^2 r_\omega
\;=\; \frac{1}{e^{2\pi\omega/\kappa} - 1},
\label{eq:we4-planck}
\end{equation}
a Planck spectrum at $T_H = \kappa/2\pi$ \citep{Hawking1975, Unruh1976}, with detailed balance
$\bar n/(\bar n + 1) = e^{-\omega/T_H}$ holding identically. Sample
numbers: for $\kappa = 0.37$ (arbitrary units), the mode
$\omega = 0.2$ carries $\bar n = 3.47\times10^{-2}$ and the mode
$\omega = 4$ carries $\bar n = 3.2\times10^{-30}$ --- the exponential
cliff of a genuine temperature. Two lessons: thermality is
\emph{produced} by the trace, not assumed; and the complex structure
never appears --- the whole computation is real covariance algebra,
with $J$ (by Lemma~\ref{lem:cloud}'s logic) untouched throughout.

\subsection{Worked example V: the CKM plaquette with no complex numbers}
\label{ssec:we-ckm}

T-P4-5 asserted that CP violation is a chirality of loop angles; here
is the computation, executed entirely in real rotation blocks. Build
the CKM matrix in the standard parameterization with angles
$(s_{12}, s_{13}, s_{23}) = (0.2250,\ 0.00369,\ 0.04182)$ and the
parent program's first-order phase $\delta = 2\pi/5$, but replace
every complex entry $z$ by its block $B(z)$ of
\eqref{eq:block} --- so $e^{-i\delta}$ is the real rotation
$R(-\delta)$, and the matrix is a real $6\times6$ array of blocks.
The rephasing-invariant quartet is the plaquette product
\begin{equation}
P \;=\; B(V_{us})\, B(V_{cb})\, B(V_{ub})^{\mathsf T} B(V_{cs})^{\mathsf T},
\qquad
J_{\rm CKM} \;=\; P_{21},
\label{eq:we5-plaquette}
\end{equation}
conjugation being block \emph{transpose} (orientation reversal,
Proposition~\ref{prop:real-C}) and the imaginary part being the
antisymmetric (sine) component. The numbers: $\delta = 2\pi/5$ gives
$J_{\rm CKM} = 3.21\times10^{-5}$, against the measured
$(3.12^{+0.13}_{-0.12})\times10^{-5}$; the achiral values
$\delta \in \{0, \pi\}$ give $J_{\rm CKM} = 0$ identically; and
re-zeroing all six family dials by random angles
($\varphi_{ij} \mapsto \varphi_{ij} + \alpha_i - \beta_j$ in block
form) leaves $P_{21}$ unchanged to machine precision --- rephasing
invariance \emph{as} holonomy invariance, watched happening. At no
step does the symbol $i$ occur: CP violation is an area-like real
invariant of a real texture, exactly as T-P4-5 states.

\subsection{Worked example VI: the dial-free door, in matrices}
\label{ssec:we-door}

Lemma~\ref{lem:weights} in coordinates, for the two cases that matter.
\emph{Weight one:} under $R(\alpha)$, the Majorana doublet transforms
as
\begin{equation}
\begin{pmatrix} a^2 - b^2 \\ 2ab \end{pmatrix}
\;\longmapsto\;
\begin{pmatrix} \cos 2\alpha & -\sin 2\alpha \\
\sin 2\alpha & \phantom{-}\cos 2\alpha \end{pmatrix}
\begin{pmatrix} a^2 - b^2 \\ 2ab \end{pmatrix},
\label{eq:we6-weight1}
\end{equation}
so the only invariant self-pairing is $a^2 + b^2 = \psi\bar\psi$: a
charged fermion can have a Dirac mass and nothing else, and lepton
number --- winding number --- is conserved at every vertex.
\emph{Weight zero:} the rotation is the identity, all three pairings
are invariant, and the reality condition \emph{is} the Majorana
condition: the $\Delta L = 2$ self-pairing is legal exactly here. The
portal arithmetic closes the story: the unique fiber-neutral trilinear
touching the door is $L H N$ with weights
$(-\tfrac12) + (+\tfrac12) + 0 = 0$. In the parent program this
one-line weight count is the constitutional basis of the seesaw and of
TeV-scale leptogenesis \citep{ArisakaA10Cosmos}; here it serves as the
sharpest possible illustration that the circle's conservation laws are
\emph{winding arithmetic} --- integers, not analysis.


\bigskip\noindent\textbf{Close of Part V, in plain terms.} Positive frequency is holomorphy: at one clock it
is the Hardy property of the analytic signal; demanded of all inertial
clocks it is analyticity in the forward tube; and granted light's full
conformal symmetry it becomes the division of twistor space into its
two halves --- the same splitting, generated by the same derived $J$,
drawn at three magnifications of symmetry. The worked examples made
each level concrete enough to redo by hand. What has been proved so
far, however, lives in the free and quasi-free world; the next Part
patrols the boundaries --- interactions, the infrared, horizons ---
and states exactly where the derivation extends, where it is
conditional, and --- at the infrared, with the door theorem --- where
a boundary long recorded as open turns out to close.

%% ===================================================================
\GGPart[Extensions and boundaries]{Extensions and Boundaries}{Interacting matter; the infrared door opened; three superselections, one custody principle}
\label{part:extensions}

\section[Part~VI in plain English]{Part~VI in plain English: past the free field, and where the account stops}
The theorems of Part~\ref{part:bridge} are kinematic and quasi-free. This Part carries them as far as they will honestly go and then marks the place where they stop.

Interacting matter is reached by the asymptotic route, and the completion is conditional: it rests on a mass gap and on asymptotic completeness, which are the standard scattering hypotheses and are named as hypotheses rather than absorbed into the proof. The infrared case is the harder one, because massless interacting theories have no gap. The theorem proved here closes that door at the rigorous-model level, exactly at the van Hove stratum, and the full relativistic dressing theory is named as the rung above and left open.

The Part ends by noticing that superselection has now been met three times --- at the charge fiber, in the infrared sectors, and in records --- and that the three are one structure met on three strata of physics. Each is written up in the same four items, and the pattern they share is then proved from the theorems already in hand rather than asserted, which is the difference between a pattern and a coincidence.

\section{Interacting matter: the asymptotic completion}
\label{sec:interacting}

\begin{theorem}[\textbf{T-P4-8}: the interacting completion; conditional]
\label{thm:p4-8}
Let interacting bulk matter satisfy: (H1) a mass gap; (H2) asymptotic
completeness; (H3) the regularity of a Wightman or Haag--Kastler
framework. Then the physical Hilbert space is canonically the Fock
space over the asymptotic real solution spaces equipped with the
derived $J$ of T-P4-2/T-P4-3: the interacting theory's complex
structure \emph{is} the free structure of its asymptotic frame, and
interactions contribute vertices --- real texture data --- and no
complex structure of their own.
\end{theorem}

\begin{proof}[Proof (by assembly)]
Under (H1) and (H3), Haag--Ruelle theory \citep{Haag1992} constructs,
from almost-local smearings of the interacting fields with
one-particle-supported test functions, creation operators whose strong
limits as $t \to \pm\infty$ exist (cluster estimates supply the Cauchy
property) and satisfy free-field commutation relations at the physical
mass; the generated in/out spaces are Fock over free one-particle
spaces, which are exactly the real-data objects of
Part~\ref{part:sources} with the polar-phase $J$. Under (H2) the
in/out spaces exhaust the physical space and the M\o{}ller operators
transport $J$ onto it. Vertices enter only through the S-matrix and
the texture (moduli and fiber angles), both real.
\end{proof}

\paragraph{Physics intuition.} An interacting world is still
interrogated by free probes: what a collider registers, long before
and long after the collision, are freely flying quanta, and it is
\emph{their} dials and clocks that define the bookkeeping. The theorem
says the bookkeeping suffices: the complex structure never belonged to
the interaction --- it belonged to the asymptotic quanta, where the
free derivation already owns it. For confined sectors the same
statement attaches to hadrons: the particles instruments actually
meet, which are gapped.

\paragraph{The hypotheses, weighed.} (H1), the mass gap, is doing
two jobs: isolating the one-particle shell so that quasi-local
creation operators can be tuned to it, and guaranteeing the cluster
estimates that make the $t \to \pm\infty$ limits Cauchy. (H2),
asymptotic completeness, is the statement that scattering exhausts the
theory --- unproven in general four-dimensional field theory for
anyone, and the honest hinge of the theorem: without it the derived
$J$ owns the scattering sector and is silent on a hypothetical
non-scattering remainder. (H3) is bookkeeping regularity. What the
theorem buys under these hypotheses is worth restating: not an
approximation scheme but an \emph{identification} --- the interacting
Hilbert space, whatever its internal complexity, carries exactly the
asymptotic frame's complex structure, so interactions renormalize
masses, couplings, and states, but never the meaning of $i$.

\paragraph{Physics intuition.} A storm at sea does not change what
``clockwise'' means. The interaction region may be arbitrarily
violent; the complex structure is an orientation convention
established where the water is calm --- at $t \to \pm\infty$ --- and
transported through the storm by unitarity. That the two calm regions
agree (in against out) is asymptotic completeness's real content here,
and it is also why the confined sector poses no special problem: one
reads the orientation off the hadrons, which are the calm water.

\section{The infrared door: sectors are ports}
\label{sec:infrared}

Where masslessness and interaction meet, the asymptotic particle
notion itself becomes subtle. The charged particle cannot shed its
soft-photon cloud; its sharp mass shell dissolves (the
\emph{infraparticle}); and clouds belonging to different velocities
are superselected: for cloud profiles $c_v/\sqrt{\omega}$ the overlap
of two dressings carries the exponent
\begin{equation}
\big|\langle\, \mathrm{cloud}_v \,|\, \mathrm{cloud}_{v'} \rangle\big|
\;=\;
\exp\!\Big(\!-\tfrac12 \int_\epsilon^{\Lambda}
\frac{|c_v - c_{v'}|^2}{\omega}\, d\omega\Big)
\;=\;
\Big(\frac{\epsilon}{\Lambda}\Big)^{|c_v - c_{v'}|^2/2}
\;\xrightarrow[\epsilon \to 0]{}\; 0,
\label{eq:ir-overlap}
\end{equation}
logarithmically divergent in the exponent and power-law vanishing in
the overlap --- the Bloch--Nordsieck signature \citep{BlochNordsieck1937}. Rigorous collision theory exists for the massless quanta
themselves \citep{Buchholz1977}, and dressed-state constructions
\citep{Chung1965, FaddeevKulish1970} organize the charged sectors, rigorously so
in model systems. Two structural facts govern the reading here.

\begin{lemma}[The cloud never touches $J$]
\label{lem:cloud}
A coherent dressing is an affine symplectic map $f \mapsto f + d$:
it shifts first moments, leaves every covariance invariant, and hence
leaves the complex structure $J = 2\Gamma\sigma$ of the sector's
Gaussian states exactly unchanged.
\end{lemma}

\begin{proof}
Covariances are second central moments; an affine shift changes only
the center. $J = 2\Gamma\sigma$ is built from the covariance and the
(fixed) symplectic form.
\end{proof}

The infrared therefore threatens the \emph{particle} notion, never
the derivation of $i$; and the sector label --- which infrared
background an instrument treats as vacuum --- is operational boundary
data, of exactly the kind AX-3 assigns to instruments. The remainder
of this section makes that reading precise and proves it, at exactly
the level the rigorous models support. The result is the \emph{door
theorem}, T-P4-9: the corner where this account was long weakest ---
and where the twistor school's commitments are strongest, massless
fields being its native habitat --- closes, and closes in a way that
strengthens the thesis, because what the infrared turns out to
be made of is real, classical, coherent bookkeeping.

\paragraph{Physics intuition.} An electron drags an unpayable debt of
arbitrarily soft light, and two electrons moving differently drag
inequivalent debts --- so ``the same state, boosted'' splits into
sectors no local operation connects. But notice what the debt is made
of: a classical, coherent, real displacement of the field. Shifting a
pendulum's rest point does not change what ``a quarter period ahead''
means. The infrared rearranges the furniture; it does not touch the
clock.



\paragraph{The model ladder.} The proof of the door theorem lives on a
ladder of three rungs, and the theorem's regime is exactly the
ladder's. The bottom rung is the \emph{van Hove stratum} --- a
massless field linearly coupled to a fixed (recoil-free) source, $H =
\mathrm{d}\Gamma(\omega) + \phi(g)$ --- where everything is exactly
solvable and the lemmas below are the whole content. The middle rung
is the \emph{Nelson and Pauli--Fierz models} --- a nonrelativistic
particle coupled to the massless scalar field or to the quantized
radiation field --- where the infraparticle structure is theorem-grade:
the fixed-momentum Hamiltonian has no ground state in Fock space
\citep{Frohlich1973}; the dressed one-particle states exist, with
H\"older regularity in the total momentum \citep{Pizzo2003}; the mass
shell survives removal of the ultraviolet cutoff
\citep{BachmannDeckertPizzo2012}; and the infraparticle scattering
states of nonrelativistic QED carry exactly the velocity-superselected
clouds the door theorem classifies \citep{ChenFrohlichPizzo2010}. The
top rung is \emph{full relativistic QED}, where the dressing theory
\citep{Chung1965, FaddeevKulish1970} is semi-rigorous and the sector
observables are the asymptotic fluxes of \citet{Buchholz1986}; the
theorem's reading carries verbatim there, and its status is displayed,
not claimed --- the field's own frontier, inherited with its name on
it.

\begin{definition}[Coherent sectors]
\label{def:coherent-sectors}
Let $W(h)$, for $h$ in the real test space, generate the Weyl algebra
of the asymptotic massless field. A \emph{coherent dressing} by a
profile $f$ is the algebra automorphism $W(h) \mapsto
e^{i\,\mathrm{Im}\langle f, h\rangle}\, W(h)$. Its GNS representation
over the vacuum state is Fock if and only if $f$ is square-integrable,
and two coherent representations with profiles $f, f'$ are unitarily
equivalent if and only if $f - f' \in L^{2}$. A \emph{sector} is an
equivalence class of profiles; its label is c-number data (the flux
configuration at infinity), and the label is invariant under every
quasi-local operation.
\end{definition}

\begin{lemma}[The cloud leaves Fock space; velocities are superselected]
\label{lem:ir-divergence}
In the massless models of the ladder, the dressing cloud of a charge
moving with velocity $v$ ($|v| < 1$) has the low-frequency profile
\begin{equation}
f_{v}(k) \;=\; \frac{-\,g}{\sqrt{2|k|}\;\,|k|\,\bigl(1 - v\cos\theta\bigr)},
\qquad |k| \le \Lambda,
\label{eq:ir-cloud}
\end{equation}
with $\theta$ the angle between $k$ and the velocity. Then, with
infrared cutoff $\epsilon$:
(i) $\displaystyle \|f_{v}\|^{2} \;=\; \frac{2\pi g^{2}}{1 - v^{2}}\,
\ln\frac{\Lambda}{\epsilon} \;+\; O(1)
\;\xrightarrow[\epsilon\to0]{}\; \infty$,
so in the cutoff-removed limit $W(f_{v})$ is not implementable on Fock
space and the dressed state generates a new coherent sector;
(ii) for $v \neq v'$,
$\|f_{v} - f_{v'}\|^{2} = c(v, v')\,\ln(\Lambda/\epsilon) + O(1)$ with
$c(v, v') > 0$, so the sector overlap obeys the power law
\eqref{eq:ir-overlap} and distinct velocities are exactly orthogonal
in the limit: velocity superselection, by the same mathematics that
superselected charge --- an unremovable divergence in a relative-phase
integral.
\end{lemma}

\begin{proof}
For (i): $|f_{v}(k)|^{2} = g^{2} / \bigl(2|k|^{3}\,
(1 - v\cos\theta)^{2}\bigr)$, so in spherical coordinates
\[
\|f_{v}\|^{2}
\;=\; \frac{g^{2}}{2} \int_{\epsilon}^{\Lambda} \frac{dk}{k}\,
      \int d\Omega\; \frac{1}{(1 - v\cos\theta)^{2}}
\;=\; \frac{g^{2}}{2}\,\ln\frac{\Lambda}{\epsilon}\;\cdot\;
      2\pi \int_{-1}^{1} \frac{du}{(1 - v u)^{2}}
\;=\; \frac{2\pi g^{2}}{1 - v^{2}}\,\ln\frac{\Lambda}{\epsilon},
\]
using $\int_{-1}^{1} (1 - v u)^{-2}\, du = 2/(1 - v^{2})$. For (ii):
the difference profile carries the same $|k|^{-3/2}$ infrared
singularity with the angular factor $(1 - v\cos\theta)^{-1} -
(1 - v'\cos\theta)^{-1}$, nonzero on a set of positive measure
whenever $v \neq v'$; the same radial integral gives
$c(v, v') = \pi g^{2} \int_{-1}^{1} \bigl[(1 - v u)^{-1} -
(1 - v' u)^{-1}\bigr]^{2} du \,>\, 0$. Substituting into the coherent
overlap formula reproduces \eqref{eq:ir-overlap} with
$|c_{v} - c_{v'}|^{2}$ read off as $c(v, v')$: the overlap is
$(\epsilon/\Lambda)^{c/2}$, vanishing as the cutoff is removed.
Disjointness is then the $L^{2}$ criterion of
Definition~\ref{def:coherent-sectors}.
\end{proof}

\begin{lemma}[The dressed generator is free-plus-c-number]
\label{lem:dressed-generator}
On the van Hove stratum, mode-wise: the coupled generator
$H = \omega\, a^{*}a + g\,(a + a^{*})$ is diagonalized by the
displacement $W(d)$ with $d = -g/\omega$, and
\begin{equation}
W(d)^{\dagger}\, H\, W(d) \;=\; \omega\, a^{*} a \;-\;
\frac{g^{2}}{\omega};
\label{eq:dressed-generator}
\end{equation}
in the multimode case, $W(d)^{\dagger} H\, W(d) =
\mathrm{d}\Gamma(\omega) - \langle g,\, \omega^{-1} g\rangle$. The
quadratic part --- the only datum the polar-phase construction of
T-P4-2 reads --- is exactly the free generator, in every sector.
\end{lemma}

\begin{proof}
$W(d)^{\dagger}\, a\, W(d) = a + d$, so
\[
W^{\dagger} H W \;=\; \omega\,(a^{*} + d)(a + d) + g\,(a + a^{*} + 2d)
\;=\; \omega\, a^{*}a \;+\; (\omega d + g)(a + a^{*}) \;+\;
\omega d^{2} + 2 g d,
\]
and the choice $d = -g/\omega$ cancels the linear term, leaving the
c-number $-g^{2}/\omega$. Together with Lemma~\ref{lem:cloud} --- the
affine shift leaves every covariance, hence $J = 2\Gamma\sigma$,
exactly invariant --- the sector's derived complex structure is the
polar phase of the \emph{same} quadratic generator in every sector.
\end{proof}

\begin{theorem}[\textbf{T-P4-9}: the infrared door --- sectors are ports; $J$ is sector-blind]
\label{thm:p4-9}
In an interacting theory with massless quanta, on the model ladder at
the levels stated there:
(i) the asymptotic soft-cloud configuration --- the flux/velocity
sector --- labels mutually disjoint representations of the asymptotic
field algebra, and the label is a c-number port assignment at AX-3:
which infrared background an instrument treats as ``vacuum'' is
operational boundary data, with the same standing as the horizon
boundary condition of the black-hole case;
(ii) within each sector the dressed asymptotic dynamics is
free-plus-c-number, and its polar phase supplies the \emph{same}
derived $J$ in every sector --- sector-blind, because its two sources,
the circle and the clock, are global;
(iii) hence the derivation of this paper extends to the infrared: the
sector multiplicity is dictionary data, the complex structure passes
through untouched, and the two never mix. The infrared problem is a
problem about which vacuum dressing an observer stands in --- not a
problem about where $i$ comes from.
\end{theorem}

\begin{proof}[Proof (by assembly)]
At the van Hove stratum the theorem \emph{is} the three lemmas: clause
(i) is Lemma~\ref{lem:ir-divergence}'s disjointness together with the
invariance of the label under quasi-local operations
(Definition~\ref{def:coherent-sectors}) --- a multiplicity of
admissible vacua, labeled by c-number boundary data, unresolvable
from within, which is precisely the kind of datum AX-3's dictionary
owns, and precisely how the horizon vacua are placed in
\S\ref{sec:superselections} below; clause (ii) is
Lemma~\ref{lem:dressed-generator} together with
Lemma~\ref{lem:cloud}; clause (iii) follows from (i) and (ii). At the
Nelson/Pauli--Fierz rung, the imported theorems supply exactly the two
existence facts the lemmas need --- the dressed one-particle states
exist \citep{Pizzo2003, BachmannDeckertPizzo2012}, and their clouds
carry the profile \eqref{eq:ir-cloud} with velocity superselection
\citep{Frohlich1973, ChenFrohlichPizzo2010} --- and the lemmas, being
statements about coherent dressings as such, apply verbatim to the
states those theorems construct. At full QED the dressing theory is
semi-rigorous \citep{Chung1965, FaddeevKulish1970}, the sector label
is physical --- by Gauss' law it is an asymptotic flux observable
\citep{Buchholz1986} --- and the reading carries with its status
displayed.
\end{proof}

\paragraph{Verification.} Four numerical checks accompany the door
theorem in the verification suite (\S\ref{sec:verification}). The
cloud norm's logarithmic slope matches the analytic coefficient
$2\pi g^{2}/(1 - v^{2})$ of Lemma~\ref{lem:ir-divergence} to
$8\times10^{-6}$ relative. The measured sector-overlap power matches
the quadrature coefficient $c(v, v')/2$ to $10^{-5}$ relative, the
overlap falling from $10^{-14}$ to $10^{-53}$ over six decades of
cutoff at strong coupling and relativistic velocity separation. The
exact dressed ground state of a coupled two-mode van Hove model
carries $\max|J - J_{\mathrm{free}}| = 5\times10^{-15}$ while its
infrared-tower sector label $\|d\|^{2}$ grows without bound at the
analytic slope. And the conjugation \eqref{eq:dressed-generator}
holds on the truncated Fock space to machine precision, the residual
falling as pure truncation error as the truncation is raised.

\paragraph{The frontier, named.} What T-P4-9 does not close is the
top rung: full relativistic QED, where the existence theory of the
dressing is the field's own open problem (the Faddeev--Kulish
program). This account inherits that problem by name rather than
hiding it --- and notes what kind of problem it is. It is an
\emph{existence} question about dressed states, not a question about
the origin of $i$: clauses (ii) and (iii) are statements about
coherent dressings as such and wait, verbatim, on any dressing the
full theory constructs. The waiting can itself be stated as a
conditional theorem. Suppose full QED admits a dressing theory
satisfying: \emph{(D1) coherent form} --- the physical charged states
are Weyl displacements of Fock states by c-number soft-photon
profiles; \emph{(D2) flux sector labels} --- two dressings are
$L^{2}$-equivalent if and only if they carry the same Gauss-law flux
class; \emph{(D3) free-plus-c-number dressed asymptotic dynamics} in
each sector. Then clauses (i)--(iii) of T-P4-9 hold verbatim in full
QED, by the same three lemmas. (D1)--(D3) are exactly what the
semi-rigorous constructions exhibit and what the rigorous program is
built to deliver; the residue is purely their existence.

\paragraph{Physics intuition.} Think of the infrared as a port fee.
Every instrument that does business at spatial infinity must declare a
convention: which configuration of arbitrarily soft light it will call
``empty.'' Two instruments moving differently make different
declarations, and no local transaction can convert one declaration
into the other --- that is all the superselection is. But the fee is
paid in a real currency: a classical, coherent displacement of the
field. Nothing about the declaration touches the clock that defines
``a quarter period ahead.'' The tellers differ; the currency is the
same.

\paragraph{For the reader of Part~\ref{part:bridge}.} The door theorem
completes a movement this paper began at the Hardy level. The massless
conformal sector is the natural home of the bridge (T-P4-6, T-P4-7),
and its interacting extension was the one corner where this account
thinned --- the corner, moreover, where a twistor-school reader keeps
the deepest commitments, massless fields being the program's native
habitat. T-P4-9 says the thinning was bookkeeping: what dissolves in
the infrared is the Fock mass shell --- a statement about which states
are reachable from which --- while the splitting the bridge is made
of, positive against negative frequency, passes through every sector
untouched. The reader who came for the twistor geometry can take the
door theorem as the statement that nothing in the infrared bends the
geometry of $\PT^{\pm}$: the division of twistor space is
sector-blind too, because it is generated by the same $J$.

\section{Three superselections, one custody principle}
\label{sec:superselections}

The derivation has now met superselection three times, and this
section writes the three encounters up as what they have turned out to
be: one structure, met on three strata of physics, each time by proof.
For uniformity, each instance is recorded in the same four items, in
the same order: the \emph{arena}; the \emph{unreadable datum}, with
the mechanism that protects it; the \emph{sector label}, and where it
lives; and the \emph{status of $J$}. The pattern the three instances
share is then stated once, as a proposition, and proved from the
theorems already in hand.

\paragraph{First instance: charge --- the port at the fiber.}
\emph{Arena:} the compact phase circle of T-P4-1, carried by every
charged field. \emph{Unreadable datum:} the absolute fiber angle. A
four-dimensional instrument couples only to angle \emph{differences}
and to windings; the absolute angle is unobservable by construction,
and the protection mechanism is geometric --- the same compactness
that quantizes charge hides the origin of its dial. Relative phases
between sectors of different winding are therefore unmeasurable: this
is charge superselection, in the form first isolated by
\citet{WickWightmanWigner1952}, here with a geometric mechanism
supplied. \emph{Sector label:} the winding number $n \in \mathbb{Z}$
--- the ledger's $M = 1$ integrality --- a global topological datum
that no quasi-local operation changes; which winding sector an
instrument's bookkeeping addresses is dictionary data at AX-3.
\emph{Status of $J$:} within each sector the circle supplies $J_{Q}$,
and on shell the clock's polar phase agrees with it exactly
(T-P4-3), so one derived $J$ acts identically on every sector
(T-P4-4). The sectors differ in an angle no one can read; the quarter
turn is the same everywhere.

\paragraph{Second instance: horizon vacua --- the port at the horizon.}
\emph{Arena:} a stationary black-hole exterior. \emph{Unreadable
datum:} the interior partner correlations --- the modes behind the
horizon that purify the exterior state. The protection mechanism is
causal: the horizon is a one-way boundary, the exterior state is the
partial trace across it, and thermality at the standard temperature is
produced by the trace itself --- the \GG{} boundary projection
(\citealp{ArisakaA5GDOS}) evaluated at a horizon, in the
horizon-as-port reading of \citealp{ArisakaC3BH}. \emph{Sector label:}
the boundary condition at the horizon --- Boulware, Unruh,
Hartle--Hawking \citep{Unruh1976, Hawking1975, Wald1994} --- a
declaration of which port state the instrument's bookkeeping assumes:
dictionary data at AX-3, not bulk ambiguity. \emph{Status of $J$:} on
the stationary exterior the Killing-frame $J$ is unique by the
positivity selection \citep{Kay1978, KayWald1991}; port choices
repopulate the states and never touch the complex structure. The
rotating case --- where the ergoregion defeats the exterior selection
and the anchor moves to the asymptotic frame, superradiance becoming a
symplectic pumping channel --- and the collapse epoch --- which
factorizes into a universal thermal channel times a decaying transient
--- extend this instance at the model level in the parent program
(\citealp{ArisakaA8QM}, entries R14--R15; the full-length treatment is
the black-hole companion, \citealp{ArisakaC3BH}).

\paragraph{Third instance: infrared clouds --- the port at infinity.}
\emph{Arena:} interacting massless matter, on the model ladder of
\S\ref{sec:infrared}. \emph{Unreadable datum:} the soft-cloud profile
at infinity, relative to another sector's. The protection mechanism is
analytic: the relative-phase integral between clouds diverges
logarithmically (Lemma~\ref{lem:ir-divergence}), so the inter-sector
overlap vanishes as a power of the cutoff --- an unremovable
divergence standing exactly where, for charge, the compactness stood.
\emph{Sector label:} the velocity/flux class --- by Gauss' law an
asymptotic flux observable \citep{Buchholz1986}; which infrared
background counts as ``vacuum'' is an instrument's port declaration at
AX-3 (T-P4-9(i)). \emph{Status of $J$:} sector-blind, by
Lemma~\ref{lem:cloud} and Lemma~\ref{lem:dressed-generator}
(T-P4-9(ii)): the dressing is affine, the quadratic generator is
untouched, and every sector's polar phase is the same $J$.

\begin{proposition}[The custody principle]
\label{prop:custody}
In each of the three instances: (a) the sector label is a c-number
datum conjugate to a \emph{global} structure --- the total winding,
the horizon interior, the field at infinity --- and is invariant under
every quasi-local operation; (b) the relative phase between sectors is
unobservable, the obstruction being in each case an unremovable
contribution to a relative-phase integral (the compact holonomy; the
traced interior; the logarithmically divergent cloud); (c) the derived
$J$ commutes with the sector decomposition and is identical in every
sector. Consequently the superselection structure is owned entirely by
the AX-3 dictionary --- all three labels are port assignments --- and
the complex structure entirely by the geometry: \emph{the dictionary
owns the sectors; the geometry owns the $i$.}
\end{proposition}

\begin{proof}
Instance by instance. \emph{Charge:} (a) is the topological invariance
of the winding (T-P4-1); (b) is the unreadability of the absolute
angle --- the geometric form of the superselection rule of
\citet{WickWightmanWigner1952}; (c) is T-P4-3 with T-P4-4: one $J$ on
shell, acting identically on every winding sector. \emph{Horizon:} (a)
the port condition is set at the horizon, a global causal boundary,
and no exterior quasi-local operation reaches through it; (b) the
partner phase sits behind the partial trace; (c) the Killing-frame $J$
is unique on the stationary exterior \citep{Kay1978, KayWald1991},
independently of the port choice. \emph{Infrared:} (a)--(c) are
T-P4-9, through Lemma~\ref{lem:ir-divergence},
Lemma~\ref{lem:cloud}, and Lemma~\ref{lem:dressed-generator}.
\end{proof}

That division of custody is, in one sentence, this preprint's account
of the quantum formalism --- and it is worth pausing on why the
pattern recurs. In all three instances the superselected quantity is
conjugate to something global, and in all three the local physics is
untouched: superselection is the boundary's bookkeeping of what no
local operation can reach. The $i$, by contrast, is built from
structures --- the circle's plane of rotation, the clock's orientation
--- that are shared \emph{within} every sector and identical
\emph{across} them. A theory whose complex structure came from
anywhere else would have no reason for this immunity; on the present
account the immunity is clause (c) of a proposition, proved three
times, on three strata, by three different mechanisms guarding the
same kind of secret.
One scope remark belongs here. The proposition is deliberately
confined to the superselection triple --- the three instances in which
an unreadable \emph{relative phase between sectors} exists to be
owned. The parent program's rotating anchor and collapse channel
(entries R14--R15 of \citealp{ArisakaA8QM}) extend the same division
of custody to five instances at the model level, but they are not
superselection structures: what the dictionary owns there is a frame
anchor and an in-channel, not a sector label, and the proposition's
clauses apply to them in the looser currency of custody rather than
the strict currency of sectors. The slogan survives unchanged in both
currencies: the dictionary owns what varies; the geometry owns the
$i$.

\begin{table}[p]
\centering
\caption[The three superselections, side by side.]{The three
superselection structures this paper meets, written uniformly: an
unreadable relative phase between sectors, an operational sector label
(a port assignment at AX-3), and a derived $J$ that passes through
untouched --- the pattern stated and proved as
Proposition~\ref{prop:custody}.}
\label{tab:three-superselections}
\small
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{@{}p{3.1cm}p{4.05cm}p{4.3cm}p{3.8cm}@{}}
\toprule
\textbf{Instance} & \textbf{Unreadable datum} & \textbf{Sector label (port)} & \textbf{$J$ status} \\
\midrule
Charge & the absolute fiber angle & winding number $n$ & $J_Q$ per sector; one $J$ on shell (T-P4-3) \\
Horizon vacua & the interior partner modes & boundary condition at the horizon & Killing-frame $J$ unique on the exterior \\
Infrared clouds & the soft profile at infinity & velocity/flux sector & T-P4-9: ports; $J$ sector-blind (theorem, model level) \\
\bottomrule
\end{tabular}
\end{table}



\bigskip\noindent\textbf{Close of Part VI, in plain terms.} \looseness=-1 The boundaries are now mapped --- and one of
them turned out to be a door. Interacting gapped matter inherits the
asymptotic frame's $i$, conditionally on the field's own standard
hypotheses. The infrared, long this account's named open corner, is
crossed at the rigorous-model level: what dissolves there is the
particle's sharp mass shell --- a fact about sector labels --- while
the complex structure passes through every sector untouched (T-P4-9).
And the three superselections the account has met --- charge, horizon
vacua, infrared clouds --- are proved to be one pattern
(Proposition~\ref{prop:custody}): an unreadable relative phase, a port
assignment at the dictionary, an untouched $J$. The dictionary owns
the sectors; the geometry owns the $i$ --- no longer a slogan but a
proposition with three instances. With the theorems proved and the
boundaries drawn, the next Part steps back to judge: what the result
signifies, to whom, what is genuinely new in it, how the three great
roads compare, and what would prove it wrong.

%% ===================================================================
\GGPart[Discussions]{Discussions}{Significance and uniqueness of the result; the three roads compared, string theory included; predictions for falsification; the open questions; and the summary of the case}
\label{part:discussion}

\section[Part~VII in plain English]{Part~VII in plain English: what the result is worth, and what would end it}
A paper that has proved its theorems still owes an account of what they are worth. This Part gives that account at length rather than in a closing paragraph, and most of it consists of comparisons the paper could have skipped.

It states what is claimed and in what regime. It sets the result beside string theory and the twistor program on the axes where the three intersect, following each program's own literature and adjudicating nothing. It then makes the twistor comparison aspect by aspect, marking the seam where the two accounts meet and naming the columns where the twistor program is ahead --- the nonlinear constructions, and the amplitudes.

Then the falsifiers and the open questions. There are six of the latter, in decreasing order of consequence, and the first two are load-bearing. The falsifiers are stated as measurements rather than as attitudes. None of the open questions threatens the theorems as stated; each bounds the regime the theorems own. A reader deciding how much weight this argument can carry should read that section before reading the conclusions.

\section{Significance of the result}
\label{sec:significance}

Three communities keep a stake in the question this paper answers, and
the significance reads differently to each; a fourth and a fifth
stake, less obvious, are argued below.

\emph{To quantum foundations}, the result relocates a century-old
postulate. Since von~Neumann's axiomatization, the complexity of the
state space has been the first assumption of the formalism ---
defended, when defended at all, by its consequences. The
reconstruction program sharpened the defense into theorems: complex
quantum mechanics is selected among operational theories by
information axioms. What no result of that kind supplies is a
\emph{referent} --- a thing in the world that the complex structure is
the structure \emph{of}. This paper supplies the referent (a circle
and a clock, both derived), a uniqueness clause (positivity on the
static arena), and an experimental fingerprint read in reverse: the
network-Bell exclusion of real quantum mechanics, which the
foundations literature received as a surprising theorem about
formalisms, becomes the expected signature of two globally shared real
sources --- the one experimental fact this account could not survive
without, found already in hand. A postulate with a referent, a
uniqueness proof, and a fingerprint is no longer a postulate in the
working sense; that conversion is the significance, whatever one
thinks of the parent program.

\emph{To the twistor school}, the result is a duality with a gift at
each end. Read toward twistor space, it supplies the motivation the
program has done without for fifty years: holomorphy is what causal
positivity looks like at conformal symmetry, so the school's central
structure acquires a physical source, and its founding question ---
why should nature be holomorphic? --- acquires an answer built from
the school's own theorems. Read back from twistor space, it explains
the program's phenomenology of strengths: why massless fields are
the native habitat (only conformal fields complexify every timelike
direction at once); why the massive sector resists (it stops at the
tube level, where the clock works but the conformal completion is
unavailable); and why gauge charge always needed separate apparatus
(the circle's half of the derivation is invisible to ray-space
geometry, by the division of labor of T-P4-3). A program that
learns, from outside, the exact boundary of its own naturalness has
gained something no internal development could give it.

\emph{To the parent program}, the result closes a ledger. The
\GGTheory{} cohort derives its physics from real axioms, and until
this line of work its own formal apparatus carried one unexamined
complex ingredient --- the state spaces of its boundary theory. The
real-bulk series settled the accounts: no complex primitive anywhere
in the constitution, with the residues named as analytic frontiers
rather than conceptual debts. And the closure is not bookkeeping
alone: three results of the cohort that looked separate --- the
antiparticle pairing, the lattice-locked CP phases, the forced
Majorana door --- are exhibited here as one family, consequences of
how the circle and the clock share, twist, and withhold the single
derived $J$. A framework whose scattered rigidities turn out to be one
mechanism has learned something about itself.

\emph{To the history and philosophy of physics}, the significance is a
case study of unusual purity. Wigner's ``unreasonable effectiveness''
has always wanted load-bearing instances examined one at a time; the
complex numbers are arguably its sharpest instance, and on this
account the instance converts from unreasonable to unrecognized: the
mathematics was effective because it was, unknowingly, a portrait of
instantiated structure. Each brute fact traded for a consistency
condition --- quantization for compactness, the antiparticle for a
handshake, antiunitarity for orientation, holomorphy for positivity
--- is a data point in the oldest methodological question physics has,
and six data points from one mechanism is a pattern.

\emph{Finally, the significance of exposure.} Foundational accounts
are cheap when nothing can refute them; this one lists its failure
modes and would not survive them. A demonstration that independent
sources can carry unshared complex structures; a measured drift of the
flavor phases off their lattice completion; an established Dirac
nature for the light neutrinos; a derivation of the circle and the
clock \emph{from} holomorphy, reversing the arrow --- any of these
retires or inverts the account. In foundations, where claims commonly
retreat to interpretation when pressed, an answer that stakes named
ways to be wrong is significant partly \emph{because} it can lose.

\section{Uniqueness and innovation}
\label{sec:uniqueness}

To our knowledge, no other framework derives both complex structures
of physics from stated axioms, and the survey substantiating that
sentence is worth a paragraph. The textbook tradition postulates the
complex Hilbert space outright and correctly declines to argue. The
reconstruction theorems select complex quantum mechanics among
operational rivals --- an existence-of-necessity result, not a
source. The real-formulation program, at its most serious in
Stueckelberg, showed a real presentation survives for single systems
and stalls at composites --- exactly where the shared-$J$ quotient of
T-P4-4 now locates the obstruction, and where experiment later dug the
grave. Geometric quantization \emph{chooses} K\"ahler polarizations
with celebrated non-uniqueness; the present account is, in that
language, a physics that forces the polarization. The twistor reading
takes holomorphy as constitutive; the string program takes the
worldsheet's complex structure as given with the formalism. Each of
these is a coherent position; none of them answers, or attempts, the
question of what the imaginary unit is made of.

The specific innovations claimed as new are five, and each can be
stated with its precise delta from the prior art. \emph{First}, the
two-sources decomposition with its coincidence theorem: that the gauge
phase and the state-space $i$ have different real sources, agreeing on
charged sectors exactly under the Feynman--St\"uckelberg pairing --- so
that antiparticles, imposed in every textbook as a positivity
condition on the output of quantization, are derived one level down as
the unique meshing of two independently forced structures.
\emph{Second}, the discrete-symmetry table as orientation bookkeeping:
unitarity versus antiunitarity, assigned by Wigner's theorem case by
case, is here read off from which of the three real orientations an
operation reverses --- antiunitary if and only if the clock is
touched. \emph{Third}, the identification of the constitutionally
selected $J$ with the generator of the Hardy splitting, which converts
the analytic signal from an engineering device into the boundary form
of the state space's own complex structure. \emph{Fourth}, the
direction-reversal at twistor space: the division $\PT^{+}/\PT^{-}$,
present in the school's literature from the beginning as a property of
primitively complex geometry, re-derived as the conformal completion
of causal positivity --- the same seam, sewn from the other side.
\emph{Fifth}, the arithmetic layer: with the integer ledger's single
circle and five-block, the CP-violating loop holonomies are
lattice-valued, making CP violation a \emph{discrete} chirality ---
a statement unavailable to any framework in which phases have no
geometry to be locked to.

Three further structural readings are claimed as new framings with
consequences --- one of them now proved in full: the uniform mechanism of
the three superselections, now stated and proved as the custody
principle (Proposition~\ref{prop:custody}), with its infrared instance
at theorem level (T-P4-9); the port classification of the vacuum and
infrared multiplicities as operational dictionary data rather than
bulk ambiguity --- for the infrared no longer a reading but a theorem;
and the
compression account of the historical sequence, which turns the
four-century gap between Cardano and Schr\"odinger from an irony into
an explanation.

Equally deliberate is what is \emph{not} claimed. The Paley--Wiener
theory, the stationary uniqueness theorems, Haag--Ruelle scattering,
the Penrose transform and its cohomology, the superselection formalism
--- every load-bearing import is credited at its point of use and
shelved in the reading guide, and the paper's contribution is stated
as assembly plus direction: imported bricks, a new building, and an
arrow of explanation that the bricks' own literature does not contain.
The nearest historical anticipations deserve the same candor, because
each held a piece: Hamilton had the real presentation and lacked the
selection; Wu and Yang had the holonomy dictionary and lacked the
dimension; Stueckelberg had the real Hilbert space and lacked the
sharing; Penrose had the division and read the arrow the other way.
Novelty, in a subject this old, is precisely a statement about which
gap remained --- and the gap this paper fills is the conjunction:
selection, dimension, sharing, and arrow, at once.

Uniqueness, last, in the structural sense: within the parent
framework this account could not have been otherwise. The arena is
forced (Theorem~0), the circle's quarter-turn is unique up to the
orientation the clock fixes (T-P4-1, T-P4-3), the clock's polar phase
is unique by positivity (T-P4-2), and the sharing is automatic because
the sources are global (T-P4-4). Nothing was available to tune. A
derivation with no adjustable element either connects to the world or
breaks; \S\ref{sec:falsifiability} lists where to strike it.

\section{Three roads: string theory, twistor theory, and \GGTheory{}}
\label{sec:threeroads}

It is orienting to place a third program beside the two this paper has compared, and the house rule of the cohort applies: structural contrasts only, stated so that each program's own logic is respected. String theory reaches its extra dimensions by an internally compelled path --- worldsheet consistency and anomaly cancellation fix the dimension and the gauge structure almost rigidly \citep{GreenSchwarz1984} --- and its relation to complex structures is intimate but different in kind from either program here: the worldsheet is a Riemann surface, so two-dimensional holomorphy is a primitive of the formalism, while the target-space quantum $i$ is inherited from ordinary quantization. The three programs thus place the complex numbers in three different positions: primitive on the worldsheet (strings), primitive in the geometry of ray space (twistors), derived from a circle and a clock (here). Table~\ref{tab:threeroads} records the comparison on the axes where the three genuinely intersect; the twelve-aspect twistor--\GG{} table of \S\ref{sec:comparison} remains the detailed instrument, and the string column is offered for orientation, not adjudication --- the two economies place their irreducible premises at opposite ends of the energy axis, and the reader is left to weigh them.

\begin{table}[htbp]
\centering
\caption[Three roads: strings, twistors, GG-Theory.]{The three roads, on the axes where they intersect. Entries follow each program's own literature; no adjudication is intended. Seven aspects, one per row: where the complex numbers enter, how many extra dimensions are carried and of what kind, why four dimensions, what is assumed about quantum mechanics, how mass arises, where each program's premises sit, and whether the complex-number question is posed at all. The three are set side by side because they are usually compared on their outputs, and what separates them is visible earlier than that, in what each takes as primitive.}
\label{tab:threeroads}
\small
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{@{}p{2.9cm}p{4.1cm}p{4.0cm}p{4.2cm}@{}}
\toprule
\textbf{Aspect} & \textbf{String theory} & \textbf{Twistor theory} & \textbf{\GGTheory{}} \\
\midrule
Where $\bC$ enters & worldsheet complex structure (primitive); target-space $i$ from quantization & holomorphy of $\PT$ (primitive) & derived: circle $+$ clock (T-P4-1--3) \\
Extra dimensions & six/seven, spatial, compactified; moduli and landscape downstream & none (an auxiliary complex ray space) & two, non-spatial, forced; no moduli \\
Why four dimensions & selected by compactification & built into the formalism & derived ($4+1+1$, 4D boundary) \\
Quantum mechanics itself & presupposed & presupposed & boundary law derived from the bulk \citep{ArisakaA5GDOS} \\
Mass & natural (compactification scales) & a named hurdle & scale fiber $+$ gap (T-P4-2, T-P4-8) \\
Premises located at & trans-Planckian consistency & conformal geometry of light & accessible-energy axioms \\
Complex-number question & not addressed & posed at full strength & answered in calibrated halves \\
\bottomrule
\end{tabular}
\end{table}



\section{What is claimed, and in what regime}
\label{sec:claims}

Claimed, as theorems at the kinematic quasi-free level (T-P4-1 through
T-P4-7): the gauge phase is circle geometry; the state-space $i$ is
the polar phase of time, unique by positivity on the static arena; the
two coincide on charged sectors exactly under the
particle--antiparticle pairing; the discrete-symmetry table is
orientation bookkeeping; CP phases are lattice-locked loop holonomies;
the multipartite complex structure costs nothing beyond the shared
$J$; and positive frequency is holomorphy at the Hardy, tube, and
twistor levels. Claimed conditionally, on the standard scattering
hypotheses (T-P4-8): the interacting completion. Claimed at the
rigorous-model level (T-P4-9): the infrared door --- sectors are
ports, the derived $J$ sector-blind --- exact at the van Hove stratum
and theorem-grade at Nelson/Pauli--Fierz on imported hypotheses. Open
and named: the full-QED completion of the door (\S\ref{sec:infrared}),
now a conditional theorem on displayed dressing hypotheses; the
exact-Kerr Hadamard analysis and the full gravitating collapse case,
whose model-level cores are closed in the parent treatment
(\citealp{ArisakaA8QM}, entries R14--R15); and the nonlinear twistor
constructions beyond the free correspondence. The register throughout is the cohort's
calibrated one: derived where provable, conditional where conditional,
open where open.

\section{Twistor theory and \GGTheory{}: a structural comparison}
\label{sec:comparison}

The two programs are natural companions rather than rivals: twistor
theory supplies the deepest available mathematics of the massless
conformal sector, and \GGTheory{} supplies real sources for the
structures twistor theory takes as given. Table~\ref{tab:twistor-gg}
records the comparison aspect by aspect, in the calibrated register:
where the twistor program itself names an open hurdle, the table
says whether the present framework addresses it, inherits it, or
leaves it untouched. The duality of Part~\ref{part:bridge} --- the
$\PT^{+}/\PT^{-}$ division as the frequency splitting --- is the
mathematical seam along which the two programs meet.

\begin{table}[p]
\centering
\caption[Structural comparison: twistor theory and GG-Theory.]{Structural
comparison of the twistor program and \GGTheory{}, aspect by aspect,
with the seam (the holomorphy bridge, T-P4-6/7) marked. Twistor-column
entries follow the program's own literature; the last column names
where each \GG{} statement lives. Twelve aspects, from the origin of the complex numbers to experimental exposure. Where the twistor program names a hurdle of its own the row says so, and where this paper leaves a question open on both sides it says that too.}
\label{tab:twistor-gg}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{3.1cm}p{5.0cm}p{5.3cm}p{1.85cm}@{}}
\toprule
\textbf{Aspect} & \textbf{Twistor theory} & \textbf{\GGTheory{}} & \textbf{Where} \\
\midrule
Origin of $\bC$ & Primitive: holomorphy is constitutive of nature & Derived: $i$ = polar phase of the clock, geometrized by the circle; holomorphy = positivity at symmetry & T-P4-1--7 \\
Positive frequency & The division $\PT^{+}/\PT^{-}$; central and exact & The same division, identified with the causal-positive splitting of the derived $J$ --- the seam & T-P4-7 \\
Massless fields & Native habitat: $H^1(\PT^{\pm},\cO(-2h-2))$ & Explained as the conformal-enhancement case; the free massless sector closed & T-P4-7 \\
Massive fields & Long-standing hurdle (multi-twistor descriptions; conformal breaking) & Natural: the clock-level $J$ needs no conformal symmetry; mass enters via the scale fiber and the gap & T-P4-2, T-P4-8 \\
Why masslessness is special & A fact of the formalism & A theorem: only conformal fields complexify every timelike direction at once & T-P4-7(iii) \\
Interactions & Renaissance via amplitudes and the twistor string \citep{Witten2004}; full interacting theory open & Conditional completion by the asymptotic route (gap + completeness); infrared door proved at the model level & T-P4-8, T-P4-9 \\
Both helicities of gravity (``googly'') & The program's named central hurdle \citep{Penrose1976} & Not addressed here; the nonlinear constructions remain open on both sides (O-P4-1) & \S\ref{sec:claims} \\
Why four dimensions & Twistor geometry is 4D-specific; taken as given & Derived: the arena is forced to $4+1+1$ with a 4D boundary & \citealp{ArisakaA2GG6} \\
Quantum state reduction & Hoped-for connection to objective reduction; unrealized & Carried by a different mechanism (open-system projection, attractor descent), with a decidable experimental fork against OR & \citealp{ArisakaA8QM} \\
Time asymmetry & Motivating concern; not delivered by the formalism & Constitutional: the boundary law is a forward semigroup; the arrow is the scale fiber's & \citealp{ArisakaA5GDOS} \\
Conformal breaking ($m$, $\Lambda$) & External to the twistor ideal & Located: mass on the scale fiber; the residual dark-energy share in the closed inventory & \citealp{ArisakaA10Cosmos} \\
Experimental exposure & Indirect (via amplitudes) & Named falsifiers: shared-$i$ tests, lattice-locked phases, neutrino nature & \S\ref{sec:falsifiability} \\
\bottomrule
\end{tabular}
\end{table}

Three lines of the table deserve prose. \emph{The seam:} on positive
frequency the two programs state the same mathematics, and the only
difference is the direction of explanation --- which is exactly what a
duality should look like from its two ends. \emph{The massive row:}
what the twistor ideal experiences as breaking (mass ruins conformal
invariance) the present framework experiences as structure (mass lives
on the scale fiber, and the clock-level derivation never needed the
conformal group) --- so the two programs are strong on complementary
sectors, and the bridge says their strong sectors agree where they
overlap. \emph{The googly row:} honesty requires the table to show at
least one hurdle that neither side has cleared; the nonlinear
constructions are it, and objection O-P4-1 records the conjecture and
the burden.



\section{Programmatic verification}
\label{sec:verification}

Every symbolic claim of Parts~\ref{part:sources}--\ref{part:extensions}
has been verified programmatically in the parent program's
verification suite, in addition to the proofs above: the circle and
clock structures square to $-\mathbf{1}$ and agree exactly on
Feynman--St\"uckelberg-paired modes (symbolically, including generic
superpositions); the Jarlskog invariant of the CKM matrix is computed
with zero complex arithmetic --- real rotation blocks, conjugation as
block transpose --- matching the standard computation to machine
precision, with rephasing invariance realized as dial re-zeroing
invariance; the ground state of an arbitrarily coupled gapped
quadratic lattice carries the polar-phase $J$ of its interacting
generator to machine precision, and M\o{}ller operators transport $J$
across a defect-scattering event at the $10^{-3}$ level; the Hardy
extension of a positive-frequency signal satisfies the Cauchy--Riemann
equation numerically, recovers its boundary value, and exhibits the
$e^{+|\omega| s}$ blow-up on the negative-frequency component; and the
infrared overlap \eqref{eq:ir-overlap} vanishes as a power of the
cutoff over twelve decades while the coherent shift leaves $J$
invariant exactly; and the door theorem carries four dedicated checks
--- the cloud norm's logarithmic slope against the analytic
coefficient of Lemma~\ref{lem:ir-divergence}, the sector-overlap
power against its quadrature value, the dressed ground state's $J$
against the free $J$ while the sector label diverges, and the
conjugation \eqref{eq:dressed-generator} to machine floor. The suite accompanies the program's working records and is
available, together with the source, through the program's site
(www.arisaka-gg.org).

\section{The Platonic concession}
\label{sec:platonic}

One boundary is drawn deliberately. Everything above concerns nature's
use of complex numbers; none of it touches their standing as
mathematics. That the complex field is algebraically closed, that
complex analyticity is rigid and consequence-rich, that the shortest
paths between real truths so often pass through $\bC$ --- these facts
are exactly as remarkable after this preprint as before it, and a
mathematical Platonist is entitled to keep their wonder intact. The
claim here is narrower and, within its regime, final: nature never
needed the invention. The cosmos carries a circle, a clock, and an
orientation; the complex numbers are how a four-dimensional observer,
blind to the circle and embedded in the clock, compresses what it
cannot see. The chronology of Part~\ref{part:history} tells the same story in
dates: the compressed format was invented three centuries before the
thing compressed was found, and the felt magic of 1926 was that gap
coming due.

\section{Predictions for falsification}
\label{sec:falsifiability}

The account is exposed on real fronts. A network-Bell violation of
complex quantum mechanics itself --- or any demonstration that
independent sources can carry \emph{unshared} complex structures ---
would break the shared-source picture (T-P4-4). A measured drift of
the flavor CP phases away from their lattice values would retire the
holonomy account of CP violation (the parent program's falsifier,
\citealp{ArisakaA9SM}). An established Dirac nature for the light
neutrinos would retire the dial-free-door consequences downstream
(\citealp{ArisakaA10Cosmos}). And within mathematics, the bridge is
checkable line by line: its components are classical theorems, and
the identification of the twistor division with the frequency
splitting is either exact or wrong.


\section{Open questions}
\label{sec:openq}

Six items, in decreasing order of consequence. (1) \emph{The full-QED completion of the infrared door}
(\S\ref{sec:infrared}): T-P4-9 closes the door at the rigorous-model
level --- exactly at the van Hove stratum, on imported theorem-grade
hypotheses at Nelson/Pauli--Fierz; what remains is the top rung, a
rigorous dressing theory for full relativistic QED (the
Faddeev--Kulish program). This is an existence question about
dressed states that the field itself owns, and on which clauses (ii)
and (iii) of the door theorem wait verbatim. (2) \emph{The
nonlinear twistor constructions}: whether the nonlinear graviton and
the Ward correspondence \citep{Ward1977} are the integrable packaging of
the same two sources \citep{MasonWoodhouse1996} --- the conjecture recorded at O-P4-1, and the one place the
bridge has not yet reached. (3) \emph{The dynamics half of the bridge}: T-P4-7 sources the
\emph{division} of twistor space in causal positivity; the twistor
program's further magic --- that holomorphy \emph{generates}
dynamics, the contour integral of an arbitrary holomorphic function
solving the field equations automatically --- is exhibited
mechanically in Course~B (homogeneity and cohomology) but not yet
\emph{sourced}. Deriving the holomorphic calculus itself from the
real data is the sharpest Penrose-specific refinement this account
admits. (4) \emph{The exact-Kerr Hadamard analysis and the full gravitating
collapse case}: their model-level cores are closed in the parent
treatment (\citealp{ArisakaA8QM}, entries R14--R15) --- the rotating
anchor moves to the asymptotic frame with superradiance as a pumping
channel, and the collapse epoch factorizes into a universal thermal
channel times a decaying transient --- while the exact curved-space
existence theory remains the field's own. (5) \emph{The modular route}: whether
Tomita--Takesaki modular flow supplies the clock structure
intrinsically, tying the CPT operator to modular conjugation --- the
long-horizon formulation. (6) \emph{The typicality target upstream}: the parent program's
measurement-theory question --- whether deterministic basin-capture
frequencies coincide with the Born measure --- on which nothing in
this paper depends. It has since been closed at the relaxation level
in the parent treatment (\citealp{ArisakaA8QM}): the Born density is
the unique conserved real density, dial-blindness and record
stability force the transport, and the boundary law's own dissipation
--- a forward semigroup anchored on Haar by the compact fiber ---
relaxes any admissible initial configuration to the Born measure,
monotonically and with no statistical postulate remaining; what
remains are rate and representation estimates, and a
relic-non-equilibrium falsifier. (The finite-volume zero-mode corner is not on this list,
because it closes by Stone--von Neumann: finitely many modes admit one
representation up to unitary equivalence, so the zero-mode $J$-choice
is convention, not structure --- the harmless finite-dimensional
shadow of the infrared sectors.) None of these threatens the theorems
as stated; each bounds the regime the theorems own.

\section{The completed program and future prospects}
\label{sec:prospects}

Within the twenty-two-paper cohort, this paper completes the
complex-variables arc: the real-bulk series RB-1--RB-12 of
\citealp{ArisakaA8QM} supplies the statements, the Standard-Model and
cosmological consequences are carried by \citealp{ArisakaA9SM} and
\citealp{ArisakaA10Cosmos}, and the present paper is the standalone
account addressed outward. The natural next steps are the full-QED
completion of the infrared door, the nonlinear-twistor conjecture, and --- on the
expository side --- the trimmed account for a broader venue once the
full preprint has weathered its referees.

\section{Summary of the Discussion}
\label{sec:disc-summary}

One derived complex structure, two real sources, three symmetry
levels, twelve compared aspects, and a comparison with the two great
alternative roads that respects both: the case, in numbers. What is
proved is proved at the kinematic quasi-free level with a conditional
interacting completion; what is open is five named items; and what is
conceded --- the mathematics of $\bC$ itself --- is conceded once and
permanently.


\bigskip\noindent\textbf{Close of Part VII, in plain terms.} The case has been weighed: significant to five
constituencies, unique in the conjunction it fills, compared without
polemic to the twistor and string roads, verified mechanically,
fenced by named falsifiers, and bounded by five open questions ---
with the Platonic remainder conceded once and permanently. The
Discussion's register has been deliberate: claims graded, credits
paid, exposure listed. The next Part sets the ledger aside and speaks
plainly --- what was established, what it would mean, and why a
half-century-old question deserved this long an answer.

%% ===================================================================
\GGPart[Conclusions]{Conclusions}{What the paper has established; the objective limitations; a personal perspective; the vision of a century in which number systems are read as geometry; and what either verdict would mean}
\label{part:conclusions}

\section[Part~VIII in plain English]{Part~VIII in plain English: what has been established, and at what cost}
The claim, stated plainly: nature's complex numbers are not primitive. They are the compression of two real structures --- a compact phase circle and a causally oriented clock with a positivity clause --- and holomorphy is what those two structures look like at sufficient symmetry. The gauge phase and the quantum $i$ coincide for charged fields, which is why a century of quantum electrodynamics could not tell them apart.

The limitations are stated without cushioning, because they are the honest measure of the claim. The theorems are kinematic and quasi-free; the interacting extension is conditional on the standard scattering hypotheses; the infrared corner is closed at the model level and not for full quantum electrodynamics; the nonlinear twistor constructions are untouched by the bridge as proved; and the selection theorem leans on the staticity of the arena, whose gravitating generalization is inherited rather than owned. Each of the four carries a named next task.

What follows is a personal section and a longer view --- the suggestion that a century which learned to read geometry as physics might come to read number systems the same way --- and both are labeled as what they are rather than presented as results. The Part closes on what either verdict would mean, because a claim worth making is one that would cost something to be wrong about.

\section{What the paper has established}
\label{sec:established}

Strip the technical casing and four sentences remain. The complex
phase of every charged particle is a real angle on a real circle ---
a sixth dimension too small and too non-spatial ever to be looked
around --- and everything quantized about charge is that circle's
compactness. The $i$ of quantum mechanics is the phase of time
itself: the unique way of marking ``a quarter cycle ahead'' that
causality and positive energy permit, real in its construction,
identical for every mode of every field. Antiparticles exist because
those two facts must agree wherever charge lives. And holomorphy ---
the property Penrose called the magic of complex numbers, up to and
including twistor space --- is what the second fact looks like when a
massless field's full symmetry is allowed to act on it: the division
of twistor space into its two halves \emph{is} the division of
histories into forward and backward turning.

Behind the four sentences stand nine theorems, three courses, six
worked examples, and a verification suite; but the four sentences are
the paper, and it is worth saying why they carry philosophical weight
beyond their proofs. Each replaces a brute fact with a consistency
condition: charge quantization was a brute integer, and is now
compactness; the antiparticle was a brute doubling, and is now a
handshake; antiunitary time reversal was a brute table entry, and is
now the only reversal that touches what $i$ is made of; twistor
holomorphy was a brute miracle, and is now positivity at full
symmetry. Explanation in physics has always meant this trade ---
fewer brute facts, more consistency conditions --- and the complex
numbers, the oldest brute fact in the quantum formalism, have now
been traded.

\section{The objective limitations and the next technical tasks}
\label{sec:limitations}

Stated without cushioning: the theorems are kinematic and quasi-free,
with the interacting extension conditional on the standard scattering
hypotheses; the infrared corner of interacting massless theories is
closed at the rigorous-model level (T-P4-9: sectors are ports, the
derived $J$ sector-blind), with the full-QED completion named; the nonlinear
twistor constructions are untouched by the bridge as proved; and the
selection theorem leans on the arena's staticity, whose gravitating
generalization is inherited, not owned. Each limitation has a named
next task, and none of the four is softer than the frontier every
neighboring program faces on the same ground.

Each limitation carries its named next task, and they are worth
listing as a program rather than a disclaimer. The interacting
completion now carries its model-level infrared theorem (T-P4-9,
Nelson/Pauli--Fierz); what it wants next is a rigorous full-QED
dressing theory. The nonlinear twistor frontier
wants a single worked case: one self-dual solution exhibited as
integrable packaging of the two sources, or a proof that it cannot be.
The rotating anchor wants the asymptotic-frame formulation written
out against the Kay--Wald obstruction. And the staticity dependence
wants the gravitating generalization stated, even conditionally.
None of these is decorative: each is a place where this account could
break, and the paper's wager is that it will not. A program that
names its own failure modes has, at minimum, earned the right to be
tested rather than dismissed.

\section{A personal perspective}
\label{sec:personal}

I have spent fifty years on the experimental front line of particle
physics, and the number system in the theory columns of my logbooks
never once appeared in the data columns. Detectors report times,
counts, angles, energies: real numbers, every one. The amplitudes were
complex; the entries never were. For most of those decades I filed
this under bookkeeping and moved on --- until the question was put to
me in Penrose's voice, as a challenge that a geometric program ought
to meet head-on: if your bulk is real, where does the $i$ live? This
paper is the long answer. The short answer, which I offer to any
experimentalist who has wondered the same thing between shifts, is
that the $i$ lives exactly where the data said it did: nowhere. What
lives are a circle we cannot look around and a clock we cannot run
backward, and the complex plane is the shorthand four-dimensional
beings invented for the view.

One more sentence of the personal register, because students ask.
The strangest part of writing this paper was not the mathematics; it
was the repeated experience of finding the key already cut. Hamilton
had the pairs; Gabor had the signal; Penrose had the division;
Stueckelberg had the real space; the experimentalists had the shared
$i$. Nobody was wrong, and almost nothing here is technically new at
the component level --- what was missing was only the insistence, from
a program built on real geometry, that the components be laid on one
table and asked whether they were a single object. They were. Fifty
years of logging real numbers from detectors made me stubborn enough
to ask; the reader may judge whether the stubbornness paid.

\section{The vision: reading number systems as geometry}
\label{sec:vision}

The geometrization of physics has climbed a familiar ladder: space
(Euclid to Riemann), time (Minkowski), gravity (Einstein), gauge force
(Weyl to Yang--Mills), scale (Wilson to the holographic renormalization
group). The parent program's wager is that the ladder ends only when
everything the dynamics cannot avoid has been made geometry. This
paper proposes that the ladder's next rung was hiding in the number
system itself: that $\bC$ --- the one piece of apparatus every physics
course installs before the physics begins --- is also a geometrization
waiting to be recognized, the circle and the clock compressed into a
symbol. If that is right, then a century from now the phrase
``complex quantum amplitude'' may read the way ``luminiferous aether''
reads to us: not wrong, but a name for structure seen before it was
located. The mathematics will not change; the understanding of what
it was about will.

\section{The trailhead, not the summit: a final reflection}
\label{sec:trailhead}

A reader should leave this paper with proportions intact. What stands
proved here is a set of identifications --- exact, checkable, and (we
believe) new --- between structures that were each already understood
alone. What stands open is the harder half of every such program:
the full-QED infrared, the nonlinear, the gravitating. And what stands entirely
outside physics is the wonder that started the question: the
uncanny excellence of complex analysis as mathematics. We have argued
that nature never needed the invention; we have not argued, and would
not, that the invention is less than miraculous. The trail continues
uphill from here, in at least five marked directions; this paper is
the trailhead sign, not the summit cairn.

And the vision has a pedagogy. If this account settles, the
curriculum inverts: a student would first meet real oscillation pairs
and the dial, then be \emph{shown} the compression --- here is the
plane, here is the quarter-turn, here is why one symbol carries both
--- and only then be handed $e^{i\theta}$, the way phasors are handed
to engineers, as owned shorthand rather than as metaphysics. The C, P,
T table would be taught as orientation bookkeeping in an afternoon.
Twistor geometry would enter as the conformal completion of the
frequency dial rather than as an isolated miracle. None of this
changes a calculation; all of it changes what a student believes
physics is made of --- and a century of students who never mistake the
portrait for the sitter is the quiet, compounding payoff of getting a
foundation right.

\section{What it would mean: mathematics, nature, and the observer}
\label{sec:meaning}

If the account survives its referees --- the human ones, and the
experimental ones listed under falsification --- then something gentle
but permanent happens to the relationship between mathematics and
physics. The question ``why is mathematics so unreasonably effective?''
loses one of its sharpest examples and gains a case study: a piece of
mathematics invented for pure algebra turned out to be effective
because it was, unknowingly, a portrait --- of a circle nature carries
at every point and a clock nature carries along every worldline. The
effectiveness was not unreasonable; it was unrecognized. And the moral
runs in both directions: toward nature, whose inventory proves
leaner and more real than its notation suggested; and toward us, the
four-dimensional observers, whose compressions are so good that it
took four centuries to notice they were compressions. Physics is the
art of telling the portrait from the sitter. This paper has tried to
perform that art on the oldest portrait in the gallery --- and to
hand the twistor school, with respect and in its own language, the
sitter its geometry has been drawing all along.

\section{The last word: why complex numbers}
\label{sec:conclusion}

Why complex numbers? Because a real circle, too small and too
non-spatial ever to be looked around, turns the harmonic analysis of
charge into complex exponentials; because a real causal order, with
energy positivity, turns every history into a positive-frequency half
that is secretly the boundary of a holomorphic function; because the
two agree wherever charge lives, at the price of antiparticles; and
because a four-dimensional observer, meeting the compressed format
everywhere and its sources nowhere, concluded that nature runs on a
number system invented for cubics. It does not. Nature runs on angles,
orderings, and orientations --- all of them real --- and the complex
numbers are the shadow those cast on the boundary where we live. The
half of the magic that was physics is hereby returned to geometry;
the half that is mathematics is returned, with gratitude, to the
mathematicians --- and to the man who insisted for fifty years that
the question deserved an answer.


\bigskip\noindent\textbf{Close of Part VIII, in plain terms.} The summit, restated once without apparatus:
nature runs on angles, orderings, and orientations, all real; the
complex numbers are the shorthand four-dimensional observers invented
for the view; and the oldest magic in the quantum formalism is
geography. For the reader who wants to own the machinery rather than
admire it, the remaining Parts are the workshop: three courses with
problem sets, the reference dictionaries, an annotated guide to the
literature --- and then the forty-two sharpest attacks we could
assemble, each answered, because a claim of this size should leave
with its defense attached.

\GGAcknowledgments

%% ===================================================================
\GGPart[Technical Appendices]{Technical Appendices}{The reference dictionaries and a pocket course --- from the analytic signal to the Penrose transform}
\label{part:appendices}

\section{Summary of abbreviations}
\label{sec:abbrev}

\GG{} (geometry--gauge); \OGGEP{} (operational geometry--gauge
equivalence principle, the dictionary axiom AX-3); \GDOS{} (geometric
dynamics of open systems, the boundary law); \GRIP{} (the gauge-runtime
invariance pipeline, the actualization layer); F--S
(Feynman--St\"uckelberg); CR (Cauchy--Riemann); $\PT$, $\PT^{\pm}$,
$\mathbb{PN}$ (projective twistor space, its halves, its null
hypersurface); $H^2$ (Hardy space); T-P4-$n$ (this paper's theorems);
RB-$n$ (the real-bulk series of \citealp{ArisakaA8QM}).

\section{Glossary of key symbols}
\label{sec:symbols}

$\Stheta$: the compact phase circle (sixth dimension). $\Ir$: the
scale interval (fifth dimension). $J$: a complex structure
(Definition~\ref{def:J}); $J_Q = \partial_\theta/Q$ the circle's, $J_A$
the clock's polar phase, coinciding on shell (T-P4-3).
$\Htransform$: the Hilbert transform \eqref{eq:hilbert}. $\sigma$, $s$:
the symplectic (bosonic) and Euclidean (fermionic) forms.
$Z^\alpha = (\omega^A, \pi_{A'})$: a twistor; $s(Z)$ the helicity form
\eqref{eq:helicity-form}. $\mathcal{T}^{+}$: the forward tube
\eqref{eq:forward-tube}. $(a, b)$: the real pair of a weight-$n$
harmonic \eqref{eq:fibre-dictionary}.

\section{A pocket course: from the analytic signal to the Penrose transform}
\label{sec:pocket}

The bridge of Part~\ref{part:bridge} can be walked in five steps with
a pencil; this appendix is the walk, condensed, with every formula on
one page per step.

\emph{Step 1 (one mode).} Take $f = \cos\omega t$. Form
$g = \Htransform f = \sin\omega t$; check $\Htransform g = -f$. Define
$F = \tfrac12(f + ig) = \tfrac12 e^{i\omega t}$ and extend
$t \mapsto t + is$: decay upstairs, growth downstairs. You have
verified T-P4-6 on one mode and met the derived $J$.

\emph{Step 2 (all modes).} Superpose. The multiplier form
\eqref{eq:hilbert} makes $\Htransform$ frequency-diagonal, so Step~1
holds mode by mode; Paley--Wiener (Theorem~\ref{thm:pw}) upgrades the
picture to the exact statement: one-sided support $=$ Hardy class.

\emph{Step 3 (all clocks).} Replace $t$ by every timelike direction:
a positive-frequency field decays under $x \mapsto x + iy$ for every
future-timelike $y$, which is holomorphy in the forward tube
\eqref{eq:forward-tube}. This is where Lorentz invariance enters and
where massive fields stop.

\emph{Step 4 (light's symmetry).} For a massless field the conformal
group acts; the tube compactifies, and its geometry is exactly
$\PT^{+}$: check on the incidence relation \eqref{eq:incidence} that
complexifying $x$ into the forward tube moves the line $L_x$ into the
region $s(Z) > 0$. This is the content of T-P4-7(iii) in coordinates.

\emph{Step 5 (the transform).} Verify the contour integral
\eqref{eq:we2-contour} on the elementary state of Worked
example~II: two poles, one contour, one rational solution of the wave
equation --- and observe that keeping the poles separable is exactly
keeping the state on one side of $\mathbb{PN}$. Frequency sign has
become pole placement; you have crossed the bridge on foot.




\section{Course A: from the Hilbert transform to holomorphy}
\label{sec:courseA}

This appendix is a self-contained graduate course on the analytic core
of Parts~\ref{part:sources}--\ref{part:bridge}: six lectures, each with
a from-scratch derivation and a worked example. A student who
completes it owns every analytic tool the paper uses.

\subsection{Lecture A.1: the transform, three ways}
\label{ssec:A1}

The Hilbert transform admits three equivalent definitions, and each
carries a different intuition. \emph{As a principal-value integral},
\eqref{eq:hilbert}: a convolution against $1/\pi t$, the odd kernel
that weighs the past against the future symmetrically --- manifestly
real, manifestly non-local. \emph{As a Fourier multiplier},
$-i\,\mathrm{sgn}(\omega)$: frequency by frequency, a quarter-period
advance --- manifestly $J^2 = -\mathbf{1}$ away from $\omega = 0$.
\emph{As the conjugate-function operator}: the map sending the
boundary values of the real part of a holomorphic function to the
boundary values of its imaginary part --- manifestly the bridge's
germ. Equivalence of the three is the content of Lecture A.2.

\emph{Worked derivation: a classical pair.} Let
$f(t) = 1/(1 + t^2)$, the Poisson kernel at height one. Close the
principal-value contour in the upper half plane around the pole at
$t' = i$:
\begin{equation}
(\Htransform f)(t)
= \mathrm{p.v.}\,\frac{1}{\pi}\!\int \frac{dt'}{(1+t'^2)(t-t')}
= \frac{t}{1+t^2},
\label{eq:A1-pair}
\end{equation}
the conjugate Poisson kernel: the transform rotates the
``cosine-like'' bump into its ``sine-like'' partner, exactly as it
rotates $\cos \mapsto \sin$ mode by mode. Check the involution:
applying \eqref{eq:A1-pair} again returns $-1/(1+t^2)$, i.e.\
$\Htransform^2 = -\mathbf{1}$ on this pair. Observe what the pair
sums to: $f + i\Htransform f = 1/(1 - it) = i/(t + i)$ --- a function
with no singularity for $\mathrm{Im}\, t > 0$. Holomorphy has already
appeared, uninvited.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Compute $\Htransform$ of the Gaussian-windowed tone $e^{-t^2/2}\cos\omega_0 t$ in the large-$\omega_0$ limit, and show it approaches $e^{-t^2/2}\sin\omega_0 t$: the transform sees the carrier, not the envelope. \emph{(Hint: multiplier form; the window's spectrum is narrow.)}
\item Show from the multiplier form that $\Htransform$ commutes with translations and dilations of $t$ but anticommutes with the reflection $t \mapsto -t$. Interpret the anticommutation in the language of T-P4-5. \emph{(Answer: reflection reverses the clock's orientation, so it must reverse $J$.)}
\item Verify $\Htransform^2 = -\mathbf{1}$ directly on the pair \eqref{eq:A1-pair} by one more contour integral.
\end{enumerate}

\subsection{Lecture A.2: Hardy space and the Paley--Wiener theorem}
\label{ssec:A2}

Following the classical treatment \citep{Titchmarsh1948}, define the
Poisson and conjugate-Poisson kernels at height $s > 0$,
$P_s(t) = s/\pi(t^2 + s^2)$ and $Q_s(t) = t/\pi(t^2 + s^2)$, and for
real $f \in L^2$ set $F(t + is) = (P_s * f)(t) + i (Q_s * f)(t)$. Three
facts, each a short computation: (i) $F$ is holomorphic in the upper
half plane ($P_s + iQ_s = i/\pi(t + is)$ is, and convolution preserves
it); (ii) as $s \downarrow 0$, $P_s * f \to f$ and
$Q_s * f \to \Htransform f$ in $L^2$ --- so the transform \emph{is}
the boundary conjugate function; (iii) on the Fourier side,
$\widehat{P_s * f} = e^{-s|\omega|}\hat f$ and
$\widehat{F(\cdot + is)} = 2 e^{-s\omega}\hat f\,\chi_{\omega > 0}$
--- the extension keeps only the positive frequencies and damps them.
Reading (iii) in both directions is the Paley--Wiener theorem
(Theorem~\ref{thm:pw}): one-sided support iff Hardy-class extension;
and the splitting projector $\tfrac12(\mathbf{1} + i\Htransform)$ is
the boundary shadow of ``extend upward and keep what survives.''

\emph{Worked example.} Take $f = \cos\omega_0 t + \tfrac13
\cos 3\omega_0 t$ (a two-mode chord). Then $F(t+is) = \tfrac12
e^{i\omega_0(t+is)} + \tfrac16 e^{3i\omega_0(t+is)}$: each overtone
decays at its own rate $e^{-\omega s}$, the chord's envelope and
instantaneous phase are read off $|F|$ and $\arg F$, and the
higher-frequency component dies faster with height --- the upper half
plane acts as a frequency microscope, which is the analyst's picture
of why one-sided regularity encodes orientation.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Prove that $\tfrac12(\mathbf{1} + i\Htransform)$ is idempotent on zero-mean signals and identify its kernel and range. \emph{(Answer: negative- and positive-frequency subspaces respectively.)}
\item For $f = \cos\omega_0 t$, compute $\int |F(t+is)|^2 dt$ as a function of height $s$ and confirm the Hardy bound is saturated by monotone decay $e^{-2\omega_0 s}$.
\item A signal has spectrum supported in $[\omega_1, \omega_2]$, $0 < \omega_1 < \omega_2$. Show its extension is entire in a strip and estimate the width. \emph{(Hint: two-sided decay.)}
\end{enumerate}

\subsection{Lecture A.3: the polar phase, computed}
\label{ssec:A3}

Lemma~\ref{lem:polar} in coordinates. Take the smallest non-trivial
case: two unit-mass oscillators with stiffness matrix
$K = \begin{psmallmatrix} 2 & -1 \\ -1 & 2 \end{psmallmatrix}$
(coupled springs; eigenfrequencies $\omega_\pm = 1, \sqrt3$). The
generator on $(\varphi, \pi)$ data is
$A = \begin{psmallmatrix} 0 & \mathbf{1} \\ -K & 0
\end{psmallmatrix}$, and the polar phase evaluates in closed form:
\begin{equation}
J_A \;=\; A\,(-A^2)^{-1/2}
\;=\;
\begin{pmatrix} 0 & K^{-1/2} \\ -K^{1/2} & 0 \end{pmatrix},
\qquad
K^{1/2} = \tfrac{1}{2}\begin{pmatrix}
\sqrt3 + 1 & 1 - \sqrt3 \\ 1 - \sqrt3 & \sqrt3 + 1
\end{pmatrix}.
\label{eq:A3-J}
\end{equation}
Check by block multiplication: $J_A^2 = -\mathbf{1}$; symplectic
compatibility from $K^{\mathsf T} = K$; and $[J_A, e^{tA}] = 0$ since
both are functions of $A$. Notice what the off-diagonal blocks say
physically: ``multiply by $i$'' mixes position into momentum with the
matrix $K^{-1/2}$ --- each normal mode advanced a quarter of \emph{its
own} period. The exact ground state of this pair has covariance
$\Gamma = \tfrac12\,\mathrm{diag}(K^{-1/2}, K^{1/2})$, and
$2\Gamma\sigma = J_A$ on the nose: the vacuum knows the polar phase.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Repeat the closed-form computation \eqref{eq:A3-J} for the antisymmetric normal-mode pair $K = \begin{psmallmatrix} 2 & 1 \\ 1 & 2\end{psmallmatrix}$ and check that only the eigenvector labels move. \emph{(Moral: the polar phase depends on the spectral planes, not on your basis.)}
\item Show that for any $\alpha > 0$, $J_{(\alpha A)} = J_A$: the polar phase is scale-free. Which physical statement in \S\ref{sec:clock} does this prove? \emph{(Answer: one $i$ for all modes regardless of frequency.)}
\item Let $A$ have a zero mode. Show no dynamics-commuting complex structure exists on the full space, and identify the obstruction with O4 of the main text.
\end{enumerate}

\subsection{Lecture A.4: from $J$ to the Hilbert space}
\label{ssec:A4}

With $(V, \sigma, J)$ in hand, the one-particle inner product
\eqref{eq:one-particle-ip} turns the real solution space into a
complex Hilbert space with $i = J$. On the oscillator pair of A.3:
the normal-mode data $(q_\pm, p_\pm)$ give
$\langle f, f\rangle = \sum_\pm \tfrac{1}{2}\big(\omega_\pm q_\pm^2 +
p_\pm^2/\omega_\pm\big)$ --- the classical energy weighted by
$1/\omega$ mode by mode, i.e.\ the \emph{action}: the quantum norm is
the classical action in disguise, which is why $\hbar$ counts quanta.
Creation and annihilation appear as the $(\mp i)$-eigenvectors of $J$,
$a_\pm \propto \omega_\pm^{1/2} q_\pm + i\,\omega_\pm^{-1/2} p_\pm$:
the textbook combination, here \emph{derived} as ``the polar phase's
own coordinates'' rather than guessed.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Verify on the oscillator pair that $\langle f, g\rangle$ of \eqref{eq:one-particle-ip} is sesquilinear with respect to $J$: $\langle Jf, g\rangle = -i\langle f, g\rangle$ under the identification $i \equiv J$.
\item Show that the classical time-average of the energy in a single mode equals $\omega$ times the one-particle norm-squared: the norm counts quanta. \emph{(This is why $\hbar$ appears where it does.)}
\item Construct the $(\pm i)$-eigenvectors of $J_A$ for the coupled pair of A.3 explicitly and confirm they diagonalize the Hamiltonian.
\end{enumerate}

\subsection{Lecture A.5: fields, the tube, and the massive/massless contrast}
\label{ssec:A5}

Promote A.3--A.4 to the Klein--Gordon field: $V$ = real solutions,
$\sigma$ = \eqref{eq:sigma-KG}, $A$ = time translation,
$-A^2 = m^2 - \Delta$ --- gapped for $m > 0$, gapless for $m = 0$ but
with $\omega = |k| > 0$ almost everywhere. The polar phase is
multiplication by $-i\,\mathrm{sgn}$ on frequencies, i.e.\ the field
Hilbert transform; positive-frequency solutions extend to the forward
tube \eqref{eq:forward-tube} because
$e^{ik\cdot x}$ with $k \in \bar V^{+}$ decays under
$x \mapsto x + iy$, $y \in V^{+}$ ($k \cdot y > 0$). The contrast that
matters for Course B: for $m > 0$ the extension exists but the tube's
conformal compactification is unavailable (mass fixes a scale); for
$m = 0$ the conformal group acts, and the tube's geometry acquires the
twistor description. Same theorem, two destinies.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item For the massless field in a periodic box, list the modes on which $J$ is undefined and count them. \emph{(Answer: the single constant mode; measure zero in infinite volume.)}
\item Show $e^{ik\cdot x}$ with $k$ future-null decays in the forward tube but not in any wider domain: the tube is exactly the shared decay region.
\item Where does the argument use the mass gap? Re-run A.5 for $m > 0$ and identify the one step that fails at $m = 0$ in finite volume.
\end{enumerate}

\subsection{Lecture A.6: where the analysis can fail, quantified}
\label{ssec:A6}

Three failure modes, each with its number. \emph{Zero modes:}
$\mathrm{sgn}(0)$ is undefined; in finite volume the constant mode of
a massless field is a genuine degree of freedom needing separate
(classical/superselected) treatment. \emph{Non-stationarity:} with a
time-dependent generator the splitting at $t = -\infty$ and
$t = +\infty$ differ by a Bogoliubov map; in the one-mode case
$\begin{psmallmatrix} \cosh r & \sinh r \\ \sinh r & \cosh r
\end{psmallmatrix}$, the ``out'' vacuum contains
$\bar n = \sinh^2 r$ ``in'' quanta --- particle creation as the
measured mismatch of two polar phases, the physics behind Worked
example~IV. \emph{The interacting infrared:} the overlap exponent
\eqref{eq:ir-overlap} evaluates to
$(\epsilon/\Lambda)^{|\Delta c|^2/2}$; for $|\Delta c|^2 = 8$ and
twelve decades of cutoff removal the overlap falls by $10^{-48}$ ---
sectors, for every practical and every principled purpose. Each
failure mode bounds the theorems of the main text exactly where
Part~\ref{part:extensions} said it would.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item For the Bogoliubov block with $r = 1$, compute $\bar n$, the two-mode entanglement entropy $S = (\bar n{+}1)\ln(\bar n{+}1) - \bar n\ln\bar n$, and the KMS temperature of the reduced state.
\item Evaluate the infrared overlap \eqref{eq:ir-overlap} for $|\Delta c|^2 = 2$ and cutoffs $\epsilon/\Lambda = 10^{-3}, 10^{-6}, 10^{-9}$; plot mentally, conclude sectorially.
\item Derive \eqref{eq:dressed-generator}: show that
$W(d)^{\dagger}\,(\omega a^{*}a + g(a + a^{*}))\,W(d) = \omega a^{*}a - g^{2}/\omega$
at $d = -g/\omega$, and conclude with Lemma~\ref{lem:cloud} that the
sector's polar-phase $J$ is the free one. Then compute the
logarithmic slope of $\|f_{v}\|^{2}$ from \eqref{eq:ir-cloud} at
$v = 0.4$ and check it against $2\pi g^{2}/(1 - v^{2})$.
\item Show that a coherent shift composed with a Bogoliubov map is again affine-plus-symplectic, and that only the Bogoliubov factor can move $J$. \emph{(This separates Worked examples IV and the cloud lemma cleanly.)}
\end{enumerate}

\section{Course B: twistor geometry from scratch}
\label{sec:courseB}

Seven lectures from two-component spinors to the frequency theorem,
assuming only special relativity and linear algebra. The destination
is the one derivation this paper most wants its reader to own: that a
spacetime point pushed into the forward tube lands its twistor line in
$\PT^{+}$ --- the bridge, computed.

\subsection{Lecture B.1: two-component spinors}
\label{ssec:B1}

Represent a spacetime point by the Hermitian matrix
\begin{equation}
x^{AA'} \;=\; \frac{1}{\sqrt2}
\begin{pmatrix} t + z & x - iy \\ x + iy & t - z \end{pmatrix},
\qquad
\det x \;=\; \tfrac12\big(t^2 - x^2 - y^2 - z^2\big),
\label{eq:B1-x}
\end{equation}
so Lorentz transformations act as $x \mapsto \Lambda x
\Lambda^\dagger$ with $\Lambda \in SL(2,\bC)$, and null vectors are
exactly the rank-one Hermitian matrices $x^{AA'} =
\kappa^A \bar\kappa^{A'}$: a null direction \emph{is} a spinor up to
phase. This factorization is the engine of everything below --- light
is spinor-valued, which is why light's geometry will complexify so
gracefully.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Verify $\det x = \tfrac12(t^2 - x^2 - y^2 - z^2)$ from \eqref{eq:B1-x} and show $x \mapsto \Lambda x \Lambda^\dagger$ preserves it: $SL(2,\bC)$ double-covers the Lorentz group.
\item Write the null vector $(1,0,0,1)$ in the form $\kappa\bar\kappa$ and exhibit the phase freedom $\kappa \mapsto e^{i\phi}\kappa$. What does this phase become in Lecture B.3? \emph{(Answer: the little-group direction the helicity form measures.)}
\item Show that two spinors $\kappa, \mu$ with $\kappa_A\mu^A \neq 0$ span every spinor: the two-component calculus needs only antisymmetric contraction.
\end{enumerate}

\subsection{Lecture B.2: twistors and incidence}
\label{ssec:B2}

A twistor is a pair $Z^\alpha = (\omega^A, \pi_{A'})$; a point $x$ and
a twistor $Z$ are \emph{incident} when \eqref{eq:incidence} holds,
$\omega = i x \pi$. Fix $Z$ with $\pi \neq 0$: the incident real
points form a null geodesic with direction $\pi^{A}\bar\pi^{A'}$ ---
a light ray, with $\omega$ recording its moment arm. Fix $x$: the
incident twistors form a two-complex-dimensional subspace, the
projective line $L_x \subset \PT$. Points are lines; rays are points;
two points are null-separated iff their lines meet. The dictionary
inverts the habits of a lifetime, which is precisely its power: what
is local in $\PT$ is global in spacetime, and vice versa.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item From \eqref{eq:incidence}, derive that the incident real points of a fixed $Z$ (with $\pi \neq 0$) form the stated null ray, and compute its direction. \emph{(Hint: difference of two solutions.)}
\item Show two points $x, y$ are null-separated iff $L_x \cap L_y \neq \emptyset$, and express their separation vector via the intersection twistor.
\item What happens to $L_x$ as $x \to \infty$ along a timelike direction? \emph{(Answer: it approaches the line of the vertex twistor at infinity --- compactification made visible.)}
\end{enumerate}

\subsection{Lecture B.3: the helicity form, and what $s(Z)$ measures}
\label{ssec:B3}

The Hermitian form \eqref{eq:helicity-form} is the unique
$SU(2,2)$-invariant on twistor space, and its physical name is
earned: in the
conventions of \citet{PenroseRindler1986} (\S6.1), a one-particle
twistor carries momentum and angular momentum
\begin{equation}
P_{AA'} = \bar\pi_A\,\pi_{A'},
\qquad
M^{AB} = i\,\omega^{(A}\bar\pi^{B)},
\qquad
M^{A'B'} = -\,i\,\bar\omega^{(A'}\pi^{B')},
\label{eq:twistor-PM}
\end{equation}
and the Pauli--Luba\'nski (helicity) invariant built from them
evaluates to exactly $s(Z)$
\citep{PenroseRindler1986}. On incident pairs the form vanishes:
substituting $\omega = ix\pi$ with $x$ Hermitian gives
\begin{equation}
s(Z) \;=\; \tfrac12\big( i x^{AA'}\pi_{A'}\bar\pi_A
- i x^{AA'}\bar\pi_A \pi_{A'} \big) \;=\; 0,
\label{eq:B3-real}
\end{equation}
so real spacetime points live entirely on the null hypersurface
$\mathbb{PN}$ --- the ``real slice'' of twistor space is where
helicity-neutral incidence happens, and the two open halves
$\PT^{\pm}$ are, so far, unclaimed territory.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Verify \eqref{eq:B3-real} in full index detail, keeping track of which reality property of $x$ is used where.
\item Compute $s(Z)$ for $Z$ incident with the complex point $x + i\epsilon e_0$ ($e_0$ the unit time direction, $\epsilon > 0$ small) to first order in $\epsilon$, and check the sign against Lecture B.4.
\item Using the momentum expression $P_{AA'} = \bar\pi_A \pi_{A'}$, show a rescaling $\pi \mapsto \lambda\pi$ rescales the null momentum: projective twistors carry rays' worth of momenta, not momenta.
\end{enumerate}

\subsection{Lecture B.4: the derivation this course exists for}
\label{ssec:B4}

Now push the point into the forward tube: $x \mapsto x - iy$ with $y$
future-timelike Hermitian ($y = \lambda\lambda^\dagger$-decomposable
with positive trace). The incident twistors of the complexified point
have $\omega = i(x - iy)\pi$, and the helicity form no longer
vanishes:
\begin{equation}
s(Z) \;=\; \tfrac12\big(\omega^A\bar\pi_A + \bar\omega^{A'}\pi_{A'}\big)
\;=\; \tfrac12\big( i(x - iy) - i(x + iy) \big)^{AA'}\pi_{A'}\bar\pi_A
\;=\; y^{AA'}\,\pi_{A'}\bar\pi_A \;>\; 0,
\label{eq:B4-key}
\end{equation}
strictly, because $y^{AA'}\pi_{A'}\bar\pi_A$ is the Minkowski pairing
of the future-timelike $y$ with the future-null direction
$\pi\bar\pi$. \emph{The entire line $L_{x - iy}$ sits in $\PT^{+}$.}
One line of spinor algebra, and the bridge is crossed: forward-tube
analyticity (Course A's destination) is exactly ``twistor lines in the
positive half'' --- T-P4-7(iii), by hand. Push $x$ into the
\emph{backward} tube and the same computation lands the line in
$\PT^{-}$: the two halves of twistor space are the two frequency
orientations, made into geography.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Re-run the computation \eqref{eq:B4-key} for $y$ \emph{past}-timelike and for $y$ spacelike; identify where each lands. \emph{(Answer: $\PT^{-}$; and straddling both halves respectively.)}
\item Deduce from \eqref{eq:B4-key} alone that no line $L_z$ of a complexified point can lie in $\mathbb{PN}$ unless $z$ is real: the null hypersurface is exactly the real slice.
\item Combine B.4 with A.5 to give a two-line proof of T-P4-7(ii): a positive-frequency field restricted to a worldline is Hardy.
\end{enumerate}

\subsection{Lecture B.5: homogeneity, the contour integral, and the residue}
\label{ssec:B5}

Penrose's original transform: for helicity $0$,
\eqref{eq:we2-contour}, with $f$ holomorphic of homogeneity $-2$.
Execute it for the elementary state
$f(Z) = [(A_\alpha Z^\alpha)(B_\beta Z^\beta)]^{-1}$. Parameterize the
line $L_x$ by $\pi_{A'} = (1, \zeta)$; the two factors become linear
in $\zeta$, with roots $\zeta_A(x), \zeta_B(x)$; the measure
$\pi\,d\pi = d\zeta$; and the contour separating the roots picks up
one residue:
\begin{equation}
\varphi(x) \;=\; \frac{1}{2\pi i}\oint
\frac{d\zeta}{a(x)\,(\zeta - \zeta_A)(\zeta - \zeta_B)\,b(x)}
\;=\; \frac{1}{\rho_A(x)\,\rho_B(x)},
\label{eq:B5-residue}
\end{equation}
with $\rho_{A,B}$ the incidence factors --- a rational, everywhere-
defined-off-singularities solution of $\Box\varphi = 0$, produced by
\emph{contour topology alone}. Homogeneity $-2$ is what makes the
integrand a well-defined one-form on the projective line (count:
$-2$ from $f$, $+2$ from $\pi\,d\pi$); for helicity $h$ the same count
forces $-2h - 2$, which is the exponent in
\eqref{eq:penrose-transform}. Frequency, meanwhile, is the
\emph{placement} of $\zeta_A, \zeta_B$: a positive-frequency state is
one whose singularities can be kept off $\PT^{+}$, so that on every
$L_{x - iy}$ the contour exists --- Lecture B.4 again, now running the
transform.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Carry out the residue computation \eqref{eq:B5-residue} keeping all factors, for the specific dual twistors $A = (0, \iota)$, $B = (0, o)$ with $\iota, o$ a spinor basis, and identify the singularity surfaces of the resulting $\varphi$.
\item Run the homogeneity count for helicity $h = \pm 2$ and match the exponents to the linearized-graviton entries of \eqref{eq:penrose-transform}.
\item Show that deforming the contour across one pole changes $\varphi$ by a solution regular at that pole's incidence region --- the germ of the coboundary freedom of B.6.
\end{enumerate}

\subsection{Lecture B.6: why cohomology, in one page}
\label{ssec:B6}

The contour formula hides an ambiguity that is actually a structure:
$f$ is not a function on $\PT^{+}$ but a representative defined on an
overlap of patches, and adding to it anything holomorphic on a single
patch changes nothing (deform the contour off the added term's
singularity-free region and the integral of that piece vanishes).
Data on overlaps, modulo single-patch data: that is exactly a \v{C}ech
cohomology class in $H^1$, computed with a two-patch cover. The
mysterious-sounding statement ``massless fields are first cohomology''
is therefore the pedestrian statement ``twistor functions are
contour-integrands, and only their patch-overlap content is
physical'' --- the same logic by which only loop holonomies of a
texture are physical (T-P4-5), transplanted to complex geometry. The
rigorous form is \citet{EastwoodPenroseWells1981}; the two-patch
picture is enough for every use in this paper.


\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item With the two-patch cover $\{\zeta \neq \zeta_A\}$, $\{\zeta \neq \zeta_B\}$ of the line, exhibit the elementary state of B.5 as a \v{C}ech cocycle and one coboundary that leaves $\varphi$ unchanged.
\item Explain, in the holonomy language of T-P4-5, why $H^0$ carries no massless field: single-patch data are re-zeroable. \emph{(One sentence suffices.)}
\item For which helicities does homogeneity permit \emph{polynomial} cocycles, and what fields do they give? \emph{(Answer: the finite-type solutions; try $h = 0$, degree $-2$.)}
\end{enumerate}

\subsection{Lecture B.7: the dictionary, closed}
\label{ssec:B7}

Assemble the course. A massless field of helicity $h$ is a class in
$H^1(\PT^{\pm}, \cO(-2h-2))$ (B.5--B.6); its frequency orientation is
which half carries it (B.4); on any timelike worldline that
orientation restricts to the Hardy splitting (Course A); and the
generator of that splitting is the polar phase of the clock --- the
derived $J$ of the main text, whose other face, on charged sectors, is
the rotation generator of the phase circle. Read left to right, the
dictionary derives twistor geography from causal positivity; read
right to left, it equips the real bulk's two structures with the most
powerful complex-geometric calculus ever built for massless fields.
The direction of \emph{explanation} is the main text's claim; the
dictionary itself, either way round, is now yours.

%% ===================================================================

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Trace one electron through the full dictionary: its dial (T-P4-1), its clock phase (T-P4-2), their pairing (T-P4-3) --- and explain why, being massive, it stops at the tube level of the bridge.
\item State exactly which step of the course fails for the googly problem, and why the failure is about \emph{nonlinearity} rather than about frequency. \emph{(Answer: B.5--B.6 linearize; self-dual deformation does not.)}
\item Write a one-paragraph referee report on T-P4-7 from a twistor-school perspective: what must be checked, and where would you probe? \emph{(Model answer: objection O-P4-1 and its response.)}
\end{enumerate}

\section{Course C: the two real sources, at lecture depth}
\label{sec:courseC}

Courses A and B built the analysis and the geometry; this course
drills the paper's own theorems, at the same depth and with the same
apparatus. Six lectures, each ending in problems; a student who
finishes all three courses has reconstructed
Parts~\ref{part:sources}--\ref{part:bridge} unaided.

\subsection{Lecture C.1: the circle sector, drilled}
\label{ssec:C1c}

Work entirely on one weight-$n$ sector \eqref{eq:fibre-dictionary}.
Write the circle action, the covariant derivative, and the holonomy
in the real pair $(a, b)$; verify by direct block computation that
the horizontal lift acts as $\partial_\mu - inA_\mu$ on
$\psi = a + ib$ and as its conjugate on $\bar\psi$; and prove
T-P4-1(iii)'s uniqueness by finding the general solution of
$RJ = JR$ for all rotations $R$ --- a two-parameter family --- and
imposing $J^2 = -\mathbf{1}$. The physical drill: compute the
Aharonov--Bohm shift for a loop enclosing flux $\Phi$ as a pure dial
rotation, with no complex number appearing until the final
repackaging.

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Show a weight-$n$ and a weight-$m$ sector admit an invariant
\emph{bilinear} pairing iff $m = -n$, and identify it with the Dirac
pairing between a field and its conjugate. \emph{(This is charge
conservation as pure representation theory.)}
\item An instrument couples to $\cos\theta$ (it reads the absolute
dial). Show its interaction violates the fiber isometry and estimate
what its existence would do to charge superselection.
\item Derive the de Broglie relation of the parent program's
dictionary: a dial winding $n$ times per fiber circuit and $p/\hbar$
times per unit length has total angle $n\theta + p\cdot x/\hbar$;
identify both windings' conservation laws.
\end{enumerate}

\subsection{Lecture C.2: the clock's uniqueness, in full}
\label{ssec:C2c}

Lemma~\ref{lem:polar} asserted uniqueness; here is the complete
argument. Let $J'$ commute with the dynamics and be positive-energy.
Decompose the solution space into the spectral planes of $A$
(nondegenerate case first); commutation forces $J'$ to preserve each
plane; on a plane, the commutant's solutions of $J'^2 = -\mathbf{1}$
are the two quarter-turns; positivity of $\sigma(f, J'f)$ selects the
same orientation as $J_A$ on every plane, because both signs are
determined by the same inequality $\omega > 0$. Degenerate
frequencies: the commutant enlarges to a unitary group on the
degenerate block, but positivity still forces $J' = J_A$ on the
block's real form --- run the argument with a two-fold degeneracy to
see the mechanism. Conclude: uniqueness is not delicacy; it is the
rigidity of ``one orientation per plane, one inequality for all
planes.''

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Carry out the degenerate-block argument explicitly for two
modes of equal frequency.
\item Show that dropping positivity yields a $2^{\#\mathrm{planes}}$-fold
family of invariant complex structures, and compute the energy
spectrum's signature for one wrong choice. \emph{(Answer: a negative
one-particle branch --- the fermionic flipped-block demonstration of
the parent program, bosonically.)}
\item Prove that any two positive-energy choices for \emph{different}
static frames of the same flat arena agree. \emph{(Hint: they share
the frequency planes of the full Poincar\'e representation.)}
\end{enumerate}

\subsection{Lecture C.3: the handshake, drilled}
\label{ssec:C3c}

T-P4-3's computation \eqref{eq:handshake}, three more times: once
with general phases (mode $\cos(n\theta - Et + \varphi)$ --- the
handshake is phase-blind); once for the antiparticle assignment
(charge $-n$, positive frequency --- agreement restored by the signed
normalization); once for the forbidden assignment, isolating the
sign. Then the discrete table as a drill: apply $C$, $P$, $T$, $CP$,
$CPT$ to the general mode and tabulate which operations flip the
relative sign of $J_Q$ and $J_A$ --- reproducing T-P4-5's table from
raw substitutions.

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Show that demanding the handshake for \emph{two} charges $n_1,
n_2$ simultaneously forces the same frequency orientation for both:
one clock serves every sector, which is T-P4-4(i) in miniature.
\item A hypothetical ``anti-handshake'' theory pairs $+n$ with
negative frequency. Show it is unitarily equivalent to the standard
one by circle reversal, and identify which discrete symmetry
implements the equivalence.
\item Where exactly does the argument use compactness of the fiber?
\emph{(Answer: nowhere in the handshake itself --- but integrality of
$n$, hence superselection's discreteness, needs it.)}
\end{enumerate}

\subsection{Lecture C.4: the shared quotient, drilled}
\label{ssec:C4c}

Build \eqref{eq:tensor-quotient} by hand for two planes: write the
four-dimensional real tensor product, the two-dimensional
identification subspace, and the induced $J$ on the quotient; verify
$J^2 = -\mathbf{1}$ \emph{on the quotient} and its failure on the full
real product. Then redo Worked example~III (the Bell state) tracking
every identification, and compute the reduced covariance of one
plane --- mixed, as entanglement requires.

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item For \emph{unshared} structures $J_1 \neq J_2$, exhibit two
representatives of the ``same'' product state whose difference is not
in the identification span: the quotient is ill-defined, which is the
Renou pathology structurally.
\item Show the swap operator descends to the quotient iff the
structures are shared. \emph{(Indistinguishability needs one $i$.)}
\item Extend \eqref{eq:tensor-quotient} to three factors and verify
associativity of the construction.
\end{enumerate}

\subsection{Lecture C.5: textures and the pentagonal lattice}
\label{ssec:C5c}

A coupling texture as a $\U(1)$-connection on the flavor graph:
edges $|y_{ij}| R(\varphi_{ij})$, vertices with dials, re-zeroing as
gauge. Fix a spanning tree of the $3\times3$ bipartite graph and null
its edges; count the surviving loop angles ($(N-1)^2 = 4$); verify on
a random texture that the four plaquette angles are re-zeroing
invariant; then impose unitarity and watch the four collapse to one
--- the Jarlskog uniqueness, by hand (Worked example~V supplies the
block arithmetic). Close with the parent program's sharpening: with
one circle and the integer $L = 5$, first-order angles live on
$(\pi/5)\bZ$, and the physical quark phase is the two-step lattice
point $2\pi/5$ \citep{ArisakaA9SM}.

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Count loop angles for $N = 2$ and conclude CP conservation:
the Kobayashi--Maskawa counting, graph-theoretically.
\item Show that a texture whose loop angles all lie in $\{0, \pi\}$
can be re-zeroed to real entries: achirality $=$ realizability.
\item On the pentagonal lattice, list every achiral angle and verify
that $2\pi/5$ is the \emph{smallest} chiral one: nature's CP
violation is minimal on its own alphabet.
\end{enumerate}

\subsection{Lecture C.6: the doors --- winding arithmetic}
\label{ssec:C6c}

Lemma~\ref{lem:weights} as a calculus. Tabulate, for a fermion of
weight $n$: allowed bilinears ($\psi\bar\psi$ always; $\psi\psi$ iff
$n = 0$), allowed trilinears with fields of weights $m_1, m_2$
(iff $n + m_1 + m_2 = 0$), and the induced conservation laws. Verify
the Standard-Model instances: every Yukawa is fiber-neutral; the
Majorana door opens only at the singlet; $LHN$ is the unique portal
touching it linearly. Conclude with the arithmetic slogan of Worked
example~VI: on the circle, conservation laws are integers.

\paragraph{Problems.}
\begin{enumerate}[leftmargin=2em]
\item Classify all fiber-neutral \emph{quartic} couplings among
fields of weights $\{0, \pm\tfrac12, \pm1\}$ and match them to the
Standard-Model quartics.
\item A weight-$\tfrac12$ field is proposed. Show single-valuedness
on the circle forbids it outright, or forces a double cover ---
and connect the alternative to the spinor double cover of
Lecture B.1.
\item Prove: any global $\U(1)$ of a fiber-neutral Lagrangian that
acts by dial shifts is automatically gauged by the circle. \emph{(In
this framework there are no ungauged dial symmetries to break.)}
\end{enumerate}

\section{An annotated reading guide}
\label{sec:guide}

The bibliography, re-shelved as five reading paths, each in
recommended order with a one-line brief. The guide assumes the
graduate reader of the Preface.

\emph{Shelf 1 --- the analysis (Course A's companions).}
\citet{PaleyWiener1934}: the source of Theorem~\ref{thm:pw}; read
\S\S 1--8 for the half-plane theory. \citet{StreaterWightman1964}:
chapters 2--3 for tube analyticity --- the field-theoretic form of
everything in Course A; the PCT chapter connects to T-P4-5.

\emph{Shelf 2 --- the twistor school (Course B's companions).}
\citet{Penrose1967}: the founding paper; read for the geometry, not
the formalism. \citet{PenroseMacCallum1973}: the readable survey ---
the positive-frequency discussion in its \S 2 is the seam this paper
builds on. \citet{EastwoodPenroseWells1981}: the rigorous transform;
theorem statements first, proofs on the second pass.
\citet{PenroseRindler1986}: the reference volume for every spinor
convention used in Course B. \citet{Penrose1976}: the nonlinear
graviton --- the open frontier, and objection O-P4-1's context.
\citet{Witten2004}: the amplitudes renaissance; skim for how twistor
variables re-entered mainstream particle theory.

\emph{Shelf 3 --- the selection theorems.}
\citet{AshtekarMagnon1975} and \citet{Kay1978}: the uniqueness of the
positive-energy complex structure, imported at T-P4-2; Kay's paper is
the sharper statement, Ashtekar--Magnon the geometric one. Read after
Lecture C.2, whose argument they complete rigorously.

\emph{Shelf 4 --- the infrared and the interacting frontier.}
\citet{Haag1992}: chapter II.4 for Haag--Ruelle theory (T-P4-8's
engine); chapter VI for superselection --- the three-superselections
pattern of \S\ref{sec:superselections} in its original habitat.
\citet{Buchholz1977} and \citet{FaddeevKulish1970}: the two original
footholds of \S\ref{sec:infrared}, in that order.
\citet{Frohlich1973}: where the infraparticle began --- the
no-ground-state theorem the door theorem rides.
\citet{Pizzo2003} and \citet{BachmannDeckertPizzo2012}: the dressing
constructions, iterated and cutoff-free --- T-P4-9's middle rung.
\citet{ChenFrohlichPizzo2010}: scattering with the clouds on, the
strongest result under the door.
\citet{Buchholz1986}: the flux observables that make the sector label
physical.
\citet{WickWightmanWigner1952}: where superselection itself began ---
Proposition~\ref{prop:custody}'s first instance, in its original
algebraic dress.

\emph{Shelf 5 --- the experiments and the sources.}
\citet{AharonovBohm1959}: the phase made physical.
\citet{Renou2021}: the shared-$i$ mandate; read the supplementary
material for the network construction. \citet{Weyl1929} and
\citet{WuYang1975}: the gauge phase's birth and its bundle reading.
\citet{GreenSchwarz1984}: the string road's anchor, for the
three-roads comparison. \citet{Penrose2004}: the question itself,
chapters 4--5 and 33 --- best read twice, once before this paper and
once after. The \GGTheory{} cohort papers are ordered for this
purpose in \S\ref{sec:cohort}; the shortest useful spine is
\citealp{ArisakaA2GG6} (the arena), then \citealp{ArisakaA8QM} (the
real-bulk series), then \citealp{ArisakaA9SM} (the flavor
holonomies).

%% ===================================================================

\bigskip\noindent\textbf{Close of Part IX, in plain terms.} The workshop closes: two courses walked the
analysis and the geometry from scratch, a third drilled the paper's
own theorems, the dictionaries fixed the symbols, and the reading
guide arranged the literature into five ordered shelves. A student
who has worked through to here holds every tool the paper uses ---
which is exactly the right condition in which to meet the next Part:
the objections, forty-two of them, from seven communities, at full
strength.

%% ===================================================================
\GGPart[Common objections]{Common Objections and Responses}{Forty-one adversarial readings --- from the twistor school, quantum foundations, the mathematician, the particle theorist, the relativist and string theorist, the historian and philosopher, and the student --- each answered with pointers into the paper}
\label{part:objections}

A paper that claims to relocate the complex numbers of physics invites
--- and should invite --- attack from every community it touches. This
Part collects the forty-two sharpest adversarial readings we could
construct or have received, grouped by the reader most likely to raise
each, and answers each with explicit pointers to the theorem, worked
example, or named open item that carries the load. Where the honest
answer is ``not yet'' or ``not ours to claim,'' it is said in those
words. The five objections of the earlier editions are retained
verbatim within their categories.

\section{The twistor school}

\bigskip
\noindent\textbf{Objection 1 (O-P4-1) (the twistor referee, on the googly)}\par\nobreak
\noindent\textbf{``Holomorphy on $\PT$ is constitutive, not derived: the nonlinear graviton and the Ward correspondence are complex-geometric through and through, and your real story covers only the free correspondence.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Correct on scope, and stated in the text: the bridge as proved covers the free massless correspondence, where the $\PT^{+}/\PT^{-}$ division is demonstrably the frequency splitting. The nonlinear constructions are named as open. But note what the free-level result already forces: if the division of twistor space is the causal-positive splitting for every free field, then the complex structure the nonlinear constructions deform is itself sourced in real causal data, and the burden of ``constitutive'' shifts. The program's conjecture is that the nonlinear twistor geometry will prove to be the integrable packaging of the same two sources; its proof or refutation is a well-posed piece of mathematics, not a matter of taste.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-P4-2) (the twistor geometer, on novelty)}\par\nobreak
\noindent\textbf{``The correspondence between $\PT^{\pm}$ and frequency sign is in Penrose--MacCallum, page one. You have discovered our abstract.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The correspondence is theirs, cited as such at every use \citep{PenroseMacCallum1973}. What is claimed as new is not the correspondence but its \emph{source}: that the division is generated by a real operator (the polar phase of the constitutional clock, T-P4-2/T-P4-6) whose own credentials --- uniqueness from causality and positivity on a derived static arena --- come from axioms that never mention $\bC$. The twistor literature treats the division as a basic structure of a primitively complex space; this paper derives the splitting first, real-ly, and then identifies the twistor division with it (T-P4-7). Same seam, opposite direction of sewing --- and the direction is the paper.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-P4-3) (the twistor geometer, on metaphysics)}\par\nobreak
\noindent\textbf{``\,`Direction of explanation' is not a mathematical statement. The equivalences are symmetric; your arrow is taste.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The equivalences are symmetric; the \emph{axiomatic dependencies} are not, and that asymmetry is checkable. The real side (circle, clock, positivity) is derived in the parent program from locality, causality, and an equivalence principle, with no complex input at any stage (Part~\ref{part:gg}); the holomorphic side has, in its own literature, no derivation from anything --- it is the starting point. An arrow between A and B is earned when A has independent support and B follows; that is a logical relation, not an aesthetic one. What would reverse it: a derivation of the circle and the clock \emph{from} holomorphy, which we would read with genuine interest.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-P4-4) (the conformal skeptic)}\par\nobreak
\noindent\textbf{``Nature is not conformally invariant --- masses, $\Lambda$, running couplings. Your twistor level is a statement about an idealized world.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Agreed, and the architecture absorbs it: the bridge is a ladder, not a single rung (Figure~\ref{fig:bridge}). The Hardy and tube levels (T-P4-6, \S\ref{sec:tube}) hold for massive, non-conformal matter --- they need only the clock --- while the twistor level is exactly the statement of what \emph{more} is bought when conformal symmetry does hold, i.e.\ for the free massless sector. Where the twistor ideal experiences mass as breaking, this framework locates mass structurally (the scale fiber; T-P4-8's gap hypothesis), so the ladder's rungs are regimes, not approximations. The idealized world is a sector of this one, and the theorem says which.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-P4-5) (the twistor program historian)}\par\nobreak
\noindent\textbf{``Penrose has emphasized the positive-frequency property of $\PT^{+}$ for fifty years as evidence that holomorphy is fundamental. You cite his observation as your theorem.''}\par\nobreak
\begin{objqabody}
\emph{Response.} His observation, his emphasis, and his evidence --- all credited. The step he did not take, because his program reads the arrow the other way, is the one this paper takes: exhibiting a \emph{real, axiomatically derived generator} for the splitting and proving its uniqueness (T-P4-2, via \citealp{AshtekarMagnon1975, Kay1978}), so that the positive-frequency property becomes evidence that \emph{causal positivity} is fundamental and holomorphy its large-symmetry face. The facts are shared; the theorem T-P4-7(iii), identifying his division with our generator's splitting level by level, is the new object on the table.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-P4-6) (the ambitwistor/amplitudes practitioner)}\par\nobreak
\noindent\textbf{``Modern twistor methods live in momentum space and amplitudes, not in your position-space Hardy theory. Your bridge connects to 1977, not to 2026.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Partly fair. The bridge as proved addresses the classical correspondence, which is also where Penrose's \emph{question} lives; the amplitudes renaissance \citep{Witten2004} is engaged in the three-roads comparison as the twistor column's modern strength on interactions. But the connection is not antiquarian: momentum-twistor variables inherit exactly the positive-energy structures whose real sourcing is this paper's content, and T-P4-8's asymptotic route is the natural place where an amplitudes-level statement would attach. We record that extension as open and worth a specialist's attention --- ideally a skeptical one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-P4-7) (the twistor geometer, on the circle)}\par\nobreak
\noindent\textbf{``Twistor space has no sixth dimension and no phase circle. If your circle were fundamental, $\PT$ would know about it.''}\par\nobreak
\begin{objqabody}
\emph{Response.} $\PT$ does not need to know about the circle, because the two sources have divided labor (T-P4-3): the circle owns the \emph{gauge} half (charge, quantization, superselection), the clock owns the \emph{state-space} half --- and twistor geometry is the conformal packaging of the clock's half only. That is why twistor methods are at their best for uncharged structure (massless free fields, self-dual geometry) and need extra apparatus for charge. On the bridge's account this is not an accident of the formalism but a portrait of the source-split itself; the circle's fingerprints appear exactly where twistor constructions import gauge bundles by hand.
\end{objqabody}

\section{Quantum foundations}

\bigskip
\noindent\textbf{Objection 8 (O-P4-8) (the real-quantum referee)}\par\nobreak
\noindent\textbf{``Real quantum mechanics is experimentally dead. A `real-bulk' account is refuted before it starts.''}\par\nobreak
\begin{objqabody}
\emph{Response.} What is dead is real quantum mechanics with \emph{unshared} complex structures --- real tensor products of separately-real spaces \citep{Renou2021}. What this preprint derives is standard complex quantum mechanics, with one globally shared $J$ built from one shared circle and one shared causal order; the complex tensor product is recovered as the real one quotiented by the shared-$J$ identification. The experiment demands exactly the sharing this account supplies, and is cited here as its strongest external support, not as a threat survived.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-P4-9) (the operator theorist, after Stone)}\par\nobreak
\noindent\textbf{``Stone's theorem already ties $i$ to time: every strongly continuous unitary group is $e^{itH}$. You have dressed a 1932 theorem in six dimensions.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Stone's theorem \citep{Stone1932} \emph{presupposes} the complex Hilbert space and then relates its given $i$ to generators; it answers ``how do $i$, $t$, and $H$ interlock?'' inside complex quantum mechanics. This paper answers the prior question: why is the state space complex at all, and what selects its $J$? The construction runs on \emph{real} data (real solution space, real symplectic form, real generator), produces $J$ by polar decomposition, and proves it unique from positivity (T-P4-2) --- and only then does Stone's theorem become available, with its $i$ now a theorem rather than an axiom. The 1932 result is downstream scaffolding, gratefully used.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-P4-10) (the reconstruction theorist)}\par\nobreak
\noindent\textbf{``Operational reconstructions (Hardy; Chiribella--D'Ariano--Perinotti; Sol\`er) already single out complex QM from information axioms. Your geometry is redundant.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The reconstructions \citep{JordanEtAl1934, Soler1995, Hardy2001, ChiribellaEtAl2011} are convergent evidence, not competition --- but note what kind of answer they give: complex QM is selected \emph{among state-space theories} by axioms about information processing. They locate $\bC$'s necessity; they do not locate $\bC$ itself in the physical world. This paper's claim is of the second kind: here is the circle, here is the clock, here is the operator (\ref{eq:polar-phase}) built from them, and here is the experimental fingerprint of their being globally shared \citep{Renou2021}. A student can hold the reconstruction theorems in one hand and Part~\ref{part:sources} in the other; the first says ``it had to be $\bC$,'' the second says ``and this is what $\bC$ was.''
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-P4-11) (the curved-space field theorist)}\par\nobreak
\noindent\textbf{``One-particle structures, taming, Ashtekar--Magnon, Kay: this is Wald's textbook, chapter 4. What exactly is yours?''}\par\nobreak
\begin{objqabody}
\emph{Response.} The apparatus is the standard one \citep{Wald1994} and cited line by line; three things are ours. The \emph{arena}: standard theory selects $J$ \emph{given} a stationary spacetime; here staticity is a theorem of the axioms (Theorem~0), so the selection acquires constitutional standing rather than background-dependence. The \emph{pairing}: the coincidence of the clock's $J$ with the circle's on charged sectors, with antiparticles as the consistency condition (T-P4-3), has no textbook counterpart. And the \emph{assembly}: carrying the same derived $J$ through discrete symmetries, flavor holonomy, tensor structure, and the twistor division as one object is the paper. Imported bricks, new building --- with the bricks' provenance stamped on each.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-P4-12) (the algebraic field theorist)}\par\nobreak
\noindent\textbf{``Your three-superselections pattern is Haag's book, chapter VI. Superselection theory needs no circles.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The theory of superselection sectors is indeed the algebraists' (\citealp{Haag1992}, gratefully shelved in the reading guide); what this paper adds is a \emph{uniform geometric mechanism} for the three instances it meets: in each case (charge, horizon vacua, infrared clouds) the unobservable is a relative phase between sectors, and in each case the obstruction is an unreadability --- of the fiber angle, of the port assignment, of the cloud at infinity --- while the derived $J$ passes through untouched (\S\ref{sec:superselections}). The abstract theory says sectors happen; the geometric account says \emph{where}, and why the $i$ never falls into the cracks between them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-P4-13) (the foundations referee, on measurement)}\par\nobreak
\noindent\textbf{``Your `real records' rest on the parent program's attractor story of measurement --- a hidden-variable theory with an unproven typicality conjecture. Your arc is only as real as that.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The dependence is disclosed in \S\ref{sec:gg-grip}, conjecture named. But mark what this paper actually needs from the measurement layer: only that registered outputs are real quantities --- counts, angles, rates --- which is an operational fact independent of any interpretation, Copenhagen's included. The GRIP account supplies the parent program's mechanism for it; the arc of \S\ref{sec:gg-arc} survives with any mechanism, because no interpretation on the market registers a complex number. The typicality conjecture bounds the parent program's claims; it does not touch the theorem that the middleware's $i$ is derived.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-P4-14) (the information theorist)}\par\nobreak
\noindent\textbf{``Real, quaternionic, and complex QM differ operationally in known ways beyond network Bell tests. Why is one experiment your anchor?''}\par\nobreak
\begin{objqabody}
\emph{Response.} Because it is the cleanest \emph{exclusion}: \citet{Renou2021} turned the real-tensor-product alternative from a foundational preference into a falsified theory, which is the strongest statement experiment ever makes. The broader landscape (single-system real simulations exist; quaternionic composites over-constrain) is consistent with, and summarized by, the same structural point the quotient theorem makes (T-P4-4): what nature demands is not ``complex numbers'' but \emph{one shared} complex structure, which is exactly what two global real sources supply and what unshared real or over-rigid quaternionic structures cannot. The one experiment is the anchor because it tests precisely the sharing.
\end{objqabody}

\section{The mathematician}

\bigskip
\noindent\textbf{Objection 15 (O-P4-15) (the analyst, on rigor)}\par\nobreak
\noindent\textbf{``These are physicist's proofs: dense domains, self-adjointness, and convergence are waved through.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Guilty as charged, deliberately and with a map. The register is stated in \S\ref{sec:claims}: kinematic-level proofs, with the functional-analytic care delegated --- explicitly, at each use --- to the classical sources where it is done once and well (\citealp{PaleyWiener1934} for the Hardy theory; \citealp{Kay1978} for the selection theorem; \citealp{Haag1992} for scattering). Every claim that is ours reduces to finite-dimensional linear algebra (weights, quarter-turns, quotients) or to an identification between two rigorously established structures. A mathematician who wants the $\epsilon$'s will find each one in the cited page ranges of the reading guide; none of the paper's content lives in the gap.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-P4-16) (the analyst, on the transform)}\par\nobreak
\noindent\textbf{``$\Htransform^2 = -\mathbf{1}$ fails on constants; your $J$ is not a complex structure on all of $L^2$-adjacent spaces you use.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Correct, tracked, and physical: the zero-frequency kernel is excluded wherever $J_A$ is asserted (the gapped hypothesis of Lemma~\ref{lem:polar}; the measure-zero statement for massless fields in infinite volume; O4 in the ledger), and Lecture~A.6 quantifies the failure modes rather than hiding them. The exclusion is not a technical embarrassment but a physical classification: the modes on which the clock defines no orientation are exactly the non-dynamical ones (constants, finite-volume zero modes), which is why they surface in physics as superselected or classical. The domain bookkeeping and the ontology agree.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-P4-17) (the algebraist, on the quotient)}\par\nobreak
\noindent\textbf{``Equation \eqref{eq:tensor-quotient} is stated for algebraic tensor products; completions, continuity of the induced $J$, and infinite dimensions need proof.''}\par\nobreak
\begin{objqabody}
\emph{Response.} For the finite-mode statements this paper makes (Worked example~III; Lecture~C.4) the algebra suffices and is complete as given. The infinite-dimensional upgrade is standard once $J$ is orthogonal/symplectic and bounded --- which the polar phase is, by construction, away from the excluded kernel --- so the quotient extends to the Hilbert completion with the induced $J$ isometric. We assert nothing sharper than that, and the second-quantized statements ride on the standard Fock functor. A careful write-up belongs in a mathematical-physics venue's appendix; nothing conceptual awaits it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-P4-18) (the geometer, on cohomology)}\par\nobreak
\noindent\textbf{``\,`\v{C}ech in one page' with two patches is not sheaf cohomology; the transform's naturality, and everything beyond $H^1$ of a line, is elided.''}\par\nobreak
\begin{objqabody}
\emph{Response.} It is exactly as much cohomology as the paper uses, and the boundary is marked: Lecture B.6 computes on the two-patch cover of a projective line, which suffices for the elementary states and for the conceptual point (only overlap data are physical), while the rigorous, natural, any-cover formulation is cited where it lives \citep{EastwoodPenroseWells1981}. We would resist the suggestion that the one-page version misleads: it is the computation every twistor course actually does first, and its moral --- cocycles modulo coboundaries as the holonomy logic of T-P4-5 transplanted --- is the honest content of the machinery at this paper's resolution.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-P4-19) (the operator theorist, on polar decomposition)}\par\nobreak
\noindent\textbf{``$A(-A^2)^{-1/2}$ requires $A$ skew-adjoint with dense range on a specified space; `generator of the dynamics' is not a definition.''}\par\nobreak
\begin{objqabody}
\emph{Response.} For every use in this paper $A$ is either a finite real matrix (the worked examples; the lattice checks) or the standard Klein--Gordon/Dirac generator on Cauchy data, whose skew-adjointness and spectral calculus are textbook; the gapped hypothesis gives bounded invertibility of $|A|$, and Lecture~A.6/O4 fence the rest. The phrase in Lemma~\ref{lem:polar} is shorthand for exactly this class, stated in its preamble. The referee's demand is met by restriction, not by evasion: we claim the formula on the class where it is classical, and the physics never asks for more.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-P4-20) (the historian of mathematics)}\par\nobreak
\noindent\textbf{``Attributions are romantic: Paley--Wiener theorems predate and postdate 1934 in many forms; `Euler wound them around a circle' is potted history.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The survey register of Part~\ref{part:history} is signposted as such, the chronology table (Table~\ref{tab:chronology}) states its episodes at one-line resolution, and the load-bearing citations are to specific results, not eras. Where a name covers a family (Paley--Wiener), the theorem statement in the text (Theorem~\ref{thm:pw}) is the precise member used. Corrections to the historical color would be received gratefully and would change no theorem; the mathematics is anchored to the cited statements, not to the anecdotes.
\end{objqabody}

\section{The particle theorist}

\bigskip
\noindent\textbf{Objection 21 (O-P4-21) (the regime referee)}\par\nobreak
\noindent\textbf{``Your theorems are free-field theorems. The world is interacting, massless-infrared-infested, and gravitating; your $i$ is proved for a world that does not exist.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The extension ledger is in Part~\ref{part:extensions} and is exactly as strong as the field's own frontier: interacting gapped matter, conditionally closed by the asymptotic route; confined sectors, covered at the hadronic level; stationary horizons, discharged with thermality as the partial trace; the interacting massless infrared, closed at the rigorous-model level by the door theorem (T-P4-9): the sector label is instrument data, the soft cloud never touches $J$, and what remains open is the full-QED existence theory, inherited by name. No claim in this preprint outruns that ledger, and the ledger outruns, at present, any competing account of where the complex structure comes from --- including the account on which it comes from nowhere.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-P4-22) (the infrared referee)}\par\nobreak
\noindent\textbf{``Your door theorem is a theorem about models. The electron of the real world is an infraparticle of full QED: no sharp mass shell, velocity superselection, and a dressing construction (Faddeev--Kulish) that after half a century is still not rigorous. You have proved the easy rungs and declared the ladder climbed.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The ladder is declared, not climbed --- that is why
the theorem states it as a ladder. What T-P4-9 claims is scoped in its
own statement: exact at the van Hove stratum; at Nelson/Pauli--Fierz,
riding imported theorems (\citealp{Frohlich1973}; \citealp{Pizzo2003};
\citealp{BachmannDeckertPizzo2012}; \citealp{ChenFrohlichPizzo2010})
whose hypotheses are displayed; at full QED, a reading carried with
its status shown, inheriting the field's own open existence problem by
name. But notice what the objection is about: whether the dressed
states \emph{exist} at full rigor --- an existence question about
particles. It is not about where $i$ comes from. The two clauses that
answer this paper's question --- the sector label is a c-number port
datum, and the coherent dressing is affine, hence $J$-blind --- are
statements about coherent dressings as such, and they hold for any
dressing the full theory will ever construct, Faddeev--Kulish
included. If the QED frontier moves, the door theorem's regime moves
with it; nothing in this paper is hostage to the schedule.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-P4-23) (the QCD skeptic)}\par\nobreak
\noindent\textbf{``\,`The hadronic spectrum is gapped' does not make Haag--Ruelle applicable to QCD: confinement, color, and the absence of a rigorous construction all stand in the way.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Correct, and the claim is scoped accordingly: T-P4-8 is conditional on (H1)--(H3) \emph{for the physical spectrum}, and its QCD reading is a consistency observation --- the asymptotic objects instruments meet are gapped hadrons, so the hypotheses attach where the operational dictionary needs them --- not a claim that four-dimensional QCD has been rigorously constructed (it has not, for anyone). The honest summary: in the regime where any interacting field theory can currently be said to have a particle interpretation, the derived $J$ is its complex structure. The construction problem itself is the millennium-grade frontier, shared by all.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-P4-24) (the flavor phenomenologist)}\par\nobreak
\noindent\textbf{``$2\pi/5 = 72^\circ$, but the fits give $\delta \approx 66$--$69^\circ$. Your `lattice-locked' phase is off by several degrees.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The pentagonal value is the parent program's \emph{first-order} statement, quoted here as such, with the quantitative-completion step and its comparison to data carried in \citealp{ArisakaA9SM} --- where the residual is treated within the program's stated theory-uncertainty model, and where the falsifier is registered: a measured drift outside the lattice's completion window retires the account. This paper's own use of the lattice is structural (T-P4-5(iv): angles live on $(\pi/5)\bZ$ at first order, so CP violation is not a continuous dial), which the few-degree completion question does not touch. Precision belongs to the parent paper; the discreteness claim is what is staked here.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 25 (O-P4-25) (the flavor theorist, on novelty)}\par\nobreak
\noindent\textbf{``Rephasing invariants and the Jarlskog construction are forty years old. `Flavor holonomy' renames known lore.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The invariants are classical; two things are not. First, the \emph{realization}: rephasing as literal dial re-zeroing on a physical circle, so that Jarlskog invariance is holonomy invariance in the same sense as Aharonov--Bohm --- a statement with content, since it makes CP violation a chirality of real texture and $CP$ itself an orientation reversal (T-P4-5), predictions of structure rather than notation. Second, the \emph{alphabet}: with one circle and $L = 5$, the loop angles are lattice-valued, which the classical lore cannot express because it has no geometry for the phases to be locked \emph{to}. Worked example~V lets a skeptic run the whole computation with no $i$ and judge whether ``renames'' survives the experience.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-P4-26) (the collider pragmatist)}\par\nobreak
\noindent\textbf{``No cross sections, no spectra, no discovery reach. What measurement does this paper change?''}\par\nobreak
\begin{objqabody}
\emph{Response.} None --- by design, and the design is stated (\S\ref{sec:claims}): this paper reorganizes the \emph{foundations} of structures whose phenomenology lives elsewhere in the cohort (the lock cluster in \citealp{ArisakaP1KK}; the flavor numbers in \citealp{ArisakaA9SM}; the cosmology in \citealp{ArisakaA10Cosmos}). Its falsifiable exposure is listed in \S\ref{sec:falsifiability} and is real but indirect: shared-$i$ tests, lattice drift, neutrino nature. A reader who counts only cross sections should read the cohort papers; a reader who has ever wondered why the Lagrangian is complex will find here the only answer on offer that could, in principle, have been false.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-P4-27) (the effective-field theorist)}\par\nobreak
\noindent\textbf{``Below any cutoff, I write the most general complex Lagrangian and match. Your real structures are invisible in the EFT, hence physically empty.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Two of the real structures are visible \emph{inside} the EFT precisely as its unexplained rigidities: the integrality of charge (why the EFT's $\U(1)$ representations are quantized at all) and the discreteness of CP phases on the pentagonal completion (why one coupling is not a free dial). A third is visible at the EFT's edge: the shared-$i$ constraint that network experiments test. EFT is the right tool for dynamics below a cutoff and the wrong tool for the question this paper asks, which is why the question survived eighty years of EFT competence untouched: matching coefficients cannot tell you what the number system is \emph{made of}.
\end{objqabody}

\section{The relativist and the string theorist}

\bigskip
\noindent\textbf{Objection 28 (O-P4-28) (the string theorist, on fairness)}\par\nobreak
\noindent\textbf{``Your three-roads table understates us: worldsheet holomorphy is not `primitive' --- it emerges from Polyakov quantization; and strings derive far more of physics than either other road.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The table's entry is scoped to this paper's single question --- where $\bC$ enters --- and on that question the worldsheet's complex structure arrives with the Riemann-surface formulation of the path integral, however motivated: it is not derived from a causal-real substrate within the program, which is all ``primitive'' means in that column. On breadth we defer without irony: the comparison paragraph and \citet{GreenSchwarz1984} record the string road's structural rigidity on its own terms, and the parent program's fuller comparison (\citealp{ArisakaA2GG6}, Remark 0.4) frames the two economies as premises placed at opposite ends of the energy axis, with the reader left to weigh them. No adjudication is made here beyond the one question asked.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-P4-29) (the relativist, on non-spatial dimensions)}\par\nobreak
\noindent\textbf{``A `dimension' with no metric role in propagation is a fancy name for an internal space. Kaluza--Klein plus fiber bundles already said everything your circle says.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The circle \emph{is} a principal fiber --- that is T-P4-1's language --- and the bundle formulation \citep{WuYang1975} is its acknowledged precedent. The added content is twofold. Constitutionally: the fiber is \emph{forced} (Theorem~0's two-sided count), not chosen, and carries the ledger's arithmetic (\S\ref{sec:gg-ledger}). Physically: unlike Kaluza--Klein's spatial circle, it is unreadable in principle (no light cone along it), which converts ``internal'' from a modeling word into a theorem-grade property with consequences --- superselection, the Born form's marginalization, the absence of angular satellites at the lock. An internal space that must exist and cannot be looked at is not merely renamed; it is located.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-P4-30) (the relativist, on rotation and collapse)}\par\nobreak
\noindent\textbf{``Kerr has no global timelike Killing field and collapse is not stationary. Your clock-selection theorem dies exactly where gravity gets interesting.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Named, fenced, and partially discharged rather than hidden: the stationary-exterior case is settled with the vacuum multiplicity as port assignments and thermality as the partial trace (\S\ref{sec:superselections}; Worked example~IV); the Kerr anchor and the collapse-epoch factorization are closed at the model level in the parent treatment (\citealp{ArisakaA8QM}, entries R14--R15): the rotating-frame selection fails exactly where the Kay--Wald obstruction \citep{KayWald1991} says it must --- physics, not bookkeeping --- so the anchor moves to the asymptotic frame and superradiance becomes a symplectic pumping channel, while the collapse epoch factorizes into a universal thermal channel times a decaying transient; what stays open, and is listed in \S\ref{sec:openq}, is the exact curved-space existence theory. Meanwhile the half of $J$ gravity cannot touch is a theorem: stage curvature preserves the fiber block, so the circle's structure --- and everything gauge about $\bC$ --- is immune. The interesting region is open on the clock's side only, and exactly as open for every neighboring program.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-P4-31) (the quantum-gravity theorist)}\par\nobreak
\noindent\textbf{``No graviton, no quantized geometry, no Planck-scale statement: a paper about the deep structure of physics that avoids quantum gravity.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Avoidance is the parent program's architecture, inherited knowingly: geometry there is not quantized --- the equivalence principle is generalized instead (AX-2), quantization enters at the boundary dictionary (AX-3), and the cohort's gravity-sector statements live in their own papers. This paper's question is well-posed and answerable \emph{within} that architecture, and its answer would survive translation: any future account in which causal order, positivity, and a compact gauge fiber persist at the fundamental level will inherit the derivation of $J$ wholesale. If quantum gravity abolishes those three, it abolishes far more than this paper.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-P4-32) (the cosmologist)}\par\nobreak
\noindent\textbf{``Your `globally static arena' is falsified by the sky: the universe expands.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The static metric of Theorem~0 is the constitutional \emph{bulk} background; observed expansion is carried, in the parent program, by the boundary/effective description whose cosmology is \citealp{ArisakaA10Cosmos}'s subject --- and the interface between the canonical bulk $J$ and observer-adapted cosmological dictionaries is exactly the port structure of \S\ref{sec:gg-gdos}, with cosmological particle creation as its channel form. What this paper needs from staticity is the uniqueness clause of T-P4-2 at the constitutional level, not a claim that comoving observers see a static sky; the distinction is the same bulk/boundary split the whole cohort runs on.
\end{objqabody}

\section{The historian and the philosopher}

\bigskip
\noindent\textbf{Objection 33 (O-P4-33) (the historian referee)}\par\nobreak
\noindent\textbf{``Paley--Wiener, tube analyticity, and the Penrose transform are all classical. You have written a review and called it a derivation.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The components are classical and cited as such; the claims that are new are identifications, and identifications are where explanations live. That the Hilbert transform --- selected by causality and positivity as the unique invariant complex structure on a static arena --- is the \emph{same} operator that generates the Hardy splitting; that the twistor division restricts on every timelike line to exactly that splitting; that the antiparticle pairing is the consistency condition between this $J$ and the gauge circle's: none of these is in the classical literature as a statement about where the complex structure of physics comes from. A review arranges known results; this argument uses them to reverse a direction of explanation that has stood since 1926.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-P4-34) (the Platonist referee)}\par\nobreak
\noindent\textbf{``You have moved the magic, not dissolved it. The circle's harmonic analysis is beautiful complex mathematics; the Platonic mystery stands.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Agreed, and conceded in \S\ref{sec:platonic} without reservation. The preprint's claim is about nature's inventory, not mathematics' worth: no complex primitive is instantiated; the structures instantiated are a circle, a clock, and orientations; the complex numbers are the observer's compression of these. Why that compression is itself a mathematical object of extraordinary depth is a question about the Platonic world, and the author declines to pretend that a physics theorem answers it. Half the magic was physics; that half is hereby geometry. The other half was never ours to dissolve.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-P4-35) (the philosopher, after Hamilton)}\par\nobreak
\noindent\textbf{``Hamilton reduced $\bC$ to ordered pairs in 1835. Your `real presentation' adds physics-flavored decoration to a done deed.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Hamilton showed the reduction is \emph{possible}; the chronology section says so and salutes him. What he could not supply --- what no one could, before there was a physics of the pairing rule --- is the \emph{selection}: which quarter-turn, why shared, why unique. Those are exactly the theorems here (T-P4-1's commutant, T-P4-2's positivity, T-P4-4's sharing), and they are where the physical content lives: an unselected real presentation is notation, a selected one is a discovery about the world. The paper's one-line relation to 1835 is stated in Part~\ref{part:history}: this is the physics that was missing from Hamilton's algebra.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 36 (O-P4-36) (the philosopher of science, on underdetermination)}\par\nobreak
\noindent\textbf{``Formulations are cheap: real, complex, quaternionic presentations of QM all exist. Choosing the real one and calling it ontology is underdetermined.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Mere reformulation would be; this is not that, on two counts. Empirically, the presentations are \emph{not} equivalent at the composite level: the unshared-real alternative is falsified \citep{Renou2021}, and the quaternionic route over-constrains composites --- the data thin the herd. Structurally, the claim is not ``the real presentation is prettier'' but ``the complex structure is a \emph{derived object} with real sources and a uniqueness theorem,'' which is a statement about explanatory architecture, testable by its consequences (antiparticle pairing, discrete-symmetry table, lattice phases) rather than by taste. Underdetermination applies to distinctions without consequences; \S\ref{sec:falsifiability} lists the consequences.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-P4-37) (the philosopher of mind, on the compression thesis)}\par\nobreak
\noindent\textbf{``\,`Humans invented $\bC$ because we are 4D-blind' is unfalsifiable psychology dressed as physics.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The thesis is offered as history's explanation, not as a theorem, and it is flagged as such (\S\ref{sec:platonic}); the theorems are the compression \emph{facts} it narrates: fiber-blind observers must package weight pairs (T-P4-1), tube-embedded observers must package frequency pairs (T-P4-6), and both packagings are unique. The psychological gloss is as falsifiable as counterfactual history ever is --- which is to say, it is an interpretation --- but it is disciplined by the mathematics: had the sources been visible, the compressed notation would have been optional, as phasors are. We commend the objection's standard and accept its cost: one paragraph of the paper is narrative, and says so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-P4-38) (the Wigner scholar)}\par\nobreak
\noindent\textbf{``Dissolving `unreasonable effectiveness' via one case study is cheap; Wigner's puzzle was about the whole pattern of mathematics' success.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Agreed, and the conclusion claims only the case: \emph{one} of the pattern's sharpest instances --- the algebraists' invention running the quantum world --- converts, on this account, from ``unreasonable'' to ``unrecognized'' (\S\ref{sec:meaning}). Whether the whole pattern so converts is the parent program's more ambitious wager, engaged in \citealp{ArisakaA11Origin} alongside Wigner, Tegmark, and Penrose's three worlds, and deliberately not spent here. A pattern is dissolved one load-bearing instance at a time; we submit this as the instance, and leave the pattern to the paper whose job it is.
\end{objqabody}

\section{The student}

\bigskip
\noindent\textbf{Objection 39 (O-P4-39) (the graduate student, practically)}\par\nobreak
\noindent\textbf{``If complex numbers are `only' compression, why does every computation use them? Should I stop?''}\par\nobreak
\begin{objqabody}
\emph{Response.} Certainly not --- compression is what good notation \emph{is}. You should keep using $\bC$ exactly as you keep using center-of-mass coordinates: because it is the optimal encoding of two real structures you would otherwise carry by hand. What the paper changes is not your calculations but your answers to ``what am I doing when I write $e^{i\theta}$?'' --- and, occasionally, your reach: the real presentation is the right frame precisely where the standard one mystifies (why antiparticles, why antiunitary $T$, why superselection, why $\PT^{\pm}$), as the worked examples are designed to show. Use the shorthand; own the longhand.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-P4-40) (the student, on predictions)}\par\nobreak
\noindent\textbf{``Does any experimental number come out different? If not, isn't this all interpretation?''}\par\nobreak
\begin{objqabody}
\emph{Response.} Within established quantum mechanics, no number changes --- deliberately: a derivation of the formalism that altered its predictions would be a refutation, not an explanation. The paper's exposure is at the edges (\S\ref{sec:falsifiability}): shared-$i$ network tests, the pentagonal completion window of the CP phases, the Majorana question --- plus one retroactive prediction of the strongest kind: that the real-Hilbert-space alternative \emph{had} to fail, for the stated geometric reason, before the experiment was run. ``Interpretation'' is the wrong shelf: this is a derivation with named ways to be wrong, which is the definition of physics done to foundations.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-P4-41) (the student, on authority)}\par\nobreak
\noindent\textbf{``My textbooks say the complex Hilbert space is a postulate. You are one preprint. Why believe you?''}\par\nobreak
\begin{objqabody}
\emph{Response.} Do not believe; check --- the paper is built for it. Every theorem reduces to computations you can do (the courses and problem sets exist for exactly this), every import is cited to a page range, and the verification suite re-derives each symbolic claim mechanically. Your textbook is right that the standard formulation postulates $\cH$ \citep{vonNeumann1932}; the question this paper adds is whether the postulate has a source, and its answer is a chain you can audit link by link in a week. Authority is the wrong tool on both sides; Lecture~C.2 and a rainy afternoon are the right ones.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-P4-42) (the student, on Penrose)}\par\nobreak
\noindent\textbf{``What do you expect Penrose himself would say?''}\par\nobreak
\begin{objqabody}
\emph{Response.} We would not presume to script him; the paper is arranged for his scrutiny rather than his applause. Our expectation, stated plainly: the two-sources split, the antiparticle handshake, and the identification of $\PT^{\pm}$ with a derived splitting would engage him, because two of the three are built from his own theorems; the direction of explanation he would contest, as objection 3 records; the googly row he would note first, as objection 1 concedes; and the Platonic remainder he would keep, with our agreement, as objection 33 grants. If the paper earns from him one sentence of the form ``the identification is correct; the reading is not mine,'' it will have done its work: the mathematics shared, the question finally joined.
\end{objqabody}

\bigskip
\noindent\emph{Closing note.} These forty-two objections are the strongest we could assemble against a paper whose thesis invites them from seven directions at once. The pattern of the answers is the paper in miniature: what is proved is pointed to; what is imported is credited; what is open is named (the googly, the full-QED infrared, the Kerr anchor, the typicality conjecture upstream); and what is conceded --- the Platonic wonder --- is conceded without regret. A reader who finds an objection missing from this list is invited to send it; the list has grown by exactly that mechanism. What remains is the paper's foundation in the literature: the references, in two groups, each with a note on what it established and how it is used here.

%% ===================================================================
\GGPart[References]{References}{Cited works, each with a relevance note}
\label{part:references}

The references are organized in two groups: the \GGTheory{} cohort
papers, among which this preprint stands in the Foundations Branch,
and all other works. Every entry carries a relevance note naming the section where it
is first used.

This paper cites its cohort siblings in completed tense: each result is
imported as established in its home paper, with hypotheses restated
here where they bear weight.

The second group is deliberately short: the classical mathematics on
which the bridge stands, the twistor school's own theorems, and the
experimental anchor.

\renewcommand{\bibsection}{\vspace{1.2em}}

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}

\begin{thebibliography}{99}
\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{K.~Arisaka.}
\emph{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 constitutional foundation of the entire program: four axioms and seven postulates from which everything downstream is derived. This paper draws on it constantly and specifically --- AX-1 supplies the causal time orientation, AX-2 the real composite connection whose complex-free character Part~\ref{part:gg} verifies clause by clause, AX-3 the operational dictionary whose positivity clause selects the sign of the derived $J$, and AX-4 the closure that forbids unforced structure. The axioms are rendered at full resolution in \S\ref{sec:gg-axioms}.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{K.~Arisaka.}
\emph{\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 arena paper: Theorem~0 derives the six-dimensional bulk $\cM_4 \times \Ir \times \Stheta$ with its globally static metric from AX-1 and AX-2 alone, establishing both of this paper's real sources --- the compact phase circle and the constitutional clock --- as forced geometry rather than postulated scaffolding. Its six-step proof skeleton, its non-spatiality classification by generators, and its two-sided minimality are summarized in \S\ref{sec:gg-theorem0} and used by every theorem of Parts~\ref{part:sources}--\ref{part:bridge}.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{K.~Arisaka.}
\emph{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 ledger paper: proves that the five integers $(M,N,L;J,K) = (1,3,5;7,35)$ are forced by the anomaly-closure architecture, each with a primary structural meaning. Two entries do direct work here: $M = 1$, the unique tensor seam, is the winding unit that makes charge quantization integral and superselection discrete (T-P4-1); and $L = 5$, the defining unified block, is the pentagonal alphabet on which the CP-violating loop holonomies of T-P4-5 are lattice-locked. Summarized in \S\ref{sec:gg-ledger}.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{K.~Arisaka.}
\emph{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 lineage paper: traces the geometrization program from Euclid through Riemann, Minkowski, Einstein, Weyl, Kaluza--Klein, and holographic renormalization to the present framework, establishing the historical pattern --- a compulsory structure recognized as geometry and absorbed --- of which this paper's move, returning the complex numbers themselves to geometry, is offered as the latest instance. The historical Part of this paper is a complex-numbers-specific companion to its broader narrative.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{K.~Arisaka.}
\emph{\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-law paper: constructs the operational dictionary demanded by AX-3 and derives the open-system runtime --- the forced projection, its always-on character, the exact memory-kernel reduction, complete positivity, the GKLS generator, and the seven textbook quantum postulates as theorems. This paper uses its architecture at every boundary: the port structure that houses the vacuum multiplicity and the infrared sectors (\S\ref{sec:infrared}, \S\ref{sec:superselections}), and the two-fibers-two-halves split summarized in \S\ref{sec:gg-gdos}.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{K.~Arisaka.}
\emph{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: builds the four-pillar system --- the total geometric--gauge object, the covariant conservation identity with its four Ward currents, the geometric free-energy functional, and GRIP, the cascade by which boundary configurations descend to the ground attractor. Its relevance here is twofold: the coupling textures whose loop holonomies measure CP chirality (T-P4-5) are GRIP-stabilized outputs rather than free functions; and the actualization story it supplies is why every registered record in this paper's arc is a real quantity (\S\ref{sec:gg-grip}).
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{K.~Arisaka.}
\emph{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 information backbone of the boundary law: the four informational pillars that complement the geometric ones of the runtime paper, fixing how boundary information is counted, conserved, and registered. It enters this paper implicitly wherever the operational dictionary's instruments are treated as information-bearing boundary data --- the reading that lets vacuum choices, port assignments, and soft-cloud sectors be classified as dictionary freedom rather than bulk ambiguity.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}

\textbf{Arisaka, K.\ (2026).}
\textit{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary}.
Arisaka-GG-A8-QM preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.}
The parent treatment and this paper's closest companion: the quantum-mechanics paper whose real-bulk series RB-1--RB-12 states, at discussion level, the results this paper expands, re-proves, and re-derives for a broader readership --- the two sources, the antiparticle handshake, the discrete-symmetry table, the interacting completion, and the holomorphy bridge. Its three-address analysis of ``the Hilbert space,'' its measurement resolution, and its Penrose engagement stand behind Parts~\ref{part:sources}--\ref{part:discussion} throughout.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{K.~Arisaka.}
\emph{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: derives the 28-parameter data set, and specifically --- for this paper's purposes --- the flavor results quoted in T-P4-5 and Worked example~V: the CKM block as a projector-misalignment cascade, the pentagonal interference structure with the first-order Dirac phase $\delta = 2\pi/5$, the discrete-symmetry theorems T-SM-46.1/46.2 (this paper's R8/R9 in their theorem form), and the dial-free-door mirror T-SM-44.1. Its theory-uncertainty model carries the precision comparison that objection O-P4-24 defers to.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{K.~Arisaka.}
\emph{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: carries the dial-free-door theorem in its cosmological form (T-Cosmos-18), the forced Majorana channel, TeV-scale resonant leptogenesis at the lock, and the observer-indexed-vacuum treatment whose port reading this paper generalizes in \S\ref{sec:superselections}. Cited in the falsifiability section for the neutrino-nature exposure: an established Dirac nature for the light neutrinos would retire its leptogenesis module and, with it, one of the named ways this account can fail.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{K.~Arisaka.}
\emph{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 philosophical summit of the cohort: the paper that engages Wigner's unreasonable effectiveness, Penrose's three worlds, and Tegmark's mathematical universe, and that carries the calibrated geometrization slogan this paper's conclusions echo. The division of labor is deliberate: this paper dissolves one load-bearing instance of the effectiveness puzzle (the complex numbers), while the pattern-level engagement --- whether all such instances so dissolve --- is that paper's more ambitious wager, deliberately not spent here (objection O-P4-38).
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}

\textbf{Arisaka, K.\ (2026).}
\textit{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 preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.}
The collider companion of the Predictions Branch, and the sharpest statement in the cohort of the contrast this paper's Part~\ref{part:gg} turns into theorems: the non-spatial circle versus Kaluza--Klein's spatial one, with the observable consequence (no angular satellites at the lock cluster) that makes non-spatiality falsifiable rather than rhetorical. Its front matter, course-style appendices, and problem-set discipline are also this paper's structural template.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}

\textbf{Arisaka, K.\ (2026).}
\textit{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability}.
Arisaka-GG-P2-N3 preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.}
The neutrino-count companion of the Predictions Branch: the two-legged derivation of exactly three generations, whose CP-combinatorics leg --- a physical CP phase requires $N \geq 3$ --- ties the existence of the chiral loop holonomies of T-P4-5 to the family count itself. Cited in the cohort map as the sister paper whose subject matter meets this one at the flavor-holonomy seam.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}

\textbf{Arisaka, K.\ (2026).}
\textit{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P3-Electron preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.}
The electron companion of the Predictions Branch: the ground-state reading of the electron's identity, stability, and pointlikeness. It meets this paper at the dial: the electron is the paradigm charged sector, its phase the canonical instance of the total dial angle $\theta + (p \cdot x - Et)/\hbar$ whose two-source decomposition is T-P4-3's subject, and its worked treatment in the parent program is the fullest single-particle illustration of the packaging this paper derives.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}

\textbf{Arisaka, K.\ (2026).}
\textit{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 preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.}
The partner paper of the Foundations Branch and the \ANine{}--\ATen{} interface of the cohort map: C, P, and T derived as orientation reversals of the three real bulk factors, the CPT theorem as total reversal, the chiral-pentagon CP phases, and the origin of the cosmic matter excess. It consumes this paper's derived complex structure --- one shared $J$ for the whole tower, carried down the map's shared-$i$ edge --- and extends the custody principle from the sectors to the mirrors.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}

\textbf{Arisaka, K.\ (2026).}
\textit{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P6-Proton preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.}
The fourth member of the Predictions Branch. It matters here because it
is the cohort's heaviest use of the same object this paper derives the
gauge phase from: baryon number is carried as a winding of the compact
circle rather than as a bookkeeping label, so proton stability is a
statement about the circle's topology and not about a symmetry imposed on
top of it. Read beside T-P4-1 and the winding integrality of the ledger,
it is the physical case in which the circle's compactness does work that
no continuous phase could do.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}

\textbf{Arisaka, K.\ (2026).}
\textit{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}.
Arisaka-GG-C1-Kaxion preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.}
The dark-matter companion of the Observations Branch: the Kaxion, with its derived mass and laboratory window. It touches this paper at the compact-angle sector: the axion-like field is the cohort's other physical exploitation of the phase circle's topology, and its strong-CP context is the one place where a would-be CP angle is dynamically relaxed rather than lattice-locked --- the complementary fate to the pentagonal phases of T-P4-5.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}

\textbf{Arisaka, K.\ (2026).}
\textit{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 preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.}
The Hubble companion of the Observations Branch: the two-ends-of-one-corridor reading of the expansion-rate measurements. It enters this paper through the cosmological-dictionary interface: the comoving observer's particle inventories and effective descriptions are port-indexed boundary data in exactly the sense of \S\ref{sec:gg-gdos}, and the corridor analysis is the cohort's fullest exercise of that observer-indexed bookkeeping.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}

\textbf{Arisaka, K.\ (2026).}
\textit{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 preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.}
The black-hole companion of the Observations Branch, and the source of the horizon-as-port reading used directly in \S\ref{sec:superselections} and Worked example~IV: the six-dimensional bulk state pure and unitary, the horizon a locus of constitutional tracing, Hawking radiation the port's output, and the Bekenstein--Hawking entropy the entanglement of the traced sector. The stationary-exterior uniqueness and the Kerr residue discussed at objection O-P4-30 live against its backdrop.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}

\textbf{Arisaka, K.\ (2026).}
\textit{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out}.
Arisaka-GG-W1-Symmetry preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.}
The first of the two white papers, and the diachronic one: it follows a
century of symmetry and conservation in the order the results were found,
and watches the conviction that symmetry causes conservation loosen at
seven successive steps until what remains conserved is an integer with
nowhere to move. That endpoint is where this paper's winding integrality
already stands, which makes W1 the historical account of why a
topological charge, and not a Noether charge, is what the compact circle
delivers. It derives nothing new; every result is carried at its parent's
declared grade.
First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}

\textbf{Arisaka, K.\ (2026).}
\textit{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology}.
Arisaka-GG-W2-Holonomy preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.}
The second of the two white papers, and the closest of the three to this
one. Its single exact statement --- a holonomy requires a closed loop, and
the interval fiber has none --- is the asymmetry between the two
non-spatial dimensions that lets the phase circle carry a gauge phase at
all, and therefore lets T-P4-1 be a derivation rather than a
redescription. W2 takes that statement to sixteen of the questions physics
has not closed, and names the dated measurements on which six of the
sixteen treatments die outright. Like its sibling it derives nothing new.
First cited in \S\ref{sec:cohort}.

\end{thebibliography}

\section*{B.~Other references}

\begin{thebibliography}{99}

\bibitem[Aharonov and Bohm(1959)]{AharonovBohm1959}
\textbf{Y~Aharonov and D~Bohm.}
Significance of electromagnetic potentials in the quantum theory.
\emph{Physical Review} \textbf{115} (1959) 485--491.
\par\noindent\textbf{Relevance.}
The experiment-defining prediction that the electromagnetic phase is physical precisely and only through holonomies: a charged beam encircling confined flux acquires a measurable shift with no local field en route. For this paper it is the empirical cornerstone of the entire gauge half: the demonstration that the dial's differences are physics while its absolute setting is not --- the fact that T-P4-1 geometrizes and that charge superselection generalizes.
First cited in
\S\ref{sec:history-physics}.

\bibitem[Argand(1806)]{Argand1806}
\textbf{J-R~Argand.}
Essai sur une mani\`ere de repr\'esenter les quantit\'es imaginaires dans les constructions g\'eom\'etriques.
\emph{Paris (1806)}.
\par\noindent\textbf{Relevance.}
The geometric interpretation that domesticated the impossible numbers: complex quantities as directed magnitudes in a plane, with $i$ the quarter-turn. On this paper's account Argand drew, without knowing it, the first portrait of the fiber plane $(a, b)$ of Proposition~\ref{prop:real-C} --- the geometry was right; only the physical referent (a circle nature carries, a rotation time performs) was missing.
First cited in \S\ref{sec:history-math}.

\bibitem[Ashtekar and Magnon(1975)]{AshtekarMagnon1975}
\textbf{A~Ashtekar and A~Magnon.}
Quantum fields in curved space-times.
\emph{Proceedings of the Royal Society of London A} \textbf{346} (1975) 375--394.
\par\noindent\textbf{Relevance.}
With \citet{Kay1978}, the uniqueness of the positive-energy complex
structure on a stationary arena --- the selection theorem behind
\S\ref{sec:clock}. First cited in \S\ref{sec:clock}.

\bibitem[Bachmann et al.(2012)]{BachmannDeckertPizzo2012}
\textbf{S~Bachmann, D-A~Deckert, and A~Pizzo.}
The mass shell of the Nelson model without cut-offs.
\emph{Journal of Functional Analysis \textbf{263} (2012) 1224--1282}.
\par\noindent\textbf{Relevance.}
Theorem-grade control of the massless Nelson model's dressed
one-particle structure with the ultraviolet cutoff removed. Together
with \citet{Pizzo2003} it supplies the existence half of the door
theorem's middle rung (T-P4-9): the dressed states whose soft clouds
the theorem classifies as port assignments do exist, with the
regularity the sector analysis needs, and with no cutoff artifact
underwriting the conclusion.
First cited in \S\ref{sec:infrared}.

\bibitem[Berry(1984)]{Berry1984}
\textbf{M~V~Berry.}
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 adiabatic phase: a quantum system transported slowly around a parameter loop returns with a phase fixed by the loop's geometry alone. Its role here is as the modern era's second demonstration (after Aharonov--Bohm) that quantum phases are holonomies --- loop quantities of a real connection --- the reading that T-P4-1 completes by promoting the fiber to a dimension.
First cited in \S\ref{sec:history-qm}.

\bibitem[Bloch and Nordsieck(1937)]{BlochNordsieck1937}
\textbf{F~Bloch and A~Nordsieck.}
Note on the radiation field of the electron.
\emph{Physical Review \textbf{52} (1937) 54--59}.
\par\noindent\textbf{Relevance.}
The original diagnosis of the infrared problem: soft-photon emission renders every scattering probability formally zero unless arbitrarily soft quanta are summed. The logarithmically divergent overlap exponent of \eqref{eq:ir-overlap} --- the mechanism by which velocity sectors become orthogonal --- is this paper's real-presentation form of their signature, and the boundary the door theorem T-P4-9 now crosses at the model level.
First cited in \S\ref{sec:infrared}.

\bibitem[Bombelli(1572)]{Bombelli1572}
\textbf{R~Bombelli.}
L'Algebra.
\emph{Bologna (1572)}.
\par\noindent\textbf{Relevance.}
The book that made the impossible numbers unavoidable: for the casus irreducibilis of the cubic, every algebraic route to the real roots passes through complex intermediaries. Real in, real out, complex in between --- the exact arc this paper finds in physics (real bulk, complex amplitudes, real records), discovered as a fact about polynomials four and a half centuries before it was located in nature.
First cited in \S\ref{sec:history-math}.

\bibitem[Born(1926)]{Born1926}
\textbf{M~Born.}
Zur Quantenmechanik der Stossvorg\"ange.
\emph{Zeitschrift f\"ur Physik \textbf{37} (1926) 863--867}.
\par\noindent\textbf{Relevance.}
The probability interpretation: $|\psi|^2$ as the measurable content of the wavefunction, with the phase left physical only through interference. In this paper's terms Born's rule is the statement that registration marginalizes the unreadable dial while relative dial angles survive --- the operational face of the fiber's unreadability, whose square-modulus form the parent program derives via Gleason.
First cited in \S\ref{sec:history-qm}.

\bibitem[Buchholz(1977)]{Buchholz1977}
\textbf{D~Buchholz.}
Collision theory for massless bosons.
\emph{Communications in Mathematical Physics} \textbf{52} (1977) 147--173.
\par\noindent\textbf{Relevance.}
The rigorous collision theory for massless bosons: asymptotic fields and scattering states constructed without a mass gap, under general locality and spectrum assumptions. In this paper it is one of the two footholds of the infrared boundary (\S\ref{sec:infrared}): it secures the massless quanta's own asymptotic structure, so that the corner T-P4-9 then closes at the model level is the charged infraparticle's dressing, not the photon sector itself.
First cited in \S\ref{sec:infrared}.

\bibitem[Buchholz(1986)]{Buchholz1986}
\textbf{D~Buchholz.}
Gauss' law and the infraparticle problem.
\emph{Physics Letters B \textbf{174} (1986) 331--334}.
\par\noindent\textbf{Relevance.}
The structural reading of the infraparticle: by Gauss' law the
asymptotic electromagnetic flux is a superselected observable, so a
charged particle's velocity is written in its cloud at infinity and
sharp mass shells are excluded. The door theorem adopts this sector
structure wholesale and adds the framework's placement --- the flux
label is dictionary data at AX-3, a port assignment, and the derived
complex structure is blind to it (T-P4-9). The flux observable is what
makes the sector label \emph{physical} rather than notational.
First cited in \S\ref{sec:infrared}.

\bibitem[Cardano(1545)]{Cardano1545}
\textbf{G~Cardano.}
Artis Magnae, sive de Regulis Algebraicis (Ars Magna).
\emph{Nuremberg (1545)}.
\par\noindent\textbf{Relevance.}
The book where the square roots of negatives first earned their keep, in the solution of the cubic, accompanied by their author's famous disparagement. The starting point of the five-century chronology of \S\ref{sec:chronology}: an invention explicitly labeled useless at birth, whose eventual physical necessity became the puzzle this paper answers.
First cited in \S\ref{sec:history-math}.

\bibitem[Cauchy(1825)]{Cauchy1825}
\textbf{A-L~Cauchy.}
M\'emoire sur les int\'egrales d\'efinies, prises entre des limites imaginaires.
\emph{Paris (1825)}.
\par\noindent\textbf{Relevance.}
The memoir that revealed the rigidity of the complex domain: integrals between imaginary limits, contour independence, and the residue calculus. That rigidity --- one derivative implying all, values in a region determining values everywhere --- is what this paper's bridge traces to one-sided frequency support: the analytic miracle as the fingerprint of an orientation of time.
First cited in \S\ref{sec:history-math}.

\bibitem[Chen et al.(2010)]{ChenFrohlichPizzo2010}
\textbf{T~Chen, J~Fr\"ohlich, and A~Pizzo.}
Infraparticle scattering states in non-relativistic QED: I. The
Bloch--Nordsieck paradigm.
\emph{Communications in Mathematical Physics \textbf{294} (2010) 761--825}.
\par\noindent\textbf{Relevance.}
The rigorous construction of infraparticle scattering states in
nonrelativistic QED, exhibiting velocity-superselected soft clouds as
the correct asymptotic objects. This is the scattering-level import of
the door theorem's middle rung: the states exist, their clouds carry
the sector label of Lemma~\ref{lem:ir-divergence}, and the dressed
asymptotic dynamics has exactly the free-plus-c-number form whose
polar phase T-P4-9 reads. The strongest rigorous result standing under
the door.
First cited in \S\ref{sec:infrared}.

\bibitem[Chiribella \emph{et~al.}(2011)]{ChiribellaEtAl2011}
\textbf{G~Chiribella, G~M~D'Ariano, and P~Perinotti.}
Informational derivation of quantum theory.
\emph{Physical Review A \textbf{84} (2011) 012311}.
\par\noindent\textbf{Relevance.}
The informational derivation of quantum theory: complex QM singled out among operational theories by axioms about information processing (purification chief among them). Cited at objection O-P4-10 as convergent evidence of a different kind from this paper's: the reconstructions prove it had to be $\bC$; the geometric account exhibits what $\bC$ was.
First cited in \S\ref{part:objections}.

\bibitem[Chung(1965)]{Chung1965}
\textbf{V~Chung.}
Infrared divergence in quantum electrodynamics.
\emph{Physical Review \textbf{140} (1965) B1110--B1122}.
\par\noindent\textbf{Relevance.}
The coherent-state resolution of the infrared divergences: charged states dressed with soft-photon clouds of definite classical profile, rendering transition probabilities finite. Its dressed sectors are exactly the coherent displacements of Lemma~\ref{lem:cloud} --- sector-moving, $J$-preserving --- and its construction anchors the port reading of the infrared in \S\ref{sec:infrared}.
First cited in \S\ref{sec:infrared}.

\bibitem[Dirac(1930)]{Dirac1930}
\textbf{P~A~M~Dirac.}
The Principles of Quantum Mechanics.
\emph{Clarendon Press, Oxford (1930)}.
\par\noindent\textbf{Relevance.}
The book that fixed quantum mechanics' modern form, transformation theory and all, weaving the $i$ into the commutation relations themselves. Cited in the history as the moment the complex unit became load-bearing formalism rather than removable shorthand --- the state of affairs this paper's two-source derivation is designed to explain rather than overturn.
First cited in \S\ref{sec:history-qm}.

\bibitem[Eastwood \emph{et~al.}(1981)]{EastwoodPenroseWells1981}
\textbf{M~G~Eastwood, R~Penrose, and R~O~Wells.}
Cohomology and massless fields.
\emph{Communications in Mathematical Physics} \textbf{78} (1981) 305--351.
\par\noindent\textbf{Relevance.}
The rigorous Penrose transform: massless fields of helicity $h$ as
$H^1(\PT, \mathcal{O}(-2h-2))$, with positive frequency supported on
$\PT^{+}$ --- the third level of the bridge. First cited in
\S\ref{sec:history-twistor}.

\bibitem[Euler(1748)]{Euler1748}
\textbf{L~Euler.}
Introductio in analysin infinitorum.
\emph{Lausanne (1748)}.
\par\noindent\textbf{Relevance.}
The Introductio, home of $e^{i\theta} = \cos\theta + i\sin\theta$: the exponential bound to the circle. On this paper's reading Euler's identity is not a curiosity but the entire mechanism in miniature --- the harmonic analysis of a compact real dimension \emph{is} the complex exponential, which is why a physical circle makes complex numbers appear wherever charge lives (T-P4-1).
First cited in \S\ref{sec:history-math}.

\bibitem[Faddeev and Kulish(1970)]{FaddeevKulish1970}
\textbf{L~D~Faddeev and P~P~Kulish.}
Asymptotic conditions and infrared divergences in quantum electrodynamics.
\emph{Theoretical and Mathematical Physics} \textbf{4} (1970) 745--757.
\par\noindent\textbf{Relevance.}
The dressed-state reformulation of QED's asymptotic dynamics: charged particles paired with their soft-photon clouds via an asymptotic interaction that never switches off, yielding infrared-finite S-matrix elements between coherent-dressed states. Here it organizes the charged sectors of \S\ref{sec:infrared} and anchors the top rung of the door theorem T-P4-9 --- the dressing is the sector label, Lemma~\ref{lem:cloud} shows it never touches $J$, and the construction's full rigor is the theorem's named frontier.
First cited in
\S\ref{sec:infrared}.

\bibitem[Fourier(1822)]{Fourier1822}
\textbf{J~Fourier.}
Th\'eorie analytique de la chaleur.
\emph{Paris (1822)}.
\par\noindent\textbf{Relevance.}
The analytical theory of heat, and with it the decomposition of arbitrary signals into exponentials --- the tool that trained physics to complexify, compute, and discard. Cited in \S\ref{sec:history-physics} as the first wave of the quiet infiltration: the $i$ as pure bookkeeping, a habit whose eventual failure (in quantum mechanics) became the puzzle.
First cited in \S\ref{sec:history-physics}.

\bibitem[Fresnel(1823)]{Fresnel1823}
\textbf{A~Fresnel.}
M\'emoire sur la loi des modifications que la r\'eflexion imprime \`a la lumi\`ere polaris\'ee.
\emph{M\'emoires de l'Acad\'emie des Sciences (1823)}.
\par\noindent\textbf{Relevance.}
Complex amplitudes for polarized light: the first sustained physical use of complex quantities, with phase and amplitude carried in one symbol and the imaginary part discarded at the end. The optics of the 1820s is where the phasor habit began, and this paper's history places it as the template for eighty years of removable $i$.
First cited in \S\ref{sec:history-physics}.

\bibitem[Fr\"ohlich(1973)]{Frohlich1973}
\textbf{J~Fr\"ohlich.}
On the infrared problem in a model of scalar electrons and massless,
scalar bosons.
\emph{Annales de l'Institut Henri Poincar\'e A \textbf{19} (1973) 1--103}.
\par\noindent\textbf{Relevance.}
The founding rigorous analysis of the infraparticle: in the massless
Nelson model the fixed-momentum Hamiltonian has no ground state in
Fock space, and the physical states live in velocity-indexed coherent
sectors. Lemma~\ref{lem:ir-divergence} is the quantitative skeleton
of Fr\"ohlich's phenomenon --- the logarithmically divergent cloud
and the power-law sector orthogonality --- and the door theorem's
middle rung stands on the existence theory this paper opened.
First cited in \S\ref{sec:infrared}.

\bibitem[Gabor(1946)]{Gabor1946}
\textbf{D~Gabor.}
Theory of communication.
\emph{Journal of the IEE \textbf{93} (1946) 429--457}.
\par\noindent\textbf{Relevance.}
The theory of communication, and the analytic signal: a real waveform completed by its Hilbert-transform quadrature into a complex signal whose modulus is the envelope and whose argument is the instantaneous phase. Cited at Worked example~I as the engineer's form of T-P4-6 --- the construction this paper shows to be, in the quantum context, not a trick but the boundary value of forward-tube holomorphy.
First cited in \S\ref{ssec:we-analytic}.

\bibitem[Gauss(1831)]{Gauss1831}
\textbf{C~F~Gauss.}
Theoria residuorum biquadraticorum, commentatio secunda.
\emph{G\"ottingen (1831)}.
\par\noindent\textbf{Relevance.}
The paper that made the plane official and coined the very name ``complex numbers,'' ending three centuries of ontological embarrassment by geometric fiat. In the chronology it marks the completion of the domestication --- and the beginning of the deeper puzzle, since a geometric interpretation within mathematics still leaves open what, in nature, the plane is a picture of.
First cited in \S\ref{sec:history-math}.

\bibitem[Gleason(1957)]{Gleason1957}
\textbf{A~M~Gleason.}
Measures on the closed subspaces of a Hilbert space.
\emph{Journal of Mathematics and Mechanics \textbf{6} (1957) 885--893}.
\par\noindent\textbf{Relevance.}
The measure-theoretic theorem: every countably additive probability assignment on the closed subspaces of a Hilbert space of dimension three or more is of trace form. In the parent program it fixes the square-modulus form of the Born weights; cited here at the actualization layer (\S\ref{sec:gg-grip}) as the reason the form of the probabilities, unlike their per-run values, is theorem-grade.
First cited in \S\ref{sec:gg-grip}.

\bibitem[Green and Schwarz(1984)]{GreenSchwarz1984}
\textbf{M~B~Green and J~H~Schwarz.}
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 result that fixes the string program's
dimension and gauge structure --- cited in the three-roads comparison
(\S\ref{sec:threeroads}) as the location of that program's
irreducible premises. First cited in \S\ref{sec:threeroads}.

\bibitem[Haag(1992)]{Haag1992}
\textbf{R~Haag.}
Local Quantum Physics: Fields, Particles, Algebras.
\emph{Springer, Berlin} (1992).
\par\noindent\textbf{Relevance.}
The standard treatise of algebraic quantum field theory. Two of its chapters are load-bearing here: the Haag--Ruelle scattering theory that powers the interacting completion T-P4-8, and the superselection theory whose abstract sector analysis the three-superselections pattern of \S\ref{sec:superselections} instantiates geometrically --- charge, horizon, and infrared sectors as three unreadabilities with one lesson.
First cited in
\S\ref{sec:interacting}.

\bibitem[Hamilton(1837)]{Hamilton1837}
\textbf{W~R~Hamilton.}
Theory of conjugate functions, or algebraic couples.
\emph{Transactions of the Royal Irish Academy \textbf{17} (1837) 293--422}.
\par\noindent\textbf{Relevance.}
The algebraic couples: $\bC$ rebuilt as ordered pairs of reals with a stipulated multiplication --- the first real presentation, and the proof that $i$ is eliminable in principle. This paper stands to Hamilton as physics to his algebra: the theorems of Part~\ref{part:sources} supply the selection his stipulation lacked --- which quarter-turn, why shared, why unique --- turning a philosophical nicety into a statement about nature.
First cited in \S\ref{sec:history-math}.

\bibitem[Hardy(2001)]{Hardy2001}
\textbf{L~Hardy.}
Quantum theory from five reasonable axioms.
\emph{arXiv:quant-ph/0101012 (2001)}.
\par\noindent\textbf{Relevance.}
The five reasonable axioms: the paper that opened the modern operational-reconstruction program, deriving quantum theory's structure from information-theoretic principles. Cited at objection O-P4-10 within the convergence argument: reconstruction locates the necessity of the complex formalism; the present account locates its physical referent.
First cited in \S\ref{part:objections}.

\bibitem[Hawking(1975)]{Hawking1975}
\textbf{S~W~Hawking.}
Particle creation by black holes.
\emph{Communications in Mathematical Physics \textbf{43} (1975) 199--220}.
\par\noindent\textbf{Relevance.}
Particle creation by black holes: the thermal flux at temperature $\kappa/2\pi$ that made horizon thermodynamics physical. Worked example~IV computes its structural core in four lines of covariance algebra --- the traced half of the horizon's squeezed pairing is exactly Planckian --- and the port reading of \S\ref{sec:superselections} places the result where the parent program's dictionary says it belongs.
First cited in \S\ref{ssec:we-horizon}.

\bibitem[Jordan \emph{et~al.}(1934)]{JordanEtAl1934}
\textbf{P~Jordan, J~von~Neumann, and E~Wigner.}
On an algebraic generalization of the quantum mechanical formalism.
\emph{Annals of Mathematics \textbf{35} (1934) 29--64}.
\par\noindent\textbf{Relevance.}
The algebraic classification of quantum formalisms: the finite-dimensional Jordan algebras admitting a quantum-mechanical interpretation, with the real, complex, and quaternionic families delimited. The classical ancestor of every ``why complex?'' discussion, cited at objection O-P4-10; on this paper's account its complex branch is the one nature instantiates because a circle and a clock source a shared $J$.
First cited in \S\ref{part:objections}.

\bibitem[Kay(1978)]{Kay1978}
\textbf{B~S~Kay.}
Linear spin-zero quantum fields in external gravitational and scalar fields.
\emph{Communications in Mathematical Physics} \textbf{62} (1978) 55--70.
\par\noindent\textbf{Relevance.}
Kay's uniqueness theorem for the stationary one-particle structure ---
the second half of the selection theorem of \S\ref{sec:clock}. First
cited in \S\ref{sec:clock}.

\bibitem[Kay and Wald(1991)]{KayWald1991}
\textbf{B~S~Kay and R~M~Wald.}
Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon.
\emph{Physics Reports \textbf{207} (1991) 49--136}.
\par\noindent\textbf{Relevance.}
The uniqueness and thermality theorems for quasifree states on spacetimes with bifurcate Killing horizons --- including the obstruction that no Hadamard state on Kerr is invariant under the horizon-generating flow. Cited at objection O-P4-30 and \S\ref{sec:openq} as the honest boundary of the stationary-case resolution: the rotating anchor is physics, and it is open.
First cited in \S\ref{part:objections}.

\bibitem[Li \emph{et~al.}(2022)]{LiEtAl2022}
\textbf{M-C~Li \emph{et~al.}.}
Testing real quantum theory in an optical quantum network.
\emph{Physical Review Letters \textbf{128} (2022) 040402}.
\par\noindent\textbf{Relevance.}
One of the 2022 experimental realizations of the network Bell test that excludes real quantum mechanics with real tensor products. Cited beside the theoretical exclusion in \S\ref{sec:history-qm}: the shared-$i$ mandate is not a thought experiment but a laboratory result, and T-P4-4 is the structure that satisfies it by construction.
First cited in \S\ref{sec:history-qm}.

\bibitem[Mason and Woodhouse(1996)]{MasonWoodhouse1996}
\textbf{L~J~Mason and N~M~J~Woodhouse.}
Integrability, Self-Duality, and Twistor Theory.
\emph{Oxford University Press (1996)}.
\par\noindent\textbf{Relevance.}
The monograph connecting twistor constructions to integrable systems: self-duality as the master integrability, with the Ward correspondence as its engine. Cited at objection O-P4-1 as the direction in which the nonlinear conjecture points: if the nonlinear twistor geometry is the integrable packaging of the two real sources, this is the mathematics in which the packaging would be exhibited.
First cited in \S\ref{part:objections}.

\bibitem[Paley and Wiener(1934)]{PaleyWiener1934}
\textbf{R~E~A~C~Paley and N~Wiener.}
Fourier Transforms in the Complex Domain.
\emph{American Mathematical Society Colloquium Publications} \textbf{19} (1934).
\par\noindent\textbf{Relevance.}
The classical source of the Paley--Wiener theorem
(Theorem~\ref{thm:pw}): one-sided frequency support is equivalent to
Hardy-space holomorphy --- the first level of the bridge. First cited
in \S\ref{sec:prelim-hardy}.

\bibitem[Penrose(1967)]{Penrose1967}
\textbf{R~Penrose.}
Twistor algebra.
\emph{Journal of Mathematical Physics} \textbf{8} (1967) 345--366.
\par\noindent\textbf{Relevance.}
The founding paper of twistor theory: the algebra of twistors, the incidence correspondence, and the proposal that light rays rather than events are the primitive objects. For this paper it defines the geometric stage of Course~B and the object --- the division of $\PT$ --- whose identification with the causal-positive frequency splitting (T-P4-7) is the central result offered to its school.
First cited in \S\ref{sec:history-twistor}.

\bibitem[Penrose(1969)]{Penrose1969}
\textbf{R~Penrose.}
Solutions of the zero-rest-mass equations.
\emph{Journal of Mathematical Physics \textbf{10} (1969) 38--39}.
\par\noindent\textbf{Relevance.}
The contour-integral form of the massless field representation: solutions of the zero-rest-mass equations as integrals of holomorphic functions over lines in twistor space. The original transform, executed concretely in Worked example~II and Lecture B.5, where its homogeneity counting and pole placement become this paper's statement that frequency sign is contour topology.
First cited in \S\ref{sec:prelim-twistor}.

\bibitem[Penrose(1976)]{Penrose1976}
\textbf{R~Penrose.}
Nonlinear gravitons and curved twistor theory.
\emph{General Relativity and Gravitation} \textbf{7} (1976) 31--52.
\par\noindent\textbf{Relevance.}
The nonlinear graviton construction, and with it the ``googly''
problem --- the twistor program's named central hurdle, recorded in
the comparison table as open on both sides. First cited in
\S\ref{sec:comparison}.

\bibitem[Penrose and MacCallum(1973)]{PenroseMacCallum1973}
\textbf{R~Penrose and M~A~H~MacCallum.}
Twistor theory: an approach to the quantization of fields and space-time.
\emph{Physics Reports} \textbf{6} (1973) 241--315.
\par\noindent\textbf{Relevance.}
The foundational exposition, including the correspondence between
positive-frequency massless fields and holomorphic data on $\PT^{+}$
--- the component of the bridge that makes the identification of
\S\ref{sec:twistor-level} available. First cited in
\S\ref{sec:history-twistor}.

\bibitem[Penrose and Rindler(1986)]{PenroseRindler1986}
\textbf{R~Penrose and W~Rindler.}
Spinors and Space-Time, Volume 2: Spinor and Twistor Methods in
Space-Time Geometry.
\emph{Cambridge University Press} (1986).
\par\noindent\textbf{Relevance.}
The standard reference for twistor geometry --- the incidence
relation, the helicity form, and the $\PT^{\pm}$ division used in
Definition~\ref{def:twistor}. First cited in
\S\ref{sec:prelim-twistor}.

\bibitem[Penrose(2004)]{Penrose2004}
\textbf{R~Penrose.}
The Road to Reality: A Complete Guide to the Laws of the Universe.
\emph{Jonathan Cape, London} (2004).
\par\noindent\textbf{Relevance.}
The canonical statement of the complex-number question this preprint
answers in calibrated form. First cited in \S\ref{sec:history-twistor}.

\bibitem[Pizzo(2003)]{Pizzo2003}
\textbf{A~Pizzo.}
One-particle (improper) states in Nelson's massless model.
\emph{Annales Henri Poincar\'e \textbf{4} (2003) 439--486}.
\par\noindent\textbf{Relevance.}
The iterated-dressing construction of the massless Nelson model's
one-particle states, with H\"older regularity in the total momentum.
This is the constructive engine behind the door theorem's middle rung:
it builds exactly the dressed states whose coherent clouds T-P4-9
classifies as port assignments, and on which the sector-blindness of
the derived $J$ (Lemma~\ref{lem:dressed-generator}) is then a
theorem rather than a hope.
First cited in \S\ref{sec:infrared}.

\bibitem[Renou \emph{et~al.}(2021)]{Renou2021}
\textbf{M-O~Renou, D~Trillo, M~Weilenmann, T~P~Le, A~Tavakoli, N~Gisin,
A~Ac\'in, and M~Navascu\'es.}
Quantum theory based on real numbers can be experimentally falsified.
\emph{Nature} \textbf{600} (2021) 625--629.
\par\noindent\textbf{Relevance.}
The experimental mandate for a single shared complex structure ---
satisfied here by construction from one circle and one causal order.
First cited in \S\ref{sec:history-qm}.

\bibitem[Riemann(1851)]{Riemann1851}
\textbf{B~Riemann.}
Grundlagen f\"ur eine allgemeine Theorie der Functionen einer ver\"anderlichen complexen Gr\"osse.
\emph{Inauguraldissertation, G\"ottingen (1851)}.
\par\noindent\textbf{Relevance.}
The inaugural dissertation: complex function theory refounded geometrically, with the surfaces that bear his name. In the chronology it marks the maturation of the rigidity discovered by Cauchy --- and the final form of the mathematics whose physical referent (an orientation of time, conformally completed) the bridge of Part~\ref{part:bridge} identifies.
First cited in \S\ref{sec:history-math}.

\bibitem[Schr\"odinger(1926)]{Schrodinger1926}
\textbf{E~Schr\"odinger.}
Quantisierung als Eigenwertproblem (Vierte Mitteilung).
\emph{Annalen der Physik \textbf{81} (1926) 109--139}.
\par\noindent\textbf{Relevance.}
The fourth communication: the paper in which the wavefunction's complexity becomes unavoidable, against its author's own sustained effort to keep $\psi$ real. Cited in \S\ref{sec:history-qm} as the founding instance of the puzzle --- the first $i$ in physics that would not cancel --- whose resolution (the shared quarter-turn of the clock, not a mode-by-mode shorthand) is T-P4-2.
First cited in \S\ref{sec:history-qm}.

\bibitem[Sol\`er(1995)]{Soler1995}
\textbf{M~P~Sol\`er.}
Characterization of Hilbert spaces by orthomodular spaces.
\emph{Communications in Algebra \textbf{23} (1995) 219--243}.
\par\noindent\textbf{Relevance.}
The lattice-theoretic characterization: infinite orthomodular spaces with the relevant completeness are Hilbert spaces over the three classical division rings. Cited at objection O-P4-10 among the reconstruction results that narrow the field to real, complex, or quaternionic --- the trichotomy within which the experiments and the shared-$J$ theorem of T-P4-4 then select.
First cited in \S\ref{part:objections}.

\bibitem[Steinmetz(1893)]{Steinmetz1893}
\textbf{C~P~Steinmetz.}
Complex quantities and their use in electrical engineering.
\emph{Proceedings of the International Electrical Congress, Chicago (1893) 33--74}.
\par\noindent\textbf{Relevance.}
The address that industrialized the phasor: alternating-current engineering rebuilt on complex quantities, with the imaginary part discarded at the end of every calculation. The paradigm of removable $i$ --- and thereby, on this paper's telling, the trap: eighty years of training that left physics without the vocabulary to say what changed in 1926.
First cited in \S\ref{sec:history-physics}.

\bibitem[Stone(1932)]{Stone1932}
\textbf{M~H~Stone.}
On one-parameter unitary groups in Hilbert space.
\emph{Annals of Mathematics \textbf{33} (1932) 643--648}.
\par\noindent\textbf{Relevance.}
The one-parameter unitary groups theorem: continuous unitary evolutions are exactly $e^{itH}$ with self-adjoint $H$. Cited at objection O-P4-9, where its role is clarified: Stone interlocks $i$, $t$, and $H$ \emph{within} complex quantum mechanics; the present account supplies the prior fact --- why the state space is complex and what selects its $J$ --- after which Stone's theorem becomes derived scaffolding.
First cited in \S\ref{part:objections}.

\bibitem[Streater and Wightman(1964)]{StreaterWightman1964}
\textbf{R~F~Streater and A~S~Wightman.}
PCT, Spin and Statistics, and All That.
\emph{W.~A.~Benjamin, New York} (1964).
\par\noindent\textbf{Relevance.}
The classic axiomatic treatment: Wightman functions, the forward-tube analyticity that this paper reads as the field-theoretic form of causal positivity (\S\ref{sec:tube}), and the PCT theorem whose orientation-bookkeeping counterpart is T-P4-5. It is the bridge's middle level in its original habitat, and the reading guide points the student to its chapters as Course~A's rigorous companion.
First cited in
\S\ref{sec:tube}.

\bibitem[Stueckelberg(1960)]{Stueckelberg1960}
\textbf{E~C~G~Stueckelberg.}
Quantum theory in real Hilbert space.
\emph{Helvetica Physica Acta \textbf{33} (1960) 727--752}.
\par\noindent\textbf{Relevance.}
The serious real-Hilbert-space program: quantum theory over $\bR$, with the complex unit replaced by a superselected operator $J$. Cited in \S\ref{sec:history-qm} as the road not taken's most rigorous milestone --- its single-system half survives (and anticipates the real-presentation spirit), while its composite structure is what the network experiments later excluded, exactly at the unshared tensor product.
First cited in \S\ref{sec:history-qm}.

\bibitem[Titchmarsh(1948)]{Titchmarsh1948}
\textbf{E~C~Titchmarsh.}
Introduction to the Theory of Fourier Integrals.
\emph{Oxford University Press, 2nd edition (1948)}.
\par\noindent\textbf{Relevance.}
The classical treatise on Fourier integrals: conjugate functions, the Hilbert transform's $L^2$ theory, and the Paley--Wiener circle of ideas in their standard form. The reading guide's analysis shelf points Course~A's student here for every $\epsilon$ the lectures wave through; the theorems used in this paper live in its chapters V and~VI.
First cited in \S\ref{ssec:A2}.

\bibitem[Unruh(1976)]{Unruh1976}
\textbf{W~G~Unruh.}
Notes on black-hole evaporation.
\emph{Physical Review D \textbf{14} (1976) 870--892}.
\par\noindent\textbf{Relevance.}
The acceleration temperature: a uniformly accelerated detector in the Minkowski vacuum thermalizes at $a/2\pi$, via the pairing of Rindler wedges. Worked example~IV's computation is its structural skeleton, and the port reading --- which modes an instrument counts is dictionary data --- is where this paper files the effect's famous observer-dependence.
First cited in \S\ref{ssec:we-horizon}.

\bibitem[von Neumann(1932)]{vonNeumann1932}
\textbf{J~von~Neumann.}
Mathematische Grundlagen der Quantenmechanik.
\emph{Springer, Berlin (1932)}.
\par\noindent\textbf{Relevance.}
The mathematical foundations: the complex separable Hilbert space installed as quantum mechanics' first postulate, where every textbook since has kept it. Cited at objection O-P4-41 as the formulation this paper does not overturn but sources: the postulate stands, and acquires a derivation with named hypotheses and named residues.
First cited in \S\ref{part:objections}.

\bibitem[Wald(1994)]{Wald1994}
\textbf{R~M~Wald.}
Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.
\emph{University of Chicago Press (1994)}.
\par\noindent\textbf{Relevance.}
The standard exposition of quantum field theory in curved spacetime: one-particle structures, quasifree states, and the algebraic formulation, in the form this paper's apparatus imports. Cited at objection O-P4-11, which states exactly what is taken (the machinery) and what is added (the constitutional arena, the circle--clock pairing, the assembly).
First cited in \S\ref{part:objections}.

\bibitem[Ward(1977)]{Ward1977}
\textbf{R~S~Ward.}
On self-dual gauge fields.
\emph{Physics Letters A \textbf{61} (1977) 81--82}.
\par\noindent\textbf{Relevance.}
The self-dual correspondence: anti-self-dual gauge fields encoded as holomorphic bundles over twistor space --- with the nonlinear graviton, the engine of twistor theory's nonlinear era. Cited at \S\ref{sec:openq} as the frontier the bridge has not reached: the conjecture that these constructions are the integrable packaging of the two real sources is recorded, and open, at objection O-P4-1.
First cited in \S\ref{sec:openq}.

\bibitem[Weyl(1918)]{Weyl1918}
\textbf{H~Weyl.}
Gravitation und Elektrizit\"at.
\emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften} (1918) 465--480.
\par\noindent\textbf{Relevance.}
Gravitation and electricity: the original gauge theory, in which local \emph{scale} invariance was to unify gravitation and electromagnetism. It failed physically (Einstein's measuring-rod objection) and succeeded structurally: its 1929 phase reinterpretation created modern gauge theory, and the parent program's two non-spatial dimensions --- scale and phase --- are, in a precise sense, both of Weyl's ideas made geometry at last.
First cited in
\S\ref{sec:history-physics}.

\bibitem[Weyl(1929)]{Weyl1929}
\textbf{H~Weyl.}
Elektron und Gravitation. I.
\emph{Zeitschrift f\"ur Physik} \textbf{56} (1929) 330--352.
\par\noindent\textbf{Relevance.}
The birth of the modern gauge principle: the 1918 scale theory reinterpreted as local phase invariance, putting $e^{i\lambda(x)}$ into electrodynamics and, unknowingly, opening the question this paper closes --- what the phase is an angle \emph{of}. On the present account Weyl's phase is the circle's coordinate, his gauge freedom the dial re-zeroing of T-P4-5, and his 1929 insight the second wave of the quiet infiltration traced in \S\ref{sec:history-physics}.
First cited
in \S\ref{sec:history-physics}.

\bibitem[Wick et al.(1952)]{WickWightmanWigner1952}
\textbf{G~C~Wick, A~S~Wightman, and E~P~Wigner.}
The intrinsic parity of elementary particles.
\emph{Physical Review \textbf{88} (1952) 101--105}.
\par\noindent\textbf{Relevance.}
The paper that introduced superselection rules: the observation that
relative phases between certain sectors of the Hilbert space are
unmeasurable in principle, so the superposition principle holds only
within sectors. The custody principle
(Proposition~\ref{prop:custody}) is this paper's geometric completion
of that observation for the three instances the derivation meets:
each superselected phase is identified with a global real datum ---
a winding, a horizon interior, a cloud at infinity --- owned by the
operational dictionary, while the derived complex structure is proved
identical across sectors.
First cited in \S\ref{sec:superselections}.

\bibitem[Witten(2004)]{Witten2004}
\textbf{E~Witten.}
Perturbative gauge theory as a string theory in twistor space.
\emph{Communications in Mathematical Physics} \textbf{252} (2004) 189--258.
\par\noindent\textbf{Relevance.}
The paper that returned twistor variables to the center of perturbative gauge theory: gauge-theory amplitudes as a string theory in twistor space, igniting the modern amplitudes program. Cited in the three-roads comparison and at objection O-P4-6 as the twistor column's contemporary strength on interactions --- and as the natural interface at which an amplitudes-level extension of the bridge would attach.
First cited in \S\ref{sec:comparison}.

\bibitem[Wu and Yang(1975)]{WuYang1975}
\textbf{T~T~Wu and C~N~Yang.}
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 bundle formulation that recognized the gauge phase as holonomy ---
the precedent for reading the phase geometrically, completed here by
promoting the fiber to a real dimension. First cited in
\S\ref{sec:history-physics}.

\end{thebibliography}

\end{document}
