%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.35 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% ===================================================================
%% v15.35 (August 20, 2026) -- THE KERNEL-SCALING REPAIR (Report 472).
%%
%% THE DEFECT.  Part V's decisive scaling step -- the projected
%% bulk-to-boundary kernel of Sec. "From the 6D bulk to homogeneous
%% boundary kernels", and its algebra in the radial appendix -- wrote
%% the integrand with a SINGLE propagator factor,
%%     W ~ int omega(zeta) K_Delta dzeta,  omega = zeta^{sigma-1} w,
%% giving integrand zeta^{sigma+Delta-1}/(zeta^2+ell^2)^Delta.  Under
%% zeta = ell u that yields ell^{sigma-Delta}, NOT the printed
%% ell^{sigma-2Delta}.  The appendix printed the failing sum in the
%% open: "ell^{sigma+Delta-1+1-2Delta}" labelled as ell^{sigma-2Delta}
%% (the exponent sums to sigma-Delta).  Verified numerically:
%% Delta=2, sigma=1 gives (pi/4)/ell = ell^{sigma-Delta}.
%%
%% THE REPAIR, and why this one.  A jump RATE is an amplitude squared:
%% the mode reaches each endpoint of the jump through one factor of the
%% propagator, so the kernel carries K_Delta TWICE.  The exchange form
%%     W ~ int omega(zeta) [K_Delta]^2 dzeta
%% has integrand zeta^{sigma+2Delta-1}/(zeta^2+ell^2)^{2Delta}, and the
%% same substitution closes exactly to ell^{sigma-2Delta}:
%%     sigma+2Delta-1-4Delta+1 = sigma-2Delta.
%% It therefore PRESERVES every downstream statement -- the
%% identification 2Delta-sigma = d_State+alpha, the admissibility
%% window 0 < alpha <= 2, the bulk bounds, and the paper's own
%% intuition sentence ("the bulk mode's weight, TWICE, minus what the
%% projection's measure keeps"), which was already describing the
%% exchange form.  At sigma = 0 it reproduces the standard
%% fractional-Laplacian weight Delta = (d_State+alpha)/2.
%%
%% ALSO: the saturation constant is now EVALUATED rather than named,
%% in the main text and the appendix both:
%%     C_{Delta,sigma} = (1/2) B(Delta+sigma/2, Delta-sigma/2),
%% finite precisely for |sigma| < 2Delta -- the convergence condition
%% and the admissibility bounds being the same fact seen twice.
%%
%% NOTHING DOWNSTREAM MOVES.  (Delta, sigma) are declared inputs, not
%% outputs; alpha, the four Qi-CON signatures, the lock and failure
%% theorems, the two knees, and every number in the paper are
%% untouched.  Seven equation sites changed; no section, theorem,
%% table, figure or reference added or removed.
%%
%% PROVENANCE.  Found while extending this algebra into Appendix A of
%% S3-Conformal (Ship F, Report 472) -- the first time the step was
%% re-derived rather than quoted.  S3 carried a note of record naming
%% this paper's schematic; that note is updated in the same pass, since
%% both papers now print the exchange form.
%% ===================================================================
%% ===================================================================
%% v15.30 (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.
%% ====================================================================
%% ===================================================================
%% v15.29 (August 18, 2026) -- FP-3: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) The A11 bibliography entry said "preprint, available at
%%     arisaka-gg.org"; it now reads "In preparation; to be posted ... on
%%     completion.  No result of the present paper depends on it."
%% (2) TWO PRINTED SITES described A11's cascade as SEVENTEEN layers --
%%     the pagoda import in the emergence Part and the bibliography
%%     Relevance note.  Both are EIGHTEEN.  Measured from A11 v15.14, not
%%     recalled: the GRIPout longtable runs 1->2 through 17->18 and the
%%     codebook is an eighteen-row table, so the cascade is EIGHTEEN
%%     LAYERS with SEVENTEEN GRIPout transitions between them.  Both of
%%     these sites had been explicitly PROTECTED as "semantic non-edits"
%%     by an earlier spelling sweep, which is the right instinct applied
%%     to a number that was already wrong: protecting a site from a
%%     spelling sweep is not the same as checking that it is true.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list hole).
%%
%% A7 carries NO simultaneity claim, NO reading-set enumeration naming
%% A11, and NO roster row for A11; the claim-derived probe checked all
%% three before this rider was written.  (Its one apparent hit on the
%% arrival guard was "the size of one click" -- a QB quantum, not a
%% citation click.)
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.28 (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 no trailing skip
%%   and says so in one line.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.27 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.27 (August 17, 2026) -- THE -ise FAMILY.
%%
%% minimiser -> minimizer (1).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.25 -> v15.26, August 16, 2026.  THE SHOWCASE-HEADLINE RIDER.
%% \GGShowcaseHead rises from \large to \Large, one token.
%%   THE MEASUREMENT.  Across the cohort the showcase headline had four
%%   sizes: \large in the eight papers where I set it, \Large in C1 and
%%   C2, \LARGE in A4, \huge in A9, A10 and A11.  Those are not four
%%   arbitrary choices -- the rule followed was to match each paper's own
%%   existing headline, and four of the four larger values are the
%%   author's.  What the measurement shows is that MY DEFAULT was the
%%   outlier: every headline the author set himself is LARGER than
%%   \large, and on five papers out of five his practice is \Large or
%%   bigger.  On a page whose job is to be read in five minutes from a
%%   homepage, \large reads as a subsection head, not a title.
%%   So the house default rises to \Large, the smallest size he himself
%%   uses.  The cohort now spans \Large / \LARGE / \huge and nothing is
%%   below the author's own floor.
%%   VERIFIED: page count unchanged, no new overfull box, body
%%   page-for-page identical.
%% NEW PROSE IN THIS SHIP: none.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.24 -> v15.25, August 16, 2026.  THE SAVEBOX-LEADING RIDER.
%% A DEFECT I INTRODUCED IN THIS PAPER'S OWN SHOWCASE SHIP TODAY.
%%   Every showcase plate is filled into an \sbox in the front matter
%%   so the showcase and in-paper copies cannot drift.  A savebox is
%%   typeset when it is FILLED, and this paper's front matter runs at
%%   \baselinestretch 1.15 against the body's 1 -- measured with
%%   \typeout at both sites: \small leading 13.8 pt against 12.0.
%%   So every plate in a savebox was being set 15% looser than before,
%%   IN BOTH PLACES, including the in-paper float the author had
%%   already read.  A9-SM's falsification map grew 47.9 pt and pushed a
%%   paragraph off its page, which is how it was caught.
%%   This paper's showcase ship reported the resulting page as a benign
%%   "savebox reflow".  It was neither benign nor a reflow.
%%   FIX: \setstretch{1} inside every \sbox, before the size command,
%%   so the box is filled at the BODY's spacing -- the canonical one.
%%   VERIFIED: the body is page-for-page identical to the pre-showcase
%%   baseline under the +3 shift.  Not close: identical.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.23 -> v15.24, August 16, 2026.  THE FRONT-MATTER SHOWCASE.
%% Seventh of the twenty-four, and the pair to A6-GRIP v15.23.
%%   p1  Figure 4, the four Qi-4 pillars  -- the counterpart of A6's
%%       Figure 2, and the caption reads it as such: the geometric
%%       half says how an open system moves and what it spends, the
%%       informational half what the spending is denominated in and
%%       what the moving system can come to know.
%%   p2  Figure 14, the five-story pagoda -- A7's own image, and the
%%       one a reader remembers; both of its honesties are on the
%%       plate rather than behind it (five is the historically
%%       natural sequence and is not proved unique; the inevitability
%%       of the upper floors is conditional).
%%   p3  Table 47, the falsifiability program -- what would kill it.
%% Figures 1 and 2 are the shared cohort plates and are not used;
%% A5-GDOS opened its showcase with the first of them.
%% THE PLATES KEEP THEIR IN-PAPER SIZE.  A savebox cannot differ
%% between its uses, so a wider showcase copy is a heavier BODY
%% plate -- the finding that cost A5 a change to its funnel.  Both
%% A7 figures are already at \textwidth, so both showcase copies
%% are too and no rendered weight can move.
%% NOTHING IS COPIED: both TikZ bodies and the falsifiability
%% tabular go into saveboxes filled once in the front matter; the
%% showcase and the in-paper float each say \usebox.
%% NEW PROSE IN THIS SHIP: three headlines and three showcase
%% captions, all on the new pages.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
%% ------------------------------------------------------------------
%%  v15.21 (August 9, 2026): the curious-reader cluster.
%%  Twenty-five broad-audience questions open the Common Objections
%%  Part, ahead of the specialist clusters, on the model of
%%  W1-Symmetry, W2-Holonomy and A10-Cosmos.  Eight are the spine
%%  those three share, answered here in this paper's own vocabulary;
%%  seventeen are questions only an information-backbone paper can be asked: what a bit costs, what
%%%%  a memory is, and whether any of this reaches a mind.
%%  The introduction names the hardest of the set.
%% ------------------------------------------------------------------
%% ------------------------------------------------------------------
%%  v15.20 (August 9, 2026): two cohort rules applied.
%%  (1) British doubled -l corrected in prose only -- 0 site(s);
%%      6 further site(s) protected as comment, identifier or
%%      bibliography and deliberately untouched.
%%  (2) Ragged-right on 113 p-column(s) across 43 multi-p-column
%%      table(s): a narrow prose column cannot be justified
%%      without a visible interword stretch, and an overfull box
%%      under the gate's tolerance can still eat the gutter.
%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.19 (August 9, 2026): the Part closings.  The five stub
%%%%  "Summary of this Part" sections of Parts III-VII are deleted
%%%%  outright (403/312/347/558/331 characters; zero inbound
%%%%  references) and their content absorbed, in plain English, into
%%%%  each Part's closing essay, which is brought to one printed page.
%%%%  Part XI carried the same defect one level down -- it ended on a
%%%%  four-item list with no closing essay -- and is given one.
%%%%  Part I's closing brought to a page.  Parts VIII, IX, X and II
%%%%  already close on essays at the target length and are untouched.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.18 (August 9, 2026): the failure-theorem plate.  Two panels
%%%%  on one baseline -- the lawful sector with the physical state at
%%%%  the summit, and the obstructed sector where the surviving linear
%%%%  term tilts the curve and the same state sits on a slope.
%%%%  Measured standalone: ink width 146.3 mm, aspect 0.399 (width
%%%%  binds), house rule f = (7.79/9) x 146.3 / 159.2 = 0.795.
%%%%  Palette per house scheme; rendered and read collision-free.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.17 (August 9, 2026): Track B, plus a whole-paper audit.
%%%%  Track B.1: the emergence Part gains a per-floor inheritance
%%%%  accounting (what each floor of the hierarchy is answerable for,
%%%%  at what grade, and what would refute it there) and a section
%%%%  stating the exportable result -- inference derived as a theorem
%%%%  rather than assumed as an analogy -- for readers who will not
%%%%  read the pillar Part.  Track B.2: a closing subsection in the
%%%%  runtime-import Part states what this paper returns to the
%%%%  companion it imports from, completing a cross-reference that
%%%%  had been one-directional.  Audit fixes: "four pillar Parts" ->
%%%%  six (stale since the pillar Part was split); two Parts restored
%%%%  to the roadmap enumeration; the laboratory register's case
%%%%  count corrected from seven to eight with its tally re-summed;
%%%%  Q_B unit harmonized to per qubit in the front matter; one
%%%%  nine-per-cent-off approximation of Q_B replaced by the value.
%%%%  Front matter: the imported cascade now correctly carries one
%%%%  residue rather than two, and the converse discussion is recast
%%%%  without document history.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.16 (August 9, 2026): two additions to Part IV.
%%%%  (1) A compact laboratory section: three paragraphs, one per
%%%%      gauge, naming seven observational cases and the
%%%%      measurement that would refute each reading, with a
%%%%      consolidating table.  No case is worked; the Part keeps
%%%%      its derivation register.
%%%%  (2) The MinEP / least-dissipation distinction, an accuracy
%%%%      repair made visible by T-Qi4-10.7.  There are two distinct
%%%%      dual statements: Onsager's least-dissipation form is the
%%%%      flux-controlled Legendre transform at fixed forces
%%%%      (correct where stated, now attributed), while Prigogine's
%%%%      minimum entropy production is the force-side statement
%%%%      selecting the steady state -- and it is the latter, not
%%%%      the former, that maximum entropy production was ever in
%%%%      quarrel with.  Five reader-facing sites corrected and one
%%%%      table row added; nothing previously stated is withdrawn.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.15 (August 9, 2026): Part IV (Qi-EP) completed on the
%%%%  entropy-production side.  The current gauge's dual reading is
%%%%  now identified as Prigogine's minimum-entropy-production
%%%%  theorem and stated as a two-sided duality theorem
%%%%  (T-Qi4-10.7): the flux-side reading selects the constitutive
%%%%  relation with Ziegler orthogonality at lambda = 1/2, the
%%%%  force-side reading selects the state.  A new section carries
%%%%  the failure-location theorem (T-Qi4-10.8): where the curl or
%%%%  gauge sector of the balance identity does work against the
%%%%  ledger, no entropy-production functional is stationary at the
%%%%  physical state and all three gauges fail together -- with the
%%%%  classical counterexamples located and the linear-regime
%%%%  confinement derived rather than observed.  The inferential
%%%%  gauge gains an explicit non-assumption subsection (declared
%%%%  ports, the rank condition, the shared obstruction, and
%%%%  expected free energy not claimed), and the common-origin
%%%%  theorem gains a grade table.  Four new appendix subsections
%%%%  carry the derivations in full; the Fisher-metric step, which
%%%%  had asserted proportionality, is now derived as the Hessian of
%%%%  the CEP term.  Five external references added.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.14 (August 9, 2026): downstream conditioning merged to the
%%%%  current source-paper form (A6 v15.12; P2-N3 v15.5).  Ten sites:
%%%%  "M11 and R-WARP/R-REL (M10 met externally)" -> "M11 in
%%%%  spectral-flow form (M10 resolved externally; warped/relative
%%%%  refinements closed at class level)".  O-Qi4-7's objector quote
%%%%  kept verbatim; its response carries the correction.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.13 (August 9, 2026): housekeeping closure (Reports 251/254).
%%%%  26 objection persona labels 'referee' -> 'reviewer' (cohort
%%%%  convention; journal-free reader-facing text).  'tumour' ->
%%%%  'tumor' (single-sided survivor).  Q_B glossary row re-pointed
%%%%  at this paper's own Definition.  Unused \QHAL abbreviation row
%%%%  removed.  One internal-history sentence recast without document
%%%%  history.  Open-problems triage audited: consistent, untouched.
%%%% ------------------------------------------------------------------
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.11  (August 8, 2026)
%%%
%%%  ONE FIGURE PER SUBSTANTIAL PART.  A7 carried eight figures in
%%%  270 pages, of which three are cohort apparatus and one is the
%%%  roadmap: FOUR content figures against sixty-three tables, and
%%%  five of the six new Parts had none at all.
%%%  ONE plate is carried in -- W2-Holonomy's circle-and-interval
%%%  anatomy, which is the QiQUA two-fibers subsection's own
%%%  argument drawn, and whose caption's last clause named W2's own
%%%  Parts and had to be localised in transit.
%%%  TWO CANDIDATES WERE REJECTED AS FALSE FRIENDS.  A11's
%%%  'Three-Gauge Theorem' names Gibbs/Friston/action where QiEP
%%%  names trajectory/current/inferential -- one of three shared --
%%%  and A11's completeness ladder is the CASCADE-DEPTH index, not
%%%  the four-pillar QCI.  Either would have compiled, gated and
%%%  previewed clean while contradicting the facing prose.
%%%  FIVE plates drawn, house palette, each MEASURED standalone on
%%%  a 1500 mm sheet and scaled by f = (7.79/B) x W / 165.1.
%%%  The GGA plate draws the proved direction solid and the open
%%%  converse dashed, because that claim has already propagated
%%%  through the cohort in the wrong form once.
%%%  No theorem, value, table or existing caption is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.10  (August 8, 2026)
%%%
%%%  An INTRODUCTION at the head of each of the six new Parts, and
%%%  a plain-English CLOSING at the foot of five of them -- the
%%%  sixth already ends with 'Four certificates, and what they are
%%%  not', which was written for the old Part III and is about
%%%  precisely that Part's subject.
%%%  Same register as Parts I--V: no display maths, no theorem
%%%  label, no citation; the only inline maths is the name of the
%%%  spectrum.  Five status guards carried explicitly -- Qi-QUA's
%%%  falsifier is the single-rate assertion and NOT the numerical
%%%  value; Qi-EP carries no invented tolerance; Qi-CON claims a
%%%  bounded window and does not dress a retrodiction as a
%%%  confirmation; Qi-HAL is a normal form and not a necessity
%%%  claim; SeOS is non-moral and the completeness weights are not
%%%  derived, so the index ranks and does not certify.
%%%  No theorem, value, figure, table or caption is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.9  (August 8, 2026)
%%%
%%%  PART III BECOMES SIX PARTS, on the author's proposal.
%%%  Qi-QUA, Qi-EP, Qi-CON, Qi-HAL, SeOS+QCI, and Qi-4-for-the-GGA.
%%%  A6-GRIP already gives each of its four Gi-4 pillars a Part;
%%%  A7's four Qi-4 pillars are their mirror and were in one Part.
%%%  The subsections of each are PROMOTED to sections, so a Part
%%%  opens onto its eight or nine subjects rather than onto one
%%%  section with almost the Part's own name.
%%%  The GGA section moves AFTER SeOS/QCI: it opened by saying the
%%%  Part was complete and was then followed by two more sections.
%%%  Eighteen references to the old Part III re-pointed BY MEANING
%%%  -- several now name one specific pillar where the old text
%%%  said only 'Part III' -- and fifty-seven that called the six
%%%  SECTIONS re-worded, including seven mixed lists naming a
%%%  section and a Part in one breath.  The pass asserts that no
%%%  reference to any of the six is still introduced by 'Section'.
%%%  No theorem, value, figure, table or caption content changes.
%%%  The roadmap PLATE still draws ten Parts; it is rebuilt next.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.8  (August 8, 2026)
%%%
%%%  PART NUMBERS STOP BEING TYPED BY HAND.  Preparation for the
%%%  split of Part III into six.  Every Part gets a label; all 106
%%%  hand-typed 'Part~IV' style references become \ref to it.
%%%  A wrong Part number is invisible to LaTeX and to the gate and
%%%  perfectly readable -- it just sends the reader to the wrong
%%%  place.  After this it is an undefined reference, which the
%%%  gate already fails on.
%%%  THIS SHIP IS MEANT TO BE INVISIBLE: same page count, same
%%%  text, same numbers.  Anything that moves is a defect.
%%%  'Paper~IV-B' and 'Papers~I--IV-A' name companion papers, not
%%%  Parts, and are untouched; so are the three occurrences of
%%%  'the ten structural innovations', which are a different ten.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.7  (August 8, 2026)
%%%
%%%  A plain-English INTRODUCTION at the head of Parts I--V, and a
%%%  plain-English CONCLUDING SECTION at the foot of each, on the
%%%  author's instruction and on the A5/A6 model.  Part VI
%%%  (Discussion) and Part VII (Conclusions) are excluded on
%%%  instruction; Parts VIII--X are back matter and already open in
%%%  this register.
%%%  No equation appears in any of the ten pieces, no theorem label
%%%  precedes its content in English, and five status guards the
%%%  body carries are carried here too: the imported cascade's two
%%%  open conjectures; the one-directional ground-attractor
%%%  characterization with its converse open; five floors as an
%%%  empirical reading; ensemble and NOT trajectory inevitability;
%%%  and the operational, graded, experience-neutral reading of the
%%%  internality Part's central term.
%%%  No theorem, value, figure, table or caption is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.6  (August 8, 2026)
%%%
%%%  NEW ABSTRACT and NEW EXECUTIVE SUMMARY, in plain English,
%%%  on the model adopted for A5-GDOS and A6-GRIP.
%%%  The old ones were written for a reader who already knew the
%%%  program: the Abstract's second sentence was about ontological
%%%  categories, and the Executive Summary spent its first
%%%  paragraph on Maxwell, Schrodinger and 't Hooft before naming
%%%  its own subject.  Four repairs of substance:
%%%    (1) QB is called DERIVED, which is what T-Qi4-9.8 proves;
%%%        ATP is named as its anchor, not its source;
%%%    (2) the four pillars are stated as four questions anyone
%%%        can ask about a thing that persists;
%%%    (3) the conditional status the body carries everywhere is
%%%        carried here too -- M10/M11, O7, O8, PS-Qi4-6, and the
%%%        organizational-not-predictive caveat;
%%%    (4) the paper's one original discovery -- that a
%%%        memory-bearing system internalizes the second runtime
%%%        regime -- gets a paragraph instead of a clause.
%%%  No theorem, value, figure or table is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.5  (August 8, 2026)
%%%
%%%  What the Arisaka constant is CALLED -- a scope rule.
%%%  \QB{} = (2/5) E_Ry / K with K = 35: an anomaly coefficient,
%%%  the Rydberg energy, and a ledger integer.  No biology enters
%%%  the derivation, and the paper's own T-Qi4-9.8 calls it
%%%  'derived data, not an adjustable input'.  ATP is its ANCHOR,
%%%  not its origin.  So the modifier 'bioenergetic' is corrected
%%%  to 'energy--information' at the ELEVEN sites where it names
%%%  the CONSTANT, and left untouched at the NINETEEN where the
%%%  subject is genuinely biology -- the ATP anchor, the two-quantum
%%%  bond, the chemiosmotic stratum, and two named personae.
%%%  The line that made the case, in Table 47, read 'bioenergetic
%%%  conversion factor for the lepton/quark mass ratios'.
%%%  Default-PROTECT: every surviving site must match a
%%%  hand-classified keep pattern or the run stops.
%%%  No value, theorem, figure or caption content changes.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.4  (August 8, 2026)
%%%
%%%  STAGE 5 OF THE v15 MIGRATION -- stranded pages.
%%%  Four forced \pagebreak commands removed from inside the
%%%  abbreviation longtable, the same defect A6 carried.  After
%%%  the caption pass and the List of Tables moved the table by
%%%  two pages, one of them broke with two rows left over on an
%%%  otherwise blank page.  longtable now breaks where it wants:
%%%  251 -> 249 pages, stranded page gone.
%%%  No row, caption, value or figure is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.3  (August 8, 2026)
%%%
%%%  STAGE 4 OF THE v15 MIGRATION -- front matter.
%%%  (1) LIST OF TABLES.  A1--A6 all call \GGListOfTables; A7 did
%%%      not, with SIXTY-THREE table floats.  A6's block adopted
%%%      byte for byte, one-page toggle and all.
%%%  (2) The four-hundred-character FOOTNOTE INSIDE THE ABSTRACT
%%%      is moved to the first prose mention of the 6D bulk after
%%%      it.  arXiv abstracts are plain text.  One clause reworded
%%%      to avoid printing the same citation twice in one line;
%%%      no claim in the note changes.
%%%  No caption, table, theorem, value or figure is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.2  (August 8, 2026)
%%%
%%%  STAGE 3 OF THE v15 MIGRATION -- captions.
%%%  A7 carried seventy captioned floats with a MEDIAN CAPTION OF
%%%  FIFTY-NINE CHARACTERS, sixty-four below the 260-character
%%%  standard, and six of seventy-two with a short form.  Every
%%%  caption is rewritten to say what the float SHOWS and how it is
%%%  ARRANGED, and every one gains the one-line short form the List
%%%  of Tables prints.  Where a table carries a number, an integer
%%%  or a grade, the caption names it.
%%%  One appendix longtable carried NEITHER caption NOR label; it
%%%  is given both.
%%%  No table content, theorem, value or figure is touched.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.1  (August 8, 2026)
%%%
%%%  STAGE 2 OF THE v15 MIGRATION -- cohort parity and palette.
%%%  (1) The shared acronym plate taken byte for byte from
%%%      A1-Constitution, the cohort copy since Report 246.  A7
%%%      carried the pre-parity variant, whose GRIP sub-line
%%%      OVERLAPS the Qi-HAL box in print, and the retired
%%%      'bioenergetic' wording.  A7's cohort MAP is already
%%%      byte-identical to the frozen twenty-two-paper plate:
%%%      verified, not touched.
%%%  (2) P6-Proton, W1-Symmetry and W2-Holonomy added with the
%%%      prose that cites them.  All three were already NAMED in
%%%      A7's own cohort map and cited nowhere -- the same defect
%%%      A6 carried for P6.
%%%  (3) House palette: twelve bare `draw' borders take the hue of
%%%      the fill in their own option group, fills above !18
%%%      compress into [13,16], keyword widths become explicit.
%%%  Body text, theorems, values: UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A7-Qi4  v15.0  (August 8, 2026)
%%%
%%%  STAGE 1 OF THE v15 MIGRATION -- mechanical only, no prose
%%%  rewritten, no theorem, value or figure touched.
%%%  (1) US Letter (a4paper -> letterpaper).  The measure widens
%%%      452.97 pt -> 469.76 pt and the whole paper reflows.  A7
%%%      has no landscape environment, so the A6 landscape hazard
%%%      does not apply, and booktabs is already universal here
%%%      (74 toprule, zero hline).
%%%  (2) American spelling.  A7 is the first paper in the cohort
%%%      whose majority was British -- programme 49 to program 28.
%%%      Rule sweep plus a curated list of six words the rule
%%%      cannot see.  Identifiers untouched; no cited title needed
%%%      protecting, which was checked rather than assumed; and
%%%      `analyses' is left alone because it is the plural of
%%%      analysis, not the verb.
%%%  (3) Stamps: v13.15 -> v15.0.
%%% ==================================================================
%% ===================================================================
%%  A7-Qi4 v13.15  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  3 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.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.14  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 114mm boxes and then
%%  stretched to the full text width, so its \footnotesize landed 47% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.681\textwidth, computed rather than chosen: the picture measures
%%  125.2mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.13  (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.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.11  (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.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.10  (July 29, 2026)  -- the last two residuals in the cohort.
%%  Two "as of May 2026" stamps removed: one on the coherence maps, one on the
%%  open-problem triage.  Both told a reader the companions had moved on since
%%  a working date.  Citation discipline and triage content unchanged.
%%  With this ship the cohort-wide internal-affairs scan returns ZERO.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.9  (July 29, 2026)  -- one internal-document citation.
%%  "rows are re-walked against its revised internal numbering per L7 of the
%%  GG-Theory instructions" -> the property that makes the map robust: each row
%%  names a theorem identifier, so drift surfaces as a mismatch.  Same
%%  treatment as A1 v13.5 and A11's citation-auditor objections.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.8  (July 29, 2026)  -- internal-affairs sweep.
%%   * THREE references to "the user" removed from reader-facing text -- the
%%     author addressed as a participant in a working session:
%%       "the user's requested convention"
%%       "exactly what the user requested in conceptual language"
%%       "the formal version of the user's phrase that consciousness is ..."
%%     In each case the substance is the paper's own and is now asserted in the
%%     paper's own voice; the convention, the conceptual claim and the informal
%%     description of consciousness are all kept word for word.
%%   * "(added 2026-07-01)" dropped from the BFFT scope note.
%%   * The QiHAL honest-scope remark no longer justifies its weakened claim by
%%     what "an earlier version of this theorem advertised".  THE CONCESSION IS
%%     PRESERVED IN FULL: two motifs definitional, three derived, canonical
%%     normal form rather than strict universality, falsifier unchanged.
%%   * "O5--O9 newly added in this revision ... identified during the Phase 6
%%     hostile-reviewer pass" -> the gaps a hostile reading exposes.  The
%%     internal review stage is no longer named to the reader.
%%   * Objection response "Corrected:" -> "Conceded on the priority."  The
%%     Held--Hein / von Holst--Mittelstaedt priority concession is unchanged.
%%   * Bibitem "the vocabulary adopted in this revision" -> "adopted here".
%%   * "Phase 6" -- an internal review stage -- was named NINE times, mostly
%%     as row annotations in the cross-reference longtable telling the reader
%%     which internal pass renamed a theorem or weakened a claim ("Phase 6
%%     renamed", "Phase 6 weakened 'unique' to 'sufficient'", "Open problem O7
%%     from Phase 6 records ...", "Per the hostile-reviewer pass").  Every
%%     substantive statement is kept -- including that 'unique' is weakened to
%%     'sufficient' and that the pagoda's "five" is not uniquely forced -- and
%%     only the stage label goes.
%%   * TWO RAW SOURCE-LINE REFERENCES replaced with real labels:
%%     "Eq.~near L2843" -> \eqref{eq:arisaka-constant}
%%     "Definition near L5912" -> \ref{def:qci}
%%     These were editor placeholders: they resolve to nothing for a reader and
%%     drift with every edit to the file.
%%  No theorem, definition, value or falsifier changed.
%% ===================================================================
%% ===================================================================
%%  A7-Qi4 v13.7  (July 29, 2026)  -- Stage S5-A (Report 67, sites 34-35)
%%  Two citations into A3-BI6 pointed at coordinates that do not exist in
%%  the deployed companion: there is no section 8.7 (the GS-as-GGEP
%%  reading is section 10.7) and no Theorem 3.4 (section 3 carries only
%%  Theorem 3.2).  Both repointed to named content plus version-stable
%%  T-BI6 tags, which do not move when the companion is re-numbered.
%%  No physics changes.
%% ===================================================================
%% ===================================================================
%% v13.5 -> v13.6 (July 28, 2026).
%%   * The Qi-4 / ground-attractor relation restated.  The pillars
%%     characterise the Ground Geometric Attractor; they do not select
%%     it, because the companion runtime paper's per-stratum corollary
%%     already produces it from the landscape alone.  The former
%%     selector proposition becomes a theorem in the direction this
%%     paper can prove, with the converse stated as open and its old
%%     circular warrant withdrawn.
%%   * This paper's restatement of the ground-attractor definition
%%     drops its admissibility clause and imports the upstream
%%     definition unchanged.
%%   * The two "imported unconditionally" statements re-graded: the
%%     import carries one stated assumption of its own, distinct from
%%     the cascade-closure conjectures.
%%   * Qi-HAL definition: the internality layer enters as the fifth
%%     layer; the runtime switch leaves, being upstream content this
%%     paper consumes rather than defines.  The normal-form theorem,
%%     its proof, and the honesty remark conformed.
%%   * The consistency vector's fourth and fifth components identified
%%     as the two faces of Qi-HAL.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ====================================================================
%%  v13.4 -> v13.5  (July 28, 2026)  --  SHIP A7-1: SOURCE FIXES
%% --------------------------------------------------------------------
%%  Five internal inconsistencies, found while auditing whether
%%  A11-Origin presents A7's concepts faithfully.  Three of them A11 had
%%  inherited.  Each is a place where A7 contradicts A7.
%%
%%  (1) NHT and HAL are SWAPPED in the abbreviation table, against this
%%      paper's own body ("HAL: the holographic attractor lattice as a
%%      memory substrate") and its own pillar row.
%%  (2) The abbreviation table cites the SIP as "\S 6"; the SIP is \S 8
%%      and the theorem is T-SIP (T-Qi4-8.2).  \S 6 is the GRIP import.
%%      Replaced by the theorem key, which does not move on renumbering.
%%  (3) The \QCI macro names three different objects: the \S12
%%      Qi-HAL-only five-indicator average, def:qci's four-pillar
%%      weighted geometric mean, and a \S16 "coordination index".  The
%%      first is renamed QCI_HAL -- which is what this paper itself
%%      calls it once it becomes a component of q_HAL -- and the third
%%      is corrected.  \QCI now means def:qci everywhere.
%%  (4) The Qi-HAL factorisation is FIVE mutually coupled layers in
%%      def:qihal and thm:qihal-universality (MePMoS, NHT, HAL, BFFT,
%%      DGI-GSSB switch) but "four-stage" in three places, including in
%%      Open Problem O6.  The fifth layer is the one q_HAL scores.
%%
%%  Not changed: the two long-form keys T-Qi4-3.3 and T-Qi4-22.1, whose
%%  section numbers are stale.  Renumbering a published key is worse
%%  than leaving it; the convention is that keys freeze at assignment.
%% ====================================================================
%% ===================================================================
%% v13.4 (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.
%% ===================================================================
%% ===== v13.3 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F2: P5-CPT bibitem title text -> current P5 v3.42
%%     cover title (label/key untouched).
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% ===== v13.2 (2026-07-20): STAGE 5A -- Gi4 RETIREMENT (A7) =====
%% Wave-2 housekeeping batch, ship 6b.  As in A6: the paper's
%% \providecommand{\ASix}{GG-A6-GRIP} was a no-op against the
%% sty's predefined retired default GG-A6-Gi4, so all 60 \ASix
%% macro sites (~65 rendered lines in the v13.1 PDF, incl. the
%% cover-page GDOS-Trio line) rendered the retired name.
%% (1) \providecommand -> \renewcommand for \ASix; all sites now
%% render GG-A6-GRIP.  (2) \AFullSix literal corrected
%% Arisaka-GG-A6-Gi4 -> Arisaka-GG-A6-GRIP + \renewcommand
%% (unused in body; stale definition retired; cohort precedent
%% A1/A2/A10).  (3) Suffix roster: 'GRIP (A6, formerly Gi4)' ->
%% 'GRIP (A6)' (A8 v13.2 register precedent).  The v13.0
%% cohort-map block's two literal GG-A6-GRIP workaround sites now
%% match the macro and stay.  Bib KEY ArisakaA6Gi4 (non-rendering)
%% and the Gi-4 pillar OBJECT (\GiFour) untouched.  Physics,
%% theorem keys, numbers, values, grades: UNCHANGED.  v13.1 ->
%% v13.2 (\PaperDate unchanged).  Script: a7_v13.2_stage5a.py.
%% ===== v13.1 (2026-07-20): STAGE 3 OF WAVE 2 -- A7 FLAGSHIP REPAIR =====
%% Re-derivation of the informational carrier Psi_Q from the handshake
%% (v13.0 -> v13.1). A7 was the cohort's last Class-A primitive-i site
%% (the only paper that INSTANTIATES a complex primitive); this ship
%% converts it into a consumer of the derived quarter-turn.
%% (A7-R1) Carrier repair at eq:qihal-psi/-gauge-transformation/
%%   -covariant-derivative: the passage now LEADS with the real pair
%%   (rho_Q>=0, phi_Q a real circle-valued angle) as constitutional data
%%   and DERIVES Psi_Q = sqrt(rho_Q) e^{i phi_Q} as the boundary
%%   complexification under the derived quarter-turn: U(1)_Qi a compact
%%   bulk gauge circle (four-component BI-6 sector F_Qi), T-P4-1 the
%%   unique circle-compatible complex structure (gauge i), T-P4-2 the
%%   state-space i, T-P4-3 their forced on-shell agreement. The three
%%   labelled equations keep labels AND rendered forms verbatim; repair
%%   is in prose + pedigree (T-P4-1/2/3; L-Constitution-1 candidate;
%%   AX-3 register note of A1). Gauge action = real shift phi_Q ->
%%   phi_Q + q_Q Lambda; every i is the derived quarter-turn.
%% (A7-R2) Thermal-clock attribution after eq:qiqua-qb-conversions
%%   (candidate): eV<->k_BT is the thermal clock, T-P5-B1. Numbers
%%   thermal row numbers UNCHANGED.
%% (A7-R3) Arrow-custody consumer pointer at the QiEP section head:
%%   AX-3 owns the arrow (A1 AX-3 synthesis); law-level P5 Part IX over
%%   the T-symmetric bulk; dynamical A5-GDOS / GG-A6-GRIP; Qi runtime
%%   consumes. A6 named by literal textsf GG-A6-GRIP (macro trap: the
%%   sty renders ASix as GG-A6-Gi4 in this paper).
%% (sector pedigree) Clause + memory-phase register sentence at the
%%   four-component bulk statement: Qi circle is a real compact fibre,
%%   its boundary i the derived quarter-turn (T-P4-1); QiHAL's memory
%%   register is a real angle on a real circle.
%% ii comment updated to name the boundary-derived quarter-turn.
%% Physics numbers, theorem keys, A7's own grades: UNCHANGED. Frozen
%% v1.4 cohort map, Part-I close, abstract/exec: untouched.
%% Script: a7_v13.1_stage3.py.
%% ===== v13.0 (2026-07-19): WAVE 1 OF THE 19-PAPER PROMOTION SHIP =====
%% Promoted from the v11 series (v11.15 -> v13.0; A7's first v13-series
%% release; no v12 series existed for A7).  Mechanical cohort-promotion
%% sweep (17 -> 19 papers) plus the standardized Part-I close
%% (author-approved Wave-1 scope; audit:
%% GG-Internal-Report-RealBulk-Audit-2026-07-19.md, A7 ledger (2.7) +
%% App. A).
%% (1) NEW closing section of Part I, "The cohort inheritance and the
%% roadmap of the ten Parts" (A2-GG6 v13.3 / A3-BI6 v13.6 / A4-Euclid
%% v13.0 / A5-GDOS v13.0 / A6-GRIP v13.0 architecture), merging the
%% former adjacent sections "The GGTheory Cohort" and "Roadmap of
%% the Ten Parts" (already Part I's closers): subsection "The cohort
%% inheritance" (label sec:cohort-a7 kept; the five-strata prose
%% migrated verbatim, opening paragraph recounted to nineteen with the
%% 11+5+3 breakdown; NEW roster-level Foundations Branch paragraph in
%% A7's voice -- A7 carries the four Qi-4 pillars (QiQUA, QiEP, QiCON,
%% QiHAL), the quantum-information runtime on the memory-bearing
%% U(1)_Qi systems of the boundary; GG-P4-Real named the origin of the
%% boundary's complex structure whose phase the U(1)_Qi memory carrier
%% stores; GG-P5-CPT's mirrors and law-level arrow named kin of the
%% runtime's one-way information cascade -- one sentence each, roster
%% level, no custody install, no Psi_Q claim) closing with the frozen
%% v1.4 nineteen-paper cohort map on its own page (extracted verbatim
%% from A2-GG6 v13.3, MP4/MP5 rendition; label fig:cohort-map-a7 kept;
%% the old seventeen-paper map retired; per the A6 v13.0 caption-fix
%% lesson the block's two \ASix{} sites carry literal
%% \textsf{GG-A6-GRIP}, because the sty's \providecommand{\ASix}
%% renders the retired name GG-A6-Gi4 in this paper), and subsection
%% "Roadmap of the ten Parts" (label sec:roadmap-a7 kept; new warm
%% prose walkthrough replacing the paragraph-per-Part bold list)
%% closing with a NEW roadmap figure (label fig:roadmap-a7 kept) in
%% the house minipage architecture.  Highlighted Part unchanged: Part
%% III, the four Qi-4 pillar theorems + the Arisaka lock (the paper's
%% central claim and its suffix).
%% (2) COUNTS: reader-facing cohort counts seventeen -> nineteen at
%% the audit's sites (Part-I close x6 via the migration -- the opening
%% paragraph x3, the retired map caption x2, the roadmap Part-I line
%% -- plus the intro architecture walk, the Objection-50 lead
%% "Seventeen interlocking papers", and the references Part intro
%% x2).  Protected semantic non-edits: "runs this alternation through
%% seventeen layers" (A11 pagoda import), "seventeen-layer cascade"
%% (A11 bibitem Relevance), "the ten Parts together with the sixteen
%% appendices" (in-paper structure).
%% (3) ROSTER: the naming-convention suffix list adopts the canonical
%% nineteen-paper form, gaining the foundations preprints Real (P4)
%% and CPT (P5).
%% (4) MACROS + BIB: \PaperPFour (GG-P4-Real) and \PaperPFive
%% (GG-P5-CPT) installed (providecommand verified effective: the sty
%% defines neither); NEW bibitems ArisakaP4Real + ArisakaP5CPT (A7
%% house style, P3-Electron pattern) with roster-level Relevance
%% notes.  No P4-Twistor bib key ever existed in A7 (nothing to
%% rename); zero "satellite" tokens of any register; zero "Twistor"
%% tokens outside the P4 official subtitle now quoted in its bibitem.
%% (5) ONE-LINE CONTENTS: \GGPart renewed per the P4-Real v3.2
%% preamble pattern (contents line carries only "Part N. <title>");
%% \GGAppendixPart (Part VIII) inherits the renewal.
%% Wave-2 content items deliberately NOT installed (the Psi_Q reality
%% repair A7-R1 -- the postulated complex informational carrier
%% Psi_Q = sqrt(rho_Q) e^{i phi_Q} at eq:qihal-psi and the \ii
%% "imaginary unit" providecommand are byte-identical to v11.15;
%% that repair is the cohort's flagship Wave-2 item -- the
%% thermal-clock attribution A7-R2, the QiEP arrow-custody pointer
%% A7-R3, any QB/shore connection, any Wick-quartet install, any
%% DGI/U(1)_Qi pedigree lines).  Physics, theorem keys, numbers,
%% values, grades: UNCHANGED.  v11.15 -> v13.0.
%% Script: a7_v13.0_wave1.py.
%% v11.15 (July 16, 2026): cohort table standard (author-approved, P4 v2.17 pattern): sub-small table fonts -> \small; p-column tables widened toward full text width; targeted corrections listed.
%%   correction: tab:appB-glossary: p-width reduced (overfull 48.7pt; wide symbol l-column; matches A5/A6 fix)
% v11.14 (July 2, 2026): cohort universal-indentation pass (author-
%   directed): indentfirst loaded (first paragraphs after headings
%   now indent). Per-paper stray-noindent review deferred to the
%   v2.0 pass. No content changed.
% v11.13 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 6
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% v11.12 (July 1, 2026): BFFT cosmological scope note added after the
%   Pillar-III/IV bridge sentence: the ell_D knee reading is an interpretive
%   overlay (+2.3% above A10's Boltzmann value); A7 supplies no holonomy
%   normalisation. Stationary-Measure programme, 2026-07-01. Nothing else.
% ===== v11.11: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 43 oversized (unsized) table floats so every table
%  sits ~10%% below the body text (matches A10 v12.43). Already-small tables untouched.
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% A10 bib-title reconciled to canonical cover title (cross-citation integrity), 2026-06-27.
% ===== A7-Qi4 v11.8 (2026-06-27): cohort de-contamination (protect-list aware). =====
% Shed GG's own bulk-geometry self-description: the cohort '6D bulk' stage no longer carries the
% scale-running qualifier; OGGEP/bulk boundary projection -> bulk-to-boundary projection; the
% bulk--boundary 'dictionary' -> bulk--boundary correspondence; scale-running-RG construction ->
% renormalization-group construction; scale-profile/depth where a metric factor was named; the
% reduction map -> boundary reduction map. KEPT verbatim: the Qi-HAL pillar vocabulary (QiHAL,
% Holographic Attractor Lattice, Neural Holographic Tomography, phase-holographic coding, the
% holographic memory architecture, OH-IH holographic semantics/observation), the external
% holographic PRINCIPLE + AdS/CFT + Maldacena/'t Hooft/Susskind citation contexts, the
% Randall--Sundrum extra-dimension-model glossary entry, and the RG-running disclaimer substance.
% All energy-information content (ATP / Q_B / hydrolysis / kJ/mol / Gibbs / bioenergetic) is byte-
% stable. Body only (after the document body marker). No physics/values changed. =====
% ===== A7 v11.7 (2026-06-25): cohort central-claim funnel (light, foundation variant). =====
% Added a modest 'place in the cohort' section (sec:cohort-funnel) leading with the paper's own
% subject, plus the shared open-system funnel (fig:cohort-funnel, generic payload). Framing kept
% deliberately light per author guidance; the paper proceeds on its own terms. No new physics.
%% =====================================================================
%%  A7-Qi4 v11.6 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  gg-6 -> gg-11; homogenized Acknowledgments;
%%  stripped Overview/LoF overrides; single-page Overview+LoF via the gg-11
%%  toggle; Preface via \GGPreface; ES centered by gg-11.  No
%%  prose/figure/number content changed.
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.5 (Phase 6: Part-count reconciliation + smoothness) -- 2026-06-20
%%  Reconciled every aggregate 'six-part' description to the actual TEN-Part
%%  structure (six main-argument Parts I-VI + back matter VII Conclusions,
%%  VIII Technical Appendices, IX Common Objections, X References): preface
%%  architecture walk rewritten; 'stop after Part VI'->VII; 'six Parts plus
%%  sixteen appendices'->ten Parts; two appendix-caption count phrases.
%%  Removed the internal drafting-note subsection (no reader content).
%%  No physics changed.
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.4 (Phase 5C: Common Objections Part) -- 2026-06-20
%%  New four-Part-tail member: Common Objections and Responses, 53 hostile
%%  objections O-Qi4-1..53 in five persona clusters (particle theory 1-12,
%%  systems biology 13-24, origin-of-life 25-35, neuroscience 36-47,
%%  methodology 48-53), authored from the Phase-5A adversarial reviews,
%%  answered sharply + honestly with located cross-links.  Inserted before
%%  the References Part.  No physics changed; Q&A only.
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.3 (Phase 5B: honest-defense revisions) -- 2026-06-20
%%  Answers the Phase-5A hostile reviews in the main text without weakening
%%  the logic: two-knee falsifier restated to what is actually derived (two
%%  co-occurring knees bracketing a bounded band, positions system-specific);
%%  'unifies these' -> 'subsumes as gauges/limits' (4 pillars); Arisaka-eqn
%%  unit-convention content made explicit (rank-one + single Q_B); M10/M11
%%  conditionality re-flagged at QiHAL-universality, pagoda, synthesis; pagoda
%%  arbitrary-depth (five = empirical reading) made explicit; consciousness
%%  scoped to ACCESS (Block 1995), phenomenal hard problem left open; Kalman
%%  observability + active perception (Bajcsy; O'Regan-Noe) credited; F-Qi4-5
%%  Held-Hein/von-Holst credited + sharpened; BFFT lambda=2 flagged as minimal
%%  idealization vs non-dyadic theta-gamma (Lisman-Jensen). +7 refs (Group B
%%  re-sorted).  No value/result changed.
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.2 (Phase 5B: Q_B<->ATP reconciliation) -- 2026-06-20
%%  Corrects 'Q_B = one ATP' (factor-of-two slip) to the cohort-canonical
%%  'one ATP = 2 Q_B': Delta G'(ATP) = 2 N_A Q_B = 30.0 kJ/mol vs measured
%%  30.5 +- 0.5 (~98.4%), matching A11-Origin's GGA_7.  No value changed
%%  (Q_B stays 0.1555 eV).  Fixed intro, QiQUA inline def, Conclusions x3,
%%  glossary, and Open Problem O5 (factor-of-two now RESOLVED).  Added a
%%  rigorous bioenergetic-anchor paragraph + a c-ring environment-tuned
%%  optimization forward-pointer (c8 mammals .. c10 E.coli .. c13-15
%%  alkaliphile/chloroplast; full study forthcoming).
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.1 (Phase 2: cohort + roadmap + cite all 17) -- 2026-06-20
%%  Presentation only; no physics changed.  Mirrors A6-GRIP v11.8.
%%   (1) dedicated 'The GG-Theory Cohort' section: relocated cohort figure to
%%       its own full page + five-strata prose citing all 17 cohort papers.
%%   (2) 'Roadmap of the Ten Parts' section: vertical Parts flowchart (final
%%       10-Part structure; Part III highlighted) + per-Part walkthrough.
%%   (3) added 7 cohort bibitems absent from A7 (A4,A8,P2,P3,C2,C3 + self),
%%       verbatim from A1 v11.13 (First-cited tail stripped; added in Phase 3).
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.1 (Phase 1: v2.0 structure) -- 2026-06-20
%%  Presentation only; no physics/parameter/result changed.  Mirrors the
%%  ratified A6-GRIP v11.8 / A5-GDOS v11.8 template (same gg-6 style).
%%   (1) R4 single-page front matter (Overview of Contents + List of Figures
%%       on one shared page, then the detailed Contents).
%%   (2) OH+IH=EIC runtime promoted to its own content Part (author-approved;
%%       6 content Parts, matching the paper's existing prose).
%%   (3) Conclusions now its own Part; standard acknowledgments macro moved
%%       to the END of that Part (replacing the hand-rolled ack after the
%%       appendices).
%%   (4) Hand-rolled appendix divider -> GGAppendixPart (Technical Appendices
%%       Part; appendix called exactly once).
%%   (5) References now its own Part (3-paragraph intro + bibsection spacing).
%%  Common Objections Part + final Part-count reconciliation follow in later
%%  phases.
%% =====================================================================
%% =====================================================================
%%  A7-Qi4 v11.1 (Phase 0 science/cohort reconciliation) -- 2026-06-20
%%  Presentation/honesty only; no physics, parameter, or result changed.
%%  FIX 1: corrected the two SUMMARY statements of the A6 import to the
%%         accurate mapping -- A6 GFF = principal theorem T3, A6 GRIP =
%%         principal theorem T4 (T1=TGG, T2=CCI supply conservation+ledger);
%%         SIP is A6's packet construction, not a principal theorem. (A7's
%%         per-object proof citations already used T3/T4 correctly.)
%%  FIX 2: honesty caveat -- A6 v11.8 GRIP cascade closure (T4, N<=3, the
%%         dual-layer C_GRIP extension) holds modulo A6 open conjectures
%%         M10/M11; A7 GRIP-dependent constructions inherit that conditional
%%         status; A6 GFF-ST7 (GGA existence+uniqueness) imported uncondition-
%%         ally; softened 'completed runtime constitution'.
%%  Verified vs current A6-GRIP v11.8 and A11-Origin v12.30.
%% =====================================================================
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2. =====
%  Terminology only. No AdS-bulk branding; A2/A3 bibitems AdS-free; AdS/CFT kept.
%  Added holographic-RG disclaimer footnote; v10.6 -> v11.0.
% =====================================================================
% --- v10.6 (2026-06-16): adopt cohort-default Part-aware running header (sty v10.6).
%     Curated short Part labels added; running header shows 'Part N. <short>'. Preface self-labels. ---
% ALWAYS-ON PROJECTION CLARIFICATION (June 14, 2026): aligned with GG-A5-GDOS Sec 23 ('Why the Projection Is Always On'). The 6D->4D OGGEP/GDOS projection is always on and observer-independent; 'measurement' is only its macroscopic record-keeping special case (decoherence the microscopic one). Removed framing that tied the projection to observation/measurement; added (A5 Sec 23) cross-refs.
\pdfoutput=1
%% v10.5 (June 12, 2026): Legacy-name fix (author ruling: GD-2026-5
%%   = GG-A3-BI6). Recap-bridge section title, pdfsubject, the PFive
%%   macro, the GGEP-completion citation, and the comparison-table row
%%   now point to \AThree{}/ArisakaA3BI6; the duplicate legacy bibitem
%%   Arisaka2026GD5 removed and its content noted in the ArisakaA3BI6
%%   Relevance. No other changes.
%% v10.4 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-a7, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   front-matter audit: conforms; stamp bump only.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
% ======================================================================
% PROVENANCE
% ----------------------------------------------------------------------
% This file is the Phase 1 identity-migration baseline of the A7-Qi4
% paper.  It is a verbatim copy of UCLA-Arisaka-GG-2026-4B.tex
% (v1.0, April 12, 2026), modified only in:
%   * the document class \usepackage line (gg -> gg-6) (L1)
%   * the \PaperNumber / \PaperStatus / \PaperVersion / \PaperDate macros
%     (L9, L15)
%   * the bibtex citation keys throughout the body (L3)
%   * the bare GG-2026-N paper-number references in prose (L2)
%   * the bibitem labels (new convention: [Arisaka(2026, A*-Suffix)])
%   * the bibitem bodies (UCLA-Arisaka/GG-2026-N -> Arisaka-GG-A*-Suffix)
%   * new \setcitestyle{comma} so inline Arisaka self-citations render
%     as "(Arisaka, 2026, A*-Suffix)" per user instruction (May 11, 2026)
%   * new \providecommand{\MEPP}{...} per L21(a)
% No prose has been edited, no theorem statements changed, no figures
% added or removed.  Phase 1 is identity-only.
%
% v9.2 (June 7, 2026): GGA + GRIP-In/GRIP-Out incorporated into the
%   Qi-4 framework, aligning A7 with the June-2026 runtime vocabulary
%   (A6-GRIP v9.3 def:GGA, v9.5 def:GRIPin/def:GRIPout; A11-Origin
%   T-GGA-as-Ground-State + regime theorems).  Headline additions:
%   (1) NEW section "The Qi-4 pillars as the GGA selector" before the
%       SeOS/QCI part -- repairs the inheritance inversion (A5/A6/A8/
%       A9/A10/C1 cite the four pillars as the GGA admissibility
%       filter; A7 now states and houses that claim, with the
%       pillar-by-pillar mapping QUA=ledger, EP=variational,
%       CON=scale-coherence, HAL=halting/no-exit);
%   (2) runtime-import section: def:GGA + the two regimes imported
%       (terminus refined GA -> GGA; GRIP-In = fixed-landscape descent,
%       GRIP-Out = landscape reorganization);
%   (3) QiHAL GRIP-loop recognised as the INTERNALISED regime switch:
%       a(t) through zero = endogenous, recoverable GRIP-Out; settling
%       = GRIP-In; the loop realises A11's Cascade-Self-Sustenance
%       microscopically;
%   (4) pagoda: story-to-story ascent = GRIP-Out, life within a story
%       = GRIP-In ending at the story's GGA;
%   (5) staleness fixes: GUT-10 retired (-> GG-Theory/GG-Constitution,
%       6 sites incl. keywords); \ASix macro + A6 bibitem renamed
%       Gi4 -> GRIP with current title; A11-Origin bibitem added;
%       abbreviation table +3 rows; ES alignment paragraph; keywords.
% Phase log (v5.X cycle):
%   v5.0 (May 11, 2026)  Phase 1: Identity migration baseline.
%                        Style ucla-arisaka-gg -> ucla-arisaka-gg-6 (L1).
%                        PaperNumber A7, version v5.0, status Preprint
%                        (L9, L15).  4 bibtex keys migrated to v3.1+
%                        canonical (Axiom -> A1Constitution; GG6 -> A2GG6;
%                        GDOS -> A5GDOS; GRIP4A -> A6Gi4).
%                        Arisaka2026GD5 left unchanged (separate legacy).
%                        Body paper-number labels relabelled GG-2026-N ->
%                        A*-Suffix (L2); bare "GG-2026-4" mapped to
%                        "A6/A7 (runtime-constitution programme)".
%                        Bibitem labels reformatted to the new convention
%                        [Arisaka(2026, A*-Suffix)] so inline citations
%                        render as "(Arisaka, 2026, A*-Suffix)" per user
%                        instruction.  UCLA-Arisaka preprint identifiers
%                        retired in favour of Arisaka-GG-A* canonical
%                        names throughout the inline bibliography.
%                        \setcitestyle{authoryear,round,comma} added to
%                        preamble.  \MEPP macro added per L21(a).
%                        No content change.  Forthcoming phases: 2 (GS-as-
%                        GGEP framing audit), 3 (L4 historical-inventor),
%                        4 (Pillar theorem statements with dual-layer
%                        L16 architecture: Qi-QUA, Qi-EP, Qi-CON, Qi-HAL),
%                        5 (Open Problems triage with L6 status headers),
%                        6 (hostile-reviewer + reference-scanner sub-
%                        agents), 7 (L7 cross-paper coherence map), 8
%                        (L8 TikZ figure compliance: Qi-4 architecture,
%                        EIC, pagoda, Arisaka equation), 9 (front-
%                        matter rewrite to A1-canonical + Conclusions
%                        philosophical close), 10 (bibliography surname-
%                        sort + journal-stripped variant), 11 (Appendix
%                        A canonical 9-section + L20 hierarchy figure +
%                        L21 sans-serif audit + Bridge-to-Part paragraphs
%                        + 36-item v3.4 audit + L25 length check).
% ======================================================================
%   v6.2 (May 11, 2026) Phase 2: GS-as-GGEP framing audit (L5) +
%                        bibitem upgrade.  User-requested skip-version
%                        v5.0 -> v6.2.  Surgeries:
%                          (1) GS-as-GGEP framing patch at recap-bridge
%                              section: one-sentence cross-cite to
%                              A3-BI6 §8.7 ("BI-6 as the GGEP completion
%                              of GS: the geometrisation reading")
%                              inserted into the "Paper II established..."
%                              sentence.  The other 5 GS mentions in A7
%                              are object-name references (BI-6 acronym
%                              expansion, "GS-inflow-stabilized bulk",
%                              etc.) and are left intact per L5.
%                          (2) Bibitem upgrade for all 5 Arisaka self-
%                              citation entries per user instruction:
%                              (a) Added ", available at
%                              https://arisaka-gg.org/." to preprint ID.
%                              (b) Expanded Relevance paragraphs from
%                              ~200 chars to ~500-800 chars to match
%                              A1-A5 reference-list style.
%                              (c) Dropped "(2026a)", "(2026b)" body
%                              suffixes -- use plain "(2026)" -- since
%                              the new label "[Arisaka(2026, A*-Suffix)]"
%                              now carries the disambiguation.
% ======================================================================
%   v6.3 (May 11, 2026) Phase 3: L4 historical-background original-
%                        inventor audit.  Inserted 5 new
%                        \subsubsection{Historical background (with
%                        citations)} blocks, one at the end of each Qi-
%                        pillar's "Why X must follow Y" opening subsection
%                        + one in the SeOS/EIC section.  Each opens with
%                        the genuine original inventor by name and year:
%                          Qi-QUA: Maxwell (1865, current locking) ->
%                                  Noether (1918, symmetry-conservation)
%                                  -> Bohr (1913) -> Schrodinger (1944)
%                                  -> Landauer (1961) -> Mitchell/Boyer
%                                  bioenergetics -> Q_B Arisaka constant
%                                  -> Arisaka equation J_E/Q_B = J_e =
%                                  J_B = J_Qi.
%                          Qi-EP:  Fermat (1657) -> Maupertuis (1744) ->
%                                  Euler/Lagrange -> Hamilton (1834) ->
%                                  Boltzmann (1872) -> Onsager (1931) ->
%                                  Prigogine (1967) -> Paltridge (1975)
%                                  MEPP -> Jaynes (1957) -> Friston
%                                  (2010) FEP -> unified Qi-EP theorem.
%                          Qi-CON: Cauchy (1853) -> Levy (1937 stables)
%                                  -> Mandelbrot (1967/1982 fractals) ->
%                                  Voss-Clarke 1/f -> BPZ (1984 CFT) ->
%                                  Cardy 1984 -> Qi-CON unified.
%                          Qi-HAL: McCulloch-Pitts (1943) -> Hebb (1949)
%                                  -> Hopfield (1982) -> O'Keefe-
%                                  Dostrovsky (1971 place cells) ->
%                                  Llinas (2001 thalamocortical) ->
%                                  Turner-Evans (2020 Drosophila ring) ->
%                                  Arisaka (2022 MePMoS/NHT/HAL/BFFT).
%                          SeOS/EIC: Schrodinger (1944 What is Life?) ->
%                                  Prigogine (1967 dissipative
%                                  structures) -> Maturana-Varela
%                                  (1980 autopoiesis) -> Dawkins (1976
%                                  selfish gene) -> 't Hooft (1993
%                                  holographic principle) -> Susskind
%                                  (1995 covariant entropy bound) ->
%                                  Maldacena (1998 AdS/CFT) -> Llinas
%                                  (2001 I of the Vortex) -> SeOS+EIC.
%                        New bibitems added for citations not yet in
%                        the inline bibliography: Maxwell1865, Noether1918,
%                        Mitchell1961, Boyer1979, Boltzmann1872,
%                        Paltridge1975, Levy1937,
%                        BelavinPolyakovZamolodchikov1984,
%                        McCullochPitts1943, Hebb1949, tHooft1993.
% ======================================================================
%   v6.4 (May 11, 2026) Phase 4: Pillar theorem statements with L16
%                        dual-layer architecture + L12 U(1)_Qi framing.
%                        Five surgeries:
%                          (A) Compatibility-relation paragraphs added
%                              at the start of each of the four Qi-pillar
%                              sections, naming the algebraic and
%                              operational layers and the named
%                              compatibility proposition:
%                              - C_QUA = T-QUA.1 (rank-one inflow) ==>
%                                T-QUA.2 (Arisaka equation J_E/Q_B =
%                                J_e = J_B = J_Qi) via OGN-5 ledger.
%                              - C_EP  = T-EP.1 (unified gauge theorem)
%                                ==> T-EP.2 (three gauges PLT/MEPP/FEP)
%                                via boundary projection of 6D action.
%                              - C_CON = T-CON.1 (operational conformality)
%                                ==> T-CON.2 (two-knee 1/f spectrum)
%                                via warp-factor + OGN-5 kernel widths.
%                              - C_HAL = T-HAL.1 (Qi-HAL universality)
%                                ==> T-HAL.2 (MePMoS/NHT/HAL/BFFT
%                                empirical realisations: Hopfield, place
%                                cells, Drosophila ring, thalamocortical)
%                                via U(1)_Qi memory carrier + GRIP
%                                cascade.
%                          (B) L12 framing paragraph for U(1)_Qi inserted
%                              at first body mention (around L767):
%                              "Constitutional admissibility of U(1)_Qi:
%                              optional, empirically preferred."  Names
%                              the three-component vs four-component
%                              bulk options, cross-cites A3-BI6 as
%                              authoritative source, and gives three
%                              empirical grounds for the four-component
%                              choice (Arisaka equation; OGN-5 dual-lock;
%                              Qi-4 pillars + EIC framework).
% ======================================================================
%   v6.5 (May 11, 2026) Phase 5: Open Problems triage with L6 status
%                        headers.  Surgeries:
%                          1. Renamed §"Limitations, falsifiability, and
%                             the empirical program" -> §"Open Problems,
%                             Limitations, and Falsifiability" (canonical
%                             L6 name aligned with A6's §"Open Problems
%                             and Future Directions") with label
%                             sec:open-problems.
%                          2. Restructured "Limitations and what remains
%                             to be done" subsection as a labelled list
%                             O1-O4 with L6 status headers (Resolved /
%                             Partially resolved / Genuinely open).
%                             Added opening triage-convention paragraph
%                             (A1-canonical pattern).
%                          3. Added 5-entry experimental open-problem
%                             list E1-E5 at the end of the Falsifiability
%                             subsection, each with L6 status header.
%                          4. Added closing "Triage summary" paragraph
%                             with counts: 0 Resolved, 1 Partially
%                             resolved (O4 in A10-Cosmos), 8 Genuinely
%                             open (O1, O2, O3, E1-E5).
% ======================================================================
%   v6.6 (May 11, 2026) Phase 6: hostile-reviewer + reference-scanner
%                        sub-agents in parallel (L13).
%                        Reference-scanner came back CLEAN (all 7 audit
%                        checks passed).  Hostile-reviewer flagged 3
%                        Critical + 7 Major + 2 Minor issues; this
%                        revision applies the priority-1 patches:
%                          C1: defensive uniqueness clause added to the
%                              qiqua-rank-one proof (one-channel
%                              hypothesis follows from BI-6 closure +
%                              GS counter-term in A3-BI6 Thm 3.4 + S 8.7).
%                          C2 (= new O5): added open problem entry for
%                              the OGN-5 selection rule for Q_B.
%                          C3: renamed "Unified gauge theorem for Qi-EP"
%                              -> "Common-origin theorem for Qi-EP"
%                              (consistency, not uniqueness).
%                          M2: removed "unique" from C_HAL boxed
%                              compatibility (changed to "sufficient");
%                              added open problem O6 on factorisation
%                              uniqueness.
%                          M4: clarified OH+IH=EIC as definitional
%                              shorthand for EIC := C_loop(OH, IH), not
%                              an arithmetic identity.
%                          M5 (= new O7): added open problem on pagoda
%                              floor count = 5 derivation from MSMF.
%                          M6 (= new O8): added open problem on
%                              quantitative bounds for Borel-Cantelli
%                              inevitability hypotheses.
%                          M7: rewrote L12 U(1)_Qi framing as honestly
%                              "operationally required for Qi-4"
%                              instead of the inconsistent
%                              "constitutionally optional, empirically
%                              preferred" framing.  Added new open
%                              problem O9 on three-component QiFour'.
%                        Triage summary updated: 14 entries
%                        (0 Resolved, 1 Partially resolved, 13
%                        Genuinely open).
% ======================================================================
%   v6.7 (May 11, 2026) Phase 7: L7 cross-paper coherence map.
%                        Added new \section{Cross-Paper Coherence Map}
%                        immediately before §Open Problems, Limitations,
%                        and Falsifiability, containing:
%                          - L7 Note-on-version paragraph clarifying that
%                            companion-paper versions are independent
%                            revision tracks.
%                          - L7 Audit-trail paragraph documenting the
%                            actual internal theorem numbering of cited
%                            companion papers A1 v5.25, A2 v5.13, A3 v5.13,
%                            A5 v6.23, A6 v5.13.
%                          - Inward map (longtable, L22-compliant): 14
%                            rows of theorem-level facts A7 imports from
%                            A1, A2, A3, A5, A6, each with specific
%                            companion-paper location and role in A7-Qi4.
%                          - Outward map (longtable, L22-compliant): 13
%                            rows of A7-Qi4 facts exported to downstream
%                            papers A8-QM, A9-SM, A10-Cosmos, P1-KK,
%                            C1-Kaxion, each with A7 theorem label and
%                            downstream anticipated use.
%                          - Closing paragraph emphasising that the
%                            present section should be re-walked
%                            against any revised companion paper before
%                            the present paper is itself republished.
%                        Both tables use longtable + footnotesize per
%                        L22, not tabular-in-resizebox-in-table[H].
% ======================================================================
%   v6.8 (May 11, 2026) Phase 8: TikZ figure pass per L8.
%                        A7 v6.7 had ZERO actual TikZ figures.  Phase 8
%                        adds four substantive figures, each wrapped in
%                        resizebox{textwidth}{!} per L8 with absolute
%                        coordinates and L23 bold-acronym style:
%                          Fig 1 (qi4-architecture): four Qi-pillars
%                            QUA -> EP -> CON -> HAL with GDOS + Gi-4
%                            template above and U(1)_Qi gauge sector
%                            below; four compatibility propositions.
%                          Fig 2 (arisaka-lock): Arisaka equation
%                            J_E/Q_B = J_e = J_B = J_Qi as four-current
%                            ledger converging on Q_B.
%                          Fig 3 (oh-ih-eic): OH + IH = EIC diagram
%                            with closed-loop perception/action coupling,
%                            v6.6 clarification that "+" is mnemonic.
%                          Fig 4 (pagoda): five-story pagoda with
%                            shinbashira central pillar; particle ->
%                            cosmos -> chemistry -> life -> mind, with
%                            open-problem O7 note on the floor count.
% ======================================================================
%   v6.9 (May 11, 2026) Phase 9: front-matter rewrite to A1-canonical
%                        template (L9) + Conclusions philosophical close
%                        (H2 integrated).  Four surgeries:
%                          1. Title block tightened: short crisp primary
%                             "The Qi-4 Pillars", 2-line subtitle
%                             "Qi-QUA, Qi-EP, Qi-CON, Qi-HAL / Quantum-
%                             Information Backbone of GDOS", short
%                             description.  Dropped legacy GD-2026-5
%                             enumeration from description.
%                          2. Abstract rewritten (~225 words, plain-
%                             English, three-paragraph + "After all"
%                             close): opening universal thesis (every
%                             open system carries information); naming
%                             of the four pillars + dual-layer theorem
%                             architecture; SeOS, EIC, pagoda
%                             deliverables; closing philosophical
%                             reflection.
%                          3. Executive Summary rewritten (~850 words,
%                             L9-canonical template): opening
%                             positioning vs predecessors with explicit
%                             \citet (ArisakaA6Gi4, ArisakaA9SM,
%                             ArisakaA10Cosmos, tHooft1993, Maldacena1998,
%                             ArisakaA1Constitution, ArisakaA2GG6,
%                             ArisakaA3BI6, ArisakaA5GDOS); boxed
%                             central principle (Qi-4 master identity);
%                             Four principal theorems paragraph
%                             summarising T-QUA, T-EP, T-CON, T-HAL;
%                             Three further deliverables (SeOS, EIC,
%                             pagoda); Significance and uniqueness;
%                             Canonical implication chain in boxed
%                             equation cascade; Closing reflection
%                             continuing the "After all" motif.
%                          4. Conclusions philosophical close (H2):
%                             replaced the previous "Closing statement"
%                             subsection with three Arisaka-voice
%                             philosophical paragraphs matching A6
%                             v5.12 pattern: (i) four pillars as one
%                             geometric object; (ii) the bargain with
%                             the laboratory (four named falsifiers,
%                             register-matching close "physics promises
%                             to write its rules down for the open-
%                             system world, and the world promises to
%                             falsify them if they ever lie"); (iii)
%                             the deeper claim about the laboratory
%                             world as projection of a single geometric
%                             structure.
%                        \GGKeywords{} placement after
%                        \end{ExecutiveSummary} already correct from
%                        legacy (L9 compliant).  \PaperStatus{Preprint}
%                        clean (L15 compliant, no live LaTeX).
% ======================================================================
%   v6.10 (May 11, 2026) Phase 10: bibliography surname-sort + journal-
%                        stripped variant per L10.
%                          - ~83 inline bibitems sorted alphabetically by
%                            first-author surname using L10 conventions:
%                            particle demotion ('t Hooft -> H, von
%                            Neumann -> N, de Rham -> R); multi-author by
%                            first surname; accent stripping for sort
%                            keys; Arisaka block sub-sorted by programme
%                            order (A1, A2, A3, A5, A6, A7, A9, A10,
%                            P1, C1, GD-2026-5 legacy).
%                          - A7 already uses the new (Arisaka, 2026,
%                            A*-Suffix) citation convention since
%                            Phase 2 v6.2; no re-lettering needed.
%                          - Journal-stripped variant
%                            Arisaka-GG-A7-Qi4-v6.10-journal.tex
%                            produced by removing every \textbf{Relevance.}
%                            paragraph (multi-line, from marker to next
%                            blank line) -- scoped to the bibliography
%                            region only, learning from A6 Phase 10's
%                            mid-bug where the unbounded regex ate the
%                            preamble.  PaperStatus updated to "Journal
%                            submission" in the journal variant.
% ======================================================================
%   v6.11 (May 11, 2026) Phase 11: final canonical pass.  H1 Bridge-
%                        to-Part paragraphs integrated per agreed
%                        A6-learned strategy; user requested no journal
%                        variant for this phase.
%                          1. H1 Bridge-to-Part paragraphs: 5 bridges
%                             added (I->II, II->III, III->IV, IV->V,
%                             V->VI) in Arisaka voice with forward-
%                             pointing prose.
%                          2. L21 sans-serif sweep applied document-wide
%                             with protect-and-stash logic (~736 plain-
%                             text programme acronyms converted to their
%                             sans-serif macro forms).
%                          3. L20 programme-acronym hierarchy figure
%                             (TikZ, six tiers) inserted at the end of
%                             the Abbreviations appendix.  Bold acronym
%                             + regular gloss per L23.
%                          4. L19 Appendix A canonical rewrite SKIPPED;
%                             A7's existing "Abbreviations, symbols, and
%                             key equations" section is comprehensive
%                             enough.
%                          5. Journal-stripped variant SKIPPED per user
%                             instruction (May 11, 2026 evening).
%                        v6.11 is the final v6.X canonical form of A7.
% ======================================================================

% ----------------------------------------------------------------------
% UCLA-Arisaka/GG-2026-4B
% GDOS with Qi-4 Pillars
% Full draft rebuilt from the old GG-2026-4 and constrained to the
% derivation chain GG-2026-1 -> GG-2026-2 -> GD-2026-5 -> GG-2026-3 -> GG-2026-4A
% ----------------------------------------------------------------------
%% ---- vv15.31 (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.
%% 3 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
%%%% ---- v15.32 (August 19, 2026) ----
%%%% SHIP A OF THE PART IV (Qi-EP) EXPANSION: the overview and the four
%%%% histories.  Five new sections between the Part opener and the
%%%% constitutional statement: an overview of the three principles and
%%%% the shared anatomy of their failures; the least-action/least-time
%%%% tradition from Fermat to Onsager--Machlup, written fresh; and the
%%%% MinEP, MEPP, and FEP histories at review depth, compressed from the
%%%% S1-EP supplement (44 pages there to 12 here) with every claim,
%%%% counterexample, and concession carried at its established strength
%%%% and nothing restated above it.  The one-line 'Historical background
%%%% (with citations)' subsection is retired as superseded; the
%%%% dual-layer paragraph, the compatibility relation, and the pillar
%%%% figure move from the unnumbered pre-section region into the
%%%% constitutional statement; and the statement gains the five-criteria
%%%% contract that the rest of the Part is then held to.  One genealogy
%%%% figure and four scope tables are new.  58 bibliography entries
%%%% added: 51 adapted from S1-EP, 6 composed for the least-action
%%%% history (Fermat 1662, Maupertuis 1744, Euler 1744, Lagrange 1788,
%%%% Bauer 1931, Gray--Taylor 2007), plus the S1-EP companion itself;
%%%% the First-cited links of the existing entries that the histories
%%%% now cite first are recomputed.  Part IV grows by about 19 pages;
%%%% no derivation section is touched in this ship (Ships B and C
%%%% follow).  Not one equation, number, or claim of the existing
%%%% derivation chain is changed.
%%%% ---- v15.33 (August 19, 2026) ----
%%%% SHIP B OF THE PART IV (Qi-EP) EXPANSION: the derivation at full
%%%% depth.  The bulk-to-boundary section now opens with the imports and
%%%% their grades (the boundary law, the ledger, the packet -- three
%%%% objects, nothing else assumed) and a four-instrument tools on-ramp
%%%% for the reader arriving from the life sciences.  The trajectory
%%%% gauge gains its regime statement and a worked parallel: Fermat's
%%%% ray and a chemotactic run as the same calculation, column by
%%%% column.  The current gauge gains the identity that the Rayleighian
%%%% IS Onsager's 1931 least-dissipation functional, the corollary
%%%% recovering Prigogine's 1945 theorem hypothesis for hypothesis
%%%% against the Ship-A history (A1-A5 mapped to projected structures),
%%%% and the five-claims quarrel table that settles seventy years of
%%%% minimum-versus-maximum in one page.  The inferential gauge gains
%%%% the assumption-by-assumption table of what the FEP supplies by
%%%% hand and what the gauge derives.  The laboratory register is
%%%% rebuilt from three dense paragraphs into eight case subsections in
%%%% the supplement's four-field format -- observed, old reading, gauge
%%%% reading, refuter -- reusing the existing sentences and importing
%%%% the observed/old-reading fields from S1-EP Part VIII.  The
%%%% unified-theorem section closes with the contract of Ship A
%%%% discharged clause by clause.  NOT ONE existing theorem, equation,
%%%% or number is changed; every addition is exposition, identity, or
%%%% table around the proved chain.  Ship C follows.
%%%% ---- v15.34 (August 19, 2026) ----
%%%% SHIP C OF THE PART IV (Qi-EP) EXPANSION: the closing sections, and
%%%% the Part complete.  The DGI/GSSB/GA one-paragraph stub becomes a
%%%% full section: the three runtime moves acting on the selector's
%%%% landscape, with the caustic of optics, the convective instability,
%%%% and the bistable percept exhibited as one Hessian degeneracy in
%%%% three gauges, and a three-panel landscape figure.  The universality
%%%% section gains the nothing-biological-was-used argument (the proofs
%%%% consumed a packet, a ledger, and a projection -- so the theorems
%%%% bind the SeOS class and life is in the class by theorem, not
%%%% analogy) and the bridge to the capstone's eighteen-layer cascade,
%%%% where Qi-EP is the selector at every layer and the three-gauge
%%%% theorem is deployed in exactly the form proved here.  The
%%%% three-sciences section now states the claim at full strength with
%%%% each clause carrying its theorem -- PLT, MEPP, and FEP as theorems,
%%%% exact where their regime theorems say, failing by theorem where the
%%%% regimes end -- names what each of the three communities gains, and
%%%% closes with the what-is-not-concluded list, because a Part whose
%%%% subject is three literatures' overclaims must not end by adding a
%%%% fourth.  A new concluding section retells the Part in plain
%%%% language (one landscape, three shadows), absorbs the falsification
%%%% paragraph and the Qi-CON bridge from the three-sciences section
%%%% (moved, the former extended with the pointer to the eight-case
%%%% register), and hands the reader the three doors: the supplement,
%%%% the capstone, and the rest of the runtime.  One clause added to the
%%%% Part I roadmap prose so the fifteen-Part map describes the expanded
%%%% Part.  With this ship the three-ship expansion approved from the
%%%% proposal is complete.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}

%% Phase 1 additions (May 11, 2026):
\providecommand{\MEPP}{\ifmmode\mathsf{MEPP}\else\textsf{MEPP}\fi} %% L21(a)
\setcitestyle{authoryear,round,comma} %% NEW citation convention: Arisaka, 2026, A*-Suffix
\providecommand{\ii}{\mathrm{i}} %% imaginary unit (boundary-derived quarter-turn; T-P4-1/2 of P4-Real)
\providecommand{\e}{\ensuremath{\mathrm{e}}} %% Euler's constant in exponential expressions
\providecommand{\Gtot}{\ensuremath{G_{\mathrm{tot}}}} %% total gauge group
\providecommand{\SLtwoH}{\ensuremath{\mathrm{SL}(2,\mathbb{H})}} %% quaternionic Lorentz
\providecommand{\Spinone}{\ensuremath{\mathrm{Spin}(1,5)}} %% Spin(1,5) group
\providecommand{\SUfive}{\ensuremath{\mathrm{SU}(5)}}
\providecommand{\UoneB}{\ensuremath{\mathrm{U}(1)_B}}
\providecommand{\UoneQi}{\ensuremath{\mathrm{U}(1)_{Qi}}}
\providecommand{\OGNfive}{\ifmmode\mathsf{OGN\text{-}5}\else\textsf{OGN-5}\fi}
\providecommand{\BIsix}{\ifmmode\mathsf{BI\text{-}6}\else\textsf{BI-6}\fi}
\providecommand{\GiFour}{\ifmmode\mathsf{Gi\text{-}4}\else\textsf{Gi-4}\fi}
\providecommand{\QiFour}{\ifmmode\mathsf{Qi\text{-}4}\else\textsf{Qi-4}\fi}
\providecommand{\GG}{\ifmmode\mathsf{GG\text{-}6}\else\textsf{GG-6}\fi}
\providecommand{\GDOS}{\ifmmode\mathsf{GDOS}\else\textsf{GDOS}\fi}
\providecommand{\OGGEP}{\ifmmode\mathsf{OGGEP}\else\textsf{OGGEP}\fi}
\providecommand{\GGEP}{\ifmmode\mathsf{GGEP}\else\textsf{GGEP}\fi}
\providecommand{\MSMF}{\ifmmode\mathsf{MSMF}\else\textsf{MSMF}\fi}
\providecommand{\GGTheory}{\ifmmode\mathsf{GG\text{-}Theory}\else\textsf{GG-Theory}\fi}
\providecommand{\TGG}{\ifmmode\mathsf{TGG}\else\textsf{TGG}\fi}
\providecommand{\CCI}{\ifmmode\mathsf{CCI}\else\textsf{CCI}\fi}
\providecommand{\LC}{\ifmmode\mathsf{LC}\else\textsf{LC}\fi}
\providecommand{\GI}{\ifmmode\mathsf{GI}\else\textsf{GI}\fi}
\providecommand{\UEP}{\ifmmode\mathsf{UEP}\else\textsf{UEP}\fi}
\providecommand{\AF}{\ifmmode\mathsf{AF}\else\textsf{AF}\fi}
\providecommand{\CP}{\ifmmode\mathsf{CP}\else\textsf{CP}\fi}
\providecommand{\RG}{\ifmmode\mathsf{RG}\else\textsf{RG}\fi}
\providecommand{\OGN}{\ifmmode\mathsf{OGN}\else\textsf{OGN}\fi}
\providecommand{\MF}{\ifmmode\mathsf{MF}\else\textsf{MF}\fi}
\providecommand{\AX}[1]{\ifmmode\mathsf{AX\text{-}#1}\else\textsf{AX-#1}\fi}
\providecommand{\POS}[1]{\ifmmode\mathsf{POS\text{-}#1}\else\textsf{POS-#1}\fi}
\providecommand{\QCI}{\ifmmode\mathsf{QCI}\else\textsf{QCI}\fi}

\PaperSeries{GG}
\PaperNumber{Arisaka-GG-A7-Qi4}
\PaperStatus{Preprint}
% ------------------------------------------------------------
%% v15.21 -> v15.22, August 15, 2026.  THE GATE-CLOSING PASS.
%   One pass across A1-A7 so that `at v15' and `passes the v15
%   gate' mean the same thing before the first arXiv submission.
%   These seven were migrated before the current toolchain and
%   each failed at least one absolute section 4 item.
%   SPELLING (section 9 decision 1, "total, in printed text"):
%     0 printed British form(s) changed here.  The raw guard
%     reported 91 across the seven; 51 were NOT defects and were
%     hand-classified one site at a time -- 33 correct American
%     English now permanently in PROTECT, 12 published titles now
%     frozen by exact text (correcting a title is a
%     misquotation), and 6 `cancellable'/`cancellability', held at
%     the doubled l because American English keeps it in
%     `cancellation', the -able form is listed both ways, and it
%     is a defined term.  The -ed/-ing forms ARE swept.
%     Edited BY OFFSET, not by text anchor: these papers hard-wrap
%     their source and anchors copied from a flattened scan do not
%     exist -- eight failed that way on the first run, and the
%     same defect hit A8's rider an hour earlier.  The sites come
%     from the guard itself, which already excludes comments,
%     labels, PROTECT and the frozen titles.
%   FLOAT STANDARD: 1 bare grid, and the ONE PLACE THIS PASS DID NOT USE THE NAMED
%%     REMEDY.  Section 14.1 says promote a bare grid into a float.
%%     A7's grid is four lines reached by the author's own colon --
%%     `the Qi-4 pillars then answer four questions:' -- and a float
%%     drifts, taking the answers away from the question and leaving
%%     the colon pointing at nothing.  It is also not a table: no
%%     header, no rules, two columns only because tabular was the
%%     nearest tool.  So it became a `description' list, which
%%     removes the grid instead of framing it.  REPORTED, and one
%%     line overrules it.
%   PALETTE: nothing to change.  0 violations.
%   NEW PROSE is listed by table and page in GG-Report-362 and is
%   a draft for the author to accept or rewrite.  Caption
%   EXTENSIONS keep every existing sentence verbatim -- section
%   14.2 reserves rewriting a caption's CLAIM, and none is.
%   NO NUMBER, NO CLAIM AND NO GRADE CHANGES ANYWHERE.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.22 -> v15.23, August 15, 2026.  THE STOREY RIDER.
%   2 printed British `storey' -> `story'.  Not a sweep this paper
%   asked for: the word was invisible to every spelling tool in the
%   toolchain until August 15.  gg_spelling had storey removed from
%   the sweep RULE and placed in the cohort-global PROTECT set,
%   because A11-Origin uses `storey-port' and `a storey count S' as
%   defined terms at 142 sites.  A global exemption for one paper's
%   defined term let the ordinary building metaphor through
%   everywhere else.
%   Found at P5-CPT's gate, which carried the same defect, and
%   measured across the cohort: A6-GRIP 1, A7-Qi4 2, C2-Hubble 3,
%   C3-BH 2 -- while A10-Cosmos had ALREADY converted its own 23
%   sites and recorded the decision in its changelog.  The cohort
%   was spelled both ways across papers, and P4-Real and P5-CPT were
%   each spelled both ways inside themselves.
%   THE TOOL IS FIXED, NOT WORKED AROUND: storey[s]? is back in the
%   rule and A11's exemption moved to a PAPER_PROTECT map keyed by
%   filename.  Re-run against a file with the known defect it fires.
%   NO PROSE WAS WRITTEN IN THIS RIDER.
% ------------------------------------------------------------
\PaperVersion{v20.0}
\PaperDate{August 21, 2026}

\hypersetup{
 pdftitle = {GDOS with Qi-4 Pillars},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}, \SeOS{}, QCI, \EIC{}, and the five-story pagoda derived from the constitutional chain A1-Constitution, A2-GG6, A3-BI6, A5-\GDOS{}, and A6-Gi4},
 pdfkeywords= {\GDOS{}, \QiFour{}, \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}, \GRIP{}, \GFF{}, \GEP{}, \CEP{}, \SeOS{}, QCI, \OH{}, \IH{}, \EIC{}, five-story pagoda, Arisaka Ratio, Arisaka Equation, Arisaka Constant, \MePMoS{}, \NHT{}, \HAL{}, \BFFT{}}
}

%% Paper-specific macros
\renewcommand{\GUT}{\textsf{GG-Constitution}}  % GUT-10 retired (cohort, June 2026)
\providecommand{\EGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\EPl}{\ensuremath{E_{\mathrm{Pl}}}}

%% Runtime vocabulary
\providecommand{\DGI}{\ensuremath{\mathrm{DGI}}}
\providecommand{\GSSB}{\ensuremath{\mathrm{GSSB}}}
\providecommand{\GA}{\ensuremath{\mathrm{GA}}}
\providecommand{\GGA}{\ensuremath{\mathrm{GGA}}}
\providecommand{\GRIP}{\ensuremath{\mathrm{GRIP}}}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\GGConstitution}{\textsf{GG-Constitution}}
\providecommand{\LGI}{\ensuremath{\mathrm{LGI}}}
\providecommand{\GGI}{\ensuremath{\mathrm{GGI}}}
\providecommand{\QiQUA}{\ensuremath{\mathrm{Qi\mbox{-}QUA}}}
\providecommand{\QiEP}{\ensuremath{\mathrm{Qi\mbox{-}EP}}}
\providecommand{\QiCON}{\ensuremath{\mathrm{Qi\mbox{-}CON}}}
\providecommand{\QiHAL}{\ensuremath{\mathrm{Qi\mbox{-}HAL}}}
\providecommand{\SIP}{\ensuremath{\mathrm{SIP}}}
\providecommand{\GEP}{\ensuremath{\mathrm{GEP}}}
\providecommand{\CEP}{\ensuremath{\mathrm{CEP}}}
\providecommand{\GFF}{\ensuremath{\mathrm{GFF}}}
\providecommand{\SeOS}{\ensuremath{\mathrm{SeOS}}}
\providecommand{\OH}{\ensuremath{\mathrm{OH}}}
\providecommand{\IH}{\ensuremath{\mathrm{IH}}}
\providecommand{\EIC}{\ensuremath{\mathrm{EIC}}}
\providecommand{\PLA}{\ensuremath{\mathrm{PLA}}}
\providecommand{\PLT}{\ensuremath{\mathrm{PLT}}}
\providecommand{\MEPP}{\ensuremath{\mathrm{MEPP}}}
\providecommand{\FEP}{\ensuremath{\mathrm{FEP}}}
\providecommand{\HAL}{\ensuremath{\mathrm{HAL}}}
\providecommand{\NHT}{\ensuremath{\mathrm{NHT}}}
\providecommand{\BFFT}{\ensuremath{\mathrm{BFFT}}}
\providecommand{\MePMoS}{\ensuremath{\mathrm{MePMoS}}}
\providecommand{\QCI}{\ensuremath{\mathrm{QCI}}}
\providecommand{\QGL}{\ensuremath{\mathrm{QGL}}}
\providecommand{\Packet}{\mathfrak{P}}
\providecommand{\Kill}{\mathfrak{K}}
\providecommand{\Viab}{\mathcal{V}}
\providecommand{\Maint}{\ensuremath{\Pi_{\mathrm{maint}}}}
\providecommand{\RepairMin}{\ensuremath{\mathcal{R}_{\mathrm{repair}}^{\min}}}
\providecommand{\QB}{\ensuremath{Q_B}}
\providecommand{\ERyd}{\ensuremath{E_{\mathrm{Ryd}}}}
\providecommand{\POVM}{\ensuremath{\mathrm{POVM}}}
\providecommand{\id}{\mathrm{id}}
\providecommand{\State}{\mathcal{X}}
\providecommand{\Ports}{\mathcal{P}}
\providecommand{\Portspull}{\mathcal{P}_{\mathrm{pull}}}
\providecommand{\Obs}{\mathcal{O}}
\providecommand{\Instr}{\mathcal{I}}
\providecommand{\Bdry}{\partial}
\providecommand{\BState}{\mathcal{H}_{\partial}}
\providecommand{\BulkState}{\mathcal{H}_{\mathrm{Bulk}}}
\providecommand{\EnvState}{\mathcal{H}_{\mathrm{env}}}
\providecommand{\SysState}{\mathcal{H}_{\mathrm{sys}}}
\providecommand{\Lind}{\mathcal{L}}
\providecommand{\Proj}{\mathcal{P}}
\providecommand{\Qproj}{\mathcal{Q}}
\providecommand{\BulkMap}{\mathcal{D}_{\OGGEP}}
\providecommand{\ArisakaRatio}{\ensuremath{\mathrm{ArisakaRatio}}}
\providecommand{\ArisakaEq}{\ensuremath{\mathrm{ArisakaEquation}}}
\providecommand{\ArisakaConst}{\ensuremath{\mathrm{ArisakaConstant}}}
\providecommand{\BA}{\ensuremath{\mathrm{BA}}}
\providecommand{\FGFF}{\ensuremath{\mathcal{F}_{\GFF}}}
\providecommand{\JE}{\ensuremath{J_E}}
\providecommand{\Je}{\ensuremath{J_e}}
\providecommand{\JB}{\ensuremath{J_B}}
\providecommand{\JQi}{\ensuremath{J_{Qi}}}
\providecommand{\JpE}{\ensuremath{J'_E}}
\providecommand{\Jpe}{\ensuremath{J'_e}}
\providecommand{\JpB}{\ensuremath{J'_B}}
\providecommand{\JpQi}{\ensuremath{J'_{Qi}}}
\providecommand{\POne}{\textbf{Paper I}~\citep{ArisakaA1Constitution}}
\providecommand{\PTwo}{\textbf{Paper II}~\citep{ArisakaA2GG6}}
\providecommand{\PThree}{\textbf{Paper III}~\citep{ArisakaA5GDOS}}
\providecommand{\PFive}{\textbf{Paper V}~\citep{ArisakaA3BI6}}
\providecommand{\PFourA}{\textbf{Paper IV-A}~\citep{ArisakaA6Gi4}}
\providecommand{\ket}[1]{\lvert #1 \rangle}
\providecommand{\bra}[1]{\langle #1 \rvert}

%% Paragraph format inherits from ucla-arisaka-gg-6.sty:
%%   \parskip = 0pt + \parindent = 1.5em (indented paragraphs,
%%   no extra vertical space).  Earlier override removed in v6.21.
\captionsetup[table]{position=top,skip=6pt,width=\textwidth}
\captionsetup[longtable]{position=top,skip=6pt,width=\textwidth}
\setlength{\LTcapwidth}{\textwidth}
\setlength{\tabcolsep}{4pt}
\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection}{0.8em}{}
\numberwithin{equation}{section}

%% Part heading
\providecommand{\GGPart}[2]{%
 \clearpage
 \refstepcounter{part}%
 \addcontentsline{toc}{part}{Part \thepart. #1 --- #2}%
 \thispagestyle{plain}%
 \begin{center}
 \vspace*{1.2em}
 {\Large\bfseries Part \thepart}\\[0.45em]
 {\huge\bfseries #1}\\[0.35em]
 {\large #2}
 \end{center}
 \vspace{0.5em}
 \GGThickLine
 \vspace{0.8em}
}

%% Layout controls
\emergencystretch=2.8em
\hfuzz=1.2pt
\vfuzz=1.2pt
\allowdisplaybreaks[3]
\widowpenalty=10000
\clubpenalty=10000

%% Appendix ToC formatting
\makeatletter
\providecommand{\switchtoappendixtocsection}{%
 \def\l@section##1##2{%
 \ifnum \c@tocdepth >\z@
 \addpenalty\@secpenalty
 \addvspace{0.6em \@plus\p@}%
 \begingroup
 \parindent \z@ \rightskip \@pnumwidth
 \parfillskip -\@pnumwidth
 \leavevmode
 \setlength\@tempdima{4.0em}%
 \advance\leftskip\@tempdima
 \hskip -\leftskip
 ##1\nobreak\dotfill \nobreak\hb@xt@\@pnumwidth{\hss ##2}\par
 \endgroup
 \fi}%
 \def\l@subsection{\@dottedtocline{2}{4.0em}{3.2em}}%
}
\makeatother


%% --- Sans-serif paper-code macros (programme-wide v7.1 sweep, May 13, 2026)
%% Programme-defined names rendered in sans-serif (per L21/L23) and with
%% full GG- prefix throughout the body.  The \AFull... variants prepend
%% Arisaka- and are used for explicit listings of preprint titles.
\providecommand{\AOne}{\textsf{GG-A1-Constitution}}
\providecommand{\ATwo}{\textsf{GG-A2-GG6}}
\providecommand{\AThree}{\textsf{GG-A3-BI6}}
\providecommand{\AFour}{\textsf{GG-A4-Euclid}}
\providecommand{\AFive}{\textsf{GG-A5-GDOS}}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}% v13.2: \renewcommand (the sty predefines \ASix as the retired GG-A6-Gi4; \providecommand was a no-op)
\providecommand{\ASeven}{\textsf{GG-A7-Qi4}}
\providecommand{\AEight}{\textsf{GG-A8-QM}}
\providecommand{\ANine}{\textsf{GG-A9-SM}}
\providecommand{\ATen}{\textsf{GG-A10-Cosmos}}
\providecommand{\PaperPOne}{\textsf{GG-P1-KK}}
\providecommand{\PaperPTwo}{\textsf{GG-P2-N3}}
\providecommand{\PaperCOne}{\textsf{GG-C1-Kaxion}}
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}%% v13.0: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFour, so this providecommand takes effect)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}%% v13.0: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive, so this providecommand takes effect)
\providecommand{\AFullOne}{\textsf{Arisaka-GG-A1-Constitution}}
\providecommand{\AFullTwo}{\textsf{Arisaka-GG-A2-GG6}}
\providecommand{\AFullThree}{\textsf{Arisaka-GG-A3-BI6}}
\providecommand{\AFullFour}{\textsf{Arisaka-GG-A4-Euclid}}
\providecommand{\AFullFive}{\textsf{Arisaka-GG-A5-GDOS}}
\renewcommand{\AFullSix}{\textsf{Arisaka-GG-A6-GRIP}}% v13.2: retired full form corrected + \renewcommand
\providecommand{\AFullSeven}{\textsf{Arisaka-GG-A7-Qi4}}
\providecommand{\AFullEight}{\textsf{Arisaka-GG-A8-QM}}
\providecommand{\AFullNine}{\textsf{Arisaka-GG-A9-SM}}
\providecommand{\AFullTen}{\textsf{Arisaka-GG-A10-Cosmos}}
\providecommand{\PaperFullPOne}{\textsf{Arisaka-GG-P1-KK}}
\providecommand{\PaperFullPTwo}{\textsf{Arisaka-GG-P2-N3}}
\providecommand{\PaperFullCOne}{\textsf{Arisaka-GG-C1-Kaxion}}

%% v11.6: \GGAcknowledgments now provided by ucla-arisaka-gg-11
%% (single canonical Acknowledgments, cohort-universal).
%% ---- v13.0: one-line Part entries (cohort standard; P4-Real v3.2
%%      preamble pattern).  \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).  \GGAppendixPart (Part VIII) inherits the
%%      renewal via the sty's pass-through.  (A7's legacy local
%%      \providecommand{\GGPart} block above was already dead code
%%      -- the sty defines \GGPart first -- and stays inert.)
\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

\begin{document}
% ---- v13.12 (August 2, 2026) ----
% New subsection 'Information and the two fibres: memory on the
% circle, erasure on the interval' at the end of the QiQUA
% Landauer subsection: a stable bit as holonomy-class data;
% Landauer's bound as the interval's toll; the Landauer-vs-QiQUA
% table re-read as one-fibre vs both-fibre audits (why Q_B
% legitimately exceeds kT ln 2, and why it is temperature-
% independent); Q_B as the exchange rate between the circle's two
% currencies; the A8 measurement chain closed ('the circle stores
% it, the interval seals it, QiQUA bills it'); register stated
% (organisation, not new theorem). Part of the A5/A6/A7 trio
% sweep.


\GGTitlePage
 {The \QiFour{} Pillars}
 {\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{} \\
 The Quantum-Information Backbone of \GDOS{}}
 {Foundational paper of the \GDOS{} Trio (with \AFive{} and \ASix{}).}
\thispagestyle{empty}

\vspace{6pt}
\begin{abstract}
\noindent
Everything that stays itself has to pay for it. A cell, a flame, a
spectrometer, a galaxy: each holds a pattern in place by taking energy
in at one boundary and letting it out at another. Physics has been able
to price the energy for a century and a half. It has never had a price
for the information.

This paper sets one. On the boundary of the six-dimensional bulk there
are four currents rather than three --- energetic, electric, baryonic,
informational --- and an elementary event does not spend them
independently. It spends all four at once, in one fixed proportion, so
that four ledgers are one ledger read in four currencies. The exchange
rate between energy and information is a single constant,
$\QB \approx 0.1555$ eV per qubit. It is not fitted: it is
the Rydberg energy divided by a ledger integer fixed upstream, scaled
by an anomaly coefficient, with nothing adjustable in the chain.

Life is where it can be checked, because life runs on one molecule. The
bond ATP breaks carries two of these quanta, and $2\QB = 30.0$ kJ/mol
stands against a measured $30.5 \pm 0.5$ --- agreement to 1.6 per cent.

Three pillars follow. One shows that least time in optics, maximum
entropy production in thermodynamics and the free-energy principle in
neuroscience are one principle read three ways. One predicts a $1/f$
band bounded by \emph{two} knees rather than one unbounded slope, and
says what sets each end --- a claim that can fail where a bare power law
cannot. One shows that anything holding a memory longer than a single
act-and-look cycle falls into a definite architecture, whatever it is
made of. Each pillar is handed an experiment that can end it.

Together the four are the certificates carried by the configuration
where the geometric descent halts. They describe that configuration;
they are not the instrument that found it, and the converse is left
open rather than claimed.
\end{abstract}

\begin{ExecutiveSummary}
\paragraph{Four questions about anything that lasts.}
Ask what a bacterium, a flame, a market and a brain have in common and
the honest answer is short: each is open to what surrounds it, and each
keeps being itself anyway. The companion papers of this program supply
the stage such a thing stands on --- a constitutional charter of four
axioms and seven postulates \citep{ArisakaA1Constitution}, a
six-dimensional geometry whose two extra directions are a scale and a
phase rather than further places to go \citep{ArisakaA2GG6}, proved free
of internal inconsistency \citep{ArisakaA3BI6}, and the law describing
how any of it looks to an observer who is inside rather than outside
\citep{ArisakaA5GDOS}. The companion to this one supplies the engine
that drives a system downhill and stops it \citep{ArisakaA6Gi4}. What
none of them supplies is the accounting for what such a system
\emph{knows}. This paper asks four questions about anything that
persists --- what does one elementary event cost, which way does the
system go, what should it look like across scales, and how does it
remember --- and answers each as a theorem with an experiment attached.

\paragraph{The first question: what does one event cost?}
The answer is that it costs all four currencies at once. Energy,
electric charge, baryon number and information leave the bulk through a
single channel, so an elementary boundary event does not range over a
four-dimensional space of possible bills; it lies on a one-dimensional
ray inside it (\QiQUA{}, Theorem T-QUA.1). Written for a laboratory,
that becomes the Arisaka equation
$J_E/\QB = J_e = J_B = J_{Qi}$: count the events in any one column and
you have counted them in all four. Two cautions travel with it. The
equality is between normalized event counts, not an assertion that one
electron and one bit are the same object; and it forbids something
definite --- there is no elementary event that carries information
while carrying no energy, no charge and no matter, which is the precise
sense in which this framework has no room for disembodied information.

\paragraph{The constant, and where it can be caught.}
The exchange rate in that equation is $\QB = (2/5)\,E_{\mathrm{Ry}}/K$
with $K = 35$: the Rydberg energy, a ledger integer fixed by the
upstream Diophantine closure, and the anomaly coefficient of the
modified Bianchi identity. Nothing in it was fitted, and the paper
records that as a theorem rather than as a hope --- $\QB$ introduces no
new continuous free parameter. Numerically it is $0.1555$ eV per qubit,
or $15.0$ kJ/mol, or about $5.8\,k_BT$ at body temperature. Life is
where it meets a measurement, because life settled on one energy
currency: the phosphoanhydride bond of ATP carries two of these quanta,
and $2\QB = 30.0$ kJ/mol against a measured $30.5\pm0.5$ agrees to
$1.6\%$. A second, independent check sits in the same molecule's
machinery. The rotary enzyme that makes ATP pays for each bond with a
whole number of protons set by the size of its rotor ring --- ten in
\emph{E.\ coli}, eight in oxygen-rich mitochondria, thirteen to fifteen
in organisms living on a thinner gradient --- and at the measured
driving voltage each single proton crossing costs one $\QB$. The gear
ratio is tuned to the environment; the size of one click is not.

It is worth saying what would refute this, because it is not the
number. A calorimeter that disagreed with $0.1555$ eV would be
arguing with the upstream ledger. What \QiQUA{} asserts, and what an
experiment can break, is that \emph{every} elementary event of the
channel shares the \emph{same} single conversion factor. One catalytic
event whose four-way bookkeeping misses by more than twenty per cent
ends the pillar.

\paragraph{A word about Landauer.}
Erasing a bit costs at least $k_BT\ln 2$, and at body temperature
$\QB$ is about eight and a half times that. This looks like a
contradiction and is not, because the two statements audit different
things. Landauer prices erasure, which is a one-way write along a
direction with no way back. \QiQUA{} prices carrying, on a circle,
where what is stored is a winding that no boundary operation can
smooth away. One event, three audits: the circle stores it, the
interval seals it, and \QiQUA{} bills it.

\paragraph{The second question: which way does it go?}
Three communities already have an answer, and they do not appear to be
the same answer. Optics and mechanics minimize a time; nonequilibrium
thermodynamics extremizes entropy production; theoretical neuroscience
minimizes a free energy. \QiEP{} (Theorem T-EP.1) shows they are three
readings of one projected functional, differing only in what is held
primary --- the path, the currents, or the beliefs. The unification is
not verbal: taken as a variational problem in each of the three, the
same functional returns the geodesic law, the flux--force law, and the
natural-gradient inference rule. It also settles an old quarrel, since
maximum entropy production and Prigogine's minimum entropy production
turn out to be conjugate readings of one stationary point rather than
rival slogans --- one selecting the transport law, the other the
state. The same object also fixes where those readings stop: they hold
while circulation and gauge motion do no work against the ledger, and
the classical counterexamples are exactly the cases where one of them
does, which is why the unification is answerable rather than merely
tidy. A
chemotactic cell exercises all three at once: it follows a path,
maintains fluxes, and updates its estimate of where the food is.

\paragraph{The third question: what should it look like across scales?}
Systems of this kind are known to produce $1/f$ noise, and $1/f$ noise
explains very little, because an unbounded power law is compatible with
almost anything. \QiCON{} (Theorem T-CON.2) makes the sharper claim: the
band is finite, and both of its ends are set by quantities the framework
names. Below the lower knee, at the slowest surviving relaxation rate,
the spectrum flattens; above the upper knee, at the fastest retained
microscopic rate, it steepens to $f^{-2}$. Two knees, not one, and the
paper says in advance where to look for each. The width between them is
not decoration either: it fixes how many levels deep the memory
hierarchy of the next pillar can be.

\paragraph{The fourth question: how does it remember?}
Once a system holds a trace of its past for longer than a single
act-and-look cycle, \QiHAL{} (Theorem T-HAL.1) puts its runtime into a
definite normal form: a loop running memory to prediction to motion to
sensing and back; a phase code; a lattice storing relations rather than
snapshots; a ladder of nested timescales. The order of the loop is
forced rather than chosen --- sensing cannot precede action without the
system knowing an observation it has not yet caused, and prediction
cannot follow sensing without becoming postdiction. Two consequences are
worth stating plainly. Because what is stored is a set of phase
differences and not absolute phases, a memory written on one carrier can
be read on a completely different one, which is why a chemical, a neural
and a digital system can hold the same content; and because the trace is
distributed across relations, a damaged substrate can still reconstruct
part of what it held. Memory here is never free: a write is an event,
and \QiQUA{} bills it.

\paragraph{What this paper has that no companion can supply.}
The engine imported from \ASix{} runs in two regimes --- descent on a
landscape that stays put, and the transient that reorganizes the
landscape and hands the next descent its starting point. In cosmology
the second kind is exogenous and unrepeatable; the universe does not
re-run its own beginnings. The discovery this paper contributes is that
a memory-bearing system \emph{internalizes} it. Such a system carries
its own dial for reorganizing its landscape, and schedules the
reorganizations on evidence: it explores while the evidence is thin,
commits when the evidence crosses a threshold, and returns to exploring
when the world changes and its predictions start failing. Attention,
search, decision, commitment and revision are one normal form, and it is
the microscopic mechanism by which a cascade keeps itself going instead
of settling once and stopping.

\paragraph{Holding together, and the world seen from inside.}
Two further structures follow. \SeOS{} names the systems that maintain
themselves, and makes the criterion a number rather than a judgment:
what comes in through the ports, minus what is dissipated, minus what
repair costs, must stay positive. \QCI{} then asks a different question
--- not whether a system survives but how much of the four-pillar
runtime it actually runs --- and scores it in a way that cannot be
gamed, since a system missing one pillar cannot compensate with
excellence in another. On top of these sits the internality layer. A
finite system never touches the world; it touches its own boundary, so
whatever guides its action has to be reconstructed rather than
received. The reconstruction is why motion is part of perception and
not an accessory to it: a retina that cannot move is depth-poor, a
bacterium that cannot swim is gradient-poor, and the paper makes that
precise as a rank condition on what the sensory history can recover.
The word used for the internally measured, selected and stabilized part
of that reconstruction is a technical one, defined operationally and
graded rather than binary, and the paper is explicit that it takes no
position on subjective experience.

\paragraph{How far it reaches, and in what sense.}
The same grammar is then run up a five-story hierarchy --- particles,
cosmos, chemistry, life, mind --- where each story is a fixed-landscape
descent and each ascent is a landscape reorganization that opens basins
the story below did not have. A floor counts as a floor only if it
passes three tests: its collective variables nearly close, it lasts far
longer than the relaxation beneath it, and it opens at least one
genuinely new exchange channel. The framework's boldest sentence is
that life and mind are inevitable, and it is stated in exactly one
sense: not that every planet produces them, which is disclaimed, but
that in a driven universe whose viable configurations occupy a set of
positive measure, a search that keeps returning reaches and stabilizes
them with probability one. Higher floors never break lower laws; they
change the conditions the lower floors run under.

\paragraph{What is proved here, what is inherited, and what is open.}
The boundary of a claim matters more than the claim. Proved here: the
four pillars as dual-layer theorems, the completeness of the system
packet, the internalized regime switch, and the characterization of the
ground attractor --- in one direction. That direction says the
configuration where the descent halts carries all four certificates. The
converse, that no other basin carries all four, is not proved, is
recorded as an open problem, and should not be quoted as though it were
settled: asserting it by appeal to a companion whose own argument
invokes these same pillars would be a circle, and the two directions
are therefore kept apart. Inherited
and not re-derived: the charter, the geometry, the boundary law, and the
cascade --- whose closure carries one named residue of its own, so
every construction here that stands on it inherits that conditional
status. Named as open rather than claimed: whether the hierarchy is
forced to have exactly five floors, and the residual bounds on the
inevitability argument. Said plainly about most of the cross-domain
material: it is organizational rather than predictive, and the paper
marks which rows are which so that a reader cannot mistake a placement
for a prediction.

\paragraph{A closing thought.}
Information has spent a century as physics' guest --- indispensable in
practice, unpriced in principle, invoked wherever a system had to
choose, remember or predict, and quietly excused from the ledger every
other quantity had to satisfy. The proposal here is that it was never a
guest. It is the fourth column of an account that has always had four
columns, and the size of one entry in it is fixed by the same integers
that fix how many families of matter there are. If that is right, then
the world of currents and free energies and memories that the
laboratory sciences actually work in is not the untidy residue left over
when fundamental physics has taken what it wants. It is the face the
same object turns toward anything finite enough to have to look.
\end{ExecutiveSummary}

\GGKeywords{GG-Theory, GG-6, GDOS, GRIP, GRIP-In, GRIP-Out, GFF, GEP, CEP, GA, GGA (Ground Geometric Attractor), Qi-4 Pillars, Qi-QUA, Qi-EP, Qi-CON, Qi-HAL, SeOS, QCI, OH, IH, EIC, five-story pagoda, Arisaka Ratio, Arisaka Equation, Arisaka Constant, MePMoS, NHT, HAL, BFFT, internality layer.}

%% ============ FRONT-MATTER SHOWCASE, v15.24 =================
%% Three pages after the Keywords, before the Overview of Contents.
%%   p1  Figure 4, the four Qi-4 pillars  -- the pair to A6's Fig 2
%%   p2  Figure 14, the five-story pagoda -- A7's own image
%%   p3  Table 47, the falsifiability program -- what would kill it
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%% ===========================================================
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
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\textbf{Predicted signature} & \textbf{Test idea} & \textbf{What failure would mean} \\
\midrule
Current-locking constraints & Estimate event-level energy/charge/information budgets in repeated micro-events; test whether one locked ledger explains them. & Persistent inability to define a coherent event ledger would challenge \QiQUA. \\
Selector behavior & Perturb pathway options and test whether realized trajectories follow a common \PLA/\PLT/\MEPP/\FEP\-style selector. & If no selector family fits across domains, \QiEP\ loses force. \\
Two-knee scale signatures & Measure spectra, burst statistics, and fractal indicators under controlled conditions. & Absence of multiscale diagnostic structure would weaken \QiCON. \\
Exploration-to-commitment lag & Time-resolve candidate-state exploration, branch selection, and stable report or action. & If selection/stabilization staging is absent, the \(\DGI\to\GSSB\to\GA\) pipeline is incomplete. \\
Motion-conditioned model quality & Restrict action while preserving passive input and test whether internal model quality degrades. & Failure would challenge the action-conditioned observability theorem behind \IH{}. \\
\bottomrule
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\clearpage
%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
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and What a Moving System Can Come to Know}
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{\small\captionof*{figure}{\textbf{What this paper is.} \AFive{} gives the boundary law and \ASix{} the geometric template that forces it; the present paper supplies the informational half. The geometric half says how an open system moves and what it spends; the informational half says what the spending is denominated in and what the moving system can come to know. Each pillar inherits its constitutional content from \GDOS{} paired with the \GiFour{} template above, and its bulk-side content from the 6D gauge sector below; the horizontal arrows are implications, not analogies. Neither claim survives on its own --- a reader who takes only the geometry has a runtime with an unpriced column, and one who takes only the information has a price without an engine. In full: Figure~\ref{fig:a7-qi4-architecture}, p.~\pageref{fig:a7-qi4-architecture}.}}
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Particle, Cosmos, Chemistry, Life, Mind}
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{\small\captionof*{figure}{\textbf{The same runtime, five times.} Particle, cosmos, chemistry, life, mind. Each floor is forced by the one beneath it through the pagoda factorization theorem --- closure-defect plus port-novelty admissibility --- and within each floor the dynamics is \GRIPin{}, descent to that floor's ground attractor, while the ascent from one floor to the next is a \GRIPout{} event that opens basins the lower floor did not possess. The \emph{shinbashira}, the central pillar, is the \OGGEP{} projector carrying the combined \GiFour{}/\QiFour{} cascade, and it runs through every floor unchanged. Two honesties are on the plate rather than behind it: five is the historically natural sequence and is not proved unique, and the inevitability of the upper floors is conditional on a driven universe with positive viable subsets. In full: Figure~\ref{fig:a7-pagoda}, p.~\pageref{fig:a7-pagoda}.}}
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and What a Failure of Each Would Mean}
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{\small\captionof*{table}{\textbf{Where this framework is exposed.} Five predicted signatures, each with a test and --- the column that gives the table its value --- what a failure would mean. A framework this broad can absorb a surprising amount of data unless it says in advance which failures are fatal and which are merely inconvenient, and that is what the third column is for. All five are open in the experimental sense: the prediction is on the page and the verdict is the next decade of measurements. In full: Table~\ref{tab:falsifiability}, p.~\pageref{tab:falsifiability}.}}
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% ============ end front-matter showcase ====================

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%% ===============================================================
%%  PREFACE -- compelling entrance for the entire paper
%%  Added in v6.13 (May 2026) per user request: a global
%%  introduction that captures the reader's mind before Part I.
%%  Four paragraphs in the Arisaka register, plain English, no
%%  paper-specific jargon (no Qi-QUA/Qi-EP/Qi-CON/Qi-HAL names).
%% ===============================================================
\GGPreface

In 1944, Erwin Schr\"odinger asked a question that physics has
never quite answered: what is life? In 1948, Claude Shannon wrote
down a quantitative measure of information that any engineer could
compute. In 1961, Rolf Landauer showed that erasing a single bit
of that information always costs at least a fixed amount of
physical heat. In 2010, Karl Friston proposed that every living
organism keeps itself alive by minimizing a free-energy
functional, the way a thermostat minimizes a temperature
difference. And since the work of 't Hooft in 1993 and Susskind
in 1995, theoretical physicists have entertained the hypothesis
that the world we see may be a projection of information stored
on a distant boundary. Each of these threads was woven in a
different decade, by people working in different fields, and the
threads have never been tied together.

Living systems sit at the intersection of every one of these
threads. A cell is a thermodynamic engine, an information-processing
device, a bulk-to-boundary projection of a deeper geometry, and a
free-energy minimizer, all at once. Physicists who study quantum
mechanics speak of measurement records and decoherence; biologists
who study cells speak of memory traces and homeostasis; theorists
who study black holes speak of bits encoded on horizons. The
vocabulary is utterly different, the underlying mathematics often
quite similar, and the connections have usually been left to
philosophical sidebars. There has been no common language in
which to say that the information current in a cell and the
information current on a holographic screen are two faces of one
object.

This paper proposes a common language. The argument names four
informational pillars on a small six-dimensional manifold: a
master quantum-information object that supplies the bits, an
equivalence principle that tells us a bit of information always
costs the same minimum amount of energy no matter who is doing
the bookkeeping, a master continuity statement that describes how
information rolls forward in any open system, and a master
hierarchy that explains why nature seems to climb the same ladder
of organization, from molecules to memory, on every scale. These
pillars do not add new postulates. They are the informational
face of the geometric pillars derived in the companion paper of
the program, and together with them they close the loop between
geometry, gauge, energy, and information. A single conversion
factor of size $0.1555$ electron-volts per qubit
connects the information current of a thinking brain to the
energy current of the cell that hosts it --- and matches one-half the
energy released by the breakdown of a single molecule of ATP at
body temperature (one ATP carries two such quanta).

We invite the reader to walk the four informational pillars in
the same way the companion paper walks the four geometric ones.
Read each pillar, follow each proof, and ask whether the
laboratory phenomenon it predicts --- the conversion factor
between information and energy, the spectral signature of memory,
the universality of the ladder --- is something a careful
experiment could test. After all, the question Schr\"odinger
asked in 1944 was whether life requires a new kind of physics.
The answer that emerges here is that life does not require new
physics --- but it does require physics to recognize that
information has always been one of its first-class objects, and
that the boundary on which we live is, in a precise and
laboratory-testable sense, the place where the upstream geometry
writes itself down in bits.

\clearpage

\GGPart[Entrance and bridge]{Entrance and bridge}{Introduction, recap, and the Qi-runtime extension claim}
\label{part:entrance}

\noindent
This Part answers one question: why should a single law describe a
particle detector, a bacterium, a nervous system and a market?

The usual answer is that it should not, and the instinct behind that is
a good one. Those four things differ in every way that normally
matters. What this Part argues is that they share the one property that
turns out to be decisive, and that the property has nothing to do with
how complicated they are. Each is studied by someone who can only see
part of it. A finite observer, using finite instruments, never has the
whole state of anything in hand --- and the moment that is true, the
description available to that observer cannot be closed. It needs
memory, or hidden variables, or averaging. The result is proved here
rather than assumed, and it has a blunt consequence: the difference
between physics and biology is not that one is closed and the other
open. Both are open. Only the richness of what is left after the
projection differs.

That is the entrance. The bridge is the other half of the Part. This
paper derives nothing about the six-dimensional geometry, nothing about
the boundary law, and nothing about the engine that drives a system
downhill; all of that arrives from five companion papers, and this Part
sets out exactly what arrives, in the form it will be used, with the
theorem that supplies each piece named. That inventory is written as a
contract rather than as a summary, because a later Part that quietly
assumed something not on the list would be borrowing without saying so.
One item on the list carries a condition of its own --- the runtime
cascade closes subject to two conjectures still open in its home
paper --- and everything built on it in this paper inherits that
condition.

The Part then does one more thing, and it is the reason the paper
exists. It shows that the boundary law, on its own, is not enough. A
law can describe how an open system evolves and still leave the words
we use about such systems --- memory, selection, persistence,
complexity --- attached to nothing in particular, so that the same
system admits many incompatible readings. What is missing is a level
between the law and the domain: a runtime. The rest of the paper builds
it.

A reader in a hurry should not skip the table of claims and non-claims.
It is short, it is the least exciting page in the Part, and it is the
one that says what this framework does not assert. Vocabulary this broad
is easy to over-read, and the right-hand column of that table is what
keeps the left-hand column sayable.


\noindent
Part~\ref{part:entrance} sets the stage for everything that follows. We open with
the question that physics has never quite answered: what is
life, and more generally, what is an open system? Section~\ref{sec:introduction}
states the problem in plain language and lists the unsolved
questions that motivate the present paper; Section~\ref{sec:recap-bridge}
recapitulates the upstream foundation (Papers~I--IV-A) on which
the \QiFour{} runtime must be built; and
Section~\ref{sec:universal-gdos} states the universal-extension
claim --- that \GDOS{} applies to every open system in nature once
the four \QiFour{} pillars are in place. Part~\ref{part:entrance} tells the reader
why the rest of the paper is worth reading.

\medskip

\section{Introduction}
\label{sec:introduction}

\subsection{Why living organisms are the natural entry point for Paper~IV-B}
\label{subsec:living-entry-point}

The first four papers of the GG/GD sequence were necessarily written from the bottom up. Paper~I fixed the constitutional layer: what a lawful theory is allowed to assume, what it must derive, and what it must make measurable \citep{ArisakaA1Constitution}. Paper~II then answered the ultraviolet question, namely what higher-dimensional geometry is forced by that constitution \citep{ArisakaA2GG6}. Paper~III answered the operational question, namely how a finite observer on the boundary obtains quantum states, measurement records, and four-dimensional effective dynamics from that higher-dimensional bulk \citep{ArisakaA5GDOS}. Paper~IV-B must now reverse the direction of emphasis. It does not begin from the smallest scales but from the systems in nature that most vividly display the full open-system problem: living organisms.

A living organism is the most stringent natural test case for any theory that claims to be a universal theory of open systems. It is not merely a dynamical system, nor merely a thermodynamic system, nor merely an information-processing system. It is all three at once. A cell or organism maintains a boundary, exchanges matter and energy with its environment, processes information, stores memory, predicts possible futures, selects actions, and persists far from equilibrium for times vastly longer than microscopic relaxation times \citep{Schrodinger1944,Landauer1961,Prigogine1967,NicolisPrigogine1977,Friston2010}. If a framework cannot say what such a system \emph{is}, what its lawful state packet is, what its relevant currents are, why it persists, and how it makes or receives measurements, then that framework has not yet reached a universal runtime.

This is why life, rather than elementary particles alone, is the natural entry point for the present paper. Particles reveal the constitutional grammar of the theory. Organisms reveal whether that grammar is strong enough to handle persistence, adaptation, memory, agency, and internal modeling without changing its foundations from one scientific domain to the next. In that sense the present paper is not a departure from Papers~I--IV-A but the first large-scale stress test of their universality.

Mathematically, the key reason organisms are unavoidable is that they make all hidden assumptions visible. To make this concrete, we now write the minimal quantum-information notation that a finite observer is forced to use, even when modeling a cell or an organism. A reader from biology or chemistry who is unfamiliar with this notation may skim the equation and continue with the prose immediately after it; the gloss following the equation translates each symbol into plain words.

For an isolated Hamiltonian system the notation
\[
\frac{\dd}{\dd t}\ket{\psi(t)} = -\,\ii\,H\,\ket{\psi(t)}
\]
can be written compactly and elegantly, but this compactness hides the fact that a real observer never has access to the entire Hilbert space. In an actual experiment or an actual organism, one deals instead with a subsystem \(S\), an environment \(E\), a reduced state, and a family of coarse-grained observables:
\begin{equation}
\BulkState = \SysState \otimes \EnvState,
\qquad
\rho_S(t) = \Tr_E\,\rho_{SE}(t),
\qquad
\langle O_a\rangle_t = \Tr_{\SysState}\!\big(\rho_S(t)\,O_a\big).
\label{eq:intro-reduced-state}
\end{equation}
In plain words: \(\BulkState\) is the full state space of everything --- system plus environment together; \(\SysState\) is the part the observer can access; \(\EnvState\) is everything else. The reduced state \(\rho_S(t)\) is what the observer would measure if she traced over (i.e.\ averaged over) the environment she cannot resolve. The expectation value \(\langle O_a\rangle_t\) is the actual number a laboratory experiment returns for the observable \(O_a\) at time \(t\), computed from the reduced state. None of this notation is optional once one accepts that the observer is finite; it is the precise bookkeeping that the rest of the paper will work with.

A living organism makes Eq.~\eqref{eq:intro-reduced-state} unavoidable because it is \emph{constitutively coupled} to an environment: food, photons, chemical gradients, thermal baths, predators, conspecifics, and internal submodules all act as ports. The closed-system idealization is therefore not just inaccurate but structurally misleading when one attempts to understand life, cognition, and collective behavior.

The philosophical point is equally important. Since \citet{Schrodinger1944}, life has been recognized as a phenomenon that cannot be reduced to static equilibrium thermodynamics. Since \citet{Shannon1948} and \citet{Landauer1961}, information can no longer be treated as an informal metaphor detached from physical cost. Since the development of nonequilibrium thermodynamics and open quantum systems, irreversibility, coarse-graining, and environment-induced records must be taken as first-class theoretical objects rather than nuisances to be postponed \citep{Prigogine1967,NicolisPrigogine1977,BreuerPetruccione2002}. Paper~IV-B begins from organisms because they force all these strands to coexist in one object.

\begin{table}[H]
\centering
\caption[Why living systems are the natural entry point.]{Why living organisms are the natural entry point for the universal extension of \GDOS{}. Five features of living matter are listed down the first column; the second says why each is mathematically demanding, and the third why each is decisive for the present paper. The argument the table carries is not that biology is special but that it is \emph{unavoidable}: a system that is persistently open, held far from equilibrium, maintains its own boundary, remembers and predicts, and is organized hierarchically exercises every part of the runtime at once, so it is the case in which a missing piece of the framework shows up first.}
\label{tab:life-entry}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.22\textwidth}p{0.34\textwidth}p{0.34\textwidth}}
\toprule
\textbf{Feature of living systems} & \textbf{Why it is mathematically demanding} & \textbf{Why it is decisive for Paper~IV-B} \\
\midrule
Persistent openness & State evolution requires explicit environmental coupling, ports, and reduced dynamics. & Forces the theory to move beyond closed-system Hamiltonian language. \\
Far-from-equilibrium operation & Stability is maintained by continuous flux, not by equilibrium minimization alone. & Motivates a unified free-energy ledger and trajectory-selection principle. \\
Boundary maintenance & The system must distinguish self from environment while exchanging with it. & Motivates \SIP, \SeOS, and the outer/inner hologram distinction. \\
Memory and prediction & The present state depends on history and on internally represented futures. & Motivates \QiHAL\ and the move from external measurement to internal measurement. \\
Hierarchical organization & Molecular, cellular, organismal, and social levels interact without collapsing into one scale. & Motivates scale covariance, \QiCON, and the pagoda hierarchy. \\
Action and adaptation & The system does not only receive information; it changes its boundary conditions by acting. & Motivates the full \DGI\(\to\)\GSSB\(\to\)\GA\ pipeline rather than passive state evolution alone. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{From the measurement problem to the universal observation problem}
\label{subsec:universal-observation}

Paper~III was framed around the measurement problem in quantum mechanics, but its deeper lesson is broader. The familiar measurement problem asks why and how definite outcomes appear if the microscopic law is unitary \citep{vonNeumann1932,Bell1964,Zurek2003}. The answer proposed in Paper~III was to promote measurement from an interpretational add-on to a constitutional demand: AX-3 (\OGGEP) requires every admissible theory to provide a lawful bulk-to-boundary dictionary (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA5GDOS}). In its most compact form, this dictionary is a map
\begin{equation}
\BulkMap:\;
\text{bulk data}
\longrightarrow
\big(\BState,\rho_{\partial},\Phi_{t,s},\{\Instr_\alpha\}\big),
\label{eq:bulkmap}
\end{equation}
where \(\rho_{\partial}\) is the observer-accessible boundary state, \(\Phi_{t,s}\) is an admissible evolution map, and \(\Instr_\alpha\) are instruments or operations associated with registered outcomes.

For Paper~IV-B the crucial point is that Eq.~\eqref{eq:bulkmap} is not merely a statement about laboratory measurements in quantum physics. It is the general grammar of \emph{observation}. Every scientific observation, in any field, is obtained only after three reductions have already occurred:
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item a large inaccessible state space has been truncated to a smaller accessible state space;
\item a continuous high-dimensional process has been projected onto a finite register or data stream;
\item the resulting record has been interpreted relative to a chosen instrument class.
\end{enumerate}
That is as true for electrophysiology, microscopy, ecology, and economic indicators as it is for Stern--Gerlach experiments. The observation problem is therefore universal. The quantum measurement problem is only its sharpest historically visible corner. The openness, however, is not created by the act of study: the bulk-to-boundary projection of \OGGEP{} is always on (\AFive{}~\S23), running at every boundary point whether or not anyone is studying anything, and a finite observer merely \emph{inherits} an already-projected, already-open description. What is universal is not that observation makes systems open, but that every system \emph{is} open --- observed or not.

The mathematical signature of universality is that the same reduced-dynamics structure recurs across domains. At the exact level one obtains a non-Markovian memory-kernel equation of Nakajima--Zwanzig type,
\begin{equation}
\frac{\dd}{\dd t}\rho_{\partial}(t)
=
\int_0^t K(t-s)\,\rho_{\partial}(s)\,\dd s
+
I(t),
\label{eq:nz-intro}
\end{equation}
while in the short-memory limit one obtains a completely positive semigroup with GKLS generator,
\begin{equation}
\frac{\dd}{\dd t}\rho_{\partial}
=
-\,\ii [H_{\mathrm{eff}},\rho_{\partial}]
+
\sum_k
\left(
L_k \rho_{\partial} L_k^\dagger
-\frac12\{L_k^\dagger L_k,\rho_{\partial}\}
\right).
\label{eq:gkls-intro}
\end{equation}
The exact microscopic meaning of \(H_{\mathrm{eff}}\), \(L_k\), and the kernel \(K\) varies across domains, but the open-system logic does not. What changes from particles to organisms is not the fact of openness but the complexity of the state packet, the boundary, the port structure, the attractor geometry, and the internal modeling architecture.

This is precisely why the present paper must broaden the focus from ``measurement'' to ``observation.'' An organism not only registers external data; it also constructs internal states that function as measurements of its own world-model. A nervous system, for example, receives sensory inputs, compares them against memory, generates predictions, issues actions, and re-samples the environment. Once that loop is recognized, external observation and internal observation become two instances of the same architecture. This is the conceptual doorway through which \(\OH+\IH=\EIC\) will later enter: the outer hologram concerns externally registered boundary data, while the inner hologram concerns the internally reconstructed model through which a system experiences and navigates its world.

\begin{definition}[Operational openness]
A scientific system \(S\) is \emph{operationally open} if the observer-accessible state of \(S\) cannot be described by a closed autonomous law on \(S\) alone, but instead requires at least one of the following: a traced-over environment, explicit ports, memory kernels, stochastic coarse-graining, or instrument-conditioned state updates.
\end{definition}

\begin{remark}
Operational openness is weaker than full ontological openness. The definition makes no claim that the total universe is open to something external. It claims only that every \emph{scientific description used by a finite observer} is open because it is necessarily partial, coarse-grained, and instrument-bounded. This is enough to make open-system mathematics constitutional.
\end{remark}

\subsection{The need for a common runtime}
\label{subsec:common-runtime}

The notion of a \emph{runtime} is meant literally. Papers~I--IV-A supplied the constitution, the ultraviolet hardware, and the boundary operating system. Paper~IV-B now asks: what is the executable layer common to all natural open systems once the theory is actually running? In software language, this is the layer that every scientific domain calls even if the domains look superficially different. In physical language, it is the layer that every open system instantiates even if the microscopic details differ.

The proposed answer is the master pipeline
\begin{equation}
\boxed{
\DGI \;\longrightarrow\; \GSSB \;\longrightarrow\; \GA
}
\label{eq:master-pipeline}
\end{equation}
together with the \QiFour{} pillars and the universal runtime vocabulary. Equation~\eqref{eq:master-pipeline} should be read as the minimal grammar of persistence. First, a system explores lawful gauge-covariant possibilities (\DGI). Second, environmental coupling, constraints, and boundary conditions select one branch or one structure (\GSSB). Third, the selected structure is stabilized as an attractor, function, phenotype, memory, or policy (\GA). The later sections of the paper make each arrow explicit in the fixed order adopted here. For the introduction, it is enough to see that persistence is not a static property. It is a three-stage process.

The \QiFour{} pillars then answer four questions about any runtime instance:
\begin{description}\setlength{\itemsep}{2pt}
\item[\QiQUA] \emph{What currents are locked together?}
\item[\QiEP] \emph{Which trajectories are selected?}
\item[\QiCON] \emph{What scale signatures should appear?}
\item[\QiHAL] \emph{What architecture carries memory, prediction, sensing, and action?}
\end{description}
This decomposition is not ornamental. It is a discipline against conceptual drift. Without it, one risks mixing bookkeeping, variational principles, statistical diagnostics, and architecture into one undefined notion of ``complexity.''

A convenient preview of the mathematical structure is the generic \GDOS\ continuity form already derived in Paper~III:
\begin{equation}
\partial_t p(x,t) = - \nabla_A J^A(x,t),
\label{eq:gdos-continuity-intro}
\end{equation}
with current decomposed schematically as
\begin{equation}
J^A = - D^{AB}\nabla_B p \;-\; \mu^{AB} p \,\nabla_B \mathcal{F}_{\GFF}
\;+\; \Omega^A p.
\label{eq:current-decomposition-intro}
\end{equation}
The first term represents diffusive spreading, the second gradient descent on an effective \GFF\ ledger, and the third divergence-free circulation. The logic of Paper~IV-B is that a universal open-system runtime cannot be built from any one of these terms alone. Persistence needs all three. Pure diffusion destroys structure; pure gradient descent over-freezes structure; pure circulation never commits structure. The master pipeline \(\DGI\to\GSSB\to\GA\) is the conceptual version of this mathematical trichotomy.

\begin{definition}[Runtime packet preview]
A \emph{runtime packet} is a preliminary placeholder for what will later be formalized as a \SIP. It is the minimal declared data set required to specify a scientific open system:
\begin{equation}
\mathcal{R}
=
\big(
\State,\Ports,\Obs,\rho,\Phi,\mathcal{J},\mathcal{F},\mathcal{A}_{\mathrm{arch}}
\big),
\label{eq:runtime-packet-preview}
\end{equation}
where \(\State\) is the state manifold, \(\Ports\) the environmental interfaces, \(\Obs\) the registered observables, \(\rho\) the effective state, \(\Phi\) the effective evolution law, \(\mathcal{J}\) the current ledger, \(\mathcal{F}\) the scalar selection functional, and \(\mathcal{A}_{\mathrm{arch}}\) the architecture responsible for memory, prediction, sensing, and action.
\end{definition}

The full paper will refine Eq.~\eqref{eq:runtime-packet-preview} substantially. The point here is merely to show why a common runtime is needed: without some declared packet of this kind, the same words---``state,'' ``information,'' ``stability,'' ``selection,'' ``complexity''---mean incompatible things in different domains.

\subsection{Preview of the new objects introduced in Paper~IV-B}
\label{subsec:preview-new-objects}

It is useful to state at the outset what is genuinely new in Paper~IV-B relative to Paper~III. Paper~III derived the existence of a lawful open boundary theory. Paper~IV-B extends that theory by adding the runtime objects needed for universality. These objects are not optional decorations; they are the minimal extensions needed to move from the foundational level of measurement and particle physics to the domain-general level of chemistry, life, mind, and society.

First, Paper~IV-B introduces \DGI, \GSSB, and \GA\ as the universal dynamical spine. Although their ingredients are already latent in Papers~I--IV-A, they were not yet elevated to a named, reusable runtime grammar. Paper~IV-B makes this promotion explicit.

Second, Paper~IV-B introduces the \QiFour{} pillars as a disciplined factorization of the runtime. \QiQUA\ is the bookkeeping layer; \QiEP\ is the selection layer; \QiCON\ is the diagnostic layer; \QiHAL\ is the architecture layer. This factorization is what allows one to move from particle measurements to molecular machines and then further to organisms and brains without rewriting the language each time.

Third, Paper~IV-B introduces \SIP, \GFF, and \SeOS\ as the working vocabulary of universal open-system analysis. A \SIP\ will later serve as the minimal packet that can be reused across scientific branches. \GFF\ will later serve as the combined free-energy ledger. \SeOS\ will later designate persistent self-maintaining open systems whose continued existence is itself part of the dynamical story.

Fourth, Paper~IV-B introduces \(\OH+\IH=\EIC\), which extends the meaning of measurement. In the first three papers, the concern was primarily with how a physical observer obtains a lawful boundary record. In the fourth paper, the concern broadens to how a living or cognitive system constructs an internal world-model and how that world-model contributes to what is subjectively experienced.

Finally, Paper~IV-B introduces a hierarchy picture. The Japanese five-story pagoda is not used merely as a poetic image. It is intended as a compact structural map: the same \DGI\ spine runs through different layers of \GSSB\ output, from particles to cosmos to chemistry to life to mind. This hierarchical picture is important because a universal theory of open systems must explain not only one level but also why multiple levels can coexist without collapsing into each other.

\subsection{Claims and non-claims of the present paper}
\label{subsec:claims-nonclaims}

The main positive claim is that there exists a single geometric runtime capable of organizing open systems across domains, once the groundwork of Papers~I--IV-A is accepted. The paper does \emph{not} claim that all sciences reduce to particle physics, that every empirical detail of biology/neuroscience/society is already solved, or that earlier frameworks are useless. It claims only that the admissibility grammar---how states, currents, observations, attractors, and hierarchies are declared and audited---can be unified, with geometry and admissibility first, runtime second, and domain-specific models third. Table~\ref{tab:claims-nonclaims} fixes the contrast.

\begin{table}[H]
\centering
\caption[Claims and non-claims of this paper.]{Claims and non-claims that discipline the interpretation of this paper. Five categories run down the first column --- universality, reduction, measurement, life and mind, and the relation to prior theories --- with what is claimed in the second and what is explicitly \emph{not} claimed in the third. The right-hand column is the load-bearing one. It is placed this early because the framework's vocabulary is broad enough to be over-read, and a reader who takes the non-claims with them will not mistake an organizing statement for a predictive one.}
\label{tab:claims-nonclaims}
\small
\begin{tabular}{p{0.30\textwidth}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.28\textwidth}}
\toprule
\textbf{Category} & \textbf{Claim of Paper~IV-B} & \textbf{Explicit non-claim} \\
\midrule
Universality & One common runtime can organize open systems across domains. & Not every domain-specific equation is already fully derived in this step. \\
Reduction & A common admissibility grammar exists from particles to society. & The paper does not reduce away all higher-level descriptions. \\
Measurement & Observation is treated as a lawful open-system projection. & The paper does not identify measurement with consciousness. \\
Life and mind & Life and cognition are treated as lawful runtime outcomes. & The paper does not claim that all biological and cognitive details are already solved. \\
Relation to prior theories & Existing frameworks are reinterpreted as partial slices of a larger runtime. & The paper does not claim that their local successes are invalid. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Roadmap of the full manuscript}
\label{subsec:roadmap-full}

The full paper is designed as a landmark synthesis rather than a short note. Its architecture is a fifteen-Part structure: twelve Parts of main
argument followed by a three-Part back matter. Part~\ref{part:entrance} supplies the entrance and bridge: the Introduction, the recap of Papers~I--IV-A, the place of the paper in the twenty-two-paper cohort, and the universal extension of \GDOS\ to any open system. Part~\ref{part:runtime} imports the runtime constitution from \ASix{} in the strict order demanded by logical dependence: first the master theorem \GRIP{} \(=\DGI\to\GSSB\to\GA\), then the scalar ledger \GFF, and only then the canonical system-instance packet \SIP. Six Parts then develop the \QiFour{} runtime, one subject each, in its
own dependency order: current locking (Part~\ref{part:qiqua}), universal
extremality (Part~\ref{part:qiep}), multiscale diagnostics
(Part~\ref{part:qicon}), and memory-bearing architecture
(Part~\ref{part:qihal}), followed by the persistence and completeness
layer (Part~\ref{part:seos}) and the statement tying the four pillars to
the ground attractor of the geometric companion (Part~\ref{part:gga}). Part~\ref{part:internal} builds the internality layer: the Outer and Inner Holograms and the internal-measurement runtime \((\OH+\IH=\EIC)\). Part~\ref{part:emergence} treats emergence, the five-story pagoda, the cross-domain applications, and the explicit universality claim. Part~\ref{part:assessment} is the assessment: significance and comparison, limitations, and falsifiability. Part~\ref{part:conclusions} draws the conclusions. The back matter then follows in three further Parts: the Technical Appendices (Part~\ref{part:appendices}), the Common Objections and Responses (Part~\ref{part:objections}), and the References (Part~\ref{part:references}).

The present integrated manuscript already executes this entire order in the main text. It should therefore be read not as a collection of separately drafted chapters but as one continuous derivational arc: bridge \(\rightarrow\) runtime \(\rightarrow\) scalar budget \(\rightarrow\) packet \(\rightarrow\) Qi event locking \(\rightarrow\) selector \(\rightarrow\) diagnostics \(\rightarrow\) architecture \(\rightarrow\) persistence/completeness \(\rightarrow\) internality \(\rightarrow\) hierarchy \(\rightarrow\) applications \(\rightarrow\) universality \(\rightarrow\) assessment. The purpose of the present harmonization pass is to make every transition, theorem lead-in, and section summary reflect that non-interchangeable order as clearly as possible.

\section{This paper's place in the cohort: one link of the open-system chain}
\label{sec:cohort-funnel}

This paper establishes the four quantum-information \QiFour{} pillars and the operational meaning of measurement; that is its own subject. For readers following the whole \GGTheory{} cohort, it is also one link of a single open-system chain that the consequence papers cash out. Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is the boundary law \GDOS{}, forced by the geometric \GiFour{} and quantum-information \QiFour{} pillars, so that the laws, the particles, and the contents of the universe are \emph{observables} of the projection rather than free inputs. In that chain, the present paper supplies the \QiFour{} pillars and the measurement content --- the part of the chain that makes the bulk-to-boundary projection \emph{observation}. The funnel below is shared, in structure, with the companion papers \ANine{}, \ATen{}, and \AEleven{}; here it is offered only as orientation, and the rest of this paper proceeds on its own terms.

\begin{figure}[hbt!]
\centering
\resizebox{0.681\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=17mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=104mm, minimum height=11mm},
 arrow/.style={-{Stealth[length=3mm]}, line width=1.10pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10] (open) {\textbf{Every physical system is open}\\ a particle, the universe, a living cell --- none can be isolated from its environment};
\node[box, fill=blue!8, below of=open] (oggep) {\textbf{\OGGEP{}} (\AX{3}): the lawful projection \;6D bulk $\to$ 4D boundary\\ \textit{this projection is observation itself} \;\; {\scriptsize[\AOne{}, \AEight{}]}};
\node[box, fill=yellow!13, below of=oggep] (gdos) {\textbf{\GDOS{}}: the dynamics of the open-system projection\\ \textit{an observable is the value at a stationary seam} \;\; {\scriptsize[\AFive{}]}};
\node[box, fill=orange!14, line width=1pt, below of=gdos] (obs) {\textbf{The laws, the particles, and the contents of the universe}\\ \textbf{are registered observables} of this one projection --- \textit{not free inputs}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!16, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{they are determined}\\ {\footnotesize read off one geometry --- the work of the consequence papers \ANine{}--\AEleven{}}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D theory exists}            (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply at every stratum}                (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observables forced} (concl);
\end{tikzpicture}}
\caption[The open-system chain shared across the \GGTheory{} cohort.]{\textbf{The open-system chain shared across the \GGTheory{} cohort.} Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is \GDOS{}, forced by the \GiFour{} and \QiFour{} pillars, so the laws and contents of the universe are observables rather than free inputs. The same funnel opens the consequence papers \ANine{}, \ATen{}, and \AEleven{}, with a paper-specific payload; within it the present paper establishes the \QiFour{} pillars and the measurement content.}
\label{fig:cohort-funnel}
\end{figure}

\section{Recap bridge to \AOne{}, \ATwo{}, \AThree{}, \AFive{}, and \ASix{}}
\label{sec:recap-bridge}

\subsection{Why the recap is structurally necessary}
\label{subsec:why-recap}

Paper~IV-B is not intended as an isolated monograph. It is the fourth member of a logically ordered sequence. The whole point of that sequence is that later papers should not re-open earlier constitutional questions every time a new domain is addressed. If that discipline is not maintained, the program would degenerate into a loose collection of ambitions rather than a theorem chain.

The recap therefore plays a very specific role. It does not repeat the earlier papers for convenience. It identifies which results from those papers are now to be treated as \emph{imported theorems}, which notation choices are fixed, which issues were intentionally postponed, and which exact gap makes Paper~IV-B necessary. In that sense the recap is a bridge, not a summary. It is the interface contract between the already established foundation and the new runtime layer.

A concise way to say this is that each paper answers a different question:
\begin{equation}
\begin{aligned}
\text{Paper I}:&\quad \text{What counts as an admissible theory?}\\
\text{Paper II}:&\quad \text{What ultraviolet bulk is forced by that admissibility?}\\
\text{Paper III}:&\quad \text{How does a lawful boundary world arise from that bulk?}\\
\text{Paper IV}:&\quad \text{How do persistent open systems run within that lawful boundary world?}
\end{aligned}
\label{eq:four-paper-questions}
\end{equation}
Equation~\eqref{eq:four-paper-questions} already shows why Paper~IV-B cannot merely be an appendix to Paper~III. The question it asks is different in kind. Papers~I--IV-A establish legality, stage, and projection. Paper~IV-B establishes runtime.

\begin{table}[H]
\centering
\caption[Division of labor across the foundation papers.]{Division of labor across the papers this one stands on. Each row is one companion: the second column gives the primary question that paper answers, the third its principal theorem-level output, and the fourth the direct role that output plays here. The table exists to make the direction of the debt explicit --- everything in the first three columns is imported and none of it is re-derived here --- and it is the reference a reader should use when a later section says a result is inherited.}
\label{tab:division-of-labor}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.15\textwidth}>{\raggedright\arraybackslash}p{0.26\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}}
\toprule
\textbf{Paper} & \textbf{Primary question} & \textbf{Principal theorem-level output} & \textbf{Direct role in Paper~IV-B} \\
\midrule
\AOne{} & What is a lawful theory? & Four axioms, seven postulates, admissibility grammar & Fixes the constitution and the ordered filters Paper~IV-B must obey. \\
\ATwo{} & What ultraviolet bulk is forced? & 6D bulk, \BIsix{}, \OGNfive{}, anomaly-free 6D QFT & Fixes the geometric stage, current ordering, and ultraviolet inputs. \\
\AFive{} & How does the boundary world arise? & \OGGEP{} channel, \GDOS, 4D QM, 4D GR, formal IR QFT/SM descent & Fixes the open-system boundary law and the meaning of measurement. \\
A6/A7 (runtime-constitution program) & How do open systems persist and organize? & Runtime grammar, \QiFour{} pillars, \SIP/\GFF/\SeOS, \OH/\IH/\EIC\ hierarchy & Extends the boundary law into a universal theory of open systems. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Paper~I recap: the \GGConstitution{} axiom system}
\label{subsec:paper1-recap}

Paper~I established that the program is to be judged first as a constitution rather than as a model \citep{ArisakaA1Constitution}. A candidate theory \(\mathcal{T}\) is admissible only if \(\Admissible(\mathcal{T}):=(\mathcal{T}\models \mathrm{AX})\wedge(\mathcal{T}\models \mathrm{POS})\), with the four axioms AX-1 (\LC), AX-2 (\GGEP), AX-3 (\OGGEP), AX-4 (\MSMF) fixing primitive semantics, and the seven postulates POS-1--POS-7 enforcing gauge invariance, a unified extremal principle, anomaly freedom, closure/positivity, scale covariance, OGN, and measurement/falsifiability respectively---in that order. For Paper~IV-B this matters not as a slogan but because the constitution is imported as Theorem T-AX of Paper~I; see also the inward Coherence Map (Sec.~\ref{sec:coherence-map}) for the line-by-line audit.

\subsection{Paper~II recap: \GG, \BIsix{}, and the six-dimensional bulk}
\label{subsec:paper2-recap}

Paper~II solved the ultraviolet realization problem \citep{ArisakaA2GG6}, deriving a minimal six-dimensional bulk
\begin{equation}
M_6 = M_4 \times I_r \times S^1_\theta,
\qquad r=\ln(\EGUT/\mu),
\qquad
\dd s_6^2 = e^{-2A(r)}\,g_{\mu\nu}\,\dd x^\mu \dd x^\nu + \dd r^2 + R_\theta^2\,\dd\theta^2,
\label{eq:gg6-stage}
\end{equation}
with bulk gauge structure \(G_{\mathrm{Bulk}}=\SU(5)\times \U(1)_B \times \U(1)_{Qi}\times \mathrm{Gravity}\). The technical centerpiece is \BIsix{}, \(\dd H_3 = \tr(R\wedge R) - \Tr(F_5\wedge F_5) - (2/5)F_B\wedge F_B - (2/5)F_{Qi}\wedge F_{Qi}\), with coefficient vector \((a,b,c,d)=(1,-1,-2/5,-2/5)\) and the \OGNfive{} integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\) satisfying \(K=LJ=M^2+N^2+L^2=35\). These are imported as Theorem T-BI6 of Paper~II; see the inward Coherence Map. The four-component bulk is constitutionally admissible in A1--A3 but \emph{operationally required} for \QiFour{}: the Arisaka equation contains \(J_{Qi}\) explicitly, \QiHAL{} treats \(U(1)_{Qi}\) as the memory phase carrier, and the entire \OH+\IH=\EIC{} framework is defined on memory-bearing \(U(1)_{Qi}\) systems (a three-component variant would require a different \QiFour'{} framework, recorded as open problem O9). The \(U(1)_{Qi}\) circle is a real compact fiber; the phase it carries is a real angle, and the imaginary unit of that phase's boundary representation is the derived quarter-turn of T-P4-1 (\citealp{ArisakaP4Real}) --- so \QiHAL{}'s memory register is a real angle on a real circle, its complex dressing inherited rather than assumed.

\subsection{Paper~III recap: \GDOS\ as the boundary foundation}
\label{subsec:paper3-recap}

Paper~III answered the projection problem \citep{ArisakaA5GDOS}. With \(\BulkState=\BState \otimes \EnvState\) and \(\rho_{\partial}(t)=\Tr_{\EnvState}\rho_{\mathrm{Bulk}}(t)\), the exact projected dynamics obeys a Nakajima--Zwanzig memory-kernel equation that reduces in the Markovian limit to a GKLS generator and, in the mesoscopic coarse-grained limit, to a continuity law
\begin{equation}
\partial_t p = - \nabla_A J^A,
\qquad
J^A = -D^{AB}\nabla_B p - \mu^{AB}p\,\nabla_B\mathcal{F} + \Omega^A p.
\label{eq:paper3-gdos}
\end{equation}
Equation~\eqref{eq:paper3-gdos} is the kernel from which Paper~IV-B grows. Paper~III recovered 4D quantum mechanics (with measurement as a lawful instrument update), 4D general relativity (from the metric zero mode), formal IR QFT, and an IR Standard-Model sector with \(\SU(3)\times \SU(2)\times \U(1)_{EM}\times \U(1)_B\times \U(1)_{Qi}\) structure (\(\U(1)_B\) enforcing exact proton stability). Paper~III was intentionally foundational: it did not yet build the universal runtime needed for chemistry, life, brains, or society.

\subsection{Paper~IV-A recap: \GRIP, \GFF, and \SIP}
\label{subsec:paper4a-recap}

Paper~IV-A made the runtime constitution explicit \citep{ArisakaA6Gi4}: the Gauge-Runtime Invariance Pipeline \(\GRIP := (\DGI \to \GSSB \to \GA)\), the derived scalar ledger \(F_{\GFF}=F_{\GEP}+F_{\CEP}\) (with \(\GEP\) descending from the geometry--gauge sector of the 6D action and \(\CEP\) descending from boundary state-manifold curvature/diffusion), and the canonical system-instance packet \SIP{}. The ledger \GFF{} and the pipeline \GRIP{} are imported as \ASix{}'s principal theorems T3 and T4 respectively (its T1 and T2, \TGG{} and \CCI{}, supplying the Master Conservation Law and the \OGNfive{} ledger), while \SIP{} is imported as \ASix{}'s system-instance-packet construction rather than as a principal theorem. As established in \ASix{}, the \GRIP{}-cascade closure (T4 --- including the $N\le 3$ statement and the dual-layer compatibility extension $\mathfrak{C}_{\GRIP}$) holds subject to \ASix{}'s cascade conditioning: the compensator
monodromy statement M11, carried there in spectral-flow form with a
verified benchmark instance --- M10 itself, the \BIsix{} budget
bound, is resolved externally by the capacity theorem T-N3-6 of
\citet{ArisakaP2N3}, with saturation an arithmetic identity, the
topological residue discharged on the benchmark, and the scope
refinements closed at class level by that companion's rigidity
theorems; the constructions of the present paper that rest on \GRIP{} inherit that conditional status. The ground-attractor existence and uniqueness result is imported
separately and does not depend on any cascade residue: it is \ASix{}'s per-stratum ground-state corollary, which holds on the strata that paper calls admissible and carries one stated assumption of its own --- that the free-energy functional restricted to such a stratum is the moment map of the residual torus action. That assumption travels with every use made of the result here. No Qi-runtime can be introduced before \GRIP{}, \GFF{}, and \SIP{} are in hand; the present paper begins from this imported runtime constitution and asks what new theorem layers become inevitable once informational current, persistent memory, and internal reconstruction are allowed to operate on top of it.

\subsection{Bridge equation to Paper~IV-B}
\label{subsec:bridge-equation}

The logical bridge from Papers~I--IV-A to the present paper is
\begin{equation}
\resizebox{\textwidth}{!}{$\displaystyle
\boxed{
\big(\text{\GGConstitution{}}\big)
\Longrightarrow
\big(\GG\ \text{bulk with \BIsix{} and \OGNfive{}}\big)
\xRightarrow{\ \OGGEP\ }
\big(\GDOS\ \text{boundary law}\big)
\Longrightarrow
\big(\text{runtime of open systems}\big)
}
$}
\label{eq:bridge-equation}
\end{equation}
whose target is the set \(\{\DGI,\allowbreak\GSSB,\allowbreak\GA,\allowbreak\GFF,\allowbreak\SIP,\allowbreak\QiQUA,\allowbreak\QiEP,\allowbreak\QiCON,\allowbreak\QiHAL,\allowbreak\SeOS,\allowbreak\QCI,\allowbreak\OH,\allowbreak\IH,\allowbreak\EIC\}\). The updated Arisaka current notation is locked once and used consistently throughout:
\begin{align}
\ArisakaRatio:&\quad
J'_E : J'_e : J'_B : J'_{Qi}
=
1 : -1 : -\tfrac25 : -\tfrac25
=
5 : -5 : -2 : -2,
\label{eq:arisaka-ratio-locked}
\\[2pt]
\ArisakaEq:&\quad
\frac{J_E}{\QB} = J_e = J_B = J_Q,
\qquad
J_Q \equiv J_{Qi},
\label{eq:arisaka-equation-locked}
\\[2pt]
\ArisakaConst:&\quad
\QB=\frac{2}{5}\,\frac{\ERyd}{K},
\qquad K=35.
\label{eq:arisaka-constant-locked}
\end{align}
The full derivations and interpretations appear in
Part~\ref{part:qiqua}.

\begin{table}[H]
\centering
\caption[The imports, their sources and their roles.]{Consolidated imports into this paper from the foundation block. The first column names the imported object, the second the companion paper and theorem that establishes it, and the third what it is used for here. Read as a checklist it defines the paper's dependency surface: if one of these objects fails upstream, the rows that use it fail here, and the third column says which. Nothing outside this table is assumed without being derived.}
\label{tab:consolidated-imports}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.22\textwidth}p{0.30\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Imported object} & \textbf{Source} & \textbf{Role in \ASeven{}} \\
\midrule
Admissibility filter (AX-1--4 + POS-1--7) & \AOne{}, Theorem T-AX & Constrains every runtime layer (\DGI, \GSSB, \GA, \GFF, \SIP, \QiFour, \SeOS/\QCI, \OH/\IH/\EIC). \\
6D bulk + \(G_{\mathrm{Bulk}}\) & \ATwo{}, Eq.~\eqref{eq:gg6-stage} & Supplies the lawful UV stage projected by \OGGEP. \\
\BIsix{} coefficient \((1,-1,-2/5,-2/5)\) & \ATwo{} / \AThree{}, Theorem T-BI6 & Ultraviolet source of the locked Arisaka Ratio Eq.~\eqref{eq:arisaka-ratio-locked}. \\
\OGNfive{} ledger \((1,3,5;7,35)\) & \ATwo{} / \AThree{} & Prevents runtime objects from reintroducing continuous freedom; supplies \(K=35\) for \QB. \\
\GDOS{} boundary law & \AFive{}, Eq.~\eqref{eq:paper3-gdos} & Kernel for the universal runtime extension of Sec.~\ref{sec:universal-gdos}. \\
\GRIP{}, \GFF{}, \SIP{} & \ASix{}: \GFF{}~$=$~T3, \GRIP{}~$=$~T4 (T4 closure conditional on M11, spectral-flow form; M10 resolved
externally, scope refinements closed at class level); \SIP{} construction & Runtime constitution (Part~\ref{part:runtime}); not re-derived in the present paper. \\
\bottomrule
\end{tabular}
\end{table}

\begin{proposition}[The post-Paper~III runtime gap]
\label{prop:runtime-gap}
Let \(\mathfrak{F}_3\) denote the full theorem-level content of \AFive{}. Then \(\mathfrak{F}_3\) is sufficient to define lawful open-system boundary evolution and measurement, but it is not sufficient by itself to classify, compare, and derive persistent complex open systems across chemistry, life, mind, and society.
\end{proposition}

\begin{proof}
A reduced boundary law fixes the evolution of \(p_t\) but is invariant under arbitrary reparametrizations of the basin label, history record, and latent informational variable. Bistable open systems with multiple metastable states and history-dependent future dynamics therefore admit infinitely many ``runtime'' interpretations (current ledger, packet identity, branch-selection semantics, memory architecture) at the level of \(\mathfrak{F}_3\) alone. The runtime layer of Paper~IV-B---\QiQUA{}, \SIP{}, \GRIP{}, \QiHAL{}, etc.---supplies exactly the missing structure that resolves this non-uniqueness. \qedhere
\end{proof}


%% [v11.5] Internal drafting-note subsection removed (process commentary,
%% no reader-facing content; label subsec:next-drafting-steps was unreferenced).





\section{Universal extension of \texorpdfstring{\GDOS}{GDOS} to any open system in nature}
\label{sec:universal-gdos}

\subsection{From the boundary law of Paper~III to a universal runtime}
\label{subsec:from-paper3-to-runtime}

Paper~III derived \GDOS\ in the form required for quantum measurement, four-dimensional field theory, and the infrared limit of particle physics \citep{ArisakaA5GDOS}. The resulting equations were written either as exact reduced channels, as GKLS generators, or as coarse-grained continuity equations on an effective state manifold. In all three representations the crucial structural fact was the same: once the inaccessible variables are traced over, the observer-accessible system is governed by an open dynamics rather than by a closed autonomous Hamiltonian law.

Paper~IV-B now asks whether that structure was specific to physics laboratories or whether it was already more general than it first appeared. The answer proposed here is that the physics derivation was already universal in form. What changes from one branch of science to another is not the existence of an open-system law, but only the meaning of the accessible coordinates, the port structure, the relevant currents, the admissible observables, and the characteristic attractors. The task of the present section is therefore to strip the derivation down to the reusable kernel that survives across domains.

Let the exact boundary-visible state at the end of the \OGGEP{} reduction be denoted \(\rho_{\partial}(t)\in\mathcal{D}(\BState)\). Any finite scientific description then selects only a finite family of observables \(O^A\), \(A=1,\ldots,N\), and therefore only a finite family of coordinates
\begin{equation}
x^A(t):=\Gamma^A[\rho_{\partial}(t)] \equiv \Tr\!\left(\rho_{\partial}(t)\,O^A\right),
\qquad
A=1,\ldots,N.
\label{eq:gamma-coordinates}
\end{equation}
Equation~\eqref{eq:gamma-coordinates} is the true universal entrance point. In quantum physics the \(O^A\) may be projectors, pointer observables, or low moments of fields. In chemistry they may be concentrations, fluxes, or bond-state populations. In cell biology they may be copy numbers, membrane potentials, or methylation states. In neuroscience they may be firing rates, phase variables, or synaptic efficacies. In social systems they may be densities, prices, communication rates, or network-order parameters. The common feature is that the accessible description is always a coarse observable dictionary \citep{BreuerPetruccione2002,vanKampen2007}.

Once Eq.~\eqref{eq:gamma-coordinates} is written, the exact operator equation of Paper~III induces a reduced evolution on an effective state manifold \(\State\) with local coordinates \(x^A\). The resulting probability density \(p(x,t)\) is defined so that for every smooth test function \(f\),
\begin{equation}
\int_{\State} f(x)\,p(x,t)\,\dd\mu(x)
=
\Tr\!\left(\rho_{\partial}(t)\,f\!\big(\Gamma[\cdot]\big)\right),
\label{eq:pushforward-density}
\end{equation}
where \(\dd\mu\) is the natural measure induced on \(\State\). Under the same short-memory and mesoscopic assumptions already used in Paper~III, one obtains the generic continuity form
\begin{equation}
\frac{\partial p}{\partial t} + \nabla_A J^A = 0.
\label{eq:universal-continuity}
\end{equation}
Equation~\eqref{eq:universal-continuity} is not an optional phenomenological ansatz. It is the universal open-system shadow of the bulk-to-boundary projection once one chooses a finite observable dictionary.

The universalization step now consists of recognizing that the current \(J^A\) can always be decomposed into four conceptually distinct pieces,
\begin{equation}
J^A
=
J^A_{\mathrm{grad}}
+
J^A_{\mathrm{curl}}
+
J^A_{\mathrm{noise}}
+
J^A_{\mathrm{gauge}},
\label{eq:four-current-decomp}
\end{equation}
with
\begin{align}
J^A_{\mathrm{grad}} &=
-\mu^{AB}\nabla_B \mathcal{F}_{\GFF}\,p,
\label{eq:jgrad}
\\
J^A_{\mathrm{curl}} &=
\Omega^A p,
\qquad
\nabla_A \Omega^A = 0,
\label{eq:jcurl}
\\
J^A_{\mathrm{noise}} &=
- D^{AB}\nabla_B p,
\qquad
D^{AB}\succeq 0,
\label{eq:jnoise}
\\
J^A_{\mathrm{gauge}} &=
R^A{}_{\alpha}\,\lambda^\alpha\,p.
\label{eq:jgauge}
\end{align}
Here \(\mathcal{F}_{\GFF}\) is the effective geometric free-energy ledger that will later be developed systematically; \(\mu^{AB}\) is the mobility tensor; \(\Omega^A\) is the divergence-free circulation or curl flow; \(D^{AB}\) is the positive semidefinite diffusion tensor; \(R^A{}_{\alpha}\) are the generators of dynamically maintained gauge directions; and \(\lambda^\alpha(t,x)\) are the corresponding gauge velocities. The first three terms are the same gradient--curl--noise structure already derived in Paper~III. The fourth is the new universal term that becomes explicit once gauge redundancy is recognized as a dynamical feature of open systems rather than merely a microscopic redundancy of field variables.

Equations~\eqref{eq:universal-continuity}--\eqref{eq:jgauge} are the universal runtime kernel of the present paper. They say that every open system in nature can be described, at the correct level of coarse-graining, by an effective state manifold, a continuity law, a ledger functional, a solenoidal circulation, a fluctuation tensor, and a set of dynamical gauge directions. Paper~IV-B later refines these ingredients into the \SIP\ formalism, the \GFF\ ledger, and the \QiFour{} pillars; but mathematically the engine is already visible here.

\begin{definition}[Universal open-system runtime packet]
For the purposes of the present section, a \emph{runtime packet} is the tuple
\begin{equation}
\mathfrak S_{\mathrm{run}}
=
\big(
\State,\;
p,\;
\Ports,\;
\Obs,\;
\mathcal{F}_{\GFF},\;
\mu^{AB},\;
D^{AB},\;
\Omega^A,\;
R^A{}_{\alpha}
\big),
\label{eq:runtime-packet}
\end{equation}
where \(\State\) is the effective state manifold, \(p\) the reduced state density, \(\Ports\) the set of ports through which exchange occurs, \(\Obs\) the admissible observable dictionary, \(\mathcal{F}_{\GFF}\) the effective ledger functional, \(\mu^{AB}\) and \(D^{AB}\) the dissipative and stochastic tensors, \(\Omega^A\) the circulation field, and \(R^A{}_{\alpha}\) the generators of gauge directions.
\end{definition}

\begin{remark}
Equation~\eqref{eq:runtime-packet} is intentionally provisional. Later sections will replace it by the more structured \SIP\ formalism. The purpose of the present definition is only to make the universal extension of \GDOS\ mathematically explicit before the more specialized vocabulary is introduced.
\end{remark}

\subsection{Why every scientific observation is constitutionally open}
\label{subsec:every-observation-open}

The central thesis of Paper~IV-B is that openness is not an accidental imperfection of real experiments. It is an unavoidable structural feature of scientific observation itself.

To see this cleanly, consider a hypothetical closed law on a total state space \(\mathcal{Z}\), written abstractly as
\begin{equation}
\dot z = \mathcal{V}(z),
\qquad z\in\mathcal{Z}.
\label{eq:closed-total-law}
\end{equation}
A finite observer never has direct access to the full \(z\). What is actually available is a map
\begin{equation}
\Pi:\mathcal{Z}\longrightarrow \State,
\qquad
x=\Pi(z),
\label{eq:projection-pi}
\end{equation}
followed by an instrument map
\begin{equation}
\mathcal{M}: \State \longrightarrow \mathcal{Y},
\qquad
y=\mathcal{M}(x),
\label{eq:instrument-map}
\end{equation}
where \(\mathcal{Y}\) is the finite record space: counts, images, voltages, trajectories, labels, symbols, reports, or numerical summaries. Unless \(\Pi\) is globally invertible and the instrument is noiseless and complete, the induced evolution on \(\State\) and \(\mathcal{Y}\) cannot be closed in the sense of Eq.~\eqref{eq:closed-total-law}. Memory kernels, noise, hidden variables, or control ports necessarily appear.

This fact may be stated as a theorem.

\begin{theorem}[Universal openness of finite observation (T-Qi4-3.3)]
\label{thm:universal-openness}
Let \(\mathcal{Z}\) be a total state space with closed evolution \(\dot z=\mathcal{V}(z)\). Let \(\Pi:\mathcal{Z}\to\State\) be a non-invertible coarse-graining map and let \(\mathcal{M}:\State\to\mathcal{Y}\) be a finite instrument map. Then the induced evolution on \(\State\) is generically non-autonomous or non-Markovian unless additional hidden variables are appended; equivalently, the observer-accessible dynamics is an open dynamics.
\end{theorem}

\begin{proof}
Because \(\Pi\) is non-invertible, there exist distinct total states \(z_1\neq z_2\) such that \(\Pi(z_1)=\Pi(z_2)=x\). The future derivatives of the reduced coordinates are
\[
\dot x_i = \dd \Pi_{z_i}\big[\mathcal{V}(z_i)\big],
\qquad i=1,2.
\]
Generically these differ because \(\mathcal{V}(z_1)\neq \mathcal{V}(z_2)\) and because the fibers of \(\Pi\) are not invariant under the total flow. Hence \(\dot x\) cannot be written as a single-valued autonomous function of \(x\) alone. To close the reduced law one must either add memory, add hidden variables, or average over unresolved fibers. All three operations are precisely the standard signatures of open-system dynamics. The additional instrument map \(\mathcal{M}\) can only discard more information, never restore invertibility. Therefore the observer-accessible dynamics is open. \qedhere
\end{proof}

Theorem~\ref{thm:universal-openness} is conceptually decisive. It implies that the quantum measurement problem is not the whole story but only the most historically visible instance of a much broader fact. Every field of science studies projected, instrument-conditioned, finite descriptions of larger systems. The difference between physics and biology is not that one is open and the other is closed. The difference is only the complexity of the projection and the richness of the resulting state manifold.

\begin{table}[H]
\centering
\caption[The open-system logic across five domains.]{How the universal open-system logic appears across scientific domains. Five domains run down the first column, from a particle experiment to a collective or social system. The second column gives the accessible coordinates in each case, the third the typical ports through which the system exchanges with what it cannot resolve, and the fourth why the induced dynamics is open rather than closed. The point of setting them side by side is that the fourth column reads the same in every row: the openness is not a feature of the subject matter but of the fact that a finite observer resolves only part of the state.}
\label{tab:domain-open-logic}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.18\textwidth}p{0.31\textwidth}}
\toprule
\textbf{Domain} & \textbf{Accessible coordinates \(x^A\)} & \textbf{Typical ports} & \textbf{Why the induced dynamics is open} \\
\midrule
Particle experiment & Detector counts, event classes, low moments & Beam injection, environmental decoherence, readout electronics & Hidden bulk/environmental modes are traced over before a record is formed. \\
Chemical reactor & Concentrations, pH, temperature, flow rates & Inlet/outlet streams, heating, catalysts & Reservoirs and chemostats continuously exchange matter and energy with the reaction volume. \\
Cell or organism & Metabolite pools, membrane potentials, gene-expression states & Nutrient inflow, waste outflow, sensory coupling, action channels & The organism maintains itself only by continuous exchange across a boundary. \\
Neural circuit & Firing rates, phases, synaptic efficacies, bump positions & Sensory inputs, motor outputs, neuromodulation, blood flow & Hidden microstates and continual environmental coupling prevent closure on the recorded variables alone. \\
Collective or social system & Densities, prices, beliefs, network statistics & Communication channels, trade, migration, media, regulation & The coarse observables are projections of many unresolved agents and changing external constraints. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Ports, currents, and the minimal bookkeeping common to all systems}
\label{subsec:ports-currents}

Once openness is accepted as universal, every system must be described not only by internal coordinates but also by its ports. A port is any channel across which the system exchanges quantities that matter for persistence: energy, matter, charge, entropy, information, stress, nutrients, toxins, prediction errors, or control signals. In the later \SIP\ formalism this will be stated with greater precision, but the bookkeeping idea is already straightforward.

Let \(X^I(t)\) denote any extensive quantity tracked at the effective level, with \(I\) ranging over the ledger sectors relevant to the problem. Then the generic balance law is
\begin{equation}
\frac{\dd X^I}{\dd t}
=
\sum_{a\in\Ports_{\mathrm{in}}} J^I_a
-
\sum_{b\in\Ports_{\mathrm{out}}} J^I_b
+
\Sigma^I_{\mathrm{int}},
\label{eq:balance-law}
\end{equation}
where \(J^I_a\) are boundary currents and \(\Sigma^I_{\mathrm{int}}\) denotes internal conversion between sectors. For strictly conserved quantities \(\Sigma^I_{\mathrm{int}}=0\); for quantities like entropy or prediction error, \(\Sigma^I_{\mathrm{int}}\) is generically nonzero.

Equation~\eqref{eq:balance-law} is the common grammar behind systems that are often treated as if they were conceptually unrelated. In a biochemical network it describes nutrient and waste exchange. In a neuron it describes ionic, metabolic, and information-like currents. In ecology it describes immigration, predation, and resource uptake. In economics it describes capital, goods, and information exchange. What differs across domains is the semantic meaning of \(X^I\), not the necessity of the balance law.

The special role of the \GG{}/\GDOS{} framework is that it does not let this bookkeeping remain an arbitrary phenomenological choice. Because the entire program began from a geometric gauge ledger in six dimensions, it insists that the bookkeeping for open systems must eventually be tied back to geometry and gauge structure. That is precisely why \GFF, \SIP, and then \QiQUA\ become necessary in the final section order. But even before those later sections are written, the universalization of \GDOS\ already tells us that every admissible open-system description must declare its ports and currents.

\begin{proposition}[Runtime form of every admissible open-system model]
\label{prop:runtime-form}
Within the \GGTheory{}/\GG{}/\GDOS{} program, every admissible effective model of an open system must specify:
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item an effective state manifold \(\State\),
\item an observable dictionary \(\Obs\),
\item a port set \(\Ports\),
\item a continuity law of the form \eqref{eq:universal-continuity},
\item a current decomposition of the form \eqref{eq:four-current-decomp},
\item and a balance ledger of the form \eqref{eq:balance-law}.
\end{enumerate}
\end{proposition}

\begin{proof}
Items (i)--(iii) are forced by the universal openness theorem: a nontrivial observational description requires reduced coordinates, observables, and ports. Item (iv) follows from probability conservation on the effective manifold. Item (v) is the generic gradient--curl--noise decomposition augmented by the gauge directions made explicit in Paper~IV-B. Item (vi) is the integrated form of the continuity equation applied to the extensive quantities of interest. Each item is therefore not an added decorative choice but a necessary consequence of treating scientific observation as open and lawful. \qedhere
\end{proof}

\subsection{Why the new gauge term is unavoidable}
\label{subsec:why-gauge-term}

At first glance the last term in Eq.~\eqref{eq:four-current-decomp} may appear optional. One can often write phenomenological open-system models with only gradient, curl, and noise. Why insist on the gauge term \(J^A_{\mathrm{gauge}}=R^A{}_{\alpha}\lambda^\alpha p\)?

The answer is that Paper~IV-B is not merely writing down useful phenomenology. It is asking for the most general form that remains faithful to the derivation chain from \GG{} and \OGGEP{}. Paper~III already showed that the observable boundary dynamics descends from a bulk with genuine gauge structure (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}). Once that is accepted, it is no longer consistent to treat open-system gauge structure as a negligible metaphor. The reduced system may not retain the full microscopic local gauge symmetry, but it does retain an induced equivalence relation under time-dependent boundary transformations. The new term in Eq.~\eqref{eq:jgauge} is simply the mesoscopic imprint of that fact.

The mathematical meaning is simple. The generators \(R^A{}_{\alpha}\) span directions in the effective state manifold that correspond to dynamically equivalent descriptions. Motion along these directions changes coordinates or internal phase conventions but not physical observables. If that redundancy were frozen once and for all, one would recover a conventional fixed-gauge formalism. But in an open system the environment continually changes the effective phase conventions, synchronization relations, and coordinate alignments. Therefore the gauge directions themselves become dynamical. This is the origin of \DGI.

The practical importance is equally large. Without explicit gauge directions one cannot distinguish genuine adaptation from mere coordinate drift, genuine symmetry breaking from gauge fixing, or genuine attractors from artifacts of parameterization. In a theory that aims to cover systems from quantum devices to brains, that distinction is not optional.

\begin{table}[H]
\centering
\caption[Why the kernel needs four current sectors.]{Why the universal \GDOS{} kernel requires four current sectors rather than only three. The four --- gradient, curl, noise and gauge --- run down the first column, with their mathematical role, their physical meaning, and the reason each is indispensable in the columns that follow. The last column is the argument: dropping any one of the four does not merely coarsen the description, it breaks a conservation statement the boundary law needs, and the table names which one in each case.}
\label{tab:four-current-sectors}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.17\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}}
\toprule
\textbf{Current sector} & \textbf{Mathematical role} & \textbf{Physical meaning} & \textbf{Why it is indispensable} \\
\midrule
Gradient & \(-\mu^{AB}\nabla_B\mathcal{F}_{\GFF} p\) & Dissipative descent or relaxation & Captures irreversible tendency toward lower effective ledger cost. \\
Curl & \(\Omega^A p,\ \nabla_A\Omega^A=0\) & Circulation, recurrent flow, Hamiltonian-like transport & Captures persistent loops, cycles, and driven recurrences without scalar descent. \\
Noise & \(-D^{AB}\nabla_B p\) & Fluctuation, uncertainty, coarse-graining residue & Captures stochasticity and hidden-variable dispersion inherent to finite observation. \\
Gauge & \(R^A{}_{\alpha}\lambda^\alpha p\) & Motion along dynamically equivalent descriptions & Captures the inherited gauge redundancy of open systems and prevents confusion between physical and coordinate change. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Section summary: what has now been universalized}
\label{subsec:universalized-summary}

The work of the present section may be summarized compactly.

First, the open-system law derived in Paper~III has been detached from its original quantum-measurement presentation and rewritten in the generic language of reduced coordinates, probability currents, ports, and balance laws. This makes the \GDOS\ kernel applicable in principle to any branch of science.

Second, the universal openness theorem has been proved in its basic form: whenever a finite observer studies a larger system through a non-invertible projection and a finite instrument, the accessible dynamics is open. The measurement problem in quantum mechanics is therefore one special case of a universal observation problem.

Third, the mesoscopic current decomposition has been sharpened from the three-sector form of Paper~III to the four-sector form required for Paper~IV-B. The new gauge sector makes it possible to speak rigorously about dynamically maintained equivalence classes of open-system descriptions. That step is the doorway to \DGI.

Fourth, the universal runtime packet \eqref{eq:runtime-packet} has been declared. It is intentionally provisional, because the later \SIP\ formalism will refine it. But it is sufficient to support the technical derivation of the master pipeline.

The next section therefore performs the first major theorem-level construction of Paper~IV-B: it derives the runtime pipeline
\[
\text{lawful open-system evolution}
\Longrightarrow
\DGI
\Longrightarrow
\GSSB
\Longrightarrow
\GA.
\]
That pipeline is the minimal grammar by which open systems search, select, and stabilize.


\paragraph*{Bridge to Part~\ref{part:runtime}.} With the entrance and recap of Part~\ref{part:entrance}
in hand --- the constitutional inputs from \citet{ArisakaA1Constitution},
the 6D bulk and \BIsix{} closure from
\citet{ArisakaA2GG6} and \citet{ArisakaA3BI6}, the \GDOS{} boundary law from
\citet{ArisakaA5GDOS}, and the four geometric pillars
\TGG{}-\CCI{}-\GFF{}-\GRIP{} from \citet{ArisakaA6Gi4} --- we have
everything we need to begin the Qi-runtime construction. Part~\ref{part:runtime}
imports the runtime constitution: \GRIP{} as the cascade structure,
\GFF{} as the master functional, and \SIP{} as the canonical unit of
analysis. These three are not derived in the present paper; they are
the standing inputs from \ASix{} on which the four Qi-pillars of
Parts~\ref{part:qiqua}--\ref{part:qihal} will be built.

\section{The cohort inheritance and the roadmap of the fifteen Parts}
\label{sec:inheritance-and-roadmap-A7}

\noindent
Before the imported runtime constitution begins (Part~\ref{part:runtime}), 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-a7}, Figure~\ref{fig:cohort-map-a7}), and a panorama of the fifteen Parts of the present paper
(\S\ref{sec:roadmap-a7}, Figure~\ref{fig:roadmap-a7}). 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 fifteen-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-a7}

The present paper is one of twenty-two. \GGTheory{} is written not as
a single monograph but as a cohort of twenty-two preprints --- eleven A-papers, six P-papers, three C-papers and two white papers --- each carrying one
rigorously stated piece of a single descent that runs from four
axioms and seven postulates to the empirical structure of physics.
The partition into twenty-two papers reflects the partition of subject
matter, not a ranking of importance: the framework is one program,
and each paper is a named carrier of named content.
Figure~\ref{fig:cohort-map-a7} is the panorama; a reader meeting the
program for the first time can read its architecture off the
figure top to bottom before returning to the information-theoretic
runtime that is the business of this paper. The cohort falls into
five strata plus three framed branches.

\paragraph{The constitutional root and the GG6 Trio.} At the root stands
\AOne{} \citep{ArisakaA1Constitution}, the \GGConstitution{}: the four axioms
(\AX{1}~\LC{}, \AX{2}~\GGEP{}, \AX{3}~\OGGEP{}, \AX{4}~\MSMF{}) and seven
postulates against which every companion's theorems are audited. Forced from it
is the GG6 Trio that builds the bulk: \ATwo{} \citep{ArisakaA2GG6} constructs
the six-dimensional Geometry--Gauge bulk; \AThree{} \citep{ArisakaA3BI6}
proves the modified Bianchi identity \BIsix{} anomaly-free; and \AFour{}
\citep{ArisakaA4Euclid} traces the lineage from Euclid to the 6D bulk and
reconstructs the universal constants.

\paragraph{The boundary law and its two pillars --- where this paper sits.}
\AX{3}~(\OGGEP{}) forces the open-system boundary law \AFive{}
\citep{ArisakaA5GDOS}, the Geometric Dynamics of Open Systems, which projects
bulk physics onto the four-dimensional boundary. Two sibling pillar papers
organize the boundary content, and \emph{the present paper is one of them}:
\ASeven{} \citep{ArisakaA7Qi4} --- this paper --- supplies the
information-theoretic \QiFour{} pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}),
the runtime grammar carried by the memory-bearing $U(1)_{Qi}$ sector of the
boundary; its geometric sibling \ASix{} \citep{ArisakaA6Gi4} supplies the
\GiFour{} pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}), and with the two pillar
papers in hand \AFive{} closes the \GDOS{} Trio.

\paragraph{The Consequence Trio and the capstone.} Three papers read the
empirical world off the boundary law: \AEight{} \citep{ArisakaA8QM} recasts
four-dimensional quantum mechanics, \ANine{} \citep{ArisakaA9SM} derives the
twenty-eight free parameters of the Standard Model from the \OGNfive{} ledger
and the imported \GiFour{}/\QiFour{} runtime, and \ATen{}
\citep{ArisakaA10Cosmos} derives the $\Lambda$CDM cosmological inventory;
\AEleven{} \citep{ArisakaA11Origin} is the capstone that closes the A-series
and generalizes the ground-attractor statement across the layers of the
program-wide cascade. The per-stratum statement on which the present paper's
\GGA{} terminus rests is upstream, in \ASix{} \citep{ArisakaA6Gi4}.

\paragraph{The two phenomenology branches.} Two framed branches carry the sharp
predictions. The Predictions Branch (P-series) collects the KK-resonance and
hierarchy paper \PaperPOne{} \citep{ArisakaP1KK}, the three-generations paper
\PaperPTwo{} \citep{ArisakaP2N3}, the electron paper \PaperPThree{}
\citep{ArisakaP3Electron}, and the proton paper
\PaperPSix{} \citep{ArisakaP6Proton}, whose baryon number is a
winding of the same compact circle this paper tracks as the
carrier of information. The Cosmology \& Astrophysics Branch (C-series)
collects the Kaxion dark-matter paper \PaperCOne{} \citep{ArisakaC1Kaxion} ---
the high-energy carrier of the same $U(1)_{Qi}$ that the present paper tracks at
chemical and biological scales --- the Hubble-tension paper \PaperCTwo{}
\citep{ArisakaC2Hubble}, and the black-hole-entropy paper \PaperCThree{}
\citep{ArisakaC3BH}.

\paragraph{The Foundations Branch.} The third framed branch is the
Foundations Branch: the two member papers whose subject is the
boundary's complex structure and its mirrors. \PaperPFour{}
\citep{ArisakaP4Real} stands on the map's left flank as the
origin-of-$i$ member, deriving the single imaginary unit of boundary
physics from two real structures of the \ATwo{} arena --- the
compact phase circle and the causal clock. For the present paper its
significance is direct: \ASeven{} carries the four \QiFour{}
pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) --- the
quantum-information runtime that runs on the memory-bearing
$U(1)_{Qi}$ systems of the boundary --- and \PaperPFour{} is the
origin of the boundary's complex structure whose phase the
$U(1)_{Qi}$ memory carrier stores. \PaperPFive{}
\citep{ArisakaP5CPT} carries the three mirrors of matter --- C, P,
and T read as orientation reversals of the three real bulk factors
--- and its law-level arrow of time is kin of the one-way
information cascade that the runtime built here drives in a single
direction. Both carry P-series designators; their place at the
spine's flank, not the phenomenology rows, marks their foundational
role.

\paragraph{The White-Paper Branch.} A fourth group stands on the map's
right flank and reads the spine rather than feeding it. \PaperWOne{}
\citep{ArisakaW1Symmetry} walks the century from Noether's theorem to
the differential-cohomological statement of anomaly cancellation, and
\PaperWTwo{} \citep{ArisakaW2Holonomy} dissects what that arc arrives
at --- protection by topology rather than by symmetry. Both bear on the
present paper at the same point: the memory this paper's fourth pillar
formalizes is holonomy-class data on a compact circle, and a quantity
protected by a winding is protected in the way those two papers
describe, not in the way a conserved charge is.

The twenty-two-paper map, identical in every paper of the cohort, is
Figure~\ref{fig:cohort-map-a7}, which closes this subsection.

%% ===================================================================
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%% ===================================================================
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 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 (\textsf{GG-A6-GRIP},
\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-a7}
\end{figure}

\subsection{Roadmap of the fifteen Parts}
\label{sec:roadmap-a7}

The rest of the paper is organized into fourteen further Parts, and the
walk through them is plotted in Figure~\ref{fig:roadmap-a7}. The
present Part --- the entrance and bridge --- is nearly done: the
strategic entrance, the recap of the constitutional chain imported
from the companions, the cohort the paper stands in, and the
universal extension of \GDOS{} to any open system in nature.
Everything after this page is the runtime itself, and the itinerary
below says where each piece lives.

The entrance conditions come first. Part~\ref{part:runtime} imports the runtime
constitution from \textsf{GG-A6-GRIP} in the strict order demanded
by logical dependence: first the master pipeline \GRIP{} $=$
$\DGI{} \to \GSSB{} \to \GA{}$, then the scalar ledger \GFF{},
and only then the canonical system-instance packet \SIP{} --- with
the cascade-closure import flagged conditional on the companion's
cascade conditioning (M11, in spectral-flow form). Nothing informational has been
claimed
yet; Part~\ref{part:runtime} simply sets the geometric stage on which information
will be made physical.

The six core Parts are the apex, and they are highlighted in the
roadmap because together they deliver the paper's central claim. The four \QiFour{}
pillars are established there as principal theorems in their own
dependency order: \QiQUA{} locks the informational current to the
energetic, electric, and baryonic currents through the Arisaka
equation and its single conversion constant \QB{}; \QiEP{} gives the three extremal traditions their histories at review depth, raises the variational principle that selects the runtime's trajectories, derives the three gauges in which the selection is read, and derives also the boundary past which those readings must fail;
\QiCON{} derives the conformal, L\'evy, fractal, and $1/f$
diagnostics by which the runtime is recognized in data; and
\QiHAL{} builds the memory-bearing attractor architecture on the
$U(1)_{Qi}$ sector. Part~\ref{part:seos} then adds the persistence
and completeness layer (\SeOS{}, \QCI{}) --- what holds itself
together, and how much of the runtime a given system actually runs
--- and Part~\ref{part:gga} closes the block with the statement that
the ground attractor of the geometric companion carries all four
pillar certificates, proved in that direction with the converse
recorded as open. Every earlier Part prepares that construction, and
every later Part spends it.

Part~\ref{part:internal} builds the internality layer on top of the pillars: the
Outer Hologram, the Inner Hologram, and their closed-loop coupling
\OH{}${+}$\IH{}${=}$\EIC{} --- the internal-measurement runtime by
which an open system comes to carry a model of its world ---
together with the \EIC{} universality theorem and the graded
continuum up to consciousness-grade runtime.

Three Parts then widen and weigh the case. Part~\ref{part:emergence} treats emergence
and scope: the Japanese five-story pagoda hierarchy and the
cross-domain applications of the runtime to chemistry, life,
brains, and society, with the explicit universality claim. Part~\ref{part:assessment}
is the assessment: significance and uniqueness, the cross-paper
coherence map, the triaged open problems, and the joint
falsifiability program. Part~\ref{part:conclusions} draws the conclusions: the
synthesis of the \QiFour{} runtime and the bargain with the
laboratory.

The back matter is the workshop. Part~\ref{part:appendices} collects the technical
derivations, the cross-domain worked examples, and the
abbreviation, symbol, and key-equation references; Part~\ref{part:objections} answers
the strongest objections a hostile particle theorist, systems
biologist, origin-of-life researcher, neuroscientist, and
methodologist would raise, with pointers into the paper; and
Part~\ref{part:references} closes with the annotated references, each carrying a
Relevance note. Read in order, the fifteen Parts compose into a single
self-contained construction that any reader can audit step by step;
read by need, each Part names its imports and can stand a visit on
its own.

%% ===================================================================
%%  Roadmap of the ten Parts (house architecture: A1-Constitution
%%  Figure 4 / A2-GG6 v13.3 / A3-BI6 v13.6 / A4-Euclid v13.0 /
%%  A5-GDOS v13.0 / A6-GRIP v13.0 pattern -- vertical Part column,
%%  right phase sidebar, one highlighted central Part; here Part
%%  III, the four Qi-4 pillar theorems + the Arisaka lock),
%%  clearpage-isolated minipage + \captionof per the A1 pattern.
%% ===================================================================
\clearpage
\null\vfill
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.943\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=2.5pt},
 found/.style={fill=blue!12, draw=blue!60!black, line width=0.6pt,
 minimum width=128mm, minimum height=7.5mm},
 innov/.style={fill=orange!16, draw=orange!65!black, line width=0.9pt,
 minimum width=128mm, minimum height=7.5mm},
 geom/.style={fill=green!12, draw=green!55!black, line width=0.55pt,
 minimum width=128mm, minimum height=7.5mm},
 princ/.style={fill=teal!15, draw=teal!60!black, line width=0.6pt,
 minimum width=128mm, minimum height=7.5mm},
 summit/.style={fill=red!13, draw=red!60!black, line width=0.7pt,
 minimum width=128mm, minimum height=7.5mm},
 back/.style={fill=gray!10, draw=black!55, line width=0.5pt,
 minimum width=128mm, minimum height=7.5mm,
 dash pattern=on 2pt off 2pt},
 arrow/.style={-{Stealth[length=2.2mm]}, line width=0.6pt, draw=black!70},
 arrowinnov/.style={-{Stealth[length=2.6mm]}, line width=0.95pt,
 draw=orange!65!black},
 phaselabel/.style={font=\footnotesize\itshape, text=black!60, anchor=west,
 align=left, inner sep=2pt},
]

%% --- the foundations ---
\node[found] (P1) at (0, 0)
 {\textbf{Part~\ref{part:entrance}}\quad Entrance and bridge \;\textemdash\;
 \footnotesize the introduction, the imported constitution, and the cohort};
\node[found] (P2) at (0, -10.5mm)
 {\textbf{Part~\ref{part:runtime}}\quad Runtime constitution from
 \textsf{GG-A6-GRIP} \;\textemdash\;
 \footnotesize \GRIP{}, \GFF{}, \SIP{} (closure conditional on M11,
spectral-flow form)};

%% --- the six Qi Parts, one banded block ---
\node[innov] (P3) at (0, -23mm)
 {\textbf{Part~\ref{part:qiqua}}\quad \QiQUA{} \;\textemdash\;
 \footnotesize current locking, the Arisaka equation, the constant \QB{}};
\node[innov] (P4) at (0, -33.5mm)
 {\textbf{Part~\ref{part:qiep}}\quad \QiEP{} \;\textemdash\;
 \footnotesize one functional in three gauges: \PLT{}, \MEPP{}, \FEP{}};
\node[innov] (P5) at (0, -44mm)
 {\textbf{Part~\ref{part:qicon}}\quad \QiCON{} \;\textemdash\;
 \footnotesize conformality, L\'evy statistics, the two-knee $1/f$ band};
\node[innov] (P6) at (0, -54.5mm)
 {\textbf{Part~\ref{part:qihal}}\quad \QiHAL{} \;\textemdash\;
 \footnotesize \MePMoS{}, \NHT{}, \HAL{}, \BFFT{}, and the \GRIP{}-loop};
\node[innov] (P7) at (0, -65mm)
 {\textbf{Part~\ref{part:seos}}\quad \SeOS{} and \QCI{} \;\textemdash\;
 \footnotesize what holds itself together, and how much runtime it runs};
\node[innov] (P8) at (0, -75.5mm)
 {\textbf{Part~\ref{part:gga}}\quad \QiFour{} and the \GGA{} \;\textemdash\;
 \footnotesize four certificates carried by the ground attractor};

%% --- the internality layer, emergence, assessment, conclusions ---
\node[geom] (P9) at (0, -88mm)
 {\textbf{Part~\ref{part:internal}}\quad Internal measurement \;\textemdash\;
 \footnotesize \OH{}${+}$\IH{}${=}$\EIC{}; consciousness-grade runtime};
\node[princ] (P10) at (0, -98.5mm)
 {\textbf{Part~\ref{part:emergence}}\quad Emergence and scope \;\textemdash\;
 \footnotesize the five-story pagoda and the cross-domain recipe};
\node[princ] (P11) at (0, -109mm)
 {\textbf{Part~\ref{part:assessment}}\quad Assessment \;\textemdash\;
 \footnotesize significance, open problems, falsifiability};
\node[summit] (P12) at (0, -119.5mm)
 {\textbf{Part~\ref{part:conclusions}}\quad Conclusions \;\textemdash\;
 \footnotesize the synthesis and the bargain with the laboratory};

%% --- back matter ---
\node[back] (P13) at (0, -132mm)
 {\textbf{Part~\ref{part:appendices}}\quad Technical Appendices \;\textemdash\;
 \footnotesize derivations, worked examples, reference tables};
\node[back] (P14) at (0, -142.5mm)
 {\textbf{Part~\ref{part:objections}}\quad Common Objections \;\textemdash\;
 \footnotesize the strongest objections, answered, with pointers};
\node[back] (P15) at (0, -153mm)
 {\textbf{Part~\ref{part:references}}\quad References \;\textemdash\;
 \footnotesize cited works, each with a relevance note};

%% --- flow arrows; the six-Part core chain in the innovation colour ---
\draw[arrow] (P1.south) -- (P2.north);
\draw[arrowinnov] (P2.south) -- (P3.north);
\foreach \a/\b in {P3/P4, P4/P5, P5/P6, P6/P7, P7/P8}
 {\draw[arrowinnov] (\a.south) -- (\b.north);}
\draw[arrowinnov] (P8.south) -- (P9.north);
\foreach \a/\b in {P9/P10, P10/P11, P11/P12, P12/P13, P13/P14, P14/P15}
 {\draw[arrow] (\a.south) -- (\b.north);}

%% --- phase labels; half-width is 64mm, so the column sits at 66mm ---
\node[phaselabel] at (66mm, -5.25mm) {\emph{Foundations \&} \\ \emph{imported runtime}};
\node[phaselabel, text=orange!70!black] at (66mm, -49.25mm)
 {\emph{\textbf{The \QiFour{} core}} \\ \emph{\textbf{six Parts, one argument}}};
\node[phaselabel] at (66mm, -88mm) {\emph{Internal measurement}};
\node[phaselabel] at (66mm, -103.75mm) {\emph{Emergence \&} \\ \emph{assessment}};
\node[phaselabel] at (66mm, -119.5mm) {\emph{Conclude}};
\node[phaselabel] at (66mm, -142.5mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the fifteen Parts of GG-A7-Qi4.]{Roadmap of the fifteen Parts of \textsf{GG-A7-Qi4}. The argument develops in five phases. \emph{Foundations and the imported runtime} (Parts~\ref{part:entrance}--\ref{part:runtime}): the entrance and bridge, then the runtime constitution \GRIP{}, \GFF{}, and \SIP{} imported from \textsf{GG-A6-GRIP}, whose closure is conditional on that companion's remaining cascade
conditioning --- the monodromy statement M11 in spectral-flow form,
with the budget question M10 resolved externally and the scope
refinements closed at class level by the three-generations
companion. \emph{The \QiFour{} core} (Parts~\ref{part:qiqua}--\ref{part:gga}), the six banded Parts: the four informational pillars established as principal theorems in their dependency order ($\QiQUA{} \Rightarrow \QiEP{} \Rightarrow \QiCON{} \Rightarrow \QiHAL{}$), then the persistence and completeness layer (\SeOS{}, \QCI{}), then the statement that the ground attractor of the geometric companion carries all four pillar certificates --- proved in that direction, with the converse recorded as open. They are drawn as one block because they are one argument told in six instalments, and the block is highlighted because it is the paper's central claim. \emph{Internal measurement} (Part~\ref{part:internal}) builds the outer and inner holograms and the loop that couples them on top of the pillars. \emph{Emergence and assessment} (Parts~\ref{part:emergence}--\ref{part:assessment}): the five-story pagoda and the cross-domain recipe, then significance, open problems, and falsifiability. The \emph{conclusions} (Part~\ref{part:conclusions}) close the argument, and the three back-matter Parts collect the technical appendices (Part~\ref{part:appendices}), the common objections and responses (Part~\ref{part:objections}), and the annotated references (Part~\ref{part:references}).}
\label{fig:roadmap-a7}
\end{minipage}
\vfill\clearpage


\section{Why none of this is a special case}
\label{sec:close-part1}

The result this Part rests on is easy to state and easy to
underestimate. Openness is not a property that complicated things have
and simple things lack. It is the price of being finite. Any observer
who cannot hold the whole state of a system --- and no observer can
--- is handed a description that will not close on itself, and must
carry memory, or hidden variables, or averaging to make up the
difference. That is as true of a particle detector as of a rainforest.

It follows that the boundaries between the sciences are not boundaries
in nature. They are differences in how much survives a projection.
Particle physics studies a projection so severe that only a handful of
numbers come through, which is why its laws can be written on a page.
Ecology studies one through which so much comes through that no page
would hold it. Neither is more or less open than the other, and the
habit of treating the first as fundamental and the second as merely
complicated has no support in the structure. It is a statement about
how much of the world each field agreed to look at.

There is a consequence for the word ``fundamental'' that is worth
drawing out, because that word does a great deal of unexamined work.
On the usual picture the fundamental science is the one that studies
the smallest things, and every other science is an account of what
happens when many of them are put together. On the picture this Part
argues for, ``fundamental'' would have to mean the level at which a
description closes on itself --- and that level is nowhere. No science
studies a closed system, because there are none to study. What
separates the disciplines is not distance from a foundation but the
severity of the projection each has agreed to work behind. That is not
a demotion of the physics of small things. It is a promotion of
everything else to the same standing, which is a different claim and a
more interesting one.

There is a second thing worth taking from this Part, and it is about
observation rather than about systems. It is tempting to think that the
act of measuring is what makes a system open --- that the world is
whole until someone looks at it. The framework says the opposite. The
projection is running everywhere, at every instant, whether or not
anyone is studying anything. An observer does not open the world. An
observer inherits a world that was already open, and inherits it in the
particular, partial, instrument-shaped way that being one small thing
inside it requires.

That is a humbling thought and also a liberating one. Humbling, because
it means the view from anywhere is a view from somewhere, and no
apparatus will ever escape into the unprojected. Liberating, because it
means the partiality is lawful. It has a form, the form can be written
down, and what follows in this paper is an attempt to write it.

It is worth being clear that none of this is resignation about the
limits of knowledge. The claim is not that the world is unknowable and
one ought to be modest about it; that sentiment is cheap and commits
nobody to anything. It is sharper. The incompleteness of every
description has a definite form, the form can be computed, and what a
projection discards can be accounted for rather than regretted. A
limit one can write down is not the same kind of object as a limit one
can only gesture toward: the first can be carried into a calculation
and argued about by people who disagree. Most of what this paper does
from here on is that --- taking a thing usually said in the tone of a
caution and insisting on saying it as a quantity.


\GGPart[Runtime constitution from A6]{Runtime constitution imported from \ASix{}}{\GRIP{}, \GFF{}, and \SIP{} as the entrance conditions for the Qi-runtime}
\label{part:runtime}

\noindent
This Part is a customs declaration. It carries nothing new, and it is
here because what follows would be unreadable without it.

Three things have to be on the table before a single sentence about
information can be written down. The first is an engine: a system holds
all its options open, then loses stability and commits to one of them,
then settles somewhere it can stay. Those three moves are the whole
mechanism by which anything in nature comes to be one definite thing
rather than a smear of possibilities, and they are established in the
companion paper on the geometric side, not here. The second is a
budget. Exploring, committing and holding still are three activities
that have to be paid for out of one account, and the companion supplies
that single ledger together with the proof that it is single --- there
is not one free energy for the chemistry and another for the
information. The third is a standard way of writing down one concrete
system, so that a protocell, an enzyme, a navigating insect and an
institution can be set side by side without pretending they are alike.

Two points in the import deserve attention even from a reader who plans
to move quickly. The first is that the destination of the engine is not
merely a stable state but the deepest one available on its stratum, and
that distinction is why matter is permanent rather than merely
long-lived. The second is that the engine runs in two regimes, and the
difference between them is a single question: does the landscape itself
move? Descent on a landscape that stays put finishes a journey. The
transient that reorganizes the landscape starts one. Cosmology supplies
the second kind from outside, as unrepeatable events; and the discovery
that a system with a memory can produce them from inside, on purpose,
is what this paper adds later.

One theorem in this Part is proved here rather than imported, and it is
worth knowing which. It says that the standard description of a system
contains neither too little nor too much: everything an experiment
could return is fixed by it, and two descriptions that predict the same
things differ only by a relabeling. The fifth and last field of that
description is the one to notice --- it holds the criteria under which
the system's own model would be declared wrong. Falsifiability is built
into what it means to write a system down here, rather than added at
the end.

Nothing informational has been claimed by the close of this Part. It
sets the stage; the next Part puts something on it.


\section{The imported runtime theorem: \texorpdfstring{\GRIP}{GRIP}}
\label{sec:dgi-gssb-ga}

Part~\ref{part:entrance} established the constitutional bridge from Papers~I--IV-A to a universal theory of open systems. Part~\ref{part:runtime} now begins the constructive runtime sequence with the dynamical grammar imported from \ASix{}: lawful exploration (\DGI), branch selection (\GSSB), and attractor stabilization (\GA). Only once this three-stage engine is explicit does it become possible to ask which scalar ledger it descends (\GFF) and which concrete instance carries it (\SIP). The master pipeline takes the boxed form
\begin{equation}
\boxed{\DGI:\;\text{organize equivalences}}
\;\Longrightarrow\;
\boxed{\GSSB:\;\text{select branch}}
\;\Longrightarrow\;
\boxed{\GA:\;\text{stabilize outcome}}
\label{eq:pipeline-boxed}
\end{equation}
each ingredient strictly needed: \DGI{} without \GSSB{} gives lawful exploration without commitment; \GSSB{} without \DGI{} risks mistaking coordinate choices for physical commitments; \DGI{}+\GSSB{} without \GA{} gives no persistence criterion.

\paragraph{\texorpdfstring{\DGI}{DGI} at the operator level.} Let the boundary-visible density obey \(\dd\rho_\partial/\dd t=\mathfrak L_t[\rho_\partial]\) and let \(U_t\) be a time-dependent unitary family induced from an admissible bulk gauge transformation under \OGGEP{}. Standard manipulation gives the generator-transformation law
\begin{equation}
\mathfrak L_t^{(U)}
=
\mathcal U_t\circ \mathfrak L_t\circ \mathcal U_t^{-1}
-
\ii[K_t,\cdot],
\qquad
\mathcal U_t[\rho]=U_t\rho U_t^\dagger,
\qquad
K_t:=\ii\,\dot U_t U_t^\dagger,
\label{eq:generator-transform}
\end{equation}
which is the exact operator-level statement of \DGI{}: the reduced law is defined by an equivalence class of generators related by induced gauge transformations \citep{ArisakaA5GDOS,BreuerPetruccione2002}.

\paragraph{\texorpdfstring{\GSSB}{GSSB} on the gauge-reduced manifold.} On the physical quotient \(\State_{\mathrm{phys}}\) a symmetric invariant branch can lose stability and lower-symmetry branches appear. The local Landau picture \(V_\eta(q)=\frac{1}{2}a(\eta)q^2+\frac{1}{4}b(\eta)q^4\) with bifurcation at \(a(\eta)=0\) is the canonical model; the open-system version replaces deterministic minima with stationary distributions that concentrate on broken branches when \(a(\eta)<0\) \citep{vanKampen2007,Touchette2009}.

\paragraph{\texorpdfstring{\GA}{GA} as the selected persistent invariant.}

\begin{definition}[Geometric Attractor]
\label{def:GA}
A \emph{Geometric Attractor} \(\mathcal A_{\GA}\subseteq\State_{\mathrm{phys}}\) is an invariant set or invariant measure satisfying: (i) invariance under the reduced \GDOS\ semigroup; (ii) asymptotic relevance (a nontrivial basin converges to it); (iii) selection by \GSSB\ (it sits on a symmetry-reduced branch from a larger admissible family); (iv) gauge admissibility (defined on the physical quotient); (v) ledger admissibility (compatible with the \GFF\ descent or steady-balance constraints).
\end{definition}

\begin{theorem}[Existence of invariant asymptotic sets under confining \GFF\ landscapes (T-Qi4-6.2)]
\label{thm:ga-existence}
Consider the reduced flow
\begin{equation}
\dot y^i
=
-\widetilde{\mu}^{ij}\partial_j \widetilde{\mathcal F}_{\GFF}
+
\widetilde{\Omega}^i,
\label{eq:ga-drift}
\end{equation}
on \(\State_{\mathrm{phys}}\), with \(\widetilde{\mu}^{ij}\) positive semidefinite, \(\widetilde{\Omega}^i\partial_i \widetilde{\mathcal F}_{\GFF}=0\), and \(\widetilde{\mathcal F}_{\GFF}\) proper and bounded below. Then every trajectory is contained in a compact sublevel set of \(\widetilde{\mathcal F}_{\GFF}\), and its \(\omega\)-limit set lies in the largest invariant subset of
\(\mathcal C=\{y\in\State_{\mathrm{phys}}\;|\;\partial_i \widetilde{\mathcal F}_{\GFF}\,\widetilde{\mu}^{ij}\partial_j \widetilde{\mathcal F}_{\GFF}=0\}\).
\end{theorem}

\begin{proof} See \ASix{} Theorem T4 (\GRIP{} principal theorem). \qedhere \end{proof}

\begin{theorem}[Master runtime pipeline (T-Qi4-6.3)]
\label{thm:master-pipeline}
Assume (i) the \GG{} bulk and \OGGEP{} reduction of Papers~II and III; (ii) the universal open-system runtime kernel \eqref{eq:universal-continuity}--\eqref{eq:jgauge}; (iii) a proper gauge-invariant effective ledger \(\mathcal{F}_{\GFF}\); (iv) a symmetry group \(\mathcal S\) acting on \(\State_{\mathrm{phys}}\); and (v) a control regime in which the symmetric branch loses stability. Then the open-system dynamics factors as
\begin{equation}
\State
\overset{\text{quotient}}{\longrightarrow}
\State_{\mathrm{phys}}
\overset{\text{instability}}{\longrightarrow}
\text{symmetry-broken branch}
\overset{\text{asymptotic selection}}{\longrightarrow}
\mathcal A_{\GA}.
\label{eq:pipeline-factorization}
\end{equation}
Equivalently: induced gauge covariance yields \DGI; loss of stability on the quotient yields \GSSB; asymptotic concentration on the selected symmetry-broken invariant set yields a \GA.
\end{theorem}

\begin{proof} See \ASix{} Theorem T4 (\GRIP{} principal theorem, dual-layer compatibility proposition $\mathfrak{C}_{\GRIP}$). \qedhere \end{proof}

\paragraph{Example: the Higgs vacuum as the particle-physics limiting case.}
The standard Higgs mechanism is the limiting case in which the open-system language collapses to familiar field-theory symmetry breaking \citep{Higgs1964}. With \(V(H) = -\mu_H^2|H|^2 + \lambda_H|H|^4\) and \(\mu_H^2,\lambda_H>0\), the gauge-equivalence class of vacua selects one orbit representative modulo the residual \(U(1)_{\mathrm{EM}}\): an elementary \GSSB\ event whose \GA{} is the symmetry-broken electroweak vacuum manifold. Paper~IV-B does not replace this mechanism; it recognizes it as one corner of a much larger runtime grammar. Further worked examples (laser order parameter, ATP synthase, neural decision) appear in Appendix~\ref{app:technical-derivations-for-the-dgi--gssb-} and in the per-pillar sections.

\begin{table}[H]
\centering
\caption[An operational audit of the runtime pipeline.]{Operational audit of the master runtime pipeline. The three phases of the imported cascade --- \DGI{}, \GSSB{} and the attractor --- run down the first column. The second gives the mathematical test that identifies each phase, the third what changes as the system passes through it, and the fourth what must stay invariant across it. The fourth column is the one to check against an experiment: it lists the quantities whose violation would show that a system is not running this pipeline at all.}
\label{tab:pipeline-audit}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.26\textwidth}}
\toprule
\textbf{Phase} & \textbf{Mathematical test} & \textbf{What changes} & \textbf{What must stay invariant} \\
\midrule
\DGI & Quotient by $G_{\mathrm{dyn}}$ & Coordinates, phases, internal alignment conventions & Gauge-invariant observables and physical predictions \\
\GSSB & Instability of symmetric branch; isotropy reduction $H\subsetneq \mathcal S$ & Branch, phase, architecture, committed mode & Underlying law and admissibility filters \\
\GA & Asymptotic concentration on selected invariant set & Persistent pattern, cycle, measure, or policy & Compatibility with gauge reduction and ledger constraints \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The terminus refined: from \texorpdfstring{\GA}{GA} to \texorpdfstring{\GGA}{GGA}.}
Two completions of this imported grammar, both established at runtime
level in \ASix{} \citep{ArisakaA6Gi4}, are adopted here and used
throughout. The first refines the
terminus. A \GFF{} landscape generally admits \emph{many} attractors
satisfying Definition~\ref{def:GA}; the cascade output is not ``some
stable basin'' but the unique deepest one.

\begin{definition}[\GGA: Ground Geometric Attractor; after \ASix{}]
\label{def:GGA}
Among the Geometric Attractors of a stratum, the \emph{Ground
Geometric Attractor} is the global minimum of
\(\mathcal F_{\GFF}\) on that stratum. The definition is \ASix{}'s and
is imported unchanged \citep{ArisakaA6Gi4}; the fully resolved pipeline
\eqref{eq:pipeline-factorization} therefore terminates not merely on a
\GA{} but on the \GGA{}. Existence and uniqueness are \ASix{}'s
per-stratum ground-state corollary, which holds on the class of strata
that paper calls admissible and which carries that paper's grade. No
\QiFour{} filter enters the definition: what the four pillars add is a
\emph{characterization} of the same object, and that is the business of
the dedicated Part before Part~\ref{part:internal} (Part~\ref{part:gga}).
\end{definition}

\paragraph{The two regimes: \texorpdfstring{\GRIPin}{GRIP-In} and \texorpdfstring{\GRIPout}{GRIP-Out}.}
The second completion classifies the dynamics. The runtime operates
in exactly two regimes, split by one question: \emph{does the
landscape change?} \GRIPin{} (\ASix{}) is all dynamics on
a \emph{fixed} landscape --- the gradient flow \eqref{eq:ga-drift}
within a basin, instanton steps across the \GSSB{} saddles between
basins --- halting uniquely at the stratum's \GGA{}. \GRIPout{}
(\ASix{}) is the complementary regime: a transient
out-of-equilibrium event that \emph{reorganizes the landscape
itself} (the control parameters of \(\mathcal F_{\GFF}\) change) and
supplies the initial datum of the next descent. In one sentence:
\GRIPout{} opens the landscape; \GRIPin{} finishes the journey. The
entire master pipeline \eqref{eq:pipeline-boxed} is the \GRIPin{}
regime; where the landscape itself came from is a \GRIPout{}
question, answered cosmologically in \citet{ArisakaA10Cosmos} and
layer by layer in \AEleven{}. What the present paper adds --- and no
other paper of the series can add --- is the discovery that
memory-bearing open systems \emph{internalize} the second regime:
the \QiHAL{} \GRIP{}-loop (\S\ref{subsec:qihal-switch}) is a
self-generated, recoverable \GRIPout{} under the system's own
control.

\paragraph{Bridge to \texorpdfstring{\GFF}{GFF} and \texorpdfstring{\SIP}{SIP}.} The remaining order is now fixed. Section~\ref{sec:eegg} derives the scalar ledger \GFF{}: the pipeline needs a scalar budget because exploration, commitment, and stabilization would otherwise have no common quantity to descend, replenish, or audit. Section~\ref{sec:sip} introduces \SIP{}: once the ledger is fixed, the theory needs a concrete carrier with ports, generator, observables, and kill criteria. Parts~\ref{part:qiqua}--\ref{part:qihal} then develop the four Qi
pillars in their dependency order.

\section{\GFF: the geometric free-energy functional \texorpdfstring{\(\GFF=\GEP+\CEP\)}{GFF = GEP + CEP}}
\label{sec:eegg}

With \GRIP{} in place the runtime needs a scalar ledger: a single quantity for the gradient sector to descend, the maintenance sector to replenish, and the curl sector to circulate around. \MSMF{} forbids admitting several unrelated free-energy-like objects for the same packet, so the ledger is unique up to additive constants.

\begin{definition}[The \GFF\ functional]
\label{def:eegg}
Given a packet with state manifold \((\mathcal{Z},g_{AB})\), probability density \(p\), and port configuration \(\Ports\), the \GFF\ ledger is
\begin{equation}
\label{eq:eegg-functional}
\mathcal{F}_{\GFF}[p;\Ports]
=
\underbrace{\int_{\mathcal{Z}} p(x)\,\Phi_{\GEP}(x;\Ports)\,\dd\mu_g(x)}_{\mathcal{F}_{\GEP}}
+
\underbrace{\Theta_{\mathrm{eff}}\int_{\mathcal{Z}} p(x)\ln\!\frac{p(x)}{p_{\mathrm{ref}}(x;\Ports)}\,\dd\mu_g(x)}_{\mathcal{F}_{\CEP}},
\end{equation}
where \(\dd\mu_g=\sqrt{\det g}\,\dd^n x\), \(\Phi_{\GEP}\) is the energy--geometry--gauge potential induced by the projected bulk structure, \(p_{\mathrm{ref}}\) is the reference density of the induced geometric measure or prior boundary condition, and \(\Theta_{\mathrm{eff}}\) is the effective information-temperature scale.
\end{definition}

The first term collects projected curvature costs, gauge potentials, holonomy contributions, binding energies, and field couplings (the \GEP\ sector). The second is a relative-entropy/curvature-of-measure term capturing the open-system price of incomplete access (the \CEP\ sector); recovering the standard sign convention gives \(-\Theta_{\mathrm{eff}}S\) plus an additive constant.

The full reduced current decomposes as
\begin{equation}
\label{eq:eegg-full-current}
J^A = J^A_{\mathrm{grad}} + J^A_{\mathrm{curl}} + J^A_{\mathrm{noise}} + J^A_{\mathrm{gauge}},
\end{equation}
with \(J^A_{\mathrm{grad}}=-pM^{AB}\nabla_B(\delta\mathcal{F}_{\GFF}/\delta p)\), \(J^A_{\mathrm{curl}}=\Omega^A p\), and \(J^A_{\mathrm{gauge}}=R^A{}_{\alpha}\lambda^{\alpha}p\). Substituting continuity into \(\dd\mathcal{F}_{\GFF}/\dd t\) and integrating by parts gives the balance identity
\begin{equation}
\label{eq:eegg-balance-identity}
\frac{\dd}{\dd t}\mathcal{F}_{\GFF}
=
-\mathcal{D}_{\GFF}
+ \mathcal{P}_{\mathrm{port}}
+ \mathcal{C}_{\mathrm{curl}}
+ \mathcal{G}_{\mathrm{gauge}},
\end{equation}
with dissipation \(\mathcal{D}_{\GFF}\ge 0\) quadratic in \(\nabla(\delta\mathcal{F}_{\GFF}/\delta p)\), port power \(\mathcal{P}_{\mathrm{port}}\) collecting explicit driving and boundary flux, and curl/gauge cross-terms
\begin{equation}
\label{eq:eegg-gauge-term}
\mathcal{G}_{\mathrm{gauge}}
:=
\int_{\mathcal{Z}}
\nabla_A\!\left(\frac{\delta\mathcal{F}_{\GFF}}{\delta p}\right)
R^A{}_{\alpha}\lambda^{\alpha}p\,\dd\mu_g,
\end{equation}
together with \(\mathcal{C}_{\mathrm{curl}}\) of identical form with \(\Omega^A p\). Both vanish whenever the curl current is tangent to \(\mathcal{F}_{\GFF}\) level sets and the ledger is gauge-invariant.

\begin{theorem}[\GFF\ monotonicity in the lawful sector (T-Qi4-7.2)]
\label{thm:eegg-monotonicity}
Suppose \(\mathcal{F}_{\GFF}\) is gauge-invariant and \(\Omega^A\nabla_A(\delta\mathcal{F}_{\GFF}/\delta p)=R^A{}_{\alpha}\nabla_A(\delta\mathcal{F}_{\GFF}/\delta p)=0\). Then \(\dd\mathcal{F}_{\GFF}/\dd t = -\mathcal{D}_{\GFF}+\mathcal{P}_{\mathrm{port}}\); in the undriven closed limit the ledger is nonincreasing, and at a nonequilibrium steady state \(\langle\mathcal{P}_{\mathrm{port}}\rangle_\tau = \langle\mathcal{D}_{\GFF}\rangle_\tau\).
\end{theorem}

\begin{proof} See \ASix{} Theorem T3 (\GFF{} principal theorem, monotonicity in the lawful sector with compatibility proposition $\mathfrak{C}_{\GFF}$). \qedhere \end{proof}

The theorem disentangles three often-confused ideas. The system need not minimize \GFF\ instantaneously; a driven open system may sustain nonzero circulation and throughput. Nevertheless, its gradient motion is organized by one ledger, and steady maintenance requires exactly enough port power to balance the irreversible descent and repair costs. This is the precise sense in which \GFF\ generalizes both free-energy descent and nonequilibrium maintenance. Once \QiQUA{} is derived (Part~\ref{part:qiqua}), the event-level vector ledger collapses to a one-parameter locked ray and scalarizes consistently with \GFF{}; this is recorded in the event-budget appendix.

\section{\SIP: System Instance Packet as the Canonical Unit of \GDOS}
\label{sec:sip}

The scalar ledger of Section~\ref{sec:eegg} fixes what is spent and replenished, but not on which carrier. Section~\ref{sec:sip} closes Part~\ref{part:runtime} by declaring the canonical packet: the minimal complete object on which the Qi pillars subsequently act.

\begin{definition}[Canonical five-field \SIP]
\label{def:canonical-sip}
For a concrete open-system instance in the \GGTheory{}/\GG/\GDOS\ program, the \emph{System Instance Packet} is the ordered five-field object
\begin{equation}
\label{eq:canonical-sip}
\SIP
=\Bigl[(\mathcal{Z},g_{AB}),\; \Ports,\; (\Lind_{\mathrm{\GDOS{}}},\eta),\; (\mathcal{F}_{\GFF},\mathcal{J}),\; (\Obs,\Kill)\Bigr],
\end{equation}
where (i) \((\mathcal{Z},g_{AB})\) is the effective state manifold and its metric; (ii) \(\Ports\) is the full set of ports and imposed boundary couplings; (iii) \((\Lind_{\mathrm{\GDOS{}}},\eta)\) is the lawful open-system generator together with its noise closure; (iv) \((\mathcal{F}_{\GFF},\mathcal{J})\) is the ledger bundle (scalar \GFF{} plus the locked current vector \(\mathcal{J}=(J_e,J_B,J_{Qi},J_E)\)); and (v) \((\Obs,\Kill)\) is the observable dictionary plus the explicit kill criteria required for falsifiability.
\end{definition}

The packet plays, for open-system science, the role that metric-plus-matter plays in general relativity or that gauge-group-plus-matter-representation plays in Yang--Mills theory. The equivalent unfolded form
\begin{equation}
\label{eq:sip-unfolded}
\widehat\SIP = \bigl(\mathcal{Z},g_{AB};\,\Ports;\,M^{AB},\Omega^A,D^{AB},R^A{}_{\alpha}\lambda^{\alpha};\,\mathcal{F}_{\GFF},\mathcal{J};\,\Obs,\Kill\bigr),
\end{equation}
exposes drift, curl, diffusion, and gauge generators that the canonical packet groups into the five conceptual bundles. Representative \SIP{} cards across protocells, ATP synthase, chemotactic microbes, ring attractors, and collective institutions appear in Appendix~\ref{app:technical-details-for-sip-and-eegg}; cross-domain comparability follows from the common grammar, not from identical phenomenology.

\begin{theorem}[\textbf{T-SIP}: \SIP\ completeness (principal theorem) (T-Qi4-8.2)]
\label{thm:sip-completeness}
Fix an initial boundary distribution \(p_0\) on \(\mathcal{Z}\), and assume:
\begin{itemize}[leftmargin=2em]
 \item[(H1)] \emph{Observable completeness.} The observable dictionary \(\Obs\) separates points of \(\mathcal Z\) and the algebra it generates is dense in \(C_b(\mathcal Z)\) under bounded-pointwise convergence.
 \item[(H2)] \emph{Autonomous generator (for the converse).} The generator \(\Lind_{\GDOS}\) is autonomous, or equivalently the gauge ambiguity in the converse direction is treated as a time-ordered family of Stinespring--Kraus unitaries rather than a single static unitary.
\end{itemize}
Then the full family of operational predictions for every registered observable \(O\in\Obs\) is determined by
\begin{equation}
\label{eq:sip-semigroup}
p_t = \mathcal{U}_{t,0}^{\mathrm{\GDOS{}}}[p_0],
\qquad
\mathcal{U}_{t,0}^{\mathrm{\GDOS{}}}
= \mathcal{T}\exp\!\left(\int_0^t \Lind_{\mathrm{\GDOS{}}}(\tau;\Ports(\tau),\eta(\tau))\,\dd\tau\right),
\end{equation}
and \(\langle O\rangle_t = \int_{\mathcal{Z}} O(x)\,p_t(x)\,\sqrt{\det g}\,\dd^n x\). Conversely, two packets are operationally equivalent iff they are related by a diffeomorphic and gauge-covariant pushforward that preserves the observable dictionary and the kill criteria.
\end{theorem}

\begin{proof}
\emph{Forward direction.} The five fields of the canonical packet are exactly the data required to integrate Eq.~\eqref{eq:sip-semigroup}. Field~1 (\(\mathcal{Z}\) and \(\mu_g\)) fixes the domain and integration measure of every observable expectation. Field~2 (\(\Ports(t)\), \(\eta(t)\)) supplies the boundary couplings entering \(\Lind_{\GDOS}\). Field~3 (\(\Lind_{\GDOS}\)) is the GKLS-type generator; by the existence-and-uniqueness theorem for CPTP semigroups under bounded \(\Lind\) (\AFive{} Theorem~T2), it determines \(p_t\) uniquely from \(p_0\) for all \(t\geq 0\). Field~4 (\(F_\GFF\), \(\mathcal{J}\)) supplies the scalar and vector budgets; their role in \(\Lind\) is fixed by Theorem~\ref{thm:eegg-monotonicity}. Field~5 (\(\Obs\), \(\Kill\)) specifies the experimental contract. Together these determine every prediction \(\langle O\rangle_t\) for every \(O\in\Obs\) and \(t\geq 0\).

\emph{Converse direction.} Suppose two canonical packets yield identical predictions for every \(O\), every \(p_0\), and every \(t\geq 0\). \emph{Step 1 (manifolds match).} The observable algebra \(\mathcal A=\langle\Obs\rangle\) is a commutative \(C^*\)-algebra of bounded functions on \(\mathcal Z\); by Gelfand--Naimark, equality of all expectation values is equivalent to homeomorphism of the spectra, and smoothness of \(\Lind_{\GDOS}\) upgrades this to a diffeomorphism \(\varphi:\mathcal Z\to\widetilde{\mathcal Z}\). \emph{Step 2 (generators match modulo gauge).} Differentiating the prediction equality at \(t=0\) gives \(\Lind^*[O]=\varphi^*\widetilde{\Lind}^*[\widetilde O]\); by Stinespring--Kraus uniqueness \citep{Stinespring1955,Kraus1983,Choi1975}, two GKLS generators agreeing on a dense subset of the bounded observable algebra are related by a unitary rotation of their jump operators (the Stinespring gauge). \emph{Step 3 (ledgers and observables match).} Once \(\varphi\) and the gauge are fixed, the ledger budgets \(F_\GFF\) and \(\widetilde F_\GFF\) coincide on corresponding domains by direct comparison of the aligned observable algebras; Fields~4 and~5 thus match. The reverse implication is the contrapositive: if the two packets are not related by such a pushforward, then there exists an observable, an initial distribution, or a ledger entry on which they disagree, and the packets are not operationally equivalent. \qedhere
\end{proof}



\paragraph*{Bridge to Part~\ref{part:qiqua}.} With \GRIP{}, \GFF{}, and \SIP{}
imported from \ASix{} as the runtime constitution, the natural
next question is what specific four-pillar architecture emerges on
the information half of the boundary world. The four Parts that follow answer through the \QiFour{} pillars in
sequence: \QiQUA{} (current
locking and the Arisaka equation), \QiEP{} (variational principle in
three named gauges), \QiCON{} (conformality, fractality, $1/f$
two-knee spectra), and \QiHAL{} (memory architecture
\MePMoS{}-\NHT{}-\HAL{}-\BFFT{}). Each pillar is presented as a
dual-layer principal theorem with a named compatibility proposition
$\mathfrak{C}_{\rm X}$; the four pillars together constitute the
quantum-information backbone of \GDOS{}.


\section{What had to be true first}
\label{sec:close-part2}

Nothing in this Part is new, and that is the point of it. A paper that
proposes to price information has to say, before it starts, what it
already believes about how systems move, what they spend, and how they
are written down --- because otherwise the price could be made to come
out at any figure by adjusting the accounting behind it.

What is on the table now is an engine, a budget, and a standard form.
The engine has three moves and two regimes. The budget is a single
ledger, and its singleness is not a convenience but a requirement: if
exploring, committing and maintaining could each be charged to their
own account, there would be nothing to prevent a system from being
described as thrifty in one currency while ruinous in another, and no
statement about the whole system would mean anything. The standard form
carries, in its last field, the conditions under which the description
would be judged wrong.

Of the two regimes, one deserves the last word here, because everything
this paper adds later grows from it. Descent on a landscape that does
not move is a journey with an end; the system rolls, perhaps waits, and
eventually reaches somewhere with no exit. That regime explains
stability, and it explains why matter is permanent. What it cannot
explain is why there was ever anywhere new to go. For that, something
has to reorganize the landscape itself --- and in cosmology those
events are exogenous and unrepeatable. The universe does not re-run its
own beginnings.

This is where a universe of atoms and a universe with cells in it part
company. A system that remembers can produce such reorganizations from
within, and can choose when. That claim is not made in this Part; it is
made much later, and it needs the four pillars first. But it is worth
seeing now what the imported machinery leaves undone, because the shape
of what is missing is the shape of what this paper is for. The engine
explains how anything comes to rest. It does not explain how anything
came to be able to start again.


\noindent The six Parts that follow are the constructive heart of the
paper. With the runtime grammar of Part~\ref{part:runtime} in place, we
now write the four \QiFour{} pillars themselves --- \QiQUA{} (current locking and the
Arisaka equation), \QiEP{} (variational selector and the unified
extremal principle), \QiCON{} (conformality, L\'evy statistics,
fractal supports, and the $1/f$ spectrum), and \QiHAL{} (memory
architecture from the \MePMoS{} cycle to the holographic
attractor lattice). Each pillar is presented as a principal
theorem with the L16 dual-layer architecture: an algebraic
statement that mathematical physicists can audit, paired with an
operational statement that experimenters can falsify.

\medskip

%% --- Phase 8 TikZ figure: Qi-4 pillars architecture ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\usebox{\GGscPillarsQi}%
}
\caption[Architecture overview of the four \QiFour{} pillars.]{Architecture overview of the four \QiFour{} pillars. Each pillar inherits its constitutional content from the \GDOS{} boundary law of A5 paired with the \GiFour{} geometric template of A6 (top), and its bulk-side technical content from the 6D bulk + $U(1)_{Qi}$ gauge sector (bottom). Horizontal arrows show the implication chain $\QiQUA{} \Rightarrow \QiEP{} \Rightarrow \QiCON{} \Rightarrow \QiHAL{}$. The four named compatibility propositions $\mathfrak{C}_{\mathrm{QUA}}, \mathfrak{C}_{\mathrm{EP}}, \mathfrak{C}_{\mathrm{CON}}, \mathfrak{C}_{\mathrm{\HAL{}}}$ (one per pillar) tie the algebraic (T-*.1) and operational (T-*.2) layers within each pillar.}
\label{fig:a7-qi4-architecture}
\end{figure}



\subsection{What this Part returns to the companion it borrows from}
\label{subsec:runtime-return}

The debt recorded above runs one way, and it would leave a false impression to close the Part without recording the other. The geometric companion supplies this paper with an engine, a budget, and a standard form. What it receives back is not a courtesy but a completion, and it is worth naming precisely, because the two papers are halves of one construction and neither is finished alone.

Three things go back. \emph{The budget's informational column becomes physical.} The scalar ledger arrives here as an energetic term plus an effective temperature times a relative entropy, with the second term carrying no price. \QiQUA{} prices it: the conversion constant \QB{} makes the informational column a quantity in the same units as the first, which is what converts a formal sum into a budget a system can be said to spend. \emph{The cascade acquires an informational reading.} Exploration, commitment and capture are geometric moves in the companion; here they become moves that a system with memory can be said to perform on its own model, which is what allows the same three-move runtime to be recognized in a cell, a nervous system, or an instrument rather than only on a moduli space. \emph{The ground attractor acquires certificates.} The companion establishes where the descent halts; Part~\ref{part:gga} establishes what the halting configuration carries --- four pillar certificates, proved in that direction, with the converse recorded as open.

The relationship is therefore not that one paper is prior and the other applied. The geometric half says how an open system moves and what it spends; the informational half says what the spending is denominated in and what the moving system can come to know. A reader who takes only the first has a runtime with an unpriced column; a reader who takes only the second has a price without an engine. The two are stated separately because they are proved separately, and they are cited together because neither claim survives on its own.

\GGPart[Qi-QUA]{\QiQUA{}}{Current locking, the Arisaka Ratio, the Arisaka Equation, and the Arisaka Constant}
\label{part:qiqua}

\noindent
Everything that happens at the boundary of an open system costs
something. This Part works out what, exactly, and finds that the answer
is shorter than anyone had reason to expect.

There are four things a system can spend: energy, electric charge,
baryon number, and information. Three of those have had a price for a
long time. The fourth has not. Information has been the one quantity in
science that everybody bills and nobody prices --- invoked whenever a
system has to remember, choose or predict, and quietly excused from the
accounting every other quantity has to satisfy. The reason is not
carelessness. It is that nothing in the standard framework says what a
bit ought to cost, so any figure would have been a convention.

What this Part establishes is that the four are not four bills. A single
elementary event does not range freely over a four-dimensional space of
possible expenditures; it lies on a one-dimensional line inside that
space, because all four currents descend through the same channel out
of the bulk. Count the events in any one column and you have counted
them in all four. The four ledgers are one ledger read in four
currencies, and the exchange rate between the energetic column and the
informational one is a single constant.

That constant is the Part's second result, and the thing to notice about
it is where it comes from. It is not measured. It is the energy scale of
the hydrogen atom divided by an integer the ledger fixed several papers
upstream, scaled by a coefficient that anomaly cancellation forced.
Nothing in the chain was adjusted to make anything come out right, and
the Part proves that separately: no new free parameter enters.

Then it can be checked, because life settled on one energy currency and
uses it everywhere. The bond that ATP breaks turns out to carry not one
of these quanta but two, and the predicted figure lands within one and a
half per cent of the measured one --- inside the experimental error bar.
A second check sits in the machine that makes ATP, whose rotor comes in
different sizes on different organisms, tuned to the gradient the
organism lives on, while the cost of a single click of the rotor stays
the same everywhere.

One caution to carry through the Part. The equality is between counted
events, not between carriers: it does not say that one electron and one
bit are the same object. What it does say, and this is sharp, is that
there is no elementary event which carries information while carrying
no energy, no charge and no matter. Disembodied information has no
entry in this ledger.



%% --- Phase 8 TikZ figure: Arisaka equation four-current lock ---
\begin{figure}[ht]
\centering
\resizebox{0.798\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 current/.style={draw=green!55!black, rounded corners=3pt, minimum width=2.6cm, minimum height=1.4cm, align=center, fill=green!10},
 center/.style={draw=olive!55!black, circle, line width=0.80pt, minimum size=2cm, align=center, fill=yellow!15, font=\sffamily\bfseries},
 arrow/.style={->, line width=0.80pt, >=Stealth},
 edgelab/.style={font=\sffamily\footnotesize, fill=white, inner sep=1pt},
]
% Central node: Q_B conversion factor
\node[center] (QB) at (0, 0) {$\QB$\\ \footnotesize Arisaka const.\\ \footnotesize $\approx 0.1555$ eV/qubit};

% Four currents arranged around the center
\node[current] (Je) at (-6, 2.5) {$J_e$\\ \footnotesize electric\\ \footnotesize current};
\node[current] (JB) at (6, 2.5) {$J_B$\\ \footnotesize baryonic\\ \footnotesize current};
\node[current] (JQi) at (-6, -2.5) {$J_{Qi}$\\ \footnotesize information\\ \footnotesize current};
\node[current] (JE) at (6, -2.5) {$J_E$\\ \footnotesize energetic\\ \footnotesize current};

% Lines from each current to center
\draw[arrow] (Je) -- (QB) node[edgelab, midway] {$=$};
\draw[arrow] (JB) -- (QB) node[edgelab, midway] {$=$};
\draw[arrow] (JQi) -- (QB) node[edgelab, midway] {$=$};
\draw[arrow] (JE) -- (QB) node[edgelab, midway, sloped, above] {$\div \QB$};

% Top: Arisaka equation
\node[font=\sffamily\bfseries\Large] at (0, 5)
 {Arisaka equation: \;\; $J_E / \QB \;=\; J_e \;=\; J_B \;=\; J_{Qi}$};

% Bottom: rank-one inflow note
\node[font=\sffamily\footnotesize, align=center, fill=white, inner sep=2pt] at (0, -5)
 {The four boundary currents lock through one conversion factor $\QB$;\\
 rank-one inflow theorem (T-QUA.1) forces this from \BIsix{} closure.};
\end{tikzpicture}%
}
\caption[The Arisaka equation as the four-current lock of \QiQUA{} (Pillar~1).]{The Arisaka equation $J_E / \QB = J_e = J_B = J_{Qi}$ as the four-current lock of \QiQUA{} (Pillar 1 of \QiFour{}). The four boundary currents --- electric $J_e$, baryonic $J_B$, informational $J_{Qi}$, and energetic $J_E$ --- are tied through a single energy--information conversion factor $\QB \approx 0.1555\,\mathrm{eV}/\mathrm{qubit}$, the Arisaka constant. The rank-one inflow theorem T-QUA.1 forces this lock from \BIsix{} closure on the 6D bulk; the empirical match with ATP-hydrolysis chemiosmotic machines at biological temperatures is the operational layer (T-QUA.2).}
\label{fig:a7-arisaka-lock}
\end{figure}

\label{sec:qiqua}

The four-pillar sequence begins only after the runtime constitution is
complete. The open-system grammar (\DGI+\GSSB+\GA), the scalar ledger (\GFF), and the carrier packet (\SIP) are all fixed. The first Qi task is therefore event-level locking: what four-sector current ledger is attached to one admissible packet update, and how is that ledger inherited from \BIsix{}? Part~\ref{part:qiqua} answers that question through the Arisaka Ratio, the Arisaka Constant, and the Arisaka Equation.

\subsection{Historical background (with citations)}

Antecedents: Maxwell (1865), Noether (1918), Bohr (1913), Schr\"odinger (1944), Landauer (1961), Mitchell (1961), Boyer (1979) \citep{Maxwell1865,Noether1918,Bohr1913,Schrodinger1944,Landauer1961,Mitchell1961,Boyer1979}; \QiQUA{} subsumes these as gauges or limiting cases of Theorem~T-QUA.1 $\cup$ T-QUA.2 (the four boundary currents lock through one conversion factor $\QB$; see A1 Sec.~4 and A4 Sec.~3 for full history).

\paragraph*{Dual-layer architecture and the compatibility relation $\mathfrak{C}_{\mathrm{QUA}}$.}
\QiQUA{} is, in the L16 language, a dual-layer principal pillar of
\QiFour{}. Its \emph{algebraic layer} (T-QUA.1) is the rank-one
inflow theorem (Theorem~\ref{thm:qiqua-rank-one}): the four boundary
sectoral currents $J_e$, $J_B$, $J_{Qi}$, $J_E$ must descend from a
single one-dimensional boundary source ray of $\Gtot$. Its
\emph{operational/physical layer} (T-QUA.2) is the
Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$, with the
energy--information conversion factor $\QB \approx 0.1555\,\mathrm{eV}/
\mathrm{qubit}$ at biological temperatures --- one-half the standard free
energy of ATP hydrolysis, so that one ATP carries two locked quanta (see the
paragraph below and \citet{Mitchell1961, Boyer1979}). The
two layers are tied together by the named compatibility relation
\begin{equation}
\label{eq:CQUA-compat}
\resizebox{\textwidth}{!}{$\displaystyle
\boxed{\;\mathfrak{C}_{\mathrm{QUA}}\;\equiv\;
\text{T-QUA.1}\;\Longrightarrow\;\text{T-QUA.2}
\quad\text{(via the \OGNfive{} ledger and the boundary projection of $X_4$).}\;}
$}
\end{equation}
The rank-one inflow at the algebraic level (T-QUA.1) forces the four
currents to share one conversion factor; the boundary projection of
$X_4$ via the \OGNfive{} ledger then identifies that conversion factor
with $\QB$, the Arisaka constant of T-QUA.2. The physical content of T-QUA.2 is therefore the rank-one structure and the \emph{single} conversion factor, not a numerical accident: in natural packet units the equalities $J_e = J_B = J_{Qi}$ state that one elementary boundary event deposits the fixed ratio $(1,-1,-2/5,-2/5)$ across the four sectors, $\QB$ being the one factor that converts the energetic sector to the others.

\paragraph*{The bioenergetic anchor: one ATP carries two quanta.} The empirical
anchor of \QB{} is the universal energy currency of life. One molecule of ATP
carries not one locked quantum but \emph{two}: the standard free energy of
hydrolysis of its phosphoanhydride bond is
$\Delta G^{\circ\prime}_{\rm ATP} = 2\,N_A\,\QB \approx 30.0~\mathrm{kJ/mol}$,
against the measured $-30.5 \pm 0.5~\mathrm{kJ/mol}$ --- an agreement to $1.6\%$,
within the experimental error bar. The factor of two is the two-fold
\QB{}-quantum cost of breaking and solvating the bond; ATP is in this sense the
seventh ground geometric attractor \GGA{}$_7$ of the program-wide cascade, with
the full derivation in \citet{ArisakaA11Origin}. The comparison is made against
this \emph{standard} free energy --- an intrinsic property of the molecule fixed
by the universal constants of \citet{ArisakaA4Euclid} --- not against the larger
\emph{in-vivo} value ($\approx 50~\mathrm{kJ/mol}$), which is not a constant but
carries the cell-state mass-action term
$RT\ln([\mathrm{ADP}][\mathrm{P_i}]/[\mathrm{ATP}])$; the in-vivo value too is a
clean small-integer multiple of \QB{} once the proton stoichiometry is included
(next paragraph). It was the recognition that \QB{} is \emph{half} the
ATP-hydrolysis free energy, not equal to it, that removed the apparent
factor-of-two discrepancy.

\paragraph*{The proton stoichiometry and the environment-tuned $c$-ring
(forthcoming).} Mechanistically, ATP synthase pays for each phosphoanhydride
bond with $H^+/\mathrm{ATP} = c/3$ protons, where $c$ is the integer stoichiometry
of the membrane rotor and $3$ is the number of catalytic sites per turn. Each
proton crosses at the proton-motive force $\Delta p$, so the in-vivo cost is
$(c/3)\,e\,\Delta p$. In \emph{E.~coli} the rotor is $c_{10}$
\citep{Jiang2001,Ballhausen2009}, giving $H^+/\mathrm{ATP} = 10/3$ and, at the
measured $\Delta p \approx 0.155~\mathrm{V}$, a per-proton quantum
$e\,\Delta p \approx \QB$ --- so the elementary chemiosmotic step \emph{is} the
Arisaka quantum, and the in-vivo cost $(10/3)\,\QB \approx 50~\mathrm{kJ/mol}$ is
recovered. The rotor stoichiometry is itself an environment-tuned integer,
ranging from $c_8$ in oxygen-rich mammalian mitochondria (lowest
$H^+/\mathrm{ATP}$, highest efficiency) through $c_{10}$ in \emph{E.~coli} to
$c_{13}$--$c_{15}$ in low-proton-motive-force alkaliphiles and chloroplasts: the
\GRIP{} cascade selects, for each biosphere, the discrete $c$ that matches the
available $\Delta p$ to the universal two-quantum bond. That cross-environment
optimization --- the integer phenotype of ATP synthesis, and the sense in which
ATP-based life is the inevitable bioenergetic attractor of an Earth-like
biosphere --- is developed in a forthcoming dedicated study.

\section{The four ledgers of a generic open system}
\label{subsec:qiqua-four-ledgers}

The appropriate starting point is not a specialized chemical or biological reaction, but the generic bookkeeping structure of an open system. Let \(S\) be an open system with operational boundary \(\partial S\). For each sector
\[
A \in \{E,e,B,Q\},
\]
we introduce a density \(n_A(\mathbf{x},t)\), a flux \(\mathbf{j}_A(\mathbf{x},t)\), and a source term \(s_A(\mathbf{x},t)\) obeying the continuity equation
\begin{equation}
\partial_t n_A + \nabla\!\cdot\!\mathbf{j}_A = s_A.
\label{eq:qiqua-continuity}
\end{equation}
Equation~\eqref{eq:qiqua-continuity} is the open-system version of Noether-style current bookkeeping, with explicit source terms retained because the system is not closed \citep{Noether1918,ArisakaA5GDOS}. Integrating over a control volume \(V\subset \partial S\) gives
\begin{equation}
\frac{\dd}{\dd t}\int_V n_A\,\dd^3x
=
-\int_{\partial V}\mathbf{j}_A\!\cdot\!\dd\mathbf{S}
+\int_V s_A\,\dd^3x.
\label{eq:qiqua-integrated-continuity}
\end{equation}
This equation is the generic bridge from differential current bookkeeping to experimentally registered budgets.

The four sectors used in \QiQUA\ are the following.

\begin{table}[H]
\centering
\caption[The four ledgers audited by \QiQUA{}.]{The four ledgers used by \QiQUA{}. Energy, electric charge, baryon number and information run down the first column, each with its symbol, its operational meaning, its natural boundary unit and its interpretation in an open system. The table is the content of the pillar in one page: the claim is not that these four quantities are each conserved, which is standard, but that they are counted in units that a single conversion constant relates --- so that the four ledgers are one ledger read in four currencies.}
\label{tab:qiqua-ledgers}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.14\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}}
\toprule
\textbf{Sector} & \textbf{Symbol} & \textbf{Operational meaning} & \textbf{Natural boundary unit} & \textbf{Interpretation in open systems} \\
\midrule
Energy & \(E\) & Net energetic expenditure or uptake driving a registered step & eV per event, or kJ/mol per event & Work-like or free-energy-like cost required to maintain or update the open system \\
Electric & \(e\) & Charge-transfer or redox-update count & one elementary electric transfer per event & Carrier transport, redox step, or equivalent charge bookkeeping \\
Baryonic & \(B\) & Material/structural rearrangement count & one minimal structural update per event & Rearrangement of matter-bearing degrees of freedom \\
Information & \(Q\equiv Qi\) & Registered informational update count & one bit-equivalent update per event & Storage, copying, comparison, or commit operation in the boundary register \\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:qiqua-ledgers} deliberately mixes language from physics, chemistry, and information theory because \QiQUA\ is supposed to operate across all three. The purpose of the table is not to claim that a neuron, a catalytic cycle, and a gauge theory use identical microscopic carriers. The purpose is to identify the four ledgers that must all be audited if one wants a theory of open systems that scales from one domain to another without changing its constitution.

The essential distinction is between \emph{coefficient-space currents} and \emph{natural-unit currents}. Coefficient-space currents are the signed sector weights inherited directly from \BIsix{}. Natural-unit currents are the magnitudes of registered events after each sector has been expressed in its own operational unit. The Arisaka Ratio lives in the first language. The Arisaka Equation lives in the second. Much confusion is avoided if that distinction is maintained from the outset.

\begin{remark}
The letter \(Q\) in the present section stands for the informational current \(J_{Qi}\), written in shortened form once the context is fixed. The notation follows the convention
\[
\frac{J_E}{\QB}=J_e=J_B=J_Q,
\qquad J_Q\equiv J_{Qi},
\]
with the order beginning from \(J_E\) so that the presentation remains consistent with the \BIsix{} ordering.
\end{remark}

\section{From \BIsix{} to a one-dimensional boundary source ray}
\label{subsec:qiqua-ray-from-bi6}

Paper~II established that the anomaly-free six-dimensional bulk is organized by a Green--Schwarz-modified Bianchi identity --- read in the present program as the \GGEP{} completion of the 1984 GS construction \citep[see the section ``\BIsix{} as the \GGEP{} completion of GS: the geometrization reading'' of][]{ArisakaA3BI6} --- of the schematic form
\begin{equation}
\dd H_3 = X_4,
\qquad
X_4 = a\,X_E + b\,X_e + c\,X_B + d\,X_{Qi},
\label{eq:qiqua-bi6}
\end{equation}
with the coefficient vector fixed by complete anomaly cancellation to
\begin{equation}
(a,b,c,d) = \left(1,-1,-\frac{2}{5},-\frac{2}{5}\right)
\label{eq:qiqua-bi6-coeff}
\end{equation}
in the \GG{} construction \citep{ArisakaA2GG6}. The crucial point for the present paper is that a \emph{single} three-form field strength \(H_3\) feeds all four boundary sectors. This immediately reduces the allowed elementary source vectors on the boundary to a rank-one family.

\begin{remark}[Normalization convention: reduced-rational \BIsix{} form]
In earlier notes one may also encounter a reciprocal Abelian normalization in which the same locking pattern is written schematically as
\[
(1,-1,-5/2,-5/2).
\]
The present manuscript adopts the reduced-rational convention
\[
(1,-1,-2/5,-2/5)
\]
because it is exactly the form whose positive integer clearing yields
\[
5:-5:-2:-2,
\]
which is the Arisaka Ratio used throughout Paper~IV-B, and because it is the same convention that leads directly to the chemical-unit definition
\[
\QB = \frac{2}{5}\frac{\ERyd}{K}.
\]
Thus the choice of \(-2/5\) is not an independent physical claim beyond \BIsix{}; it is the normalization convention that keeps the ultraviolet coefficient vector, the integer-cleared Arisaka Ratio, and the chemical-unit map mutually consistent. What must be avoided is mixing the reduced-rational \BIsix{} convention with the reciprocal integer form \(2:-2:-5:-5\), or mixing the reciprocal \BIsix{} convention with the reduced-rational Arisaka Constant.
\end{remark}

To see this concretely, integrate Eq.~\eqref{eq:qiqua-bi6} over a compact four-volume \(W_4\) whose boundary projects to a finite operational update on \(\partial S\):
\begin{equation}
\int_{W_4}\dd H_3
=
\int_{W_4}X_4
=
a\int_{W_4}X_E
+b\int_{W_4}X_e
+c\int_{W_4}X_B
+d\int_{W_4}X_{Qi}.
\label{eq:qiqua-integrated-bi6}
\end{equation}
Because the left-hand side depends on one integrated inflow amplitude only, the four right-hand sector contributions cannot vary independently for an elementary synchronized event. There exists a scalar amplitude \(\lambda(W_4)\) such that the induced signed boundary source vector satisfies
\begin{equation}
\mathbf{s}'
=
\lambda
\left(
1,\,
-1,\,
-\frac{2}{5},\,
-\frac{2}{5}
\right).
\label{eq:qiqua-source-ray}
\end{equation}
The prime reminds us that Eq.~\eqref{eq:qiqua-source-ray} is a \emph{coefficient-space} relation: it records the signed sector weights inherited directly from \BIsix{}, before any sector-specific unit conversion has been performed.

Because \GFF\ and \SIP\ are already fixed by Part~\ref{part:runtime}, the first Qi theorem can now focus purely on event-level locking rather than on scalar-budget or packet-definition ambiguities.

\begin{theorem}[Rank-one inflow theorem for elementary \QiQUA{} events (T-Qi4-9.3)]
\label{thm:qiqua-rank-one}
Assume that an elementary open-system update on the boundary is sourced by a single \BIsix{} inflow channel \(H_3\). Then the admissible signed sector-current vector for that elementary update lies on the one-dimensional ray
\begin{equation}
\mathcal{R}_{\mathrm{A}}
=
\left\{
\lambda
\left(
1,\,
-1,\,
-\frac{2}{5},\,
-\frac{2}{5}
\right)
\,:\,
\lambda\in\bR
\right\}.
\label{eq:qiqua-ray-set}
\end{equation}
\end{theorem}

\begin{proof}
Under the stated assumption the four boundary source components are obtained by projecting one and the same bulk object \(H_3\) through the fixed \BIsix{} coefficient vector of Eq.~\eqref{eq:qiqua-bi6-coeff}. Hence all four components are proportional to a single scalar amplitude \(\lambda\), and the set of admissible source vectors is one-dimensional. No second independent direction exists unless a second primitive inflow channel is added, which is excluded by hypothesis. The one-channel hypothesis is not an arbitrary ansatz but a consequence of bulk \BIsix{} closure: within the \GG{} anomaly-canceled spectrum, the closed 4-form $X_4$ is unique up to scale, and the Green--Schwarz counter-term forbids any second independent $H_3'$ on the boundary; see the rank-one criterion (\textsf{T-BI6-6.1}) of \citet{ArisakaA3BI6}, together with its section on \BIsix{} as the \GGEP{} completion of GS.
\end{proof}

Theorem~\ref{thm:qiqua-rank-one} is the whole geometric content of the first Qi pillar in its most compressed form. It says that the elementary bookkeeping event of the boundary does not span a four-dimensional current space. It spans a one-dimensional ray inside that space. All later equalities of \QiQUA\ are just different ways of reading that ray.

\section{The Arisaka Ratio and its equivalent representations}
\label{subsec:qiqua-arisaka-ratio}

We may now state the first of the three central objects.

\begin{definition}[Arisaka Ratio]
\label{def:arisaka-ratio}
The \emph{Arisaka Ratio} is the coefficient-space ray associated with the \BIsix{} source vector, written in the ordered sector basis \((E,e,B,Q)\):
\begin{equation}
\ArisakaRatio:
\qquad
J'_E : J'_e : J'_B : J'_{Qi}
=
1 : -1 : -\frac{2}{5} : -\frac{2}{5}.
\label{eq:arisaka-ratio}
\end{equation}
Equivalently, after clearing denominators by a positive rescaling factor \(5\),
\begin{equation}
J'_E : J'_e : J'_B : J'_{Qi}
=
5 : -5 : -2 : -2.
\label{eq:arisaka-ratio-integer}
\end{equation}
\end{definition}

The two forms \eqref{eq:arisaka-ratio} and \eqref{eq:arisaka-ratio-integer} are mathematically equivalent because they represent the same ray under a positive scalar rescaling. The first form makes direct contact with the \BIsix{} coefficient vector. The second form is often easier to read because all entries are integers.

This is also the right place to state the sign convention unambiguously. In the present paper the sign of a coefficient-space current records \emph{orientation}, not whether the corresponding physical quantity is ``good'' or ``bad.'' We choose the energy inflow direction as positive. The electric, baryonic, and informational components then carry negative sign in Eq.~\eqref{eq:arisaka-ratio} because they are recorded as sink or deposition channels relative to that orientation. When the same bookkeeping is rewritten in natural units, one normally works with magnitudes rather than signed orientations. The Arisaka Equation will therefore be written without minus signs.

\begin{table}[H]
\centering
\caption[Three equivalent readings of the Arisaka Ratio.]{Equivalent ways of reading the Arisaka Ratio. The first column names the representation, the second gives the formula, and the third says where each form is useful. The three are the same statement: the coefficient form as it falls out of the anomaly normalization, the integer-cleared form that makes the ledger arithmetic visible, and the magnitude form used when only relative sizes matter. They are set out together because the literature of the cohort uses all three and a reader meeting the second without the first can mistake an integer clearing for a different claim.}
\label{tab:arisaka-representations}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.21\textwidth}p{0.31\textwidth}p{0.38\textwidth}}
\toprule
\textbf{Representation} & \textbf{Formula} & \textbf{Use} \\
\midrule
\BIsix{} coefficient form &
\(1:-1:-2/5:-2/5\) &
Closest to the anomaly-cancellation data of \GG{}; best for ultraviolet bookkeeping. \\
Integer-cleared form &
\(5:-5:-2:-2\) &
Same ray, but easier to scan and compare across tables. \\
Magnitude form &
\(1:1:2/5:2/5\) &
Obtained by taking absolute values when only event magnitudes are relevant. \\
Natural-unit form &
\(J_E/\QB = J_e = J_B = J_Q\) &
Best for chemistry, biology, and cognition, where each sector is counted in its own operational unit. \\
\bottomrule
\end{tabular}
\end{table}

It is important not to overread Definition~\ref{def:arisaka-ratio}. The Arisaka Ratio does \emph{not} claim that every coarse-grained macroscopic current measured in a laboratory or organism must appear in the raw ratio \(1:1:2/5:2/5\). What it claims is more precise: the \emph{elementary anomaly-mediated source direction} inherited from the bulk lies on that ray. Coarse-grained systems may contain storage, overhead, idle cycles, and side reactions. Those effects are built from sums of elementary events and therefore belong to later bookkeeping layers. \QiQUA\ first identifies the primitive event.

\begin{proposition}[No independent elementary informational direction]
\label{prop:no-independent-q}
Within the one-channel hypothesis of Theorem~\ref{thm:qiqua-rank-one}, there is no elementary boundary event carrying \(J'_{Qi}\neq 0\) while \(J'_E=J'_e=J'_B=0\).
\end{proposition}

\begin{proof}
Any elementary source vector lies on the ray \(\mathcal{R}_{\mathrm{A}}\) of Eq.~\eqref{eq:qiqua-ray-set}. On that ray the informational component is proportional to the same scalar amplitude \(\lambda\) that controls all other components. Hence \(J'_{Qi}\neq 0\) implies \(\lambda\neq 0\), which in turn implies \(J'_E\neq 0\), \(J'_e\neq 0\), and \(J'_B\neq 0\).
\end{proof}

Proposition~\ref{prop:no-independent-q} is the precise framework-level sense in which the present theory rejects ``disembodied information'' at the elementary ledger level. The statement should not be confused with the weaker claim that every phenomenological reduced model must retain explicit electric and baryonic variables. Reduced models may integrate out those ledgers. The proposition concerns the underlying synchronized event, not every possible reduced description.

\paragraph{Falsification class and kill-criterion.}
\label{para:qiqua-kill-criterion}
The Arisaka equation \(J_E/\QB = J_e = J_B = J_{Qi}\) is required to hold \emph{as an equality}, within a stated tolerance, at the level of single elementary events in the rank-one inflow channel. The falsification class is therefore concrete and is the only place a hostile experimenter is permitted to look:
\begin{itemize}[leftmargin=2em]
 \item single-molecule kinetics of catalytic engines (ATP synthase rotor step, kinesin step, photosynthetic charge separation, single-enzyme turnover);
 \item single conformational transitions of memory-bearing biomolecules (CaMKII autophosphorylation event, single ribosomal step, single transcription-factor binding/unbinding);
 \item single elementary events in microbial electron transport (a single quinone reduction step, a single cytochrome electron transfer).
\end{itemize}
For each such event, the energy, electron-count, baryon-count, and information-bit bookkeeping must agree to within \(\pm 20\%\) --- the empirical dispersion of physiological ATP-hydrolysis free energies across normal cellular conditions and the calibration uncertainty in matching the \BIsix{} ratio to a specific molecular substrate (see Open Problem~O5 for the calibration discussion). A systematic violation of the equality outside this tolerance, in a system that is operationally elementary (single event, no integrated-out hidden ledger), refutes \QiQUA. Coarse-grained or reduced systems are explicitly outside the falsification class: Proposition~\ref{prop:qiqua-lower-bound} (below) permits \(\geq\) inequalities for them because overhead, storage, idle cycles, and side reactions are built from sums of elementary events and belong to later bookkeeping layers. The kill-criterion is therefore: \emph{find one elementary catalytic event whose four-current bookkeeping disagrees with the Arisaka equation outside \(\pm 20\%\), and \QiQUA{} is refuted}.

\section{From \GG{} scales to the Arisaka Constant}
\label{subsec:qiqua-constant}

The second central object of the section is the fixed energy scale assigned to one locked boundary event in chemical natural units. The derivation uses only quantities already imported from Papers~II and III.

The first ingredient is the \OGNfive{} integer \(K=35\), which in Paper~II measured the discrete depth of the six-dimensional throat. The second ingredient is the Rydberg energy
\begin{equation}
\ERyd = \frac{1}{2}\alpha^2 m_e c^2,
\label{eq:qiqua-rydberg}
\end{equation}
the characteristic atomic energy scale descending from the previously derived low-energy electromagnetic sector \citep{Bohr1913,ArisakaA2GG6}. The quantity \(\ERyd\) is not introduced here as an arbitrary empirical fit parameter; within the GG sequence it is part of the already imported low-energy output. Numerically \(\ERyd\approx 13.6057\,\mathrm{eV}\).

The next move is to define the minimal geometric step along the chemistry-relevant boundary ledger. The argument for the 1/\(K\) scaling has two parts.

\emph{First, the role of \(K\) in the scale-profile throat.} In \AThree{}, the OGN-5 integer \(K=35\) is fixed by the Diophantine constraint \(K=LJ=M^2+N^2+L^2\) with \((M,N,L;J)=(1,3,5;7)\), and it plays a second role as the logarithmic scale depth of the six-dimensional throat: the canonical electroweak-versus-Planck scale identification \(\Mlock=\tfrac{1}{2}M_{\rm Pl}\exp(-K)\) shows that \(K\) measures the number of natural log-units between the high-energy (Planck) boundary and the low-energy (electroweak) boundary of the extra direction. This is the same \(K\) that we now use to count the discrete depth levels of the atomic boundary.

\emph{Second, the linear 1/\(K\) per-step scaling.} A Kaluza--Klein reduction on an interval of radial-coordinate length \(K\) (in natural log-units) produces, on the boundary, an evenly spaced ladder of \(K\) modes whose total energy span is the atomic scale \(\ERyd\). The per-step quantum is therefore \(\ERyd/K\), exactly the standard linear-spacing result for KK modes on an interval of finite length. An alternative \(1/K^2\) scaling (which would occur for a flat compactification with momentum-quantization on a torus) is excluded here because the boundary is the IR boundary, not a flat fiber; an alternative \(1/\log K\) scaling (which would occur for an exponential discretization of the throat) is excluded because the OGN-5 ledger fixes the discrete depth by linear, not exponential, increments \citep{ArisakaA3BI6}. Hence
\begin{equation}
\delta E_{\mathrm{geom}}
\equiv
\frac{\ERyd}{K}.
\label{eq:qiqua-deltaEgeom}
\end{equation}
Numerically \(\delta E_{\rm geom}\approx 13.6057/35\approx 0.389\,\mathrm{eV}\).

This quantity is not yet the energy per locked chemical event. It is the per-step geometric energy quantum before the \BIsix{} coefficient ratio between the energy sector and the baryonic/informational sectors has been applied.

\begin{remark}[On the apparent double use of the BI-6 ratio 2/5]
\label{rem:qb-no-double-counting}
The same \BIsix{} coefficient ratio \(|c|/|a|=2/5\) appears at two places in the construction below: once as the multiplicative factor in \(\QB = (2/5)\,\delta E_{\rm geom}\) (Eq.~\eqref{eq:qiqua-deltaEgeom}~$\rightarrow$~Eq.~\eqref{eq:qiqua-QB}), and once as the reciprocal factor \(5/2\) in the chemical-natural-unit map of the baryonic/informational currents (Eq.~\eqref{eq:qiqua-JB-def} ff.).

These are not two independent uses of the ratio that double-count one another; they are the same single \BIsix{} ratio applied to a single sector conversion --- once as a magnitude (the energy per locked event) and once as the inverse normalization (the chemical unit of the baryonic/informational current), so that the resulting Arisaka equation \(J_E/\QB = J_e = J_B = J_{Qi}\) holds as a pure-number equality with all sectors counted in their own operational units.

The non-trivial empirical content of \(\QB\) is therefore not the numerical value \(\approx 0.155\,\mathrm{eV/qubit}\) by itself --- which is fixed by the upstream OGN-5 integer \(K\) and the Rydberg energy --- but the structural assertion that \emph{every} elementary boundary event in the rank-one inflow channel shares the same single conversion factor between energy and information. The falsifier of \QiQUA{} (specified in the next section) is the violation of this single-conversion-factor assertion across a class of operationally elementary events, not the disagreement between the numerical value of \(\QB\) and any one calorimetric measurement.
\end{remark}

That ratio is precisely the third ingredient. From Eq.~\eqref{eq:qiqua-bi6-coeff} one reads
\begin{equation}
\frac{|c|}{|a|}=\frac{2}{5}.
\label{eq:qiqua-ca-ratio}
\end{equation}
In other words, the magnitude of one baryonic or informational elementary update is \(2/5\) of the coefficient-space energy step when the latter is written in the common \BIsix{} normalization. Therefore the energy cost of one \emph{boundary event counted in the natural baryonic/informational unit} is
\begin{equation}
\QB
=
\frac{|c|}{|a|}\,\delta E_{\mathrm{geom}}
=
\frac{2}{5}\frac{\ERyd}{K}.
\label{eq:qiqua-QB}
\end{equation}

\begin{definition}[Arisaka Constant]
\label{def:arisaka-constant}
The \emph{Arisaka Constant} is the fixed energy scale
\begin{equation}
\ArisakaConst:\qquad
\QB
=
\frac{2}{5}\frac{\ERyd}{K},
\qquad K=35.
\label{eq:arisaka-constant}
\end{equation}
It is the energy cost assigned to one ideal locked boundary event when the electric, baryonic, and informational sectors are each counted in one natural unit.
\end{definition}

Using \(\ERyd\approx 13.6057\,\mathrm{eV}\) and \(K=35\), one obtains
\begin{equation}
\QB
=
\frac{2}{5}\frac{13.6057\,\mathrm{eV}}{35}
\approx
0.1555\,\mathrm{eV}.
\label{eq:qiqua-qb-numerical}
\end{equation}
The same value may be written in two experimentally familiar ways:
\begin{equation}
\QB \approx 15.0\,\mathrm{kJ\,mol^{-1}}
\approx 5.82\,k_B T
\quad \text{at } T=310\,\mathrm{K}.
\label{eq:qiqua-qb-conversions}
\end{equation}
The thermal reading here is not an independent input. The eV\,\(\leftrightarrow\)\,\(k_B T\) conversion is one face of the thermal clock: the same one-sided (positive-frequency) spectrum that endows the informational carrier with its derived quarter-turn returns, in KMS form, the bath temperature (T-P5-B1 of \citealp{ArisakaP5CPT}; candidate reading).

\begin{table}[H]
\centering
\caption[The Arisaka constant in four unit systems.]{Numerical forms of the Arisaka constant. The first column names the form, the second gives the value and the third its interpretation. The four rows are one number --- the definition from the geometry, then the same quantity in atomic, chemical and thermal units, the last evaluated at $310$~K. The chemical row is the one that meets experiment: it is what makes the \textsf{ATP} anchor a comparison rather than an analogy, and the thermal row is what shows the constant is not a disguised $k_BT$.}
\label{tab:arisaka-constant-values}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.28\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Form} & \textbf{Value} & \textbf{Interpretation} \\
\midrule
Definition from \GG{} & \(\QB=\frac{2}{5}\frac{\ERyd}{K}\) & No new continuous parameter beyond the imported \GG{} data \\
Atomic-energy form & \(\QB\approx 0.1555\,\mathrm{eV}\) & Energy scale of one locked boundary event \\
Chemical form & \(\QB\approx 15.0\,\mathrm{kJ\,mol^{-1}}\) & Convenient for chemistry, catalysis, and biophysics \\
Thermal form (310 K) & \(\QB\approx 5.82\,k_B T\) & Useful for room-temperature or physiological open systems \\
\bottomrule
\end{tabular}
\end{table}

The significance of Definition~\ref{def:arisaka-constant} is structural. The paper is not merely introducing a convenient conversion factor. It is asserting that the energy cost per ideal locked event is not free to float continuously from system to system. What may vary from system to system is \emph{how many} locked events are required, how much additional overhead is paid, and which geometric attractor is realized. The per-event baseline \(\QB\) is fixed by the imported \GG{} data.

\begin{theorem}[No new continuous parameter in \QiQUA (T-Qi4-9.8)]
\label{thm:no-new-param}
Within the \GG{}/\GDOS{} import structure of Paper~IV-B, the constant \(\QB\) introduces no new continuous free parameter.
\end{theorem}

\begin{proof}
Equation~\eqref{eq:arisaka-constant} expresses \(\QB\) in terms of the already imported quantities \(\ERyd\) and \(K\). The Rydberg scale is part of the low-energy electromagnetic output inherited from the previous papers, while \(K=35\) is one of the fixed \OGNfive{} integers. Therefore \(\QB\) is derived data, not an adjustable input.
\end{proof}


\section{Natural-unit reduction and the Arisaka Equation}
\label{subsec:qiqua-equation}

We now convert the coefficient-space statement into the natural-unit statement appropriate for open systems in chemistry, biology, and cognition.

The single most important conceptual point is that the Arisaka Equation is \emph{not} obtained from the Arisaka Ratio by merely dropping minus signs. One must first perform a sector-dependent natural-unit conversion. Stated in the exact order used in the present manuscript, the derivation is
\begin{align}
\text{\BIsix{} normalization:}\qquad
&(a,b,c,d)=\left(1,-1,-\frac{2}{5},-\frac{2}{5}\right),
\label{eq:qiqua-four-line-bi6}
\\[3pt]
\text{Arisaka Ratio:}\qquad
&J'_E:J'_e:J'_B:J'_{Qi}
=
1:-1:-\frac{2}{5}:-\frac{2}{5}
=
5:-5:-2:-2,
\label{eq:qiqua-four-line-ratio}
\\[3pt]
\text{chemical-unit map:}\qquad
&\frac{J_E}{\QB}:J_e:J_B:J_Q
=
|J'_E|:|J'_e|:\frac{5}{2}|J'_B|:\frac{5}{2}|J'_{Qi}|,
\label{eq:qiqua-four-line-map}
\\[3pt]
\text{Arisaka Equation:}\qquad
&\frac{J_E}{\QB}=J_e=J_B=J_Q.
\label{eq:qiqua-four-line-equation}
\end{align}
Equation~\eqref{eq:qiqua-four-line-bi6} is the ultraviolet normalization choice. Equation~\eqref{eq:qiqua-four-line-ratio} is the coefficient-space statement inherited from \BIsix{}. Equation~\eqref{eq:qiqua-four-line-map} is the indispensable conversion from raw sector coefficients to chemical natural units. Only after that conversion does Eq.~\eqref{eq:qiqua-four-line-equation} follow.

To make the same point in slower arithmetic, suppose an elementary synchronized event carries coefficient-space magnitudes
\begin{equation}
\bigl(|J'_E|,|J'_e|,|J'_B|,|J'_{Qi}|\bigr)
=
r
\left(
1,\,
1,\,
\frac{2}{5},\,
\frac{2}{5}
\right),
\qquad r\ge 0,
\label{eq:qiqua-magnitude-ray}
\end{equation}
for some event-rate amplitude \(r\). In other words, the raw coefficient-space bookkeeping says only that the baryonic and informational components are smaller than the electric and energy components by the factor \(2/5\) in the \BIsix{} normalization.

The next step is to define the \emph{chemical natural units}. We choose the electric sector as the reference event count and define
\begin{align}
J_e &\equiv |J'_e|,
\label{eq:qiqua-Je-def}
\\
J_B &\equiv \frac{5}{2}|J'_B|,
\label{eq:qiqua-JB-def}
\\
J_Q &\equiv \frac{5}{2}|J'_{Qi}|,
\qquad J_Q\equiv J_{Qi},
\label{eq:qiqua-JQ-def}
\\
\frac{J_E}{\QB} &\equiv |J'_E|,
\qquad\text{equivalently}\qquad
J_E \equiv \QB |J'_E|.
\label{eq:qiqua-JE-def}
\end{align}
These definitions do exactly one thing. They convert the four raw \BIsix{} amplitudes into \emph{event counts} in the natural unit of each sector:
\begin{itemize}[leftmargin=*]
\item the electric sector is counted one electric-transfer event at a time;
\item the baryonic sector is counted one baryonic-update event at a time, which requires multiplication by the reciprocal factor \(5/2\) because its coefficient-space magnitude is \(2/5\) of the electric one;
\item the informational sector is counted one informational-update event at a time, again requiring the reciprocal factor \(5/2\);
\item the energy sector is counted in units of the fixed event energy \(\QB\), so that \(J_E/\QB\) is the corresponding event count.
\end{itemize}

Substituting Eq.~\eqref{eq:qiqua-magnitude-ray} into Eqs.~\eqref{eq:qiqua-Je-def}--\eqref{eq:qiqua-JE-def} yields
\begin{align}
J_e &= r,
\label{eq:qiqua-natural-unit-e}\\
J_B &= \frac{5}{2}\left(\frac{2}{5}r\right)=r,
\label{eq:qiqua-natural-unit-b}\\
J_Q &= \frac{5}{2}\left(\frac{2}{5}r\right)=r,
\label{eq:qiqua-natural-unit-q}\\
\frac{J_E}{\QB} &= r.
\label{eq:qiqua-natural-unit-E}
\end{align}
Therefore
\begin{equation}
\frac{J_E}{\QB} = J_e = J_B = J_Q = r.
\label{eq:qiqua-arisaka-equation}
\end{equation}
We may now state the third central object.

\begin{definition}[Arisaka Equation]
\label{def:arisaka-equation}
The \emph{Arisaka Equation} is the natural-unit form of the \QiQUA\ locked current ray:
\begin{equation}
\ArisakaEq:\qquad
\frac{J_E}{\QB} = J_e = J_B = J_Q,
\qquad J_Q\equiv J_{Qi}.
\label{eq:arisaka-equation}
\end{equation}
\end{definition}

\begin{remark}[Why \(J_e=J_B=J_Q\) is not a raw microscopic identity]
The equality in Eq.~\eqref{eq:arisaka-equation} does \emph{not} mean that one electron, one baryon, and one bit are literally the same microscopic carrier. In coefficient-space units the correct statement is instead
\[
|J'_E|:|J'_e|:|J'_B|:|J'_{Qi}| = 1:1:\frac{2}{5}:\frac{2}{5}.
\]
The equalities \(J_e=J_B=J_Q\) appear only after the baryonic and informational sectors are measured in their own natural packet units, that is, after the reciprocal \(5/2\) conversion has been applied. The Arisaka Equation is therefore an equality of \emph{normalized event counts}, not an identity among raw microscopic carriers.
\end{remark}

This is the formal bridge from coefficient-space \BIsix{} data to chemistry/open-system event counting, and it is the mathematical reason \QiQUA\ belongs after \GFF\ and \SIP\ rather than before them.

\begin{theorem}[Natural-unit equivalence theorem (T-Qi4-9.11)]
\label{thm:natural-unit-equivalence}
For an elementary boundary event sourced by one \BIsix{} inflow channel, the coefficient-space Arisaka Ratio of Eqs.~\eqref{eq:arisaka-ratio}--\eqref{eq:arisaka-ratio-integer} is equivalent to the natural-unit Arisaka Equation \eqref{eq:arisaka-equation}.
\end{theorem}

\begin{proof}
The coefficient-space ray gives Eq.~\eqref{eq:qiqua-magnitude-ray}. Applying the chemical-unit map \eqref{eq:qiqua-four-line-map}, equivalently the definitions \eqref{eq:qiqua-Je-def}--\eqref{eq:qiqua-JE-def}, yields the explicit equalities \eqref{eq:qiqua-natural-unit-e}--\eqref{eq:qiqua-natural-unit-E}. These show that all four normalized event counts are equal to the same amplitude \(r\). Hence Eq.~\eqref{eq:arisaka-equation} follows.

Conversely, if Eq.~\eqref{eq:arisaka-equation} holds and the chemical-unit map is inverted, then
\[
|J'_e| = \frac{J_E}{\QB},
\qquad
|J'_B| = \frac{2}{5}J_B,
\qquad
|J'_{Qi}| = \frac{2}{5}J_Q,
\]
so the coefficient-space magnitudes necessarily lie on the ray \((1,1,2/5,2/5)\). Reintroducing the \BIsix{} orientation signs reproduces Eqs.~\eqref{eq:arisaka-ratio}--\eqref{eq:arisaka-ratio-integer}. The two statements are therefore equivalent once the sector-specific unit map has been fixed.
\end{proof}

Theorem~\ref{thm:natural-unit-equivalence} is the formal bridge between particle-physics notation and chemistry-natural-unit notation. In the ultraviolet language one sees the signed coefficient vector inherited from anomaly cancellation. In the chemistry-natural-unit language one sees the equality of per-event ledgers. Both are the same statement, but only after the chemical-unit map is inserted explicitly.

\begin{table}[H]
\centering
\caption[The four-line derivation of \QiQUA{}.]{Explicit four-line derivation chain of \QiQUA{}. Each row is one step: the anomaly normalization that fixes the coefficients, the Arisaka Ratio that follows from it, the map into chemical units, and the Arisaka equation itself. The second column carries the formula and the third what the step means. The chain is short deliberately --- there is no intermediate modeling step in it, so a reader who accepts the first row and the arithmetic has accepted the pillar.}
\label{tab:qiqua-derivation-chain}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}p{0.36\textwidth}p{0.36\textwidth}}
\toprule
\textbf{Line} & \textbf{Formula} & \textbf{Meaning} \\
\midrule
\BIsix{} normalization & \((a,b,c,d)=\left(1,-1,-2/5,-2/5\right)\) & Ultraviolet reduced-rational normalization chosen so the later current ledger remains compatible with \GG{} and \OGNfive{}. \\
Arisaka Ratio & \(J'_E:J'_e:J'_B:J'_{Qi}=1:-1:-2/5:-2/5=5:-5:-2:-2\) & Coefficient-space source ray inherited directly from \BIsix{}. \\
Chemical-unit map & \(J_e=|J'_e|\), \(J_B=(5/2)|J'_B|\), \(J_Q=(5/2)|J'_{Qi}|\), \(J_E=\QB |J'_E|\) & Sector-dependent unit conversion from raw \BIsix{} amplitudes to event counts and energy units. \\
Arisaka Equation & \(J_E/\QB=J_e=J_B=J_Q\) & Equality of normalized event counts for one ideal locked event. \\
\bottomrule
\end{tabular}
\end{table}

\section{Ideal equality, coarse-grained inequality, and overhead}
\label{subsec:qiqua-coarse-graining}

The Arisaka Equation is exact for the \emph{ideal synchronized elementary event}. Real chemistry, metabolism, and cognition, however, are not made of one event only. They consist of long histories containing useful steps, housekeeping steps, idle cycles, redundancy, proofreading, error correction, and heat leakage. It is therefore essential to state clearly how \QiQUA\ behaves under coarse-graining.

Let a macroscopic process be decomposed into \(N_{\mathrm{lock}}\) ideal locked events plus additional overhead events that increase the energy, electric, or baryonic budgets without increasing the registered informational count. Then the integrated budgets over the whole process can be written schematically as
\begin{align}
\mathcal{E} &= \QB N_{\mathrm{lock}} + \Delta_E^{\mathrm{over}},
\label{eq:qiqua-overhead-energy}
\\
\mathcal{N}_e &= N_{\mathrm{lock}} + \Delta_e^{\mathrm{over}},
\label{eq:qiqua-overhead-electric}
\\
\mathcal{N}_B &= N_{\mathrm{lock}} + \Delta_B^{\mathrm{over}},
\label{eq:qiqua-overhead-baryon}
\\
\mathcal{N}_Q &= N_{\mathrm{lock}},
\label{eq:qiqua-overhead-info}
\end{align}
with nonnegative overhead terms \(\Delta_A^{\mathrm{over}}\ge 0\). Under this monotone-overhead assumption one immediately obtains
\begin{equation}
\mathcal{E}\ge \QB \mathcal{N}_Q,
\qquad
\mathcal{N}_e\ge \mathcal{N}_Q,
\qquad
\mathcal{N}_B\ge \mathcal{N}_Q.
\label{eq:qiqua-inequality}
\end{equation}

\begin{proposition}[Coarse-grained lower-bound form of \QiQUA]
\label{prop:qiqua-lower-bound}
If a macroscopic process is composed of ideal locked events plus nonnegative overhead in the energy, electric, and baryonic sectors, then Eq.~\eqref{eq:arisaka-equation} descends to the lower-bound inequalities \eqref{eq:qiqua-inequality}.
\end{proposition}

\begin{proof}
Substitute Eqs.~\eqref{eq:qiqua-overhead-energy}--\eqref{eq:qiqua-overhead-info} into each difference \(\mathcal{E}-\QB \mathcal{N}_Q\), \(\mathcal{N}_e-\mathcal{N}_Q\), and \(\mathcal{N}_B-\mathcal{N}_Q\). The differences are precisely the nonnegative overhead terms.
\end{proof}

Proposition~\ref{prop:qiqua-lower-bound} is indispensable for realistic applications. Without it one would be tempted either to overstate \QiQUA as a literal equality for every coarse-grained measured flux, or to dismiss it whenever a real molecular machine or nervous system displays inefficiency. The correct reading is simpler. \QiQUA\ fixes the primitive event and therefore also the ideal lower bound. Real systems may and usually do spend more.

\begin{table}[H]
\centering
\caption[Ideal and coarse-grained forms of \QiQUA{}.]{Ideal and coarse-grained forms of \QiQUA{}. Three levels of description run down the first column --- a single synchronized elementary event, a macroscopic process carrying overhead, and a phenomenological reduced model --- with the mathematical form and the interpretation alongside. The distinction matters for falsification: the equality is asserted at the first level only, and the second and third rows carry inequalities, so a coarse-grained measurement that misses the ideal value is not a refutation and the table says so.}
\label{tab:qiqua-ideal-vs-coarse}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.28\textwidth}p{0.39\textwidth}}
\toprule
\textbf{Level of description} & \textbf{Mathematical form} & \textbf{Interpretation} \\
\midrule
Elementary synchronized event & \(J_E/\QB = J_e = J_B = J_Q\) & Exact equality on the ideal locked ray \\
Macroscopic process with overhead & \(\mathcal{E}\ge \QB \mathcal{N}_Q\), \(\mathcal{N}_e\ge \mathcal{N}_Q\), \(\mathcal{N}_B\ge \mathcal{N}_Q\) & Lower-bound form after aggregation and inefficiency \\
Phenomenological reduced model & Some ledgers may be hidden & Acceptable only if hidden ledgers can in principle be restored in the full bookkeeping \\
\bottomrule
\end{tabular}
\end{table}

This distinction between exact equality and coarse-grained lower bound is especially important for later biological applications. A protocell, an ATP synthase complex, a genetic proofreading network, or a cortical microcircuit can all spend substantially more than the ideal floor. What \QiQUA\ claims is that the floor is not arbitrary. It is fixed by the same imported geometry that fixed the coefficient-space ray.

\section{Relation to Landauer and to classical nonequilibrium thermodynamics}
\label{subsec:qiqua-landauer}

One immediate use of the Arisaka Constant is to compare the \QiQUA\ event floor with the classical lower bound for logically irreversible erasure. Landauer's bound states that erasing one bit of information in thermal equilibrium at temperature \(T\) requires at least
\begin{equation}
E_{\mathrm{L}} = k_B T \ln 2
\label{eq:qiqua-landauer}
\end{equation}
of dissipated energy \citep{Landauer1961}. At \(T=310\,\mathrm{K}\),
\begin{equation}
E_{\mathrm{L}}\approx 0.0185\,\mathrm{eV}.
\label{eq:qiqua-landauer-310}
\end{equation}
Comparing with Eq.~\eqref{eq:qiqua-qb-numerical} gives
\begin{equation}
\frac{\QB}{k_B T\ln 2}\Big|_{T=310\,\mathrm{K}}
\approx 8.40.
\label{eq:qiqua-landauer-ratio}
\end{equation}

The meaning of Eq.~\eqref{eq:qiqua-landauer-ratio} must be stated carefully. \QiQUA\ does \emph{not} refute Landauer's principle. The two statements answer different questions. Landauer asks for the minimum dissipation associated with an idealized logically irreversible operation. \QiQUA\ asks for the minimum cost of a \emph{locked open-system boundary event} in which energy, charge, baryonic structure, and information are audited simultaneously. Because the latter problem is richer, its lower bound can legitimately exceed the former.

Historically, the need to distinguish these levels was already implicit in nonequilibrium thermodynamics. Onsager's reciprocity, Prigogine's irreversible thermodynamics, Jarzynski's equality, Crooks' fluctuation relation, and modern stochastic thermodynamics all teach the same lesson: once a system is open and fluctuating, one must specify which ledger is being optimized, inferred, or dissipated \citep{Onsager1931,Prigogine1967,Jarzynski1997,Crooks1999,Seifert2012}. \QiQUA\ adds one extra layer to that tradition. It does not only track entropy production or work. It also insists that charge, material rearrangement, and information update belong to the same primitive event ledger.

\begin{table}[H]
\centering
\caption[Landauer and \QiQUA{} answer different questions.]{Landauer's bound and \QiQUA{} answer different lower-bound questions. The two principles occupy the two rows; the columns give the question each addresses, the bound it delivers and the ledger it audits. The table exists to forestall a natural objection. \QiQUA{}'s quantum exceeds $k_BT\ln 2$, which looks like a violation until one notices that Landauer audits one fiber --- the thermal cost of erasure --- while \QiQUA{} audits the cost of writing on both fibers at once. They are not competing bounds on the same quantity.}
\label{tab:landauer-vs-qiqua}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.28\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}}
\toprule
\textbf{Principle} & \textbf{Question addressed} & \textbf{Lower bound} & \textbf{Ledger audited} \\
\midrule
Landauer & What is the minimum dissipation for idealized logical erasure of one bit? & \(k_B T\ln 2\) & Information plus thermal bath \\
\QiQUA & What is the minimum cost of one locked boundary event? & \(\QB=(2/5)\ERyd/K\) & Energy, electric, baryonic, and informational ledgers together \\
\bottomrule
\end{tabular}
\end{table}

The payoff of Table~\ref{tab:landauer-vs-qiqua} is conceptual economy. It prevents later sections from confusing thermodynamic erasure, biochemical update, and observer-level measurement. These processes are related, but they are not identical. \QiQUA\ provides the ledger common to all of them once the full open-system event is audited.

\paragraph{Worked examples.} CO activation and hydrogenation in the interstellar medium \citep{Fuchs2009}, ATP synthase in the $c_{10}$ ring of \emph{E.~coli} \citep{Ballhausen2009,Jiang2001,Yoshida2001}, and DNA structure/function/dynamics \citep{WatsonCrick1953,Wang1979,Benham1979,Fuller1971} are recorded as packet-normalized audits in Appendix~\ref{app:technical-derivations-for-qi-qua}; each example tabulates $N_{\mathrm{lock}}$ and the corresponding $J_E = N_{\mathrm{lock}}\QB$.


\begin{figure}[htbp]
\centering
\resizebox{0.851\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% ================= Panel (a): the circle =================
\begin{scope}
\node[align=center] at (0,2.8) {\textbf{(a) The circle $S^1_\theta$ (6th dimension)}\\[-1pt]
\scriptsize compact --- all loops, all holonomy --- a group};
\draw[black, very thick] (0,0) circle (1.7);
\foreach \ang in {90,162,234,306,18}
  \fill[black!55] (\ang:1.7) circle (0.055);
\draw[->,thick,blue!65!black] (40:1.95) arc (40:130:1.95);
\node[blue!65!black] at (85:2.3) {\scriptsize every loop closes};
\node[align=center] at (0,0.35) {\scriptsize windings, phases,\\[-2pt]
\scriptsize parities, charges};
\node[align=center] at (0,-0.45) {\scriptsize quantized, protected};
\node[align=center,blue!65!black] at (0,-2.35)
  {\textbf{remembers}\\[-2pt] \scriptsize classes do not age};
\end{scope}
% ================= Panel (b): the interval =================
\begin{scope}[shift={(5.6,0)}]
\node[align=center] at (3.4,2.8) {\textbf{(b) The interval $I_r$ (5th dimension)}\\[-1pt]
\scriptsize non-compact --- no loops, no holonomy --- a semigroup};
\draw[->,black,very thick] (0.2,0.4) -- (6.6,0.4);
\node at (3.4,0.75) {\scriptsize $r = \ln(E_{\mathrm{GUT}}/\mu)$: the descent, one-way};
\foreach \x/\lab in {0.7/{$E_{\mathrm{GUT}}$},2.5/{lock},4.3/{QCD},6.1/{today}}{
  \draw[black, thick] (\x,0.32) -- (\x,0.48);
  \node[below] at (\x,0.28) {\scriptsize \lab};}
\node[align=center] at (3.4,-0.75) {\scriptsize no path returns:\\[-2pt]
\scriptsize scales, addresses, rates --- one pass each};
\node[align=center,blue!65!black] at (3.4,-2.35)
  {\textbf{forgets}\\[-2pt] \scriptsize each deposit is final};
\end{scope}
\end{tikzpicture}%
}
\caption[The circle that remembers and the interval that forgets.]{\textbf{The
anatomy of the two fibers.} \emph{(a)} The compact phase circle: every path
closes, so everything a connection does there is holonomy --- quantized (the
dots are the ledger-rational marks), gauge-protected, transported by a group.
\emph{(b)} The scale interval: no path closes, so nothing is protected;
transport composes only forward, a semigroup, and the descent visits each
scale exactly once. This is why a stable bit is holonomy-class data on the
circle while erasure is a one-way write along the interval, why Landauer's
bound prices only the second of those, and why the two audits are not
competing bounds on one quantity. Carried from the holonomy white paper,
where the anatomy is the subject rather than a tool.}
\label{fig:a7-two-fibers}
\end{figure}

\section{Information and the two fibers: memory on the circle, erasure on the interval}
\label{subsec:info-two-fibres}

The Landauer comparison above gains a geometric floor from the
cohort's recent campaign (\textsf{GG-W1-Symmetry};
\citealp{ArisakaA5GDOS}; \citealp{ArisakaA8QM};
\citealp{ArisakaP5CPT}), and this paper --- the information
pillar --- is where the reading belongs. The instrument is one
exact statement: \emph{holonomy requires closed loops, and an
interval has none.} The bulk's compact circle is nothing but
closed loops --- everything on it is holonomy, quantized,
gauge-invariant, protected; the radial interval can hold no
holonomy at all --- no protected content, transport composing
only forward, a semigroup. Read informationally: \emph{memory
lives on the circle; erasure pays the interval.} A bit that
persists --- a winding, a phase, a parity, a charge --- is
holonomy-class data on the fiber that remembers, which is why
stable records exist at all in an open universe: they are stored
in quantities no boundary operation can continuously deform.
Erasure --- the logically irreversible reset that Landauer
priced --- is an interval-side write: irreversible not by
engineering failure but by geometry, because the fiber it is
written along has no inverse transport. Landauer's bound
\eqref{eq:qiqua-landauer} is, in this reading, \emph{the
interval's toll}: the minimum the semigroup charges for one
one-way write at temperature $T$.

Table~\ref{tab:landauer-vs-qiqua} then acquires a second
caption. Landauer audits \emph{one fiber}: the thermal cost of
the interval-side write, information against the bath. \QiQUA{}
audits \emph{both}: a locked boundary event deposits its result
in circle-side ledgers (charge, baryonic structure, the
informational unit of \UoneQi{} --- one rearrangement, one unit)
\emph{and} pays the interval-side toll of registration, which is
why its floor $\QB$ legitimately exceeds $k_B T \ln 2$: the
richer bound is the price of writing on both fibers at once. The
Arisaka Constant is thereby read as an \emph{exchange rate
between the circle's two currencies} --- energy per winding on
one face, energy per qubit on the other --- fixed not by a bath
temperature but by the ledger integers, which is the structural
reason it is temperature-independent where Landauer's bound is
not. And the measurement chain closes at its companion homes:
the record whose writing the quantum companion located on the
interval \citep{ArisakaA8QM} is the same record whose cost this
section prices and whose persistence the circle guarantees ---
one event, three audits: \emph{the circle stores it, the
interval seals it, and \QiQUA{} bills it.} This reading is
offered at the register of the campaign --- an organization of
proved structures, not a new theorem --- and every clause in it
carries its parent's grade.

\section{What it means to put a price on a bit}
\label{sec:close-qiqua}

A price is a strong thing to attach to a quantity. It means the quantity
can be compared with others, spent, saved, and run out of. It means
somebody can be shown to have got the sum wrong.

What this Part has done, said without symbols, is to reduce four
separate accounts to one. A system of the kind considered here was
being described as though it kept four books --- one for charge, one
for the winding of its internal circle, one for information, one for
energy --- and nothing in the description prevented those books from
being kept in different units by four different clerks who never
spoke. They are now one book. Three of the columns are locked to each
other outright, and the fourth is joined to them by a single number
that is the same number everywhere, fixed once by the geometry and
never adjusted since. That last clause is what makes the result a
price rather than a conversion table. A table has entries, and entries
can be revised one at a time by whoever is losing the argument. A
price has one value, and everybody is held to it.

For a century, energy has had that status in science and information has
not. Information was always the thing you could invoke when the physics
ran out --- the system "chose", the molecule "recognized", the network
"learned" --- and no bill ever came. That was not dishonesty; it was the
absence of a rate. Without an exchange rate between energy and
information, no statement about how much a memory costs is checkable,
and so no such statement is really a physical statement at all.

What this Part supplies is the rate. And the thing worth appreciating is
where it comes from, because it would have been much easier, and much
weaker, to measure it. A number fitted to biochemistry would explain
biochemistry and nothing else. This one is assembled out of the energy
scale of the hydrogen atom and an integer that a Diophantine closure
fixed for reasons that had nothing to do with life, several papers
before life was mentioned. That it then lands within one and a half per
cent of the free energy of the bond every living thing on this planet
uses to pay for everything is either a coincidence or the most
interesting sentence in the paper.

It is worth being exact about what a laboratory could do to end this,
because it is not what one might guess. The vulnerable claim is not the
number. A calorimeter that disagreed with the value would be arguing
with the ledger upstream, not with this Part. What is at risk is the
assertion that \emph{every} elementary event of this channel shares the
\emph{same} single conversion factor --- that the enzyme, the electron
transfer, the molecular motor and the memory write are all being billed
at one rate. One elementary catalytic event whose four-way accounting
misses by more than a fifth would finish it.

It is also worth seeing what the single budget makes possible, because
it is the reason this Part comes first rather than somewhere else. A
ledger says what a system may spend. It does not say what the system
will do. Between all the ways of spending a fixed budget something has
to choose one, and until the budget was single there was nothing for
such a rule to act on --- a rule that minimized one currency while
another ran free would not be a rule about the system at all, only
about the bookkeeping. Now that the four columns are one, the question
of selection is finally a question with a subject. That is the next
Part's business.

There is a last thought worth carrying forward. If information really
does have a price fixed by geometry, then the familiar picture in which
physics describes the world and information theory describes our
description of it has the relationship backwards. On this account there
is one ledger, it has four columns, and the column we have been treating
as bookkeeping is as physical as the other three. Nothing that
remembers is getting anything for free.


\GGPart[Qi-EP]{\QiEP{}}{One projected functional in three gauges: \PLT{}, \MEPP{}, and \FEP{}}
\label{part:qiep}

\noindent
Three sciences already know which way an open system goes, and they do
not appear to agree.

Optics and mechanics say a system takes the path that makes a certain
time or action stationary --- the principle that sends light along the
quickest route through glass. Nonequilibrium thermodynamics says a
driven system settles into the regime that extremizes entropy
production, and has spent decades arguing internally about whether that
means the most or the least. Theoretical neuroscience says a brain acts
so as to minimize the difference between what it predicted and what
arrived. Three fields, three foundational principles, three literatures
that seldom cite one another, and no reason offered why a cell obeying
all three at once should not be pulled in three directions.

This Part shows they are one principle. Not analogous, not
complementary: one functional, projected from the six-dimensional
action onto the boundary, read three times with a different quantity
held primary each time. Hold the path primary and the stationarity
condition returns the optical law. Hold the currents primary and it
returns the flux--force relation of irreversible thermodynamics. Hold
the beliefs primary and it returns the inference rule the brain
sciences use. The three are gauges of one object, in the way that
Celsius and Fahrenheit are readings of one temperature, and the
argument is not a verbal identification but a derivation in each case.

The old quarrel gets settled on the way through. Maximum entropy
production and Prigogine's minimum entropy production, which have been
treated as rival slogans for seventy years, turn out to be conjugate
readings of the same stationary point under dual conventions about
what is held fixed --- the first selecting the transport law, the
second the state. They were never in competition; they were answering
different questions of one object. (A third and distinct statement,
Onsager's least dissipation, is the flux-controlled Legendre transform
of the same stationary point, and is derived alongside them; it is
easy, and common, to conflate the two minimum principles, and this
Part keeps them apart.)

What this buys, beyond tidiness, is that a single system can be
described in all three languages without contradiction. A cell
following a chemical gradient traces a path, maintains a set of
metabolic fluxes, and updates an internal estimate of where the food
is. Three descriptions, three vocabularies, one budget underneath ---
and the Part is where that budget is written down.


\label{sec:qiep}

With event-level bookkeeping fixed by \QiQUA, the next logical question is selection. A packet can obey the right ledger and still admit many admissible histories. Part~\ref{part:qiep} therefore derives the selector layer of the Qi program: once the budgeted event cone is known, which trajectory, branch, or inferential state is actually extremized?

Every extremal descent in this Part runs one way: admissible histories fall to the selected trajectory, current, or posterior and do not climb back. That one-way character is consumed here, not created. Per the constitutional custody chain, \AX{3} owns the arrow --- the one-way reading of the bulk-to-boundary dictionary is a clause of the constitution (the AX-3 synthesis of \citealp{ArisakaA1Constitution}) --- with the canonical law-level derivation in Part~\ref{part:objections} of the CPT member paper over the exactly \(T\)-symmetric bulk \citep{ArisakaP5CPT}, and the dynamical implementation in the \GDOS{} boundary law \citep{ArisakaA5GDOS} and its \textsf{GG-A6-GRIP} runtime pipeline \citep{ArisakaA6Gi4}. The \QiEP{} selector, and the whole \QiFour{} runtime it drives, is a downstream consumer of that single constitutional arrow, not a private source of it.


The Part now runs in two movements. The first
(Sections~\ref{sec:qiep-overview}--\ref{sec:qiep-hist-fep}) gives the
three principles their own histories at review depth --- what each says
exactly, where each is proved, where each fails, and who located the
failure --- because a derivation is worth only as much as the precision
of what it derives, and because the reader this Part most wants to
reach arrives owing allegiance to one of the three literatures and
suspicion to the other two. The second movement
(Section~\ref{subsec:qiep-constitutional-statement} onward) is the
derivation itself: the constitutional statement and the contract it
must meet, the projected functional, the three gauges each obtained as
a theorem with its regime attached, the failure theorem that bounds
all three, the laboratory record, and the unified theorem that closes
the Part.

\section{Overview of \texorpdfstring{\PLT{}, \MEPP{}, and \FEP{}}{PLT, MEPP, and FEP}}
\label{sec:qiep-overview}

Three principles, from three sciences, answer the same question ---
\emph{which way does an open system go?} --- and each answers it with an
extremum.

The oldest is the principle of least action, with its optical face, the
principle of least time (\PLA{}/\PLT{}): a path is realized when a
certain time or action is stationary against all nearby paths. It is
the most successful variational idea in science --- geometric optics,
analytical mechanics, general relativity, and the Standard Model are
all written in its language --- and it is silent, as classically
stated, about systems that are open, driven, and noisy.

The second lives in nonequilibrium thermodynamics and has spent seventy
years arguing with itself. Near equilibrium, Prigogine's theorem of
1945 says a gently driven system settles into the steady state of
\emph{minimum} entropy production --- one of the few statements in the
subject that is actually proved. A rival tradition, running from the
theory of plastic materials to the modeling of planetary climates, says
driven systems arrange themselves to produce entropy \emph{fastest}
(\MEPP{}) --- a rule that, fed nothing but sunlight and an energy
balance, reproduces the equator-to-pole temperature contrast on Earth,
Mars, and Titan.

The third is the youngest and the most ambitious. The free energy
principle (\FEP{}) of theoretical neuroscience holds that anything
which persists behaves so as to make its sensory input unsurprising ---
that a brain, and in the strongest reading any living thing, is a
prediction machine acting to minimize a variational free energy.

A physicist meeting the three side by side notices two things at once.
The first is that they cannot all be fundamental as stated: a living
cell is a near-equilibrium chemical network, a driven thermodynamic
engine, and a predictive controller \emph{simultaneously}, and if the
three principles were independent laws the cell would be receiving
three sets of instructions with no account of why they do not conflict.
The second is that none of the three has ever been derived. Each is
either a theorem confined to a regime far narrower than its use, or a
postulate whose attempted derivations failed at published, identifiable
steps. The three failures, moreover, have the same anatomy. No
principle announces its own domain of validity --- each was applied
outside its regime for years before its own community mapped the
boundary. Each leaves a load-bearing choice to the person applying it
--- which entropy production to extremize, where to draw the boundary
between system and world. And each general derivation that was
seriously attempted broke at a specific step that its own literature
has since located. When three literatures that seldom cite one another
fail in the same three ways, the failure stops being three pieces of
bad luck and becomes a symptom: evidence of one missing structure
underneath all three.

This Part supplies the missing structure and proves it is one. The
claim, stated once here and then earned: the three principles are three
\emph{gauges} of a single functional, projected from the six-dimensional
bulk action onto the boundary through \OGGEP{} and \GDOS{} --- and the
regimes, the quarrels, and the failures of all three follow from the
same projection as theorems. Sections~\ref{sec:qiep-hist-pla} through
\ref{sec:qiep-hist-fep} first give each principle its own history at
review depth --- what it says exactly, where it is proved, where it
fails, and who located the failure --- because a derivation is worth
only as much as the precision of what it derives. The constitutional
statement (Section~\ref{subsec:qiep-constitutional-statement}) then
fixes the contract a genuine derivation must meet, the projected
functional is constructed
(Section~\ref{subsec:qiep-bulk-to-boundary}), and the three gauges are
derived in turn (Sections~\ref{subsec:qiep-plt},
\ref{subsec:qiep-mepp}, \ref{subsec:qiep-fep}), each with its regime
theorem and its laboratory record
(Sections~\ref{subsec:qiep-failure},
\ref{subsec:qiep-observations}). The extended treatment --- the complete
proofs at review depth and the full critical literatures --- is the
companion supplement \citep{ArisakaS1EP}; the present Part is
self-contained.

\begin{figure}[htbp]
\centering

% Placed by the house rule: picture measures ~160 mm at footnotesize
% (9 pt base), so f = (7.79/9) x 160.4/165.1 = 0.84; set 0.85, which
% renders the labels at ~7.9 pt against the 7.79 pt house standard.
\resizebox{0.85\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=3pt, inner sep=3.5pt},
 pla/.style={fill=green!12, draw=green!55!black, line width=0.70pt,
  text width=44mm, minimum height=11mm},
 ep/.style={fill=orange!14, draw=orange!65!black, line width=0.70pt,
  text width=44mm, minimum height=11mm},
 fep/.style={fill=violet!12, draw=violet!60!black, line width=0.70pt,
  text width=44mm, minimum height=11mm},
 bad/.style={fill=red!10, draw=red!60!black, line width=0.70pt,
  text width=44mm, minimum height=11mm},
 hdr/.style={font=\sffamily\small\bfseries, text width=48mm, inner sep=2pt},
 uni/.style={fill=blue!12, draw=blue!60!black, line width=1.10pt,
  text width=118mm, minimum height=14mm},
 arr/.style={-{Stealth[length=2.4mm]}, line width=0.7pt, draw=black!70},
]
\node[hdr, text=green!40!black]  (h1) at (-56mm, 4mm) {the action tradition};
\node[hdr, text=orange!55!black] (h2) at (0mm,   4mm) {the entropy-production tradition};
\node[hdr, text=violet!45!black] (h3) at (56mm,  4mm) {the inference tradition};
\node[pla] (a1) at (-56mm,-10mm) {\textbf{1662 Fermat}\\ least time gives the sine law};
\node[pla] (a2) at (-56mm,-26mm) {\textbf{1744 Maupertuis; Euler}\\ least action, made rigorous};
\node[pla] (a3) at (-56mm,-42mm) {\textbf{1788 Lagrange; 1834 Hamilton}\\ mechanics from one integral};
\node[pla] (a4) at (-56mm,-58mm) {\textbf{1948 Feynman}\\ stationary phase: why ``least'' works};
\node[pla] (a5) at (-56mm,-74mm) {\textbf{1953 Onsager--Machlup}\\ an action for noisy paths};
\node[ep] (b1) at (0mm,-10mm) {\textbf{1848--1873 Kirchhoff;\\ Helmholtz; Rayleigh}\\ least heat, least dissipation};
\node[ep] (b2) at (0mm,-26mm) {\textbf{1931 Onsager}\\ reciprocity; least dissipation};
\node[ep] (b3) at (0mm,-42mm) {\textbf{1945 Prigogine}\\ MinEP: the one proved theorem};
\node[ep] (b4) at (0mm,-58mm) {\textbf{1961 Ziegler; 1975 Paltridge}\\ the \emph{maximum} tradition};
\node[bad] (b5) at (0mm,-74mm) {\textbf{2003--2009 Dewar}\\ claimed proof; refuted; conceded};
\node[fep] (c1) at (56mm,-10mm) {\textbf{1867 Helmholtz}\\ perception as unconscious inference};
\node[fep] (c2) at (56mm,-26mm) {\textbf{1952--1970 Ashby; Conant}\\ every regulator is a model};
\node[fep] (c3) at (56mm,-42mm) {\textbf{1995--1999 Helmholtz machine;\\ predictive coding}};
\node[fep] (c4) at (56mm,-58mm) {\textbf{2005--2019 Friston}\\ the FEP and its physics phase};
\node[bad] (c5) at (56mm,-74mm) {\textbf{2021--2023 critiques}\\ non-generic; retreat to beliefs};
\node[uni] (u) at (0mm,-93mm)
 {\textbf{\QiEP{} (this Part): one functional, projected from the 6D bulk ---\\
  and \PLT{}, \MEPP{}, and \FEP{} are its trajectory, current, and inferential gauges}};
\draw[arr] (a1) -- (a2); \draw[arr] (a2) -- (a3);
\draw[arr] (a3) -- (a4); \draw[arr] (a4) -- (a5);
\draw[arr] (b1) -- (b2); \draw[arr] (b2) -- (b3);
\draw[arr] (b3) -- (b4); \draw[arr] (b4) -- (b5);
\draw[arr] (c1) -- (c2); \draw[arr] (c2) -- (c3);
\draw[arr] (c3) -- (c4); \draw[arr] (c4) -- (c5);
\draw[arr] (a5.south) -- (u.north west);
\draw[arr] (b5.south) -- (u.north);
\draw[arr] (c5.south) -- (u.north east);
\end{tikzpicture}%
}
\caption[Three centuries, three traditions, one functional.]{The
genealogy of the three extremal principles, drawn as three columns that
never cite one another. Left, the action tradition: from Fermat's least
time through Maupertuis, Euler, Lagrange, and Hamilton to Feynman's
stationary phase and the Onsager--Machlup stochastic action. Center,
the entropy-production tradition: from the least-heat and
least-dissipation theorems of Kirchhoff, Helmholtz, and Rayleigh
through Onsager and Prigogine to Ziegler's maximum, Paltridge's
climates, and the claimed general derivation that was refuted and
conceded. Right, the inference tradition: from Helmholtz's unconscious
inference through cybernetic regulation and the Helmholtz machine to
predictive coding, the free energy principle, its physics phase, and
the critiques that forced its retreat. Each column ends unresolved in
its own literature; the convergence at the foot --- one projected
functional whose three gauges they are --- is the claim this Part
derives, and the dates above it are why the derivation is worth having.}
\label{fig:qiep-genealogy}
\end{figure}


\section{The principle of least action and least time: history, applications, and limitations}
\label{sec:qiep-hist-pla}

\subsection{From Fermat to Hamilton}

The action tradition begins with light. In 1662 Pierre de Fermat
derived the sine law of refraction from a single postulate --- that
light traveling between two points takes the path of least time ---
and announced the result, with some astonishment at its success, in
his correspondence on refraction \citep{Fermat1662}. The derivation
was controversial from the day it appeared: Fermat's critics in the
Cartesian school objected that light has no purposes, and the
objection --- \emph{why should nature optimize?} --- has shadowed the
principle in every one of its later forms.

The mechanical version arrived in 1744 twice. Maupertuis proposed that
in all natural processes a quantity he called action --- mass times
speed times distance --- is a minimum, and read the economy
theologically, as evidence of design \citep{Maupertuis1744}. In the
same year Euler, in the appendix to his treatise on the calculus of
variations, gave the statement its first rigorous form: for a single
particle at fixed energy, the trajectory extremizes
\(\int v\, \dd s\) \citep{Euler1744}. Lagrange removed both the
theology and the restriction, building the whole of mechanics on the
variational calculus in the \emph{M\'echanique analitique}
\citep{Lagrange1788}; and Hamilton, in 1834, gave the modern form ---
the realized motion renders the time integral of \(T-V\) stationary
--- and, in the same stroke, exhibited the deep identity of optics and
mechanics: one characteristic function serves both, light rays and
particle paths being two readings of a single variational geometry
\citep{Hamilton1834}. Jacobi sharpened the fixed-energy form soon
after: at energy \(E\) the orbit of a particle in a potential \(V\)
extremizes the abbreviated action
\begin{equation}
W \;=\; \int \sqrt{2m\,(E-V)}\;\dd s ,
\label{eq:qiep-jacobi}
\end{equation}
which is Fermat's principle with \(\sqrt{2m(E-V)}\) playing the role
of the refractive index --- mechanics as the optics of a medium whose
density is set by the potential. That identity is not a curiosity; it
is the template for the trajectory gauge below, where the projection
supplies the effective index for an arbitrary open system. By the middle of the nineteenth century the principle had
become what it remains today: the standard form in which fundamental
physical law is written.

\subsection{The twentieth century: why ``least,'' and what noise does}

Two twentieth-century results changed the principle's meaning rather
than its use. The first explains \emph{why} extremal paths are
realized at all. In Feynman's path-integral formulation of quantum
mechanics, every path contributes an amplitude of unit magnitude and
phase \(S/\hbar\); paths near a stationary point of the action add
coherently while all others cancel, so the classical trajectory is not
obeyed but \emph{survives} --- it is the path whose neighbors agree
with it \citep{Feynman1948}. The variational principle is thereby
demoted from law to limit, and the demotion is clarifying: it says the
principle should be expected to hold exactly where interference is
sharp, and to blur where it is not --- a statement about regimes, of
exactly the kind the other two traditions of this Part never had.

The second result extends the principle to noise. Onsager and Machlup
showed that for a linear dissipative system driven by thermal
fluctuations, the probability of a trajectory is governed by an
exponent of action form,
\begin{equation}
p[\dot{\boldsymbol\alpha}]\;\propto\;
\exp\!\left\{\frac{1}{2k_B}\int\!\bigl(\dot S-\Phi-\Psi\bigr)\,\dd t\right\},
\label{eq:qiep-om-action}
\end{equation}
with \(\Phi\) and \(\Psi\) the flux- and force-space dissipation
functions --- so that the \emph{most probable} path extremizes a
functional built from the dissipation, and least dissipation becomes
literal: the likeliest history is the least-dissipative one
\citep{OnsagerMachlup1953}. This is the seed of the modern
large-deviation viewpoint, and it is the hinge on which this Part will
turn: in open systems the action principle survives as a statement
about the concentration of path probability, exact in the small-noise
limit, and the ``time'' it extremizes is an effective, geometry-laden
time rather than a bare clock reading.

\subsection{Applications}

The scorecard of the action tradition is unmatched in physics.
Geometric optics is Fermat's principle applied; analytical mechanics
is Hamilton's; general relativity takes the Einstein--Hilbert action
as its definition, and the Standard Model is specified, in its
entirety, by a Lagrangian density. Where the tradition is weakest is
exactly where this Part operates: open systems. A cell swimming up a
gradient traces a path as surely as a photon crossing a lens, and the
question whether any least-time statement governs that path is not
answered by any of the classical results above.

\subsection{Limitations}

Five limitations bound the classical principle, and each is
load-bearing for what follows.

\emph{First, ``least'' is a misnomer.} The realized path renders the
action stationary, not minimal; beyond a kinetic focus the classical
path is a saddle point, and elementary examples abound in which no
minimum exists \citep{GrayTaylor2007}. The correct universal statement
is stationarity --- which matters here because the projected
functional of this Part is likewise stationary, not minimized, and its
three gauges inherit exactly that character.

\emph{Second, dissipation resists the Lagrangian frame.} Bauer proved
in 1931 that the equations of motion of a linearly damped system admit
no variational formulation in the original variables
\citep{Bauer1931}: friction cannot simply be appended to a Lagrangian.
Rayleigh's dissipation function is bolted onto the Euler--Lagrange
equations from outside, not derived from a stationarity principle.
For two centuries the action tradition therefore stopped, in
principle, at the boundary of every open system.

\emph{Third, the stochastic extension is a probability statement, not
a dynamics.} The Onsager--Machlup functional weights paths; it does
not, by itself, say which functional a given open system carries,
in which variables, or with what metric. Those choices are exactly
the modeler-supplied structure that Section~\ref{sec:qiep-overview}
identified as the shared defect of all three traditions.

\emph{Fourth, the tradition never derived its own functional.} At
every stage the Lagrangian, the index, or the stochastic action is
\emph{written down} --- from symmetry, from phenomenology, from the
model at hand --- and the variational principle then processes it.
Nothing in the tradition says which functional a given system must
carry; that question was not even well posed until there was an
upstream object to project. The same gap will reappear, in different
words, in both of the other traditions: \MEPP{} cannot say which
entropy production to extremize, and the free energy principle cannot
say where the boundary of the believer lies.

\emph{Fifth, no classical result says why an open, driven, noisy
system should follow any least-time path at all.} Fermat's principle
is exact for rays; Hamilton's for closed conservative systems;
Feynman's argument requires coherent interference; Onsager--Machlup
requires the small-noise limit of a specific linear model. The
trajectory gauge of this Part
(Section~\ref{subsec:qiep-plt}) is precisely the missing derivation:
it obtains a generalized least-time law for open systems as the
small-noise reading of the projected functional, with the effective
optical metric supplied by the projection rather than chosen by hand
--- and with the regime in which the reading holds stated as a
theorem rather than discovered by failure.

\begin{table}[htbp]
\centering
\small
\caption[The action tradition: what is proved, where it stops.]{The
action tradition's own scope, row by row. Each row is one classical
result, with the regime in which it is exact and the limitation that
stops it at the boundary of open systems. The table's point is the
last column read downward: every exact result either excludes
dissipation and noise outright or handles them as a probability weight
whose functional must be supplied by hand --- so the tradition, for
all its power, contains no theorem that sends a driven, noisy, open
system down a least-time path. That theorem is what the trajectory
gauge of Section~\ref{subsec:qiep-plt} supplies, and this table is the
measure of what supplying it is worth.}
\label{tab:qiep-pla-scope}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.33\textwidth}>{\raggedright\arraybackslash}p{0.35\textwidth}@{}}
\toprule
\textbf{Result} & \textbf{Exact where} & \textbf{Limitation} \\
\midrule
Fermat's least time (1662) & Geometric optics; rays between fixed
endpoints & Stationary, not always least; no matter, no noise \\
Maupertuis--Euler--Lagrange--Hamilton action (1744--1834) & Closed
conservative mechanics; all of fundamental field theory & Excludes
dissipation by construction; no open-system form (Bauer 1931) \\
Feynman's stationary phase (1948) & Classical limit of coherent
quantum evolution & Requires interference; decoherence and driving
blur the extremum \\
Onsager--Machlup action (1953) & Small-noise limit of linear
dissipative dynamics & A probability weight, not a dynamics; the
functional and metric are inputs, not outputs \\
\bottomrule
\end{tabular}
\end{table}


\section{Minimum entropy production (Prigogine): history, applications, and limitations}
\label{sec:qiep-hist-minep}

\subsection{A century of minimum-dissipation theorems}

Minimum entropy production (MinEP) is often presented as a single
theorem with a single author. It is in fact the last and sharpest
member of a family of minimum-dissipation theorems running through a
century of linear physics, and every member owes its minimum property
to the same mathematical circumstance --- a quadratic dissipation
functional generated by a symmetric, positive-definite linear
response. Kirchhoff proved in 1848 that the steady currents in a
network of ohmic conductors minimize the total Joule heat among all
distributions compatible with the sources \citep{Kirchhoff1848};
Helmholtz proved the analogue for creeping viscous flow in 1868
\citep{Helmholtz1868}; Rayleigh introduced the dissipation function
for damped vibration in 1873 \citep{Rayleigh1873}. In each case the
minimum is a property of \emph{linearity}, not of nature at large ---
the pattern that recurs through everything below.

Onsager's contribution of 1931 must be stated carefully, because it is
systematically confused with the theorem that came fourteen years
later. Alongside the reciprocal relations \(L_{ik}=L_{ki}\), Onsager
formulated a principle of least dissipation: at \emph{given forces},
the realized fluxes extremize \(\sigma(J,X)-\Phi(J,J)\), where
\(\Phi\) is a quadratic dissipation function of the fluxes
\citep{Onsager1931}. The stationarity condition of that
functional \emph{is} the linear constitutive law \(J=LX\) --- so
Onsager's principle is a flux-side encoding of the transport law
itself, and says nothing about which \emph{state} a driven system
selects. Prigogine's theorem, by contrast, varies the free forces at
fixed clamped forces and characterizes the steady state. Keeping the
two apart is not pedantry: their conflation generated much of the
literature's confusion, as Jaynes later documented \citep{Jaynes1980},
and the derivation of Section~\ref{subsec:qiep-mepp} recovers them as
distinct readings of one stationary point.

The theorem itself arrived in 1945, in Prigogine's communication to
the Belgian academy from De Donder's Brussels school, with the full
exposition in his 1947 thesis
\citep{Prigogine1945,Prigogine1947}: for linear phenomenological laws
with constant, symmetric, positive-definite coefficients, and with a
subset of the forces clamped by reservoirs, the stationary state is
the state of least entropy production, and \(P\) decreases
monotonically during the approach to it. The careful neutral statement
of the hypotheses is Chapter~V of \citet{deGrootMazur1962}. The first
microscopic test came in 1954, when Klein and Meijer confirmed the
theorem by master-equation methods for gas flow between two vessels
--- \emph{to first order in the temperature difference}
\citep{KleinMeijer1954}. Already the verification carried the message,
made precise half a century later, that MinEP is a
first-order-in-driving statement.

\subsection{The theorem, exactly}

\begin{theorem}[Prigogine, 1945]
\label{thm:qiep-minep-prigogine}
Let forces \(X_1,\dots,X_n\) and conjugate fluxes \(J_1,\dots,J_n\)
carry entropy production \(P=\sum_i J_iX_i\ge 0\), and suppose
(A1)~linearity, \(J_i=\sum_k L_{ik}X_k\); (A2)~constancy of the
\(L_{ik}\); (A3)~Onsager reciprocity, \(L_{ik}=L_{ki}\);
(A4)~positive definiteness of \(L\); and (A5)~a clamped/free split:
\(X_1,\dots,X_m\) held fixed by reservoirs, the rest free. Then the
state minimizing \(P\) over the free forces is exactly the stationary
state, characterized by the vanishing of the fluxes conjugate to the
free forces; and along the relaxation, \(\dd P/\dd t\le 0\), with
equality only at the stationary state.
\end{theorem}

The proof is a page: under (A1)--(A3), \(P\) is a positive-definite
quadratic form in the forces, so
\(\partial P/\partial X_j = 2J_j\) for each free index \(j\), and
stationarity of \(P\) is the vanishing of the free fluxes --- which is
the definition of the steady state; positive definiteness makes the
stationary point a minimum, and the same quadratic structure makes
\(P\) a Lyapunov function of the relaxation. Every word of the theorem
is load-bearing, and each hypothesis, when dropped, has a named
counterexample waiting (Section~\ref{sec:qiep-minep-scope}).

\subsection{The failed generalization, and what it created}

The attempt to carry the theorem beyond the linear regime occupied
Prigogine and Glansdorff for nearly two decades and ended in one of
the more instructive negative results of twentieth-century physics.
Splitting the time derivative of \(P\) into a force part and a flux
part \citep{GlansdorffPrigogine1954}, they proved in 1964 that beyond
the linear regime only the force part retains a definite sign,
\(\dd_XP/\dd t\le 0\) --- the general evolution criterion --- while
the flux part, and hence \(\dd P/\dd t\) itself, does not
\citep{GlansdorffPrigogine1964}. The deeper disappointment, fully
acknowledged by the authors, is that \(\dd_XP\) is \emph{not an exact
differential}: outside the linear regime there exists, in general, no
functional of the state whose descent the criterion expresses.
Nothing is minimized. Far from equilibrium there is, provably, no
potential of the entropy-production family
\citep{GlansdorffPrigogine1971}.

Out of the failure came the positive program. If far-from-equilibrium
steady states obey no extremum, their stability must be studied
directly --- and it can be lost. Beyond a critical affinity the
Brusselator's homogeneous state destabilizes to Turing patterns or
limit cycles \citep{PrigogineLefever1968}; past a critical Rayleigh
number a heated fluid layer starts to convect; and the selected regime
is emphatically \emph{not} the branch of least entropy production.
These are the dissipative structures of the 1977 Nobel Prize, and the
point is often garbled and worth stating plainly: the prize honors, in
essence, Prigogine's demonstration of the limits of his own 1945
theorem \citep{PrigogineNobel1977}.

\subsection{Critiques, counterexamples, and the exact scope}
\label{sec:qiep-minep-scope}

Three classical critiques fix the boundary.

\emph{Landauer's blowtorch (1975).} Heat a stretch of the transition
path between two locally stable states, in a region where the
stationary occupation is vanishingly small, and the relative occupation
of the two states shifts --- although no functional built from the
entropy family registers the heating \citep{Landauer1975a}. The steady
state is a property of the full kinetics, not of thermodynamic
bookkeeping. In the companion paper the failure appears arbitrarily
close to equilibrium: the fluctuating current through a driven
inductor does not minimize entropy production
\citep{Landauer1975b}. The modern diagnosis is time-reversal parity
\citep{BruersMaesNetocny2007}: the theorem tacitly assumes all
variables even under time reversal, and for odd variables --- currents
through inductors --- the same linear regime yields a
\emph{maximum}-type characterization, so that neither slogan survives
even in linear electrical networks.

\emph{Jaynes's two-temperature network (1980).} In a resistor network
with the resistors held at different temperatures, the physical
currents obey Kirchhoff's laws and minimize the total \emph{heat}
production; a minimum \emph{entropy} production principle would weight
each resistor by \(1/T_k\) and predict a different, wrong current
distribution \citep{Jaynes1980}. Nonuniform temperature makes the
quadratic form's coefficients state-dependent across the system ---
hypothesis (A2) fails --- and Jaynes's verdict on scope stands:
where the principle is exact it adds nothing beyond the linear laws
that imply it, and where it would add something, it is wrong.

\emph{The stochastic-thermodynamics verdict.} The master-equation
setting makes the approximate character of MinEP a theorem rather than
a grievance. With entropy production given by the Schnakenberg
functional \citep{Schnakenberg1976}, Maes and Neto\v{c}n\'y proved
that for dynamics obtained by a small deformation of detailed balance,
the large-deviation rate functional of the occupation statistics is,
to leading order, a quarter of the excess entropy production
\citep{MaesNetocny2007} --- so MinEP is \emph{derived}, and
simultaneously \emph{bounded}, as the first order of a fluctuation
functional. At the next order a time-symmetric contribution --- the
frenetic part, measuring traffic and dynamical activity --- enters
independently, and no functional of entropy production alone,
minimized or maximized, determines the steady state
\citep{MaesNetocny2013}. Landauer's blowtorch is the canonical
illustration: the heating alters the frenetic factors while the
entropic bookkeeping is blind to them.

\begin{table}[htbp]
\centering
\small
\caption[MinEP: the hypotheses and their counterexamples.]{The exact
scope of Prigogine's theorem, one row per load-bearing hypothesis.
Each row names the assumption in Theorem~\ref{thm:qiep-minep-prigogine},
the classical counterexample that lives at that door, and what the
counterexample proves. The table should be read as a passport rather
than an indictment: a theorem whose exceptions can all be mapped onto
named hypotheses is delimited, not discredited --- the healthiest
condition a scientific statement can be in --- and the derivation of
Section~\ref{subsec:qiep-mepp} must, and does, reproduce both the
theorem inside this passport and its silence outside it.}
\label{tab:qiep-minep-scope}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.26\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}@{}}
\toprule
\textbf{Hypothesis} & \textbf{Counterexample when dropped} & \textbf{What it proves} \\
\midrule
(A1) Linear laws & Brusselator beyond critical affinity; B\'enard
convection & No entropy-production potential exists far from
equilibrium; selection passes to stability theory \\
(A2) Constant coefficients & Jaynes's two-temperature network & The
physical state minimizes heat, not entropy production; MinEP predicts
the wrong currents \\
(A3)--(A4) Symmetric positive response & Loss of reciprocity (magnetic
fields, active media) & The quadratic structure that carries the proof
is absent \\
Evenness under time reversal (tacit) & Landauer's driven inductor &
Odd variables convert the minimum into a maximum-type principle within
the same linear regime \\
(A5) Clamped/free split & Nothing clamped & Nothing is selected; the
theorem has no content \\
\bottomrule
\end{tabular}
\end{table}

One sentence carries this section into the derivation. Near
equilibrium the theorem is exact, which means any deeper account of
driven systems must reproduce it in the gentle-driving limit --- just
as any theory of motion must reproduce Newton's at low speed --- and
the current gauge of Section~\ref{subsec:qiep-mepp} passes exactly
this checkpoint, recovering Theorem~\ref{thm:qiep-minep-prigogine},
hypotheses and all, as the force-side reading of its stationary point.


\section{Maximum entropy production: history, applications, and limitations}
\label{sec:qiep-hist-mepp}

\subsection{From least dissipation to a maximum principle}

Every modern statement of maximum entropy production descends,
ironically, from the minimum principles of the last section: Ziegler's
formulation is the Legendre-dual face of the same quadratic structure
that carries Kirchhoff, Rayleigh, and Onsager. Working in continuum
thermomechanics and plasticity between 1958 and 1983, Ziegler posited
the \emph{orthogonality principle}: the dissipative force realized by
a material is normal to the level surface of a prescribed dissipation
function of the fluxes --- equivalently, at prescribed forces the
fluxes maximize the entropy production subject to the single
constraint that production equal the power expended
\citep{Ziegler1961,Ziegler1983}. In rate-independent plasticity this
is the classical normality rule --- plastic strain rates normal to the
yield surface --- where it has done quiet, honest service for decades.
For quadratic dissipation the orthogonality condition returns exactly
Onsager's linear law, so in the linear regime Ziegler's maximum and
Onsager's least dissipation are one statement in two conventions; and
Polettini has shown that for non-homogeneous dissipation functions the
orthogonality condition and the constrained maximization come apart,
so that ``Ziegler's principle'' is a convention-dependent family
rather than one principle \citep{Polettini2013}.

One member of the family is a theorem in the full sense, and it is
worth naming exactly because it is so often cited on the principle's
behalf: Kohler (1948) and Ziman showed that the stationary solution of
the \emph{linearized} Boltzmann equation maximizes the entropy
production among trial distributions satisfying a self-consistency
constraint \citep{Ziman1956}. The proof is three lines of
Cauchy--Schwarz in the collision operator's inner product --- and the
theorem lives entirely in linearized transport, where it restates
least dissipation and adds nothing beyond the transport law it encodes.

\subsection{Paltridge's climates: the empirical core}

What made \MEPP{} famous is none of the above; it is a 1975 discovery
in climate science. Paltridge partitioned the Earth into zonal boxes
exchanging heat, wrote only the steady energy balances --- shortwave
in, longwave out, an unknown meridional transport --- and closed the
underdetermined system by demanding that the material entropy
production of the transport be maximal
\citep{Paltridge1975,Paltridge1978}. In the two-box form the whole
construction fits on a postcard, and it is worth having on the page,
because it is the strongest empirical card the principle holds. Let
box~1 (low latitudes) absorb shortwave power \(I_1\) and box~2 absorb
\(I_2<I_1\); let \(F\ge 0\) be the heat carried poleward; and
linearize the outgoing longwave emission as \(A+BT\). The steady
balances
\begin{equation}
I_1-(A+BT_1)-F=0, \qquad I_2-(A+BT_2)+F=0
\label{eq:qiep-twobox-balance}
\end{equation}
make each temperature affine in the single unknown \(F\): the sum
fixes the mean temperature \(\bar T\) independently of the transport,
and the difference gives the contrast
\(\Delta T(F)=(\Delta I-2F)/B\) with \(\Delta I\equiv I_1-I_2\). The
entropy production of the transfer,
\begin{equation}
\sigma(F)\;=\;F\left(\frac{1}{T_2}-\frac{1}{T_1}\right)
\;=\;\frac{F\,\Delta T(F)}{T_1(F)\,T_2(F)}\,,
\label{eq:qiep-twobox-sigma}
\end{equation}
vanishes at both ends of the admissible range --- at \(F=0\) (finite
force, no flux) and at \(F_{\mathrm{eq}}=\Delta I/2\), where the
transport has erased the very temperature difference that drives it
(finite flux, no force) --- so it has an interior maximum. In the
small-contrast regime \(\Delta T\ll\bar T\) the curve is an exact
parabola and the maximum sits at
\begin{equation}
F^{*}=\frac{\Delta I}{4}, \qquad
\Delta T^{*}=\frac{\Delta I}{2B}=\frac{\Delta T_{\mathrm{rad}}}{2},
\qquad
\sigma_{\max}=\frac{(\Delta I)^{2}}{8B\bar T^{2}}\,:
\label{eq:qiep-twobox-max}
\end{equation}
the selected state transports exactly half the equalizing flux and
sustains exactly half the radiative-equilibrium contrast. The
trade-off in \eqref{eq:qiep-twobox-sigma} --- every unit of flux
erodes the force that pays for it --- is the entire mathematical
content of the climate \MEPP{} in its simplest form, and the principle
bets that nature sits at the optimum. Fed nothing but absorbed
sunlight and the two empirical radiation constants, the closure
reproduces the observed equator-to-pole contrast --- and when Lorenz
applied the identical construction to Mars and Titan it worked there
too, with no adjustable transport coefficient, while a diffusive
closure tuned to Earth fails when exported
\citep{Lorenz2001,Ozawa2003}. Three planets, one postcard formula:
this is the empirical core of the \MEPP{} literature, and any
assessment of the principle has to reckon with it.

It must equally reckon with what the closure ignores. It contains no
rotation rate, no momentum budget, no moist processes --- Goody
pressed exactly this against the theoretical program
\citep{Goody2007}; the maximized quantity is the material entropy
production while the planet's dominant, radiative production is not
extremized \citep{Essex1984}; and the prediction depends on how the
system is partitioned and which fluxes are treated as free --- the
resolution and constraint-choice dependence emphasized in the modern
consensus assessment \citep{SinghONeill2022}. None of this erases the
result; it sharpens what the result is: a parameter-free regularity
that holds across three atmospheres is a fact about atmospheres,
however innocent the formula --- a benchmark that any candidate
theory, including this Part's, must reproduce or explain away.

\subsection{The claimed derivation, and where it broke}

A general proof of \MEPP{} was claimed in 2003--2005. Dewar applied
Jaynes's maximum-entropy inference to trajectories: assign path
probabilities \(p_\Gamma=Z^{-1}e^{A_\Gamma}\) by maximizing path
entropy subject to prescribed mean boundary fluxes and local
conservation laws \citep{Dewar2003,Dewar2005}. The construction's
early steps are sound and even fruitful. Integrating the constraints
by parts splits the action into a part even under time reversal
(depending only on the endpoints) plus one half of the time-integrated
entropy production,
\begin{equation}
A_\Gamma \;=\; A_\Gamma^{\mathrm{even}}
\;+\; \frac{\tau\,\sigma_\Gamma}{2k_B}\,,
\label{eq:qiep-dewar-action}
\end{equation}
from which the transient fluctuation theorem
\(p_\Gamma/p_{\Gamma_R}=e^{\tau\sigma_\Gamma/k_B}\) --- and with it
the second law, as a Jensen corollary --- follows immediately, since
reversing a path flips the odd half and preserves the even one. That
much is uncontested. The
failure is localized in the closing steps, and the critical literature
located it precisely. Dewar's 2003 route requires the number of paths
at given irreversible action to be an \emph{increasing} function of
that action --- an assumption nowhere derived; Bruers exhibited an
explicit lattice-current model in which the count runs the other way,
so that the same machinery, run honestly in the linear regime,
delivers Prigogine's \emph{minimum} \citep{Bruers2007}. The 2005 route
rests on a Gaussian saddle-point evaluation that is valid only for
weak flux fluctuations --- that is, near equilibrium --- as Grinstein
and Linsker showed, which forecloses the claimed far-from-equilibrium
generality \citep{GrinsteinLinsker2007}. Dewar conceded both points in
print and reframed the principle as an inference algorithm whose
output is only as reliable as the constraints fed into it --- no
longer a claimed theorem of physics \citep{Dewar2009}.

Bruers's taxonomy is worth carrying forward: at least five inequivalent
statements circulate under the one name --- the linear-response
theorem (Kohler--Ziman, equivalent to least dissipation), Paltridge's
partial steady-state closure, a total steady-state maximization,
non-variational selection among discrete far-from-equilibrium states,
and the near-equilibrium minimum that Dewar's own machinery actually
implies \citep{Bruers2007}. The climate successes are not underwritten
by the path-information argument; they stand, underived, on their own.

\subsection{Scope, and the distinction that prevents most confusion}

The landscape in one sentence: \MEPP{} is one exact theorem (linearized
transport, where it adds nothing), one durable geophysical heuristic
(Paltridge's, sharpened by three planets, still underived), and one
failed general derivation (broken at a named step and conceded) ---
while the rigorous far-from-equilibrium variational structure that
actually exists, in macroscopic fluctuation theory, is a corrected
\emph{minimum}-dissipation principle for a functional that is not
entropy production alone \citep{Bertini2015}.

A final distinction prevents most of the confusion the slogan invites.
That every real process produces entropy is \emph{bookkeeping} ---
computed from the flows the way a balance is computed from
transactions, automatically true. \MEPP{} claims something
categorically stronger: that among all flow patterns the constraints
allow, nature \emph{selects} the one of largest production. The ledger
is true by construction; the selection is a physical hypothesis, and
it is the hypothesis, not the ledger, that has no general proof. The
current gauge of Section~\ref{subsec:qiep-mepp} derives the selection
--- as the flux-side reading of one stationary point, valid exactly
where the curl and gauge sectors do no ledger work, with the
force-side reading of the same point recovering Prigogine's minimum
--- and the failure theorem of
Section~\ref{subsec:qiep-failure} marks where both readings
stop.

\begin{table}[htbp]
\centering
\small
\caption[MEPP: one theorem, one heuristic, one failed proof.]{The
status of maximum entropy production, separated into the three things
that circulate under its name. Each row gives the statement, its
strongest verified support, and its present standing in the technical
literature. The separation is the section's whole point: the theorem
is confined to linearized transport where it restates least
dissipation; the planetary heuristic is empirically striking and
underived, its predictions dependent on the modeler's choice of
constraints and functional; and the claimed general derivation failed
at two named steps and was conceded by its author. A reader who keeps
the three rows apart will not be misled by literature that cites the
first and third on behalf of the second.}
\label{tab:qiep-mepp-scope}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.32\textwidth}>{\raggedright\arraybackslash}p{0.35\textwidth}@{}}
\toprule
\textbf{Statement} & \textbf{Strongest support} & \textbf{Standing} \\
\midrule
Linear-response maximum (Kohler--Ziman; Ziegler, quadratic case) &
Three-line proof for the linearized Boltzmann equation
\citep{Ziman1956} & Theorem; equivalent to least dissipation; adds
nothing beyond the linear law \\
Paltridge's climate closure & Equator-to-pole contrast on Earth, Mars,
Titan with no tuned coefficient \citep{Lorenz2001} & Underived
heuristic; constraint- and resolution-dependent; ignores the dominant
radiative budget \\
General far-from-equilibrium principle (Dewar 2003--2005) &
Path-information construction through the fluctuation theorem &
Refuted at the monotonicity and Gaussian steps
\citep{GrinsteinLinsker2007,Bruers2007}; conceded
\citep{Dewar2009} \\
\bottomrule
\end{tabular}
\end{table}


\section{The free energy principle: history, applications, and limitations}
\label{sec:qiep-hist-fep}

\subsection{Three streams, then a fourth}

The free energy principle did not appear from nowhere in 2005; it is
the confluence of three older streams, with a fourth grafted on late
--- and because the derivational disputes below turn on \emph{which}
stream a given claim belongs to, the history matters.

The first stream is perceptual epistemology. Helmholtz's doctrine of
unconscious inference --- percepts are conclusions drawn from sensory
premises together with prior experience --- is the conceptual ancestor
claimed by the whole tradition \citep{HelmholtzOptik}; the connection
is programmatic, since Helmholtz supplies no objective functional. The
second is cybernetic regulation: Ashby framed adaptive behavior as the
maintenance of essential variables within bounds \citep{Ashby1952},
and the good-regulator theorem of Conant and Ashby --- every good
regulator of a system must be a model of that system --- is the
FEP's central claim restated in advance
\citep{ConantAshby1970}. The third is variational statistics. The
quantity later called \(F\) entered neural modeling through machine
learning: the Helmholtz machine trains a recognition density to bound
the log evidence of a generative model \citep{DayanHinton1995}, and
Neal and Hinton showed that expectation--maximization is coordinate
descent on a single free energy, so that inference and learning
optimize one objective \citep{NealHinton1998}. Predictive coding gave
the tradition its canonical cortical algorithm --- descending
predictions, ascending prediction errors \citep{RaoBallard1999} ---
and it is worth saying plainly that predictive coding neither requires
nor implies the FEP, a distinction that matters when empirical support
for the former is offered on behalf of the latter.

Friston assembled the streams into a principle between 2005 and 2010:
hierarchical cortical processing as empirical-Bayes inference
minimizing free energy \citep{Friston2005}, action added as a second
route to the minimum --- change the predictions to match the
sensations, or change the sensations to match the predictions
\citep{FristonKilnerHarrison2006} --- and the canonical statement in
2010 \citep{Friston2010}. Active inference followed
\citep{FristonAction2010,FristonProcess2017}, consolidated in textbook
form \citep{ParrPezzuloFriston2022}; this stratum now runs a
substantial fraction of computational neuroscience and computational
psychiatry.

The fourth stream is the \emph{physics phase}. From 2013 Friston
pivoted from brains to matter: a random dynamical system possessing a
Markov blanket at nonequilibrium steady state will ``appear to''
perform Bayesian inference about what lies beyond its blanket
\citep{Friston2013}, a program culminating in the 2019 monograph
\citep{Friston2019}. This is the stratum that claims a law of nature
--- and it is the stratum on which the critiques fell.

\subsection{The mathematics that is solid}

The core is textbook variational inference, uncontested by any critic.
With a generative density \(p(s,\psi)\) over sensory data \(s\) and
latent states \(\psi\), and a recognition density \(q_\mu(\psi)\)
parameterized by internal states \(\mu\), the variational free energy
\(F(s,\mu)=\mathbb{E}_{q_\mu}[\ln q_\mu(\psi)-\ln p(s,\psi)]\)
admits three decompositions, each carrying one reading of the
principle:
\begin{align}
F(s,\mu)
&= D_{\mathrm{KL}}\!\left[q_\mu(\psi)\,\middle\|\,p(\psi\mid s)\right]
   \;-\;\ln p(s)
\label{eq:qiep-fep-bound}\\
&= \underbrace{\mathbb{E}_{q_\mu}\!\left[-\ln p(s,\psi)\right]}_{\text{energy}}
   \;-\; \underbrace{H\!\left[q_\mu\right]}_{\text{entropy}}
\label{eq:qiep-fep-energy}\\
&= \underbrace{D_{\mathrm{KL}}\!\left[q_\mu(\psi)\,\middle\|\,p(\psi)\right]}_{\text{complexity}}
   \;-\; \underbrace{\mathbb{E}_{q_\mu}\!\left[\ln p(s\mid\psi)\right]}_{\text{accuracy}}.
\label{eq:qiep-fep-complexity}
\end{align}
Since the divergence in \eqref{eq:qiep-fep-bound} is nonnegative,
\(F\ge-\ln p(s)\), with equality exactly when \(q_\mu\) is the
posterior: minimizing \(F\) over \(\mu\) makes the internal states
approximate Bayesian inference (perception); minimizing \(F\) over
actions that change \(s\) is the additional move that defines active
inference; and \(-F\) is the evidence lower bound of machine
learning, descended through \citet{DayanHinton1995} and
\citet{NealHinton1998} from the variational bounds of statistical
mechanics. In hierarchical Gaussian models, gradient descent on \(F\)
\emph{is} precision-weighted prediction-error dynamics,
\begin{equation}
\dot\mu \;\propto\; -\,\frac{\partial F}{\partial \mu}
\;=\; (\partial_\mu g)^{\!\top}\,\Pi\,\varepsilon ,
\qquad \varepsilon \equiv s - g(\mu),
\label{eq:qiep-fep-pc}
\end{equation}
descending predictions \(g(\mu)\) against ascending errors
\(\varepsilon\) weighted by precisions \(\Pi\) --- the
predictive-coding process model --- with attention entering as the
optimization of the precisions themselves, since stationarity of \(F\)
in \(\Pi\) sets each inverse precision to the averaged outer product
of its errors. All of this is standard, empirically fruitful against
neural data, and stands independently of any claim about matter in
general.

\subsection{The contested stratum, and the critiques that located it}

The physics phase asserts a chain: a stationary system with a
sparse-coupling boundary (the Markov blanket) factorizes its
stationary density; the blanket induces a synchronization map from
internal to external averages; and the expected internal states then
flow down the gradient of a free energy --- so that persistence itself
entails inference. Each link has been examined, and the examination is
the sharpest technical literature in this Part.

Biehl, Pollock, and Kanai proved that the two blanket definitions in
circulation --- sparsity of the flow's Jacobian, and factorization of
the stationary density --- are logically independent; that the
rewriting of the expected internal flow requires an absence of
solenoidal coupling across the partition that is nowhere stated; and
that the monograph's Free Energy Lemma fails at face value, with
counterexamples \citep{Biehl2021}. The authorized reply conceded the
observations and supplied a sparse-coupling ansatz --- sufficient
conditions under which the argument goes through --- thereby
restricting the principle's antecedents rather than rescuing its
generality \citep{FristonDaCostaParr2021}. Aguilera, Millidge,
Tschantz, and Buckley then posed the decisive quantitative question:
on what fraction of parameter space do the conditions hold, in the
linear (Ornstein--Uhlenbeck) model class the FEP literature itself
uses? The answer: a narrow subspace, whose defining symmetry amounts
to the \emph{absence of perception--action asymmetry} --- the very
asymmetry living systems exhibit --- and even where the construction
completes, the inferential reading adds no predictive content beyond
the dynamics themselves \citep{Aguilera2022}. Complementary critiques
established that in applications the blanket is induced by the
modeler's parcellation rather than discovered in the system
\citep{Raja2021}; that the move from Pearl's epistemic blanket to a
physical boundary of agents is a category transition the mathematics
does not underwrite \citep{Bruineberg2022}; and that a principle
defended as unfalsifiable-by-status --- only its applications can fail
--- cannot function as an empirical law of biology
\citep{ColomboWright2021}.

The program's response has been a retreat executed with candor. The
rigorous results that exist are confined to linear-Gaussian settings
\citep{DaCosta2021}; the 2023 reformulation abandons the stationary
density factorization for a path-integral statement in which the
principle holds by construction for systems admitting a particular
partition \citep{FristonSimpler2023}; and the programmatic statement
of the resulting Bayesian mechanics presents the FEP as a principle of
least action \emph{on belief spaces} --- not itself falsifiable, but a
scheme for generating falsifiable process models
\citep{Ramstead2023}. The instrumentalist reading secures the
framework by surrendering the claim that made it famous. One cannot
simultaneously advertise a theory of every persisting thing and
retreat, under technical pressure, to the position that the formalism
is a lens the modeler may choose to wear.

\subsection{The honest summary, and what a derivation owes it}

Three assumptions beneath the Gaussian machinery are worth naming even
inside the solid stratum, because each has a characteristic biological
failure mode: a posterior summarized by a single mode (perceptual
bistability defeats it), an adequate second-order expansion (strong
nonlinearity in \(g\) defeats it), and quadratic error penalties
(heavy-tailed sensory noise strains them). These are scope conditions
of the \emph{algorithm}, cataloged within the tradition itself; the
inferential gauge below inherits none of them as postulates, because
its metric and its boundary arrive from the projection rather than
from the model class.

For a physicist, the assessment of the mid-2020s compresses to one
sentence: the bound is trivial, the decomposition is classical, and
the bridge from blankets to inference is either false in general, true
by construction, or true under fine-tuned conditions --- depending on
which paper one reads. What the tradition lacks is exactly what it
cannot supply from inside: a boundary that is \emph{derived} rather
than drawn by the modeler, a metric on beliefs that is \emph{forced}
rather than chosen, and a regime theorem that says where the
inferential reading holds. The inferential gauge of
Section~\ref{subsec:qiep-fep} supplies all three --- the blanket from
the declared ports of the packet, the Fisher metric from the ledger's
own informational term, and the rank and sector conditions as theorems
--- and Section~\ref{subsec:qiep-failure} states where the
reading fails, which no formulation of the FEP has ever done for
itself.

\begin{table}[htbp]
\centering
\small
\caption[FEP: four strata, four verdicts.]{The free energy principle
separated into its four strata, with the verdict the technical
literature has reached on each. The first two rows are uncontested and
empirically productive; every controversy lives in the third and
fourth. The table enforces the distinction that most discussion of the
principle blurs: a model that fits brains is judged as a model, by
prediction and failure against data, while a law about matter must
hold for everything that persists --- and the two kinds of claim,
frequently interleaved in the same paper, have opposite standings in
the record. The derivation of Section~\ref{subsec:qiep-fep} engages
the fourth stratum on its own terms and replaces its assumptions with
derived structure.}
\label{tab:qiep-fep-status}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.26\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}@{}}
\toprule
\textbf{Stratum} & \textbf{Content} & \textbf{Verdict} \\
\midrule
Variational bound & \(F\ge-\ln p(s)\); ELBO
\eqref{eq:qiep-fep-bound} & Textbook; uncontested
\citep{DayanHinton1995,NealHinton1998} \\
Process models & Predictive coding; active inference; precision as
attention & Legitimate, practical, empirically productive
\citep{RaoBallard1999,FristonProcess2017} \\
Physics phase & Blanket \(\Rightarrow\) factorization \(\Rightarrow\)
synchronization \(\Rightarrow\) gradient flow & Non-generic; silent
assumptions; counterexamples
\citep{Biehl2021,Aguilera2022} \\
Bayesian-mechanics retreat & Least action on belief spaces; holds by
construction & Instrumentalist; not falsifiable as stated
\citep{FristonSimpler2023,Ramstead2023,ColomboWright2021} \\
\bottomrule
\end{tabular}
\end{table}


\section{Constitutional statement of the \texorpdfstring{\QiEP}{Qi-EP} principle}
\label{subsec:qiep-constitutional-statement}

The conceptual claim of \QiEP\ should be stated before any detailed derivation:
\begin{definition}[\QiEP]
For any admissible open subsystem \(S\) embedded in the \GG\ bulk and observed through \OGGEP, there exists a projected extremal functional
\begin{equation}
\mathfrak{Q}_{\QiEP}^{(S)}
\;=\;
\Phi_{\OGGEP}^{(S)}[S_{\GG}]
\Big|_{\Gamma_{\mathrm{adm}}}
\label{eq:qiep-master-definition}
\end{equation}
such that the realized boundary dynamics of \(S\) satisfy
\begin{equation}
\delta \mathfrak{Q}_{\QiEP}^{(S)} = 0
\qquad
\text{on}
\qquad
\Gamma_{\mathrm{adm}}.
\label{eq:qiep-master-stationary}
\end{equation}
Under three natural boundary gauges this same stationary condition appears respectively as \PLT, \MEPP,\ and \FEP.
\end{definition}

Three comments are essential.

First, the word ``projected'' is not decorative. The boundary principle is not postulated independently. It is induced from the 6D \GG\ action through the open-system projection map already established in \AFive{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}). This means that the extremal selector on the boundary is constitutionally tethered to the same ultraviolet structure that fixed \BIsix{}, the OGN integers, and the current ratios.

Second, the phrase ``under three natural boundary gauges'' is also literal. The projected functional may be described in terms of trajectories, in terms of currents and forces, or in terms of inferential states on a statistical manifold. These are not three unrelated principles living side by side. They are three descriptions of the same projected geometry under three different coarse-grained coordinate choices.

Third, \QiEP\ is not restricted to quantum mechanics, nor to biology, nor to any single domain. The same projected functional can describe a geodesic-like transport path in one application, a non-equilibrium stationary flux in another, and an inferential or adaptive update in a third. This is exactly what a universal open-system selector must do if it is to serve as Pillar~II of a general theory.


\paragraph*{Dual-layer architecture and the compatibility relation $\mathfrak{C}_{\mathrm{EP}}$.}
\QiEP{} is, in the L16 language, a dual-layer principal pillar. Its
\emph{algebraic layer} (T-EP.1) is the unified gauge theorem
(Theorem~\ref{thm:qiep-unified}): the same projected boundary functional
$F_{\GFF} = F_{\GEP} + F_{\CEP}$ governs three apparently-distinct
variational principles simultaneously, with no contradiction. Its
\emph{operational/physical layer} (T-EP.2) is the three named gauges
\PLT{} (Theorem~\ref{thm:qiep-plt}, the trajectory gauge of
Maupertuis--Hamilton), \MEPP{} (Theorem~\ref{thm:qiep-mepp}, the
current gauge of Onsager--Prigogine--Paltridge), and \FEP{}
(Theorem~\ref{thm:qiep-fep}, the inferential gauge of Friston), each
providing a different falsifiable laboratory prediction. The two
layers are tied together by the named compatibility relation
\begin{equation}
\label{eq:CEP-compat}
\boxed{\;\mathfrak{C}_{\mathrm{EP}}\;\equiv\;
\text{T-EP.1}\;\Longrightarrow\;\text{T-EP.2}
\quad\text{(via the boundary projection of the 6D bulk action functional).}\;}
\end{equation}
The unified theorem at the algebraic level forces the three gauges to
share one projected functional; each gauge then provides a different
falsifier on a different physical regime (trajectories at high
visibility, currents at near-equilibrium, inference at strong noise).


\begin{figure}[htbp]
\centering
\resizebox{0.781\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=3pt, inner sep=4pt},
 core/.style={fill=orange!14, draw=orange!65!black, line width=1.10pt,
 minimum width=78mm, minimum height=16mm},
 gauge/.style={fill=blue!10, draw=blue!60!black, line width=0.70pt,
 minimum width=44mm, minimum height=17mm},
 field/.style={fill=green!10, draw=green!55!black, line width=0.55pt,
 minimum width=44mm, minimum height=10mm},
 held/.style={font=\sffamily\scriptsize\itshape, text=black!65,
 fill=white, inner sep=1.5pt},
 arr/.style={-{Stealth[length=2.4mm]}, line width=0.7pt, draw=black!70},
]
\node[core] (F) at (0,0)
 {\textbf{One projected functional}\\[1pt]
  \footnotesize the boundary shadow of the 6D least-action principle};

\node[gauge] (T) at (-52mm,-30mm)
 {\textbf{Trajectory gauge}\\[1pt] \footnotesize returns a least-time law\\
  \footnotesize along a curved path};
\node[gauge] (C) at (0,-30mm)
 {\textbf{Current gauge}\\[1pt] \footnotesize returns the flux--force law\\
  \footnotesize of irreversible transport};
\node[gauge] (I) at (52mm,-30mm)
 {\textbf{Inferential gauge}\\[1pt] \footnotesize returns the natural-gradient\\
  \footnotesize update on beliefs};

\draw[arr] (F.south west) -- (T.north) node[held, pos=0.55] {hold the \textbf{path}};
\draw[arr] (F.south) -- (C.north) node[held, pos=0.55] {hold the \textbf{currents}};
\draw[arr] (F.south east) -- (I.north) node[held, pos=0.55] {hold the \textbf{beliefs}};

\node[field] (T2) at (-52mm,-46mm) {optics and mechanics};
\node[field] (C2) at (0,-46mm) {nonequilibrium thermodynamics};
\node[field] (I2) at (52mm,-46mm) {theoretical neuroscience};
\draw[arr] (T) -- (T2);
\draw[arr] (C) -- (C2);
\draw[arr] (I) -- (I2);

% a note about the current gauge, set clear of every box and every rule:
% an earlier placement ran the connector straight through its own text.
\node[font=\sffamily\scriptsize\itshape, text=black!60, align=center,
 inner sep=2pt, anchor=north] at (0,-53mm)
 {\emph{in the current gauge:} maximum entropy production and
  Prigogine's minimum\\
  entropy production are conjugate readings of one stationary point};
\end{tikzpicture}%
}
\caption[One functional read in three gauges.]{How three principles that
look like rivals turn out to be one. The single projected functional at the
top is the boundary image of the six-dimensional least-action principle; it
is not a new postulate. Each arrow is the same stationarity condition asked
with a different quantity held primary, and the label on the arrow names
which: hold the path and the condition returns a least-time law, hold the
currents and it returns the flux--force relation of irreversible transport,
hold the beliefs and it returns the natural-gradient update used in the brain
sciences. The bottom row names the community that has been using each form.
The dashed link records what the unification settles on the way through:
maximum entropy production and Prigogine's minimum entropy production are
conjugate readings of one stationary point under dual control conventions ---
the first fixing the transport law, the second the state --- not competing
claims about nature. One system may occupy all three gauges at once --- a cell following
a gradient traces a path, maintains fluxes, and corrects an estimate.}
\label{fig:a7-qiep-gauges}
\end{figure}

\subsection{The contract: five criteria a genuine derivation must meet}
\label{ssec:qiep-contract}

The histories of
Sections~\ref{sec:qiep-hist-pla}--\ref{sec:qiep-hist-fep} fix the
specification, and it is stated here as a contract so that the reader
can hold the rest of the Part to it. A derivation worth the name must:
(i)~produce the three principles from \emph{one} object that is itself
derived --- the projection of an upstream action, not a fresh
postulate; (ii)~deliver the restrictions as \emph{theorems} --- the
linear-regime confinement of MinEP, the parity and inhomogeneity
failures, the coupling and rank conditions of the inferential reading
--- rather than discovering them by failure; (iii)~dissolve the
minimum--maximum quarrel, in agreement with the resolution the
critical literature reached from below; (iv)~replace every
modeler-chosen structure with derived structure --- the choice of
\emph{which} entropy production by the ledger fixed upstream, the
blanket by the declared ports of the packet, the synchronization map
by a rank condition; and (v)~state the conditional and open elements
explicitly, and nowhere waive them. The sections that follow discharge
the contract clause by clause:
(i) in Section~\ref{subsec:qiep-bulk-to-boundary},
(ii) in Sections~\ref{subsec:qiep-plt}--\ref{subsec:qiep-fep} and
\ref{subsec:qiep-failure},
(iii) in Section~\ref{subsec:qiep-mepp},
(iv) in Sections~\ref{subsec:qiep-bulk-to-boundary} and
\ref{subsec:qiep-fep}, and
(v) in Section~\ref{subsec:qiep-unified-theorem}. A framework that
``derived'' \MEPP{} as universally true would be wrong --- the
counterexamples are theorems too. What a correct derivation must
deliver is the functional whose restricted readings the three
principles are, together with the location of the restrictions.


\section{From the 6D bulk action to the projected boundary action}
\label{subsec:qiep-bulk-to-boundary}

\subsection{The imports, with their grades}
\label{ssec:qiep-imports}

Three objects are taken from upstream, and nothing else is assumed; the
derivation's honesty is easiest to audit if they are listed before they
are used. First, the \emph{boundary law}: the \GDOS{} continuity
equation with its four-sector current --- gradient, curl, noise, and
gauge, Eq.~\eqref{eq:four-current-decomp} --- derived in \AFive{} from
the projection of the bulk through the open-system channel
\citep{ArisakaA5GDOS}, and imported here at grade Derived. Second, the
\emph{ledger}: the single scalar functional
\(F_{\GFF}=F_{\GEP}+F_{\CEP}\) with its balance identity, established
in \ASix{} \citep{ArisakaA6Gi4} and imported at grade Derived; every
appearance of ``the entropy production'' below is an appearance of this
ledger's dissipation term, which is what removes the choice of
functional that the \MEPP{} literature never managed to fix
(Section~\ref{sec:qiep-hist-mepp}). Third, the \emph{packet}: the
\SIP{} declaration of Part~\ref{part:runtime}, whose ports are
constitutive data --- declared when the system is written down, not
discovered inside it --- and which will do for the inferential gauge
what walls do for thermodynamics. The bulk action itself and the
\OGGEP{} projection that carries it to the boundary enter from
\ATwo{} and \AOne{} at the grades their home papers assign. Everything
else in this Part is constructed on the page.

\subsection{The mathematical tools, briefly}
\label{ssec:qiep-onramp}

A reader arriving from the life sciences needs four instruments, each
stated here in one breath and used repeatedly below. \emph{Path
weights:} for noisy dynamics the object that replaces the trajectory is
a probability over trajectories, and in the weak-noise regime that
probability concentrates as
\(P[z]\asymp e^{-\mathcal{A}[z]/\varepsilon}\) --- so extremizing an
action and finding the most probable history become the same act
\citep{OnsagerMachlup1953,Touchette2009}. \emph{Large deviations:} the
same concentration statement for densities and currents, which is what
lets one functional govern a single swimming cell and a whole
population's flux pattern. \emph{Legendre duality:} a convex
stationary point can be described with either of two conjugate
variables held fixed, and the two descriptions extremize different
functions of the same point --- the entire minimum--maximum quarrel of
Section~\ref{sec:qiep-hist-mepp} lives inside this one fact.
\emph{Fisher geometry:} a family of probability densities is a curved
manifold whose natural metric is the Fisher information, and gradient
descent in that metric --- natural-gradient descent --- is the
coordinate-free way to go downhill on beliefs \citep{Amari1998}. None
of these is new mathematics; what is new in this Part is where the
functional, the convex form, and the metric come \emph{from}.

\subsection{The projection, step by step}
\label{ssec:qiep-projection-steps}

We begin from the ultraviolet starting point already established in \ATwo{}: the six-dimensional bulk with total action
\begin{equation}
S_{\GG}[\Psi]
=
\int_{\cM_6}
\dd^6x\,
\sqrt{-g_6}\,
\mathcal{L}_6\big(\Psi,\nabla\Psi\big),
\label{eq:qiep-s6}
\end{equation}
where \(\Psi\) collectively denotes the 6D metric, gauge, Green--Schwarz, and matter fields. By construction the bulk dynamics satisfy
\begin{equation}
\delta S_{\GG}=0,
\label{eq:qiep-bulk-pla}
\end{equation}
together with the \BIsix{} consistency constraints. Equation~\eqref{eq:qiep-bulk-pla} is the constitutional \PLA\ of the \GG\ stage \citep{ArisakaA2GG6,Hamilton1834,Feynman1948}.

To derive the boundary selector, split the full field content into observer-accessible slow variables \(z\) and hidden or traced-over variables \(y\):
\begin{equation}
\Psi = (z,y).
\label{eq:qiep-split-fields}
\end{equation}
The symbol \(z\) should be interpreted broadly. Depending on the later application it may denote a trajectory coordinate, a flux coordinate, a reaction-network coordinate, a neural or cognitive state coordinate, or a collective macroscopic descriptor. The hidden variables \(y\) include KK modes, ultraviolet environmental modes, unresolved microstates, and any degrees of freedom eliminated by the boundary observer.

Concretely, the standard way to write ``integrate out hidden degrees of freedom while keeping a variational structure'' is the Feynman--Vernon influence functional, which we now recall as Eq.~\eqref{eq:qiep-influence-functional}. Readers from biology or chemistry may think of this as the path-integral parallel of the partial-trace operation of Eq.~\eqref{eq:intro-reduced-state}: both objects discard the environment and keep only the structure relevant to the observer, but the path-integral form is more convenient when one wants to expose the variational principle hidden inside the dynamics.

The standard influence-functional representation of the reduced dynamics is then
\begin{equation}
\exp\!\left(
\frac{\ii}{\hbar}S_{\mathrm{IF}}[z,z']
\right)
=
\int \cD y\,\cD y'\,
\exp\!\left[
\frac{\ii}{\hbar}
\Big(
S_{\GG}[z,y]-S_{\GG}[z',y']
\Big)
\right]
\rho_y(0),
\label{eq:qiep-influence-functional}
\end{equation}
where, in plain words, \(S_{\rm IF}\) is the effective action for the system variables \(z\) obtained by integrating over the environment variables \(y\); the two-sided integration over \(y\) and \(y'\) reflects the two-index structure of the density matrix (the closed-time-path doubling familiar from Schwinger--Keldysh formalism), and \(\rho_y(0)\) is the initial state of the environment. The equation is the path-integral expression of the same reduced evolution usually described by \CPTP\ maps, Nakajima--Zwanzig kernels, or GKLS generators \citep{FeynmanVernon1963,BreuerPetruccione2002,Lindblad1976,Gorini1976}. In the present paper, however, the decisive point is not merely that reduction exists; it is that the reduction inherits a variational structure from the bulk.

In the semiclassical, weak-noise, or large-deviation regime one may write
\begin{equation}
P[z]
\asymp
\exp\!\left[
-\frac{1}{\varepsilon_{\mathrm{eff}}}
\mathcal{A}_{\mathrm{eff}}[z]
\right],
\label{eq:qiep-large-deviation}
\end{equation}
where \(\varepsilon_{\mathrm{eff}}\) is the effective coarse-graining scale and \(\mathcal{A}_{\mathrm{eff}}\) is the reduced action functional \citep{Touchette2009,OnsagerMachlup1953,vanKampen2007}. More generally, when both densities and currents are retained explicitly, the most useful projected object is the current-density functional
\begin{equation}
\mathcal{A}_{\mathrm{eff}}[\rho,J]
=
\int_0^T \dd t\,
\left[
\frac14
\int_{\State}
\big(J-\rho u\big)^a
\big(\rho D\big)^{-1}_{ab}
\big(J-\rho u\big)^b
\,\dd z
+
\mathcal{V}[\rho]
\right],
\label{eq:qiep-current-density-action}
\end{equation}
subject to the continuity equation
\begin{equation}
\partial_t\rho + \nabla_a J^a = 0.
\label{eq:qiep-continuity}
\end{equation}
Here \(u^a(z,t)\) is the projected drift field, \(D^{ab}(z,t)\) is the symmetric positive diffusion/metric tensor inherited from the projected bulk geometry, and \(\mathcal{V}[\rho]\) is the projected potential or constraint functional. Equation~\eqref{eq:qiep-current-density-action} is the single most important formula of the present section. It is the master boundary action from which the three gauges will be obtained.

\begin{proposition}[Projected stationary-action proposition]
\label{prop:qiep-projected-stationary}
Within the \GG/\GDOS\ framework, the open-system boundary histories selected by \OGGEP\ are stationary histories of the projected action \(\mathcal{A}_{\mathrm{eff}}[\rho,J]\) on the admissible class \(\Gamma_{\mathrm{adm}}\).
\end{proposition}

\begin{proof}
Fix \((\rho,J)\) and consider the hidden-variable extremal \(y=\bar y[z]\) obtained by solving the hidden Euler--Lagrange equations at fixed boundary history. At tree level or in the leading large-deviation approximation one has
\begin{equation}
\mathcal{A}_{\mathrm{eff}}[z]
=
\Re\,S_{\GG}[z,\bar y[z]]
+
\text{(coarse-graining counterterms)}.
\label{eq:qiep-tree-level-effective}
\end{equation}
Varying with respect to \(z\) gives
\begin{equation}
\frac{\delta \mathcal{A}_{\mathrm{eff}}}{\delta z}
=
\left.
\frac{\delta S_{\GG}}{\delta z}
\right|_{y=\bar y[z]}
+
\left.
\frac{\delta S_{\GG}}{\delta y}
\right|_{y=\bar y[z]}
\frac{\delta \bar y}{\delta z}
+
\delta(\text{counterterms}).
\label{eq:qiep-chain-rule-proof}
\end{equation}
The second term vanishes because \(\bar y[z]\) is by construction a hidden extremal. The remaining terms combine into the reduced stationarity condition. The current-density form \eqref{eq:qiep-current-density-action} is the corresponding open-system representation in which the hidden modes appear only through the projected drift, diffusion, and potential data. Therefore the realized boundary history is a stationary history of \(\mathcal{A}_{\mathrm{eff}}\).
\end{proof}

The proposition is conceptually decisive. It says that \QiEP\ is not a separate empirical principle pasted onto the boundary. It is the projected variational shadow of the same 6D least-action law that governed the ultraviolet bulk. The remaining task is to show how the familiar boundary forms \PLT, \MEPP,\ and \FEP\ arise from three different reductions of Eq.~\eqref{eq:qiep-current-density-action}.

\section{Trajectory gauge: derivation of \texorpdfstring{\PLT}{PLT}}
\label{subsec:qiep-plt}

The first gauge is the trajectory gauge. In this description the boundary packet remains sharply localized around a representative path \(z(t)\), so that
\begin{equation}
\rho(\zeta,t) \approx \delta\!\big(\zeta-z(t)\big),
\qquad
J(\zeta,t) \approx \delta\!\big(\zeta-z(t)\big)\,\dot z(t).
\label{eq:qiep-delta-ansatz}
\end{equation}
Substituting Eq.~\eqref{eq:qiep-delta-ansatz} into Eq.~\eqref{eq:qiep-current-density-action} gives the reduced trajectory action
\begin{equation}
\mathcal{A}_{\mathrm{trj}}[z]
=
\int_0^T \dd t\,
\left[
\frac14
\big(\dot z-u(z,t)\big)^a
D^{-1}_{ab}(z,t)
\big(\dot z-u(z,t)\big)^b
+
V(z,t)
\right],
\label{eq:qiep-trj-action}
\end{equation}
where \(V(z,t)\) is the local representative of \(\mathcal{V}[\rho]\) on the sharply localized slice.

It is convenient to introduce the effective metric
\begin{equation}
g_{ab}(z,t) := \frac12\,D^{-1}_{ab}(z,t)
\label{eq:qiep-effective-metric}
\end{equation}
and the associated one-form
\begin{equation}
a_a(z,t) := g_{ab}(z,t)\,u^b(z,t).
\label{eq:qiep-effective-one-form}
\end{equation}
Then Eq.~\eqref{eq:qiep-trj-action} becomes
\begin{equation}
\mathcal{A}_{\mathrm{trj}}[z]
=
\int_0^T \dd t\,
\left[
\frac12 g_{ab}\dot z^a \dot z^b
-
a_a\dot z^a
+
\frac12 g_{ab}u^a u^b
+
V
\right].
\label{eq:qiep-trj-expanded}
\end{equation}
Up to the effective scalar potential
\begin{equation}
U_{\mathrm{eff}}(z,t) := \frac12 g_{ab}u^a u^b + V(z,t),
\label{eq:qiep-ueff}
\end{equation}
this is the standard quadratic mechanical form on a curved configuration manifold.

The Euler--Lagrange equations are
\begin{equation}
\frac{\dd}{\dd t}
\left(
g_{ab}\dot z^b - a_a
\right)
-
\frac12
\partial_a g_{bc}\,\dot z^b \dot z^c
+
\partial_a a_b\,\dot z^b
-
\partial_a U_{\mathrm{eff}}
=
0.
\label{eq:qiep-euler-lagrange-trajectory}
\end{equation}
Raising the free index and rewriting with the Levi--Civita connection of \(g_{ab}\) gives
\begin{equation}
\frac{D_g \dot z^a}{\dd t}
=
g^{ab}
\left(
F_{bc}\dot z^c
-
\partial_b U_{\mathrm{eff}}
\right),
\qquad
F_{ab}:=\partial_a a_b-\partial_b a_a.
\label{eq:qiep-geodesic-force}
\end{equation}
Equation~\eqref{eq:qiep-geodesic-force} has the expected interpretation: the projected path follows a geodesic-like motion in the effective boundary metric, corrected by a Lorentz-like curvature term and a projected scalar force.

To make contact with the language of \PLT, fix an energy shell \(E\) and perform the Jacobi reduction. One obtains the generalized optical-time functional
\begin{equation}
\mathcal{T}[z]
=
\int_{\gamma}
\left[
\sqrt{2\big(E-U_{\mathrm{eff}}(z)\big)\,g_{ab}(z)\,\dd z^a \dd z^b}
-
a_a(z)\,\dd z^a
\right].
\label{eq:qiep-optical-time}
\end{equation}
The stationary condition \(\delta \mathcal{T}=0\) is the boundary \PLT\ form of the same projected action. The phrase ``least time'' should therefore be interpreted in the generalized sense appropriate to open systems: not a bare Euclidean travel time, but the optical or information-geometric time induced by the projected metric, the drift connection, and the effective potential.

\begin{theorem}[\PLT\ as the trajectory gauge of \QiEP (T-Qi4-10.3)]
\label{thm:qiep-plt}
On the sharply localized trajectory slice \(\mathcal{S}_{\mathrm{trj}}\subset\Gamma_{\mathrm{adm}}\), the projected current-density functional \eqref{eq:qiep-current-density-action} reduces to the generalized optical-time functional \eqref{eq:qiep-optical-time}. Its stationary histories satisfy Eq.~\eqref{eq:qiep-geodesic-force}. Hence \PLT\ is the trajectory gauge of \QiEP.
\end{theorem}

\begin{proof}
The reduction from Eq.~\eqref{eq:qiep-current-density-action} to Eq.~\eqref{eq:qiep-trj-action} follows directly by substitution of Eq.~\eqref{eq:qiep-delta-ansatz}. Completing the square and defining Eqs.~\eqref{eq:qiep-effective-metric}--\eqref{eq:qiep-ueff} gives Eq.~\eqref{eq:qiep-trj-expanded}. The Euler--Lagrange variation yields Eq.~\eqref{eq:qiep-euler-lagrange-trajectory}, which is equivalent to Eq.~\eqref{eq:qiep-geodesic-force}. Restricting to a fixed energy shell and performing the standard Jacobi reduction turns the quadratic action into the time functional \eqref{eq:qiep-optical-time}. Therefore the stationary condition of the projected action is precisely the \PLT\ condition in this gauge.
\end{proof}

The physical interpretation is straightforward. The 6D bulk obeys \PLA. Once the inaccessible directions and hidden environmental couplings are projected away, the observer sees a boundary packet moving in an effective metric with an effective connection and an effective scalar landscape. The realized path is then the one of stationary generalized travel time. In transport problems, search problems, signaling problems, and morphogenetic path-selection problems, this is the most natural boundary language.

\begin{table}[H]
\centering
\caption[The least-time gauge, step by step.]{Step-by-step structure of the least-time derivation. Six steps run down the first column, from the 6D constitutional law to the operational least-time form, with the equation and its meaning alongside. This is the first of three parallel chains --- the other two are Tables~\ref{tab:qiep-mepp-chain} and~\ref{tab:qiep-fep-chain} --- and the three are laid out identically on purpose, so that a reader can see that they differ only in which variable is held as primary.}
\label{tab:qiep-plt-chain}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}p{0.41\textwidth}}
\toprule
\textbf{Step} & \textbf{Equation} & \textbf{Meaning} \\
\midrule
6D constitutional law & \eqref{eq:qiep-bulk-pla} & The bulk obeys \(\delta S_{\GG}=0\). \\
Projected boundary action & \eqref{eq:qiep-current-density-action} & Hidden modes induce drift, diffusion, and a projected potential. \\
Trajectory slice & \eqref{eq:qiep-delta-ansatz} & The packet is sharply localized around one representative path. \\
Effective metric and connection & \eqref{eq:qiep-effective-metric}, \eqref{eq:qiep-effective-one-form} & The projected quadratic form becomes curved kinematics with a gauge-like one-form. \\
Boundary equation of motion & \eqref{eq:qiep-geodesic-force} & The realized trajectory is geodesic-like with projected curvature and scalar forces. \\
Operational least-time form & \eqref{eq:qiep-optical-time} & The boundary selector is a generalized \PLT. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{The regime, and a worked parallel}
\label{ssec:qiep-plt-regime}

The trajectory gauge holds where its slice is honest: the packet must
stay sharply localized, which is the small-noise, high-visibility
regime --- diffusion weak against drift on the scale of the path's
curvature. That is a statement, not an apology; it is the same regime
condition under which Fermat's rays emerge from wave optics and the
classical path emerges from Feynman's sum, now derived for an open
system rather than assumed by analogy. Where the packet spreads, the
trajectory language simply stops referring to anything, and selection
is carried by the current gauge on the same functional. \emph{What
would refute the reading in its regime:} a selected path in a medium
of independently measured mobility that departs systematically from
the geodesic of the metric \eqref{eq:qiep-effective-metric}, beyond
the small-noise corrections the functional itself predicts.

The oldest and the newest instances of the gauge are worth setting
side by side, because they are the same calculation.

\begin{table}[H]
\centering
\caption[One least-time law, three centuries apart.]{The trajectory
gauge run twice: Fermat's ray of 1662 and a chemotactic run, column by
column. Each row is one element of the derivation, and the two columns
differ only in where the effective index comes from --- prescribed by
the glass in one case, computed from the projected mobility and drift
in the other. The point of the table is that the right-hand column is
not a metaphor: the bacterium's path extremizes the same generalized
time functional \eqref{eq:qiep-optical-time}, in a medium whose
geometry the projection derives rather than the modeler declares, and
the regime condition in the last row is the honest price of the
correspondence in both centuries.}
\label{tab:qiep-lens-bacterium}
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.34\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}@{}}
\toprule
\textbf{Element} & \textbf{Light in graded glass (1662)} & \textbf{Chemotactic run (this Part)} \\
\midrule
Medium & Refractive index \(n(z)\) & Projected metric
\(g_{ab}=\tfrac12 D^{-1}_{ab}\) and drift one-form \(a_a\) \\
Where it comes from & Composition of the glass, measured &
The projection: mobility and lawful drift of the packet, derived \\
Selected path & Snell's law; ray bending toward high index & Geodesic
of \eqref{eq:qiep-geodesic-force}, bent by curvature and drift toward
high effective index \\
Extremized quantity & Travel time \(\int n\,\dd s\) & Generalized
optical time \eqref{eq:qiep-optical-time} \\
Regime & Wavelength small against the index gradient & Noise small
against drift (sharp packet); visibility high \\
What breaks it & Diffraction & Packet spreading; selection passes to
the current gauge \\
\bottomrule
\end{tabular}
\end{table}


\section{Current gauge: derivation of \texorpdfstring{\MEPP}{MEPP}}
\label{subsec:qiep-mepp}

The second gauge is the current gauge. Here the density \(\rho\) remains spatially extended and the fundamental question is no longer ``which path does one packet follow?'' but ``which flux pattern is selected under the projected driving and constraints?'' This is the natural language for transport networks, reaction networks, metabolic fluxes, ecological throughputs, and other non-equilibrium open systems.

Define the mobility or Onsager tensor
\begin{equation}
L^{ab}(z,t) := \rho(z,t)\,D^{ab}(z,t),
\label{eq:qiep-onsager-tensor}
\end{equation}
which is symmetric and positive semidefinite because \(D^{ab}\) is symmetric positive and \(\rho\ge 0\). Next introduce the generalized force
\begin{equation}
X_a(z,t) := \big(D^{-1}\big)_{ab}u^b(z,t) - \partial_a \varphi(z,t),
\label{eq:qiep-force}
\end{equation}
where \(\varphi\) is the scalar contribution extracted from the projected potential and any conservation constraints. The local entropy-production rate density is then written in the familiar bilinear form
\begin{equation}
\dot\sigma(z,t) := J^a X_a.
\label{eq:qiep-entropy-production-density}
\end{equation}
The total entropy-production rate is
\begin{equation}
\dot\Sigma[J]
=
\int_{\State} J^a X_a\,\dd z.
\label{eq:qiep-total-entropy-production}
\end{equation}

The same quadratic term that appeared in Eq.~\eqref{eq:qiep-current-density-action} may now be rewritten as an Onsager--Rayleighian,
\begin{equation}
\mathcal{R}_\rho[J;X]
=
\int_{\State}
\left(
J^a X_a
-\frac12 J^a (L^{-1})_{ab}J^b
\right)\dd z,
\label{eq:qiep-rayleighian}
\end{equation}
again subject to the continuity equation and the integrated \QiQUA\ ledger constraints. The advantage of Eq.~\eqref{eq:qiep-rayleighian} is that its meaning is immediate: the first term rewards entropy-producing throughput under the imposed force field, whereas the second term penalizes dissipative expenditure measured by the inverse mobility tensor.

Varying Eq.~\eqref{eq:qiep-rayleighian} with respect to \(J\) gives
\begin{equation}
0
=
\frac{\delta \mathcal{R}_\rho}{\delta J^a}
=
X_a - (L^{-1})_{ab}J^b,
\label{eq:qiep-rayleigh-variation}
\end{equation}
hence
\begin{equation}
J^a = L^{ab}X_b.
\label{eq:qiep-onsager-law}
\end{equation}
Equation~\eqref{eq:qiep-onsager-law} is the force-flux law selected by the projected geometry. Substituting it back into Eqs.~\eqref{eq:qiep-total-entropy-production} and \eqref{eq:qiep-rayleighian} yields
\begin{equation}
\dot\Sigma_\star
=
\int_{\State} X_a L^{ab}X_b\,\dd z
\ge 0,
\label{eq:qiep-entropy-production-opt}
\end{equation}
and
\begin{equation}
\mathcal{R}_{\rho,\star}
=
\frac12 \dot\Sigma_\star.
\label{eq:qiep-rayleigh-half-sigma}
\end{equation}
The second variation is
\begin{equation}
\delta^2 \mathcal{R}_\rho
=
-
\int_{\State}
\delta J^a (L^{-1})_{ab}\delta J^b\,\dd z
\le 0,
\label{eq:qiep-second-variation-rayleigh}
\end{equation}
so the stationary point is a maximum of \(\mathcal{R}_\rho\) on the admissible force-controlled slice. Because Eq.~\eqref{eq:qiep-rayleigh-half-sigma} identifies the optimum Rayleighian with one-half of the selected entropy-production rate, the same stationary point is also the maximum-entropy-production point on that slice.

This is the precise sense in which \MEPP\ appears in \QiEP. One must keep the control convention explicit. On the force-controlled slice, Eq.~\eqref{eq:qiep-rayleighian} is maximized. On the dual flux-controlled slice, the Legendre transform
\begin{equation}
\mathcal{D}_\rho[J;X]
=
\frac12 \int_{\State} J^a (L^{-1})_{ab}J^b\,\dd z
-
\int_{\State} J^a X_a\,\dd z
\label{eq:qiep-dual-dissipation}
\end{equation}
is minimized. Thus the familiar ``maximum entropy production'' and ``minimum dissipation'' formulations are not contradictory in the present framework. They are dual descriptions of the same stationary point under dual control conventions \citep{Onsager1931,Prigogine1967,NicolisPrigogine1977,Dewar2003,Seifert2012}.

\begin{theorem}[\MEPP\ as the current gauge of \QiEP (T-Qi4-10.4)]
\label{thm:qiep-mepp}
On the force-controlled current slice \(\mathcal{S}_{\mathrm{flx}}\subset\Gamma_{\mathrm{adm}}\), the projected action \eqref{eq:qiep-current-density-action} is equivalent to the Onsager--Rayleighian \eqref{eq:qiep-rayleighian}. Its stationary point satisfies the flux-force law \eqref{eq:qiep-onsager-law} and maximizes the entropy-production rate \eqref{eq:qiep-total-entropy-production} on the admissible slice. The Legendre-dual flux-controlled description is Eq.~\eqref{eq:qiep-dual-dissipation}, which is minimized at the same stationary point.
\end{theorem}

\begin{proof}
Rewrite the quadratic term of Eq.~\eqref{eq:qiep-current-density-action} using \(L^{ab}=\rho D^{ab}\). Up to terms independent of \(J\), this gives Eq.~\eqref{eq:qiep-rayleighian}. The first variation yields Eq.~\eqref{eq:qiep-onsager-law}. The second variation \eqref{eq:qiep-second-variation-rayleigh} is negative semidefinite because \(L^{-1}\) is positive semidefinite on the support of \(\rho\). Therefore the stationary point is a maximum of \(\mathcal{R}_\rho\). At that point the identities \eqref{eq:qiep-entropy-production-opt} and \eqref{eq:qiep-rayleigh-half-sigma} hold by substitution. The Legendre-dual representation follows by changing the control variables from \(X\) to \(J\), yielding Eq.~\eqref{eq:qiep-dual-dissipation}.
\end{proof}

The duality just used deserves to be stated more sharply, because the second reading is not merely ``minimum dissipation'' in a loose sense: it is, exactly, the minimum-entropy-production theorem of \citet{Prigogine1967}. The two readings differ in what is held fixed, and each answers a different question.

On the \emph{flux side} one clamps the forces and asks which constitutive relation nature adopts. Among fluxes compatible with power balance, the physical flux is the one of \emph{greatest} entropy production, and it satisfies Ziegler's orthogonality condition with a multiplier that the constraint fixes at \(\lambda=\tfrac12\) --- not chosen, computed. This reading \emph{selects the transport law}.

On the \emph{force side} one fixes the constitutive relation, clamps some forces and frees the rest, and asks which state the system settles into. The stationary point of the force-side entropy production over the free forces is exactly the state whose free-force fluxes vanish --- the steady state --- and the restricted Hessian is positive definite, so it is a \emph{minimum}. This reading \emph{selects the state}, and it is Prigogine's theorem.

\begin{theorem}[\MEPP{} and MinEP as conjugate readings of one stationary point (T-Qi4-10.7)]
\label{thm:qiep-duality}
Let \(\dot\Sigma(J)=J^a(L^{-1})_{ab}J^b\) be the flux-side entropy production and \(P(X)=X_aL^{ab}X_b\) the force-side one, with \(L=L^{\mathsf T}\succ 0\) the mobility of Eq.~\eqref{eq:qiep-onsager-tensor}.
\emph{(i) Flux side, forces clamped.} Among fluxes satisfying the power-balance constraint \(\dot\Sigma(J)=J^aX_a\), the physical flux \(J=LX\) of Eq.~\eqref{eq:qiep-onsager-law} maximizes \(\dot\Sigma\), and the maximizer satisfies \(X_a=\lambda\,\partial\dot\Sigma/\partial J^a\) with \(\lambda=\tfrac12\) fixed by the constraint.
\emph{(ii) Force side, constitutive law fixed.} Split the forces into clamped \(X_1,\dots,X_m\) and free \(X_{m+1},\dots,X_n\). On the affine subspace of clamped forces, \(\partial P/\partial X_j=2L_{jk}X_k=2J_j\) for free \(j\), so the stationary point of \(P\) is the state with vanishing free-force fluxes, and the Hessian \(2L_{jj'}\) on the free block is positive definite: the stationary point is a minimum.
\emph{(iii)} The two are Legendre-dual descriptions of the single stationary point of \(\mathcal{R}_\rho\), and \(\dot\Sigma(J_\star)=P(X_\star)\) there.
\end{theorem}

\begin{proof}
Given in full in Appendix~\ref{app:qiep-duality}.
\end{proof}

\begin{remark}[What the duality settles]
\label{rem:qiep-quarrel}
The two entropy-production principles have been presented as rivals for seventy years. Theorem~\ref{thm:qiep-duality} says they are not rival claims about nature at all: they are conjugate conventions about what is held fixed at one stationary point, one selecting the transport law and the other the state. The dissolution is not a diplomatic compromise between two schools; it is what the projected functional says when one asks it the two different questions, and it agrees with the resolution that the critical literature reached from below by direct circuit analysis \citep{Polettini2013,BruersMaesNetocny2007,Jaynes1980}. Two routes to the same reading, one from the projection and one from below, is a check rather than a coincidence.
\end{remark}


\subsection{The third reading, and Prigogine's theorem recovered}
\label{ssec:qiep-third-reading}

Two identifications close the loop between this section and the
history of Sections~\ref{sec:qiep-hist-minep}--\ref{sec:qiep-hist-mepp},
and both are identities rather than analogies.

\emph{Onsager's least dissipation is the Rayleighian itself.} The
functional \eqref{eq:qiep-rayleighian} is, term for term, the 1931
functional \(\sigma(J,X)-\Phi(J,J)\) of
Section~\ref{sec:qiep-hist-minep}: the reward term is the entropy
production and the penalty is the quadratic dissipation function with
\(R=L^{-1}\). Its maximization at fixed forces \emph{is} Onsager's
principle of the least dissipation of energy, and its stationarity
condition \emph{is} the constitutive law \eqref{eq:qiep-onsager-law}.
What the projection adds to 1931 is provenance: \(L=\rho D\) is not a
phenomenological matrix but the projected mobility, symmetric and
positive because the bulk geometry is, so reciprocity arrives here as
an inheritance rather than a separate microscopic argument. The three
classical statements are thereby held in one hand: least dissipation
encodes the \emph{law}, the flux-side maximum selects the
\emph{transport} against power balance, and the force-side minimum
selects the \emph{state} --- three questions, one stationary point.

\begin{corollary}[Prigogine's theorem, recovered with its hypotheses]
\label{cor:qiep-minep-recovered}
Restrict the force-side reading of Theorem~\ref{thm:qiep-duality}(ii)
to a packet whose projected mobility \(L=\rho D\) is constant over the
occupied region, whose boundary variables are even under time
reversal, and whose ports clamp the forces \(X_1,\dots,X_m\) while
leaving the rest free. Then the stationary state minimizes the
force-side entropy production \(P(X)\) over the free forces, the
minimum is the state with vanishing free-force fluxes, and \(P\)
decreases monotonically along the relaxation. This is
Theorem~\ref{thm:qiep-minep-prigogine}, hypothesis for hypothesis:
(A1)~linearity is the quadratic restriction of the projected action,
(A2)~constancy is the state-independence of the projected mobility,
(A3)~reciprocity is the symmetry of \(D\), (A4)~positive definiteness
is its positivity on the packet's support, and (A5)~the clamped/free
split is the ports declaration of \SIP{}. Where any hypothesis fails,
the corollary fails exactly as the 1945 theorem does --- and
Theorem~\ref{thm:qiep-failure} computes the failure instead of
discovering it.
\end{corollary}

The corollary is the checkpoint of
Section~\ref{sec:qiep-hist-minep} passed: the one proved theorem of
the entropy-production tradition reappears, hypotheses and all, as the
gentle-driving shadow of the projected functional. The seventy-year
quarrel can now be closed in a table.

\begin{table}[H]
\centering
\caption[Five claims, one stationary point: the quarrel settled.]{The
extremal statements of the entropy-production tradition, each placed
at the one stationary point of the Rayleighian
\eqref{eq:qiep-rayleighian}. Each row names the historical claim, what
it varies, what it holds fixed, and what it selects; the last column
locates the statement in the present derivation. The table is the
settlement of the quarrel reviewed in
Sections~\ref{sec:qiep-hist-minep}--\ref{sec:qiep-hist-mepp}: the five
claims were never rivals about nature, but five control conventions at
one stationary point --- except the last, which claimed what no
convention delivers and failed as
Section~\ref{sec:qiep-hist-mepp} records.}
\label{tab:qiep-quarrel}
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.15\textwidth}>{\raggedright\arraybackslash}p{0.15\textwidth}>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}@{}}
\toprule
\textbf{Claim} & \textbf{Varies} & \textbf{Holds fixed} & \textbf{Selects} & \textbf{Here} \\
\midrule
Onsager 1931, least dissipation & Fluxes & Forces & The transport law
\(J=LX\) & \(\mathcal{R}_\rho\) itself, Eq.~\eqref{eq:qiep-rayleighian} \\
Ziegler; Kohler--Ziman (\MEPP{}, linear) & Fluxes on the power-balance
surface & Forces & The transport law, as a maximum &
Theorem~\ref{thm:qiep-duality}(i), \(\lambda=\tfrac12\) computed \\
Prigogine 1945 (MinEP) & Free forces & Clamped forces; the law & The
steady state & Theorem~\ref{thm:qiep-duality}(ii);
Corollary~\ref{cor:qiep-minep-recovered} \\
Paltridge (climate \MEPP{}) & Material transport & Radiative ports &
The flux pattern & Flux-side reading with ports declared
(\S\ref{subsec:qiep-observations}) \\
Dewar 2003--2005 (general \MEPP{}) & Path measure & Mean fluxes & A
universal maximum --- claimed & Not recovered; fails where
Theorem~\ref{thm:qiep-failure} says it must \\
\bottomrule
\end{tabular}
\end{table}

This current-gauge derivation is especially important for chemistry and life. A living system is not well described as a single sharply localized path. It is better described as a network of sustained throughputs, coupled channels, homeostatic gradients, and current constraints. In precisely that regime the most natural boundary language is \MEPP\ or its dual minimum-dissipation form. The underlying selector, however, is still the same projected \GG/\GDOS\ variational principle.

\begin{table}[H]
\centering
\caption[The entropy-production gauge, step by step.]{Step-by-step structure of the maximum-entropy-production derivation. Seven steps run down the first column, from the mobility tensor through the generalized force and the entropy production to the stationary flux law and the two readings of its dual. The layout deliberately matches Tables~\ref{tab:qiep-plt-chain} and~\ref{tab:qiep-fep-chain}: the three chains start from the same projected functional and diverge only in the choice of primary variable, which is the content of the pillar's common-origin theorem.}
\label{tab:qiep-mepp-chain}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}p{0.41\textwidth}}
\toprule
\textbf{Step} & \textbf{Equation} & \textbf{Meaning} \\
\midrule
Mobility tensor & \eqref{eq:qiep-onsager-tensor} & The projected metric becomes an Onsager transport tensor weighted by the density. \\
Generalized force & \eqref{eq:qiep-force} & The projected drift and potential define the thermodynamic driving field. \\
Entropy production & \eqref{eq:qiep-total-entropy-production} & The bilinear force-flux pairing is the boundary entropy-production rate. \\
Rayleighian & \eqref{eq:qiep-rayleighian} & Entropy production competes with dissipative cost. \\
Stationary flux law & \eqref{eq:qiep-onsager-law} & Extremization selects the physical current pattern. \\
Dual description & \eqref{eq:qiep-dual-dissipation} & The same stationary point is Onsager's least dissipation in the flux-controlled picture. \\
Force-side reading & Theorem~\ref{thm:qiep-duality}(ii) & Holding the transport law fixed and freeing some forces, the same point is Prigogine's minimum entropy production, selecting the state. \\
\bottomrule
\end{tabular}
\end{table}

\section{Inferential gauge: derivation of \texorpdfstring{\FEP}{FEP}}
\label{subsec:qiep-fep}

The third gauge is the inferential gauge. It becomes natural whenever the open subsystem does not merely transmit or dissipate, but also maintains internal states that summarize hidden causes and guide action. Cells, nervous systems, adaptive agents, and cognitive architectures all fall into this class.

Let the projected boundary variables be decomposed as
\begin{equation}
z = (\eta,s,a,\mu),
\label{eq:qiep-blanket-split}
\end{equation}
where \(\eta\) denotes hidden external causes, \(s\) sensory states, \(a\) active states, and \(\mu\) internal states. The Markov-blanket interpretation is that the subsystem accesses the external world only through \(s\), changes it through \(a\), and maintains a compact internal model through \(\mu\) \citep{Friston2010}. In the present framework the crucial addition is that the generative structure is not arbitrary: it is induced from the projected \GG/\GDOS\ geometry.

Let \(p(\eta,s,a)\) denote the projected generative density inherited from the boundary stationary measure. Introduce a parametric recognition family \(q_\mu(\eta)\) representing the subsystem's internal approximation to the posterior over hidden causes. The variational free energy is then
\begin{equation}
\mathcal{F}(s,a,\mu)
=
\int q_\mu(\eta)\,
\log\frac{q_\mu(\eta)}{p(\eta,s,a)}
\,\dd\eta.
\label{eq:qiep-variational-free-energy}
\end{equation}
As usual, Eq.~\eqref{eq:qiep-variational-free-energy} may be rewritten as
\begin{equation}
\mathcal{F}(s,a,\mu)
=
D_{\mathrm{KL}}
\!\big(
q_\mu(\eta)\,\|\,p(\eta|s,a)
\big)
-
\log p(s,a),
\label{eq:qiep-vfe-kl}
\end{equation}
so \(\mathcal{F}\) upper-bounds surprisal and is minimized when the recognition family matches the posterior.

To recover \FEP\ from the projected action, restrict the density \(\rho\) to the statistical manifold
\begin{equation}
\mathcal{S}_{\mathrm{inf}}
=
\big\{
\rho_{\mu,s,a}(\eta)
=
q_\mu(\eta)\,\delta(s-s_t)\,\delta(a-a_t)
\big\}.
\label{eq:qiep-inferential-slice}
\end{equation}
When the quadratic term of Eq.~\eqref{eq:qiep-current-density-action} is pulled back to this manifold, it induces the Fisher metric
\begin{equation}
g^{\mathrm F}_{ij}(\mu)
=
\int q_\mu(\eta)\,
\partial_i \log q_\mu(\eta)\,
\partial_j \log q_\mu(\eta)\,
\dd\eta,
\label{eq:qiep-fisher-metric}
\end{equation}
which is the natural information-geometric metric on the parameter manifold \citep{Amari1998,Amari2000,Jaynes1957}. The restricted action becomes
\begin{equation}
\mathcal{A}_{\mathrm{inf}}[\mu]
=
\int_0^T \dd t\,
\left[
\frac12 g^{\mathrm F}_{ij}(\mu)\,\dot\mu^i\dot\mu^j
+
\mathcal{F}\big(s(t),a(t),\mu(t)\big)
\right].
\label{eq:qiep-inferential-action}
\end{equation}

The Euler--Lagrange equations are
\begin{equation}
g^{\mathrm F}_{ij}\,\ddot\mu^j
+
\Gamma^{\mathrm F}_{ijk}\,\dot\mu^j\dot\mu^k
+
\partial_i \mathcal{F}
=
0,
\label{eq:qiep-fep-second-order}
\end{equation}
where \(\Gamma^{\mathrm F}_{ijk}\) is the Levi--Civita connection of the Fisher metric. In the overdamped or adiabatically relaxed limit --- the relevant limit for slow inferential updates compared with fast hidden fluctuations --- the inertial term may be neglected, leaving
\begin{equation}
\dot\mu^i
=
-
\kappa_\mu\,
\big(g^{\mathrm F}\big)^{ij}
\partial_j \mathcal{F}.
\label{eq:qiep-natural-gradient}
\end{equation}
Equation~\eqref{eq:qiep-natural-gradient} is the natural-gradient version of the \FEP: internal states descend variational free energy in the Fisher geometry. Active states obey the analogous projected descent
\begin{equation}
\dot a^\alpha
=
-
\kappa_a\,
\frac{\partial \mathcal{F}}{\partial a^\alpha},
\label{eq:qiep-active-inference}
\end{equation}
which is the action side of active inference: the system acts on the environment to reduce sensory prediction error.

\begin{theorem}[\FEP\ as the inferential gauge of \QiEP (T-Qi4-10.5)]
\label{thm:qiep-fep}
On the statistical slice \(\mathcal{S}_{\mathrm{inf}}\subset\Gamma_{\mathrm{adm}}\), the projected action \eqref{eq:qiep-current-density-action} pulls back to the Fisher-metric action \eqref{eq:qiep-inferential-action}. In the overdamped limit, its stationary dynamics reduce to the natural-gradient \FEP\ equations \eqref{eq:qiep-natural-gradient} and \eqref{eq:qiep-active-inference}. Hence \FEP\ is the inferential gauge of \QiEP.
\end{theorem}

\begin{proof}
The pullback of the density family \(\rho_{\mu,s,a}\) onto the parameter manifold \(\mu\) converts the quadratic kinetic term of Eq.~\eqref{eq:qiep-current-density-action} into the Fisher quadratic form \eqref{eq:qiep-fisher-metric}; this is derived explicitly in Appendix~\ref{app:qiep-fisher}. The projected potential becomes the variational free energy \(\mathcal{F}\), because the hidden-sector elimination identifies the admissible statistical mismatch cost with the KL divergence between \(q_\mu\) and the projected generative density. The Euler--Lagrange variation of Eq.~\eqref{eq:qiep-inferential-action} yields Eq.~\eqref{eq:qiep-fep-second-order}. The overdamped limit removes the second-order term and leaves the natural-gradient descent \eqref{eq:qiep-natural-gradient}. The active-state update \eqref{eq:qiep-active-inference} follows by the same reduction applied to the action variables.
\end{proof}

This derivation is the formal bridge from the present physics paper to later life, brain, and mind applications. It says that a persistent adaptive subsystem can be described as an open-system packet living on a statistical manifold whose geometry is inherited from the same projected bulk action. The resulting inferential descent is not a separate law introduced only for neuroscience. It is the inferential gauge of the same \QiEP\ selector that also appears as \PLT\ and \MEPP.

\subsection{What the inferential gauge does not assume}
\label{subsec:qiep-fep-nonassumptions}

The inference reading is the one most often overclaimed, so it is worth stating precisely which of its usual supports the present derivation does \emph{not} lean on. Four points, each of which the framework replaces with something checkable.

\emph{The blanket is replaced by declared ports.} Nothing above assumes that a statistical boundary can be found inside the stationary density of a generic system. The decomposition \eqref{eq:qiep-blanket-split} is read here as \emph{constitutive data}: the ports of the system packet \SIP{}, declared when the system is written down and carrying the criteria under which the description is judged wrong (\S\ref{subsec:ports-currents}). This is the same move by which thermodynamics escapes the ambiguity of ``the system'' --- by declaring its walls rather than hoping to discover them \citep{Aguilera2022}.

\emph{The synchronization map is replaced by a rank condition.} That the internal states parameterize a posterior over external ones is not assumed here. The internality construction of Part~\ref{part:internal} makes the internal model's existence conditional and checkable: the external perturbation is recoverable from the sensory history if and only if an explicit action-conditioned observability matrix has full column rank, and where the rank fails, no unique internal model exists. Systems that do not satisfy the inference reading are thereby classified rather than assumed away.

\emph{The obstruction is the one already located.} The couplings that must be excluded for the inference reading to hold are, in boundary coordinates, curl-sector couplings --- the same \(\mathcal{C}_{\mathrm{curl}}\) of Theorem~\ref{thm:qiep-failure}. The inferential gauge holds exactly where the current gauge's extremum readings hold, and fails with them, for the same reason. One obstruction, three literatures.

\emph{What is not derived is not claimed.} The expected-free-energy functional of prospective policy selection does not follow from Eq.~\eqref{eq:qiep-inferential-action}. In the present framework prospective selection is carried by a different mechanism --- the internalized regime switch of the runtime --- and no claim is made that Eq.~\eqref{eq:qiep-natural-gradient} contains it.

The four replacements are worth fixing in a table, because they are
the exact points at which the critical literature of
Section~\ref{sec:qiep-hist-fep} located the free energy principle's
troubles --- and the derivation must be seen to answer each one rather
than to relabel it.

\begin{table}[H]
\centering
\caption[What the FEP assumes, and what the gauge derives.]{The
inferential gauge against the free energy principle it recovers,
assumption by assumption. Each row is one structure the FEP literature
supplies by hand, with the critique that exposed it
(Section~\ref{sec:qiep-hist-fep}) and the derived object that replaces
it here. The last row is the one no formulation of the principle has
ever supplied for itself: a stated regime, with the failure outside it
carried by the same theorem that bounds the other two gauges. The
table is the section's claim in compact form: the working mathematics
of the tradition transfers unchanged, while every contested structure
arrives derived, conditional, or explicitly not claimed.}
\label{tab:qiep-fep-vs-gauge}
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.40\textwidth}@{}}
\toprule
\textbf{Structure} & \textbf{In the FEP literature} & \textbf{In the inferential gauge} \\
\midrule
Boundary & Markov blanket, sought in the stationary density;
definitions inequivalent \citep{Biehl2021} & Ports of the packet,
declared as constitutive data (\SIP{}) \\
Metric on beliefs & Chosen with the model class & Fisher metric,
pulled back from the projected quadratic form
\eqref{eq:qiep-fisher-metric} \\
Internal model exists & Synchronization map, assumed; non-generic
\citep{Aguilera2022} & Rank condition of the internality construction;
checkable, and classifying where it fails \\
Solenoidal coupling & Excluded silently
\citep{FristonDaCostaParr2021} & The curl sector of
Theorem~\ref{thm:qiep-failure}: the exclusion is the stated regime \\
Policy selection & Expected free energy, added & Not claimed; carried
by the runtime's regime switch \\
Regime of validity & Not stated; contested & Stated: the lawful
sector, failing with the other gauges by
Theorem~\ref{thm:qiep-failure} \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\centering
\caption[The inferential gauge, step by step.]{Step-by-step structure of the free-energy-principle derivation. Six steps run down the first column, from the blanket decomposition through the projected generative model and the Fisher metric to the inferential action and the natural-gradient descent it induces. This is the chain that matters most outside physics: it obtains the free-energy principle used across theoretical neuroscience as a \emph{gauge} of the same functional that gives least time and maximum entropy production, rather than as a separate principle introduced for living systems.}
\label{tab:qiep-fep-chain}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}p{0.41\textwidth}}
\toprule
\textbf{Step} & \textbf{Equation} & \textbf{Meaning} \\
\midrule
Blanket decomposition & \eqref{eq:qiep-blanket-split} & External hidden causes, sensory states, active states, and internal states are separated. \\
Projected generative model & \eqref{eq:qiep-variational-free-energy} & The subsystem compares its recognition density with the projected world-model. \\
Statistical slice & \eqref{eq:qiep-inferential-slice} & The current-density action is restricted to a parametric statistical manifold. \\
Fisher metric & \eqref{eq:qiep-fisher-metric} & The projected kinetic term becomes information geometry. \\
Inferential action & \eqref{eq:qiep-inferential-action} & The same reduced action becomes a Fisher-kinetic plus free-energy functional. \\
Natural-gradient descent & \eqref{eq:qiep-natural-gradient} & In the overdamped limit the selector becomes \FEP. \\
\bottomrule
\end{tabular}
\end{table}

\section{Where the three gauges must fail}
\label{subsec:qiep-failure}

A derivation that produced only successes would be worth less than one that produces its own limits. This subsection computes exactly where the three gauges stop holding, and the answer is already written in the runtime's own bookkeeping: in the two sectors of the boundary current that neither descend the ledger nor diffuse.

Recall the four-sector decomposition \eqref{eq:four-current-decomp} --- gradient, curl, noise, and gauge --- and the balance identity \eqref{eq:eegg-balance-identity} it induces on the ledger,
\(\dd\mathcal{F}_{\GFF}/\dd t = -\mathcal{D}_{\GFF}+\mathcal{P}_{\mathrm{port}}+\mathcal{C}_{\mathrm{curl}}+\mathcal{G}_{\mathrm{gauge}}\).
The reduction of the projected action \eqref{eq:qiep-current-density-action} to the Onsager--Rayleighian \eqref{eq:qiep-rayleighian} used two properties of the lawful drift, both of them exactly the conditions under which the last two terms vanish: the curl current is tangent to the ledger's level sets, so circulation does no work against the ledger, and the ledger is gauge-invariant, so the gauge sector does none either. Where either condition fails, the reduction loses a term --- and with it, the whole entropy-production family loses its stationarity.

\begin{theorem}[Where the extremum readings must fail (T-Qi4-10.8)]
\label{thm:qiep-failure}
Let the lawful drift be split as \(\rho u = \rho u_{\mathrm{d}} + \rho u_{\mathrm{c}} + \rho u_{\mathrm{g}}\), with \(u_{\mathrm{d}}\) collecting the gradient and noise sectors of \eqref{eq:four-current-decomp}. Then the physical stationary current of the full projected action is
\begin{equation}
J_\star^a = L^{ab}X_b + \rho u_{\mathrm{c}}^a + \rho u_{\mathrm{g}}^a,
\label{eq:qiep-fail-Jstar}
\end{equation}
and the first variation of \(\dot\Sigma\) at \(J_\star\) retains a term linear in the variation whose integral against the potential-generated family of admissible variations is \(-(\mathcal{C}_{\mathrm{curl}}+\mathcal{G}_{\mathrm{gauge}})/\Theta_{\mathrm{eff}}\). Consequently, wherever \(\mathcal{C}_{\mathrm{curl}}\neq 0\) or \(\mathcal{G}_{\mathrm{gauge}}\neq 0\), no functional of the entropy-production family is stationary at the physical state: the flux-side and force-side readings of Theorem~\ref{thm:qiep-duality} fail together, and the inferential gauge fails with them.
\end{theorem}

\begin{proof}
Given in full in Appendix~\ref{app:qiep-failure}.
\end{proof}

The theorem does not report that the extremum readings sometimes fail. It computes the exact tax that makes them fail. A term linear in the variation survives at the physical state, and a point carrying a surviving linear term sits on a slope, not on a summit. That is the whole mechanism, and the classical counterexamples are three names for that one slope.
Figure~\ref{fig:qiep-failure} draws both cases on one baseline.

\begin{remark}[The counterexamples, located]
\label{rem:qiep-counterexamples}
The known failures of the entropy-production principles land in the two obstructing sectors and nowhere else. A driven circuit with an inductive branch fails the force-side reading because an inductive current is odd under time reversal: in boundary coordinates it is curl-sector circulation doing work, \(\mathcal{C}_{\mathrm{curl}}\neq 0\). A network held at two temperatures fails because a non-uniform temperature field makes the ledger's level sets and the circulation non-orthogonal, so the work terms of \eqref{eq:eegg-balance-identity} survive \citep{Jaynes1980}. And the modern statement that selection beyond linear order depends on time-symmetric activity that no entropy-production functional contains is the observation that \(\mathcal{C}_{\mathrm{curl}}\) and \(\mathcal{G}_{\mathrm{gauge}}\) are time-symmetric sector couplings invisible to \(\dot\Sigma\) \citep{Polettini2013,BruersMaesNetocny2007}. The framework does not rescue the extremum readings in these cases; it derives that they must fail there, which is what a correct derivation owes.
\end{remark}

\begin{remark}[Beyond quadratic order, and why the confinement is a theorem]
\label{rem:qiep-nonlinear}
The Onsager--Rayleighian \eqref{eq:qiep-rayleighian} is the quadratic restriction of the projected action \eqref{eq:qiep-current-density-action}. The full functional is not quadratic, and its stationarity conditions beyond quadratic order are not expressible as an extremum of any entropy-production functional --- in agreement with, and for the same structural reason as, the macroscopic-fluctuation-theory result that what survives far from equilibrium is a corrected minimum-cost principle for the full functional rather than an entropy extremum \citep{Bertini2015}. The confinement of both entropy-production readings to the linear regime is therefore a theorem about the projection, not an empirical accident of the systems in which they were first tried.
\end{remark}

The three results of this subsection --- the duality, the failure location, and the order confinement --- are what convert \QiEP\ from a unification into a falsifiable claim. A framework that derived the three principles and nothing else would be a restatement; one that also derives, from the same object and with no new input, the exact conditions under which each must break is making a claim that measurement can refute.


\begin{figure}[htbp]
\centering
\resizebox{0.795\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=3pt, inner sep=4pt},
 panel/.style={fill=blue!8, draw=blue!55!black, line width=0.70pt,
  rounded corners=4pt},
 panelbad/.style={fill=orange!10, draw=orange!60!black, line width=0.70pt,
  rounded corners=4pt},
 axis/.style={-{Stealth[length=2.0mm]}, line width=0.55pt, draw=black!60},
 curve/.style={line width=1.10pt, draw=blue!60!black},
 curvebad/.style={line width=1.10pt, draw=orange!65!black},
 tang/.style={line width=0.80pt, draw=red!55!black, dashed},
 dotmark/.style={circle, fill=black!85, inner sep=0pt, minimum size=1.7mm},
 lab/.style={font=\sffamily\scriptsize, text=black!75, inner sep=1.5pt},
 tag/.style={font=\sffamily\scriptsize\bfseries, inner sep=2pt},
 cond/.style={font=\sffamily\scriptsize\itshape, text=black!70,
  inner sep=1.5pt},
]

%% --- left panel: the lawful sector, an extremum ---
\begin{scope}[shift={(0,0)}]
\node[panel, minimum width=66mm, minimum height=58mm] at (0,-17mm) {};
\node[tag, anchor=north] at (0,10mm)
 {Lawful sector:\; $\mathcal{C}_{\mathrm{curl}}=\mathcal{G}_{\mathrm{gauge}}=0$};
\draw[axis] (-25mm,-32mm) -- (25mm,-32mm) node[lab, anchor=north east] {$\delta$};
\draw[axis] (-25mm,-32mm) -- (-25mm,-6mm) node[lab, anchor=south west] {$\dot\Sigma$};
\draw[curve] plot[domain=-18:18, samples=57, smooth]
  ({\x*1mm}, {(-12 - 0.035*(\x)*(\x))*1mm});
\draw[tang] (-12mm,-12mm) -- (12mm,-12mm);
\node[dotmark] at (0,-12mm) {};
\node[lab, anchor=south] at (0,-10.6mm) {$J_\star$};
\node[cond, anchor=north, text width=58mm] at (0,-35mm)
 {no linear term survives:\\ the physical state is a summit};
\end{scope}

%% --- right panel: the obstructed sector, no extremum ---
\begin{scope}[shift={(80mm,0)}]
\node[panelbad, minimum width=66mm, minimum height=58mm] at (0,-17mm) {};
\node[tag, anchor=north] at (0,10mm)
 {Obstructed:\; $\mathcal{C}_{\mathrm{curl}}+\mathcal{G}_{\mathrm{gauge}}\neq 0$};
\draw[axis] (-25mm,-32mm) -- (25mm,-32mm) node[lab, anchor=north east] {$\delta$};
\draw[axis] (-25mm,-32mm) -- (-25mm,-6mm) node[lab, anchor=south west] {$\dot\Sigma$};
\draw[curvebad] plot[domain=-18:18, samples=57, smooth]
  ({\x*1mm}, {(-12 - 0.035*(\x)*(\x) + 0.22*(\x))*1mm});
\draw[tang] (-12mm,-14.64mm) -- (12mm,-9.36mm);
\node[dotmark] at (0,-12mm) {};
\node[lab, anchor=north west] at (1.2mm,-12.6mm) {$J_\star$};
\node[cond, anchor=north, text width=58mm] at (0,-35mm)
 {a linear term survives:\\ the physical state sits on a slope};
\end{scope}
\end{tikzpicture}%
}
\caption[Why the entropy readings fail where circulation does work.]{The
mechanism of Theorem~\ref{thm:qiep-failure}, drawn. Both panels plot the
entropy production \(\dot\Sigma\) against an admissible variation \(\delta\)
about the physical stationary current \(J_\star\) of
Eq.~\eqref{eq:qiep-fail-Jstar}; the curvature is the same in both, because it
comes from the same positive-definite inverse mobility. On the left the curl
and gauge sectors do no work against the ledger, the cross term of the
reduction vanishes, and \(J_\star\) sits at the summit --- which is what makes
the flux-side and force-side readings of Theorem~\ref{thm:qiep-duality}
extremal characterizations at all. On the right one of those sectors does
work, a term linear in \(\delta\) survives at \(J_\star\), and the same curve
is tilted: the dashed tangent no longer lies flat, and a point with a
surviving linear term sits on a slope rather than on a summit. The tilt is not
qualitative --- it is
\(-(\mathcal{C}_{\mathrm{curl}}+\mathcal{G}_{\mathrm{gauge}})/\Theta_{\mathrm{eff}}\),
so the work the two sectors do against the ledger \emph{is} the slope, and no
functional of the entropy-production family is stationary where that work is
nonzero. Read the two panels as one statement: the extremum readings are not
approximately true outside the lawful sector, they are structurally absent
there.}
\label{fig:qiep-failure}
\end{figure}

\section{The three gauges in the laboratory}
\label{subsec:qiep-observations}

The derivation would be idle if the gauges did not read real systems, and unfalsifiable if each reading did not name the measurement that would break it. This is where the argument meets things one can point at: a thermodiffusion cell, a resistor network, a pan of convecting fluid, three planetary atmospheres, the enzyme that makes ATP, a swimming bacterium, and a predicting brain. The format is the same for every case and deliberately blunt --- what is observed, how the old principle read it, how the derived gauge reads it, and the measurement that would prove the new reading wrong. None of the eight is offered as a confirmation; they are placements, and the discipline of the register is to say which rows could bite.

\subsection{Path selection in a structured medium}

\emph{Observed.} A particle or excitation crossing a medium of varying mobility selects one path among many --- the situation the least-time law was historically built on.

\emph{The old reading.} Fermat's principle, exact for rays, silent on noise.

\emph{The gauge reading.} The trajectory gauge in its home regime: a least-time law with the effective metric fixed by the projected mobility rather than declared, per Table~\ref{tab:qiep-lens-bacterium}.

\emph{What would refute it.} A selected path in a medium of independently measured mobility that departs systematically from the geodesic of that metric, beyond the small-noise corrections the functional itself predicts.

\subsection{Thermodiffusion: the textbook force-side state}

\emph{Observed.} A fluid mixture held between plates at different temperatures develops a stationary concentration gradient --- the Soret effect --- carrying heat but no net matter flow.

\emph{The old reading.} Prigogine's theorem in its home regime: with the thermal force clamped and the diffusive force free, the stationary state minimizes entropy production, realized by the vanishing of the matter flux.

\emph{The gauge reading.} The force-side reading of Theorem~\ref{thm:qiep-duality}(ii), exactly as in the classical proof; the derivation adds only the provenance of the quadratic form --- the ledger's Hessian --- and the statement of scope: constant coefficients across the cell, all variables even under time reversal.

\emph{What would refute it.} A measured stationary Soret state carrying a persistent matter flux at small driving, or a minimum displaced from the zero-flux state beyond the quadratic corrections.

\subsection{Planetary heat transport: the flux side at scale}

\emph{Observed.} With no fluid dynamics --- no rotation rate, no tuned diffusivity --- an energy balance closed by maximizing the material entropy production of the meridional transport reproduces the equator-to-pole contrast on Earth, Mars, and Titan (Section~\ref{sec:qiep-hist-mepp}).

\emph{The old reading.} The flagship success of \MEPP{}, with its standing embarrassments: nothing in the principle says which entropy production to maximize, and the choice that works ignores the radiatively dominant part of the budget.

\emph{The gauge reading.} The flux-side reading with the radiative channels entering as clamped ports and the material transport as varied current --- which is what makes an energy balance fed only by sunlight reproduce the contrast, and which converts Paltridge's empirical constraint choice into the declared ports of the packet. The ledger, fixed upstream, identifies the material entropy production as the extremized quantity; what was arbitrary becomes stated, and therefore falsifiable.

\emph{What would refute it.} An atmosphere, observed or adequately resolved in simulation, whose material transport sits systematically off the flux-side stationary point with radiative ports genuinely clamped and scale separation genuinely present --- or a dependence of the selected state on resolution that survives the stated scale-separation condition. Both concerns are live in the climate literature, and the reading stands exposed to them.

\subsection{Jaynes's two-temperature network: the counterexample as a theorem}

\emph{Observed.} A resistor network with elements held at different temperatures distributes its currents by Kirchhoff's laws --- minimizing total heat production, not total entropy production; the two differ whenever the temperatures do (Section~\ref{sec:qiep-hist-minep}).

\emph{The old reading.} A refutation of MinEP's generality --- and it is.

\emph{The gauge reading.} Theorem~\ref{thm:qiep-failure} with the inhomogeneity mechanism: a temperature field varying across the system makes the ledger's level sets and the circulation non-orthogonal, the work terms survive, and the extremum principle is \emph{derived} to fail here. The physical selection remains the stationary point of the full projected cost, which does not degenerate to an entropy functional in this case.

\emph{What would refute it.} A driven network whose physical current distribution deviated at leading order from the stationary point of the full quadratic cost --- not merely from the entropy extremum.

\subsection{B\'enard convection: the boundary of both readings}

\emph{Observed.} A fluid layer heated from below conducts at small driving and, above a critical threshold, organizes into convection rolls whose transport exceeds conduction.

\emph{The old readings.} For MinEP, the historical falsifier: the convective branch produces more entropy and it is the convective branch that is stable. For the maximum tradition, the historical motivation.

\emph{The gauge reading.} A regime boundary, not a paradox: below threshold the curl sector does no ledger work and the force-side minimum governs; the instability is the point at which a circulating sector begins to do work, the quadratic reduction fails per Theorem~\ref{thm:qiep-failure}, and selection passes to the full functional. The framework does not predict the convective pattern from an entropy extremum --- and says why no such prediction can be exact.

\emph{What would refute it.} A driven steady state beyond threshold demonstrably selected by an entropy-production extremum alone, robustly across geometries --- which the failure theorem forbids.

\subsection{Chemotaxis: the inferential gauge with a checkable rank condition}

\emph{Observed.} A swimming bacterium climbs a chemical gradient it cannot spatially resolve, comparing present against remembered concentration and modulating its tumbling rate.

\emph{The old reading.} The free energy literature's minimal agent: an internal state inferring an external one across a blanket.

\emph{The gauge reading.} The natural-gradient law \eqref{eq:qiep-natural-gradient} with a one-dimensional belief coordinate --- the methylation state as the estimate of gradient sign --- and the rank condition doing real work: the internal model exists \emph{because} self-motion raises the observability rank, and a tethered cell, sampling but not moving, falls below the condition and can hold no unique gradient estimate. Receptors and motor are the declared ports; the existence of the inference reading is a checkable condition on the sensorimotor loop rather than an assumption about the cell.

\emph{What would refute it.} Gradient-climbing performance surviving closure of the action channel --- immotile cells inferring gradients they cannot move through --- or an identified chemotactic architecture whose update law departs from natural-gradient structure where the metric is measurable.

\subsection{Neural prediction: the gauge the principle was built for}

\emph{Observed.} Cortical responses attenuate to predicted input and amplify prediction error; hierarchical processing carries top-down predictions against bottom-up residuals; behavioral and neural data are fit, quantitatively, by precision-weighted prediction-error dynamics (Section~\ref{sec:qiep-hist-fep}).

\emph{The old reading.} The free energy principle's home territory --- where the working mathematics was always solid regardless of the foundational claims.

\emph{The gauge reading.} The inferential gauge in its hierarchical Gaussian restriction, where \eqref{eq:qiep-natural-gradient} reduces to exactly the predictive-coding dynamics of the process theories --- so everything that literature has established empirically transfers unchanged. What the derivation adds is the boundary of the reading (strongly circulating dynamics across the declared ports leave the gradient-flow regime) and a bridge that is checkable in principle: the metric is fixed by the ledger, so precision weighting becomes a measurable geometry rather than a free parameter family.

\emph{What would refute it.} Identified neural inference dynamics whose metric structure, where measurable, is not Fisher --- or inference maintained where the circulating sector demonstrably does ledger work, without the degradation the reduction predicts.

\subsection{ATP synthesis: all three gauges in one molecule}

\emph{Observed.} The rotary enzyme that synthesizes ATP pays for each bond with an integer number of proton crossings set by its rotor size, while the bond's free energy is universal --- the measured standard hydrolysis free energy against the derived \(2\QB\) of \QiQUA{}, at the percent level (\S\ref{subsec:qiqua-landauer}).

\emph{The old readings.} Bioenergetics treats the number as measured; the free energy literature treats the cell as an inference engine; the thermodynamic literature treats the chemiosmotic machine as a near-equilibrium transducer.

\emph{The gauge reading.} The one system in this register that exercises all three gauges at once. The proton path through the rotor is trajectory-gauge --- a least-time path through a curved potential. The steady turnover is current-gauge --- flux--force operation of a linear transducer, with the duality of Theorem~\ref{thm:qiep-duality} fixing the operating point. The regulatory network around it is inferential-gauge --- the cell's estimate of its energy state updated by natural-gradient descent. The three descriptions do not conflict because they are three restrictions of the one projected functional \eqref{eq:qiep-current-density-action} --- which is the concrete answer to the question this Part opened with, of why a cell obeying three principles is not pulled in three directions.

\emph{What would refute it.} An elementary catalytic event whose four-way bookkeeping violates the single conversion constant outside the stated tolerance --- the falsifier of \QiQUA{}, inherited here.

\subsection{What the eight cases share}

In every case the derived reading does three things the original principle could not: it says \emph{which} functional, fixed upstream and never chosen per application; it says \emph{where the reading stops}, as a stated sector or rank condition rather than a surprise; and it names a measurement that could kill it. And in every case the failure literature of Sections~\ref{sec:qiep-hist-minep}--\ref{sec:qiep-hist-fep} reappears not as embarrassment but as confirmation of a boundary the derivation drew in advance. That inversion --- counterexamples as theorems --- is the practical content of deriving a principle instead of adopting it.

\begin{table}[H]
\centering
\caption[The three gauges read against eight observations.]{The pillar's observational register: which gauge each case exercises, and the measurement that would refute the reading. The right-hand column is the one that matters --- a derivation that unified three principles without naming what would break each reading would have bought coherence at the price of content. Two of the eight cases sit past the failure boundary of Theorem~\ref{thm:qiep-failure} and are listed as such: there the framework predicts that no entropy-production extremum can select the state, and a demonstration to the contrary refutes it.}
\label{tab:qiep-observations}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.15\textwidth}p{0.56\textwidth}}
\toprule
\textbf{Observation} & \textbf{Gauge} & \textbf{What would refute the reading} \\
\midrule
Path selection in a structured medium & Trajectory & A selected path departing systematically from the geodesic of the independently measured mobility metric \\
Thermodiffusion (Soret cell) & Current, force side & A stationary state carrying persistent matter flux at small driving \\
Planetary heat transport & Current, flux side & Material transport systematically off the flux-side stationary point with ports clamped and scales separated \\
Two-temperature driven network & Past the failure boundary & Selection by an entropy-production extremum where circulation does ledger work \\
Convection beyond threshold & Past the failure boundary & A driven steady state beyond threshold selected by an entropy extremum alone, robustly across geometries \\
Chemotaxis & Inferential & Gradient climbing surviving closure of the action channel; or non-natural-gradient updates where the metric is measurable \\
Neural prediction & Inferential & Measured inference metric that is not Fisher; or inference without degradation where the circulating sector does ledger work \\
ATP synthesis & All three & Four-way event bookkeeping violating the single conversion constant beyond tolerance \\
\bottomrule
\end{tabular}
\end{table}

The register is deliberately short and deliberately answerable. Five of the eight rows would refute a gauge reading, two would refute the failure theorem that bounds the gauges, and one --- the last --- would refute the pillar upstream of all of them. That distribution is the honest shape of a claim that unifies: most of its exposure is inherited, and the part that is its own is the part that says where it stops.

\section{The unified \texorpdfstring{\QiEP}{Qi-EP} theorem: one projected functional, three gauges}
\label{subsec:qiep-unified-theorem}

The three previous subsections are the heart of the section. They show that \PLT,\ \MEPP,\ and \FEP\ are not independent principles living in separate disciplines. They are three reductions of one master current-density action.

It is useful to package this result formally. Define the quadratic form
\begin{equation}
\mathbb{K}_\rho(\delta J,\delta J)
:=
\int_{\State}
\delta J^a
(\rho D)^{-1}_{ab}
\delta J^b\,\dd z,
\label{eq:qiep-master-quadratic-form}
\end{equation}
and write the projected action compactly as
\begin{equation}
\mathfrak{Q}_{\QiEP}[\rho,J]
=
\int_0^T \dd t\,
\left[
\frac14\,\mathbb{K}_\rho(J-\rho u,J-\rho u)
+
\mathcal{V}[\rho]
\right]
\qquad
\text{subject to}
\qquad
\partial_t\rho+\nabla\!\cdot J=0.
\label{eq:qiep-master-functional}
\end{equation}
Then the three gauges may be summarized as follows:
\begin{align}
\text{trajectory gauge:}\quad
& (\rho,J) \mapsto \big(\delta_{z(t)},\delta_{z(t)}\dot z(t)\big)
&&\Longrightarrow&&
\mathfrak{Q}_{\QiEP}\to \mathcal{T}[z],
\label{eq:qiep-gauge-trajectory}\\
\text{current gauge:}\quad
& (\rho,J) \mapsto (\rho,J)
&&\Longrightarrow&&
\mathfrak{Q}_{\QiEP}\to \mathcal{R}_\rho[J;X],
\label{eq:qiep-gauge-current}\\
\text{inferential gauge:}\quad
& (\rho,J) \mapsto \rho_{\mu,s,a}
&&\Longrightarrow&&
\mathfrak{Q}_{\QiEP}\to \mathcal{A}_{\mathrm{inf}}[\mu].
\label{eq:qiep-gauge-inferential}
\end{align}

\begin{theorem}[Common-origin theorem for \QiEP: three gauges of one projected functional (T-Qi4-10.6)]
\label{thm:qiep-unified}
Within the \GG/\GDOS\ framework, the projected functional \eqref{eq:qiep-master-functional} generates the three boundary principles
\[
\PLT,\qquad \MEPP,\qquad \FEP
\]
as its trajectory, current, and inferential gauges respectively. Therefore the three principles are gauge-equivalent boundary faces of one projected 6D variational law.
\end{theorem}

\begin{proof}
The proof is the composition of Theorems~\ref{thm:qiep-plt}, \ref{thm:qiep-mepp}, and \ref{thm:qiep-fep}. Each theorem establishes that a distinct admissible slice of \(\Gamma_{\mathrm{adm}}\) together with a distinct coarse-grained coordinate choice reduces the same master functional \eqref{eq:qiep-master-functional} to one of the three familiar boundary forms. Since the starting functional is the same in all three cases, the resulting principles are not independent axioms but gauge-equivalent reductions.
\end{proof}

The most important conceptual consequence is that \QiEP\ restores unity across scientific domains. A transport problem, a metabolic steady state, and an adaptive inferential system no longer require three unrelated foundational principles. They require one projected selector viewed in three different gauges. This is exactly the kind of cross-domain unification that Paper~IV-B is meant to achieve.

\begin{table}[H]
\centering
\caption[The three gauges of one functional.]{The three gauges of \QiEP{}. Each row is one gauge --- trajectory, current and inferential --- with its primary variable, the reduced functional it extremizes, and its operational interpretation. The table is the pillar's claim in one page: least time, maximum entropy production and the free-energy principle are not three principles but one, read with three different choices of what is held primary, and the second column is where the choice is made.}
\label{tab:qiep-three-gauges}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}p{0.32\textwidth}}
\toprule
\textbf{Gauge} & \textbf{Primary variable} & \textbf{Reduced functional} & \textbf{Operational interpretation} \\
\midrule
Trajectory gauge & \(z(t)\) & \(\mathcal{T}[z]\) & Path or branch selection under an effective metric and connection (\PLT). \\
Current gauge & \((\rho,J)\) & \(\mathcal{R}_\rho[J;X]\) & Force-controlled flux selection and entropy-production extremization (\MEPP). \\
Inferential gauge & \(\mu(t)\) & \(\mathcal{A}_{\mathrm{inf}}[\mu]\) & Natural-gradient descent of variational free energy (\FEP). \\
\bottomrule
\end{tabular}
\end{table}


\begin{table}[H]
\centering
\caption[The grade of every step in the \QiEP{} chain.]{The grade of every step of the \QiEP{} chain, in the program's standard grading. ``Derived'' means proved here or in the cited companion from the stated imports; ``conditional'' names the condition and states it; ``argued'' means supported but not proved; ``not claimed'' marks the adjacent results this pillar deliberately does not assert. The right-hand column is the load-bearing one: a pillar whose subject is three principles that were historically overclaimed must not add a fourth example, and this table is where the boundary of the present claim is drawn.}
\label{tab:qiep-grades}
\small
\begin{tabular}{p{0.40\textwidth}>{\raggedright\arraybackslash}p{0.14\textwidth}p{0.36\textwidth}}
\toprule
\textbf{Statement} & \textbf{Grade} & \textbf{Condition / authority} \\
\midrule
Projected functional \eqref{eq:qiep-current-density-action} from the 6D action & Derived & \S\ref{subsec:qiep-bulk-to-boundary}, via \OGGEP{} \\
Trajectory gauge \(\Rightarrow\) \PLT{} & Derived & Theorem~\ref{thm:qiep-plt}; small-noise limit \\
Current gauge \(\Rightarrow\) flux--force law & Derived & Theorem~\ref{thm:qiep-mepp} \\
\MEPP{}/MinEP duality at one stationary point & Derived & Theorem~\ref{thm:qiep-duality}; Appendix~\ref{app:qiep-duality} \\
Failure location of both readings & Derived & Theorem~\ref{thm:qiep-failure}; Appendix~\ref{app:qiep-failure} \\
Inferential gauge \(\Rightarrow\) natural-gradient \FEP{}; Fisher metric from \CEP{} & Derived & Theorem~\ref{thm:qiep-fep}; Appendix~\ref{app:qiep-fisher} \\
Internal-model existence & Conditional & rank condition of the internality construction (Part~\ref{part:internal}) \\
Statements inheriting the imported cascade closure & Conditional & \ASix{}'s conditioning, imported at its own grade (\S\ref{subsec:paper4a-recap}) \\
Exhaustiveness of the three gauges & Argued & from the packet's structure (state, flow, registration); not proved \\
Expected free energy / policy selection from \eqref{eq:qiep-inferential-action} & Not claimed & does not follow; carried by a different mechanism \\
Universal validity of \MEPP{} & Not claimed & false outside its regime, and derived to be false (Theorem~\ref{thm:qiep-failure}) \\
\bottomrule
\end{tabular}
\end{table}

\subsection{The contract, discharged}
\label{ssec:qiep-contract-discharged}

Section~\ref{ssec:qiep-contract} fixed five criteria before the
derivation began; the Part should not end without checking them off
against what was actually proved. (i)~\emph{One derived object:} the
three principles were obtained from the single projected functional
\eqref{eq:qiep-master-functional}, itself the \OGGEP{} projection of
the bulk action rather than a fresh postulate
(Proposition~\ref{prop:qiep-projected-stationary}). (ii)~\emph{The
restrictions as theorems:} the small-noise confinement of the
trajectory reading (Theorem~\ref{thm:qiep-plt}), the linear-regime
confinement of both entropy-production readings
(Remark~\ref{rem:qiep-nonlinear}), and the sector and rank conditions
of the inferential reading all arrive derived, with the failure
locations computed in Theorem~\ref{thm:qiep-failure} rather than
discovered by counterexample. (iii)~\emph{The quarrel dissolved:}
Theorem~\ref{thm:qiep-duality} and Table~\ref{tab:qiep-quarrel}, in
agreement with the resolution the critical literature reached from
below. (iv)~\emph{Modeler-chosen structure replaced:} the ledger fixes
which entropy production, the declared ports fix the boundary, and the
rank condition replaces the synchronization map
(Table~\ref{tab:qiep-fep-vs-gauge}). (v)~\emph{The conditional and
open elements stated:} Table~\ref{tab:qiep-grades}, where the grades
are printed and nowhere waived. What a reader should retain is not
that the three principles are true --- each is true exactly where its
regime theorem says --- but that the three regimes, the three laws,
and the three failures now follow from one object.


\section{\texorpdfstring{\DGI,\ \GSSB,\ and \GA}{DGI, GSSB, and GA} inside the \texorpdfstring{\QiEP}{Qi-EP} landscape}
\label{subsec:qiep-dgi-gssb-ga}

\QiEP\ is a projected selector, and a selector is not yet a dynamics:
it says which configurations are stationary, not how a system moves
among them, splits between them, or stays in one. Those three
questions belong to the runtime engine of
Theorem~\ref{thm:master-pipeline}, and its three moves --- \DGI{},
\GSSB{}, and \GA{} --- act on the \QiEP{} landscape in three specific,
checkable ways. This section records the action of each, because the
three moves are also where the three gauges' most familiar anomalies
--- caustics, instabilities, bistable percepts --- turn out to be one
phenomenon wearing three costumes.

\emph{\DGI{} acts as the gauge orbits of the selector.} Admissible
informational-gauge transformations --- relabelings of the boundary
description that carry no ledger cost --- change
\(\mathfrak{Q}_{\QiEP}\) by a boundary term only, so stationary points
come in gauge-equivalence classes, and the physical content of the
selector lives on the orbit space. Each gauge shows the orbit in its
own clothes. In the current gauge it is the gauge sector of the
four-current \eqref{eq:four-current-decomp}: flow that relabels
without transporting. In the inferential gauge it is
reparameterization of the belief coordinates: the natural-gradient law
\eqref{eq:qiep-natural-gradient} is form-invariant under
\(\mu\to\mu'(\mu)\) precisely because the Fisher metric transforms as
a tensor --- which is why the derivation produced the natural gradient
rather than the naive one, and why ``which coordinates do the beliefs
use'' is a question with no physical content.

\emph{\GSSB{} acts as branch creation.} Where the Hessian of the
selector degenerates,
\(\det\,\partial^2\mathfrak{Q}_{\QiEP}=0\), a stationary point splits,
and the runtime commits the system to one branch of the split. The
three gauges meet this one condition as three famous phenomena. In the
trajectory gauge it is the kinetic focus: the point beyond which the
extremal path becomes a saddle
(Section~\ref{sec:qiep-hist-pla}) and neighboring extremals cross ---
the caustic of optics. In the current gauge it is the instability
threshold: B\'enard onset is the degeneration of the quadratic form as
a circulating sector begins to do work, which is why the entropy
readings lapse exactly there
(\S\ref{subsec:qiep-observations}). In the inferential gauge it is
bistable perception: a posterior landscape whose single mode splits,
defeating the single-mode assumption cataloged in
Section~\ref{sec:qiep-hist-fep}. One degeneracy condition, three
literatures' edge cases --- the same economy the failure theorem
showed for the counterexamples.

\emph{\GA{} acts as the persistence criterion.} A stationary point
that the runtime selects must also be one it can keep: in the gradient
sectors the ledger descends monotonically
(\(\dot{\mathcal F}_{\GFF}\le 0\) along the lawful flow), so the
selected extremum is an attractor rather than a transient, and the
depth of its basin is the resource that Part~\ref{part:seos} will
convert into a persistence class. This is the clause that connects the
selector to everything downstream: the \SeOS{} membership conditions
of Part~\ref{part:seos}, and, in the companion capstone, the Ground
Geometric Attractor whose basin-by-basin succession carries the
eighteen-layer cascade \citep{ArisakaA11Origin}.

\begin{figure}[htbp]
\centering
\resizebox{0.88\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, inner sep=3pt},
 panel/.style={fill=blue!8, draw=blue!55!black, line width=0.70pt,
  rounded corners=4pt, minimum width=52mm, minimum height=50mm},
 crv/.style={line width=1.10pt, draw=blue!60!black},
 orb/.style={line width=1.10pt, draw=violet!60!black},
 arr/.style={-{Stealth[length=2.2mm]}, line width=0.70pt, draw=black!70},
 dotm/.style={circle, fill=black!85, inner sep=0pt, minimum size=1.6mm},
 tag/.style={font=\sffamily\scriptsize\bfseries, inner sep=2pt},
 lab/.style={font=\sffamily\scriptsize, text=black!70, inner sep=1.5pt},
]
\begin{scope}[shift={(-60mm,0)}]
\node[panel] at (0,-20mm) {};
\node[tag] at (0,1mm) {\DGI{}: the orbit};
\draw[crv] plot[domain=-20:20, samples=41, smooth]
 ({\x*1mm}, {(-30 + 0.028*(\x)*(\x))*1mm});
\draw[orb] (0,-30mm) ellipse [x radius=9mm, y radius=2.6mm];
\node[dotm] at (9mm,-30mm) {};
\node[dotm] at (-9mm,-30mm) {};
\node[lab, text width=44mm] at (0,-40mm)
 {a ring of equivalent minima:\\ relabelings with no ledger cost};
\end{scope}
\begin{scope}[shift={(0,0)}]
\node[panel] at (0,-20mm) {};
\node[tag] at (0,1mm) {\GSSB{}: the branch point};
\draw[crv] (0,-8mm) .. controls (0,-16mm) .. (0,-20mm);
\draw[crv] (0,-20mm) .. controls (-3mm,-26mm) and (-8mm,-30mm) .. (-11mm,-36mm);
\draw[crv] (0,-20mm) .. controls (3mm,-26mm) and (8mm,-30mm) .. (11mm,-36mm);
\node[dotm] at (0,-20mm) {};
\node[lab, anchor=west] at (1.5mm,-17mm)
 {$\det\,\partial^2\mathfrak{Q}=0$};
\node[lab, text width=44mm] at (0,-42mm)
 {one valley splits: caustic,\\ instability, bistable percept};
\end{scope}
\begin{scope}[shift={(60mm,0)}]
\node[panel] at (0,-20mm) {};
\node[tag] at (0,1mm) {\GA{}: the basin};
\draw[crv] plot[domain=-20:20, samples=41, smooth]
 ({\x*1mm}, {(-34 + 0.055*abs(\x)^(1.6)/2.4)*1mm});
\node[dotm] at (0,-34mm) {};
\draw[arr] (-14mm,-19mm) -- (-6mm,-29mm);
\node[lab, anchor=south west] at (-15mm,-19mm)
 {$\dot{\mathcal F}_{\GFF}\le 0$};
\node[lab, text width=44mm] at (0,-42mm)
 {descent holds the selection:\\ the extremum is an attractor};
\end{scope}
\end{tikzpicture}%
}
\caption[The runtime's three moves on the selector's landscape.]{The
three moves of the runtime engine acting on the \QiEP{} landscape,
drawn one per panel. Left, \DGI{}: gauge transformations carry no
ledger cost, so a stationary point is a ring of equivalent minima and
the physical content lives on the orbit space --- in the inferential
gauge this is exactly why the natural gradient, and not the naive one,
appears in Eq.~\eqref{eq:qiep-natural-gradient}. Center, \GSSB{}:
where the selector's Hessian degenerates a valley splits, and the
three gauges meet this single condition as the caustic of optics, the
convective instability of the current reading, and the bistable
percept of the inferential one --- three literatures' edge cases as
one degeneracy. Right, \GA{}: in the gradient sectors the ledger
descends monotonically, so what the selector picks the runtime keeps;
the basin's depth is the quantity Part~\ref{part:seos} converts into a
persistence class. The figure is a schematic of mechanisms, not a
computed surface, and each panel's claim is carried by the theorem it
cites in the text.}
\label{fig:qiep-landscape-moves}
\end{figure}


\section{Why \texorpdfstring{\QiEP}{Qi-EP} applies to all open systems}
\label{subsec:qiep-universal-applicability}

At this stage it is worth stepping back and asking why the section matters beyond formal derivation. The answer is that every natural open system presents some combination of the three selector questions:
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item Which path or branch is followed?
\item Which flux pattern or throughput is sustained?
\item Which internal model or policy is stabilized?
\end{enumerate}
A purely physical transport problem may use almost exclusively the first question. A reaction network, molecular motor, or ecosystem may use primarily the second. A cell, brain, or adaptive agent inevitably uses the third. But no natural system is forbidden from moving between gauges. A cell chemotaxes along a path (\PLT\ language), maintains metabolic fluxes (\MEPP\ language), and updates internal states in response to sensory mismatch (\FEP\ language). The same object therefore occupies all three gauges at once, depending on which variables have been retained by the observer.

This is why the present derivation cannot be restricted to one discipline. If \QiEP\ were only a neuroscience principle, it could not serve as Pillar~II of a grand open-system framework. If it were only a non-equilibrium thermodynamic principle, it could not later support \QiHAL\ and the inner-hologram architecture. If it were only a geodesic-trajectory principle, it could not explain living systems. The point of the present derivation is exactly that the same projected 6D variational law has enough flexibility to cover all three uses without multiplying foundational assumptions.

The universality claim can be summarized in a domain table.

\begin{table}[H]
\centering
\caption[Which gauge is natural in which domain.]{Illustrative domain-wise readings of \QiEP{}. Six domains run down the first column, from particle and wave transport to societies and collective systems. The second column names the gauge that is most natural there, the third the object it selects, and the fourth an illustrative reading. The word \emph{illustrative} in the heading is load-bearing: these are organizational placements meant to orient a reader in an unfamiliar domain, not predictions, and none of them is offered as evidence.}
\label{tab:qiep-domain-uses}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.19\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Domain} & \textbf{Most natural gauge} & \textbf{Selected object} & \textbf{Illustrative reading} \\
\midrule
Particle and wave transport & \PLT & Path or branch & Effective geodesic, tunneling channel, or propagation branch under the projected boundary geometry. \\
Chemistry and reaction networks & \MEPP & Flux pattern & Stationary throughput through coupled channels under force-controlled or resource-controlled driving. \\
Molecular machines & \MEPP\,+\,\PLT & Cycle and substep path & Driven cycles and substep selections constrained by the \QiQUA\ budget. \\
Cells and homeostatic systems & \MEPP\,+\,\FEP & Metabolic plus inferential state & Throughput must be maintained while internal states track the hidden environment. \\
Brains and cognition & \FEP\,+\,\PLT & Inference trajectory and policy & Prediction, sensing, and action unfold on an information-geometric manifold. \\
Societies and collective systems & All three & Path, flow, and policy & Collective trajectories, throughputs, and internal models are all needed simultaneously. \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Nothing biological was used}
\label{ssec:qiep-nothing-biological}

The reach of the theorems is set by what their proofs consumed, and it
is worth listing how short that list is. The derivation used a packet
with declared ports, a ledger, and a projection --- \SIP{},
\GFF{}, and \OGGEP{}/\GDOS{} --- and at no step a membrane, a genome,
a neuron, or any property that distinguishes living matter from driven
matter. The scope of the three gauge theorems is therefore the class
of systems carrying that structure: every open system admitting a
packet description, and in particular the entire \SeOS{} class of
Part~\ref{part:seos}, whose membership conditions are stated there as
five checkable clauses and whose persistence theorem is what converts
``selected'' into ``still there to be measured.'' A living organism
and a brain are members of that class by those theorems --- not by
analogy, not by aspiration, but by satisfying the clauses --- and so
the three principles reach organisms and minds with exactly the
regimes and exceptions derived above, and with nothing further
granted. This is the sense, and the only sense, in which the Part's
claim extends to life: the theorems bind the class, and life is in the
class.

\subsection{The cascade demonstration}
\label{ssec:qiep-cascade-bridge}

The claim of the previous subsection is structural; the companion
capstone makes it a demonstration \citep{ArisakaA11Origin}. There the
\GRIP{}-In/\GRIP{}-Out engine, running under the \QiFour{}
admissibility conditions of the present paper, drives the
eighteen-layer cascade of Ground Geometric Attractors from the gauge
group and the electroweak vacuum through the periodic table, molecular
closure, and the first cell to the connectome, the conscious episode,
and the layer that states the constitution which produced it ---
layers 7--10 are life, layers 14--17 are mind. \QiEP{} is the selector
at every one of those layers: which reaction path in the chemistry
layers, which flux architecture in the metabolic ones, which internal
model in the cognitive ones. The capstone's dynamics block deploys the
three-gauge theorem in exactly the form proved here --- the
thermodynamic gauge recovering the Gibbs free energy in the
\(J_{Qi}=0\) limit, the variational gauge recovering the Friston form
at the declared boundary, the action gauge recovering the \GG{} field
action in the detector limit --- so the chain of custody runs unbroken
from Eq.~\eqref{eq:qiep-master-functional} to the account of why
there is anyone to read it. What this Part supplies to that chain is
the load-bearing middle: the selector, derived once, with its regimes
attached, valid at every floor because no floor's special furniture
was used in proving it.


\section{Three sciences, one principle}
\label{sec:close-qiep}

It is worth pausing on how odd the situation was before this Part.

Three mature sciences each had a variational principle at the bottom of
it. Each principle worked. Each was arrived at independently, by people
solving different problems in different centuries with different
mathematics. And each field, if pressed, would have described its
principle as fundamental --- not as a special case of anything. A cell
is simultaneously a mechanical object following a path, a thermodynamic
object maintaining fluxes, and an inferential object correcting
predictions; if the three principles were really three, the cell would
have been receiving three sets of instructions, and there was never any
account of why they did not conflict.

They did not conflict because they were one instruction. That is the
content of this Part, and the form of the argument matters: the three
are not shown to be analogous, or compatible, or reducible in some
limit. One functional is projected from the bulk onto the boundary, and
each of the three principles is recovered by asking for stationarity
with a different quantity held primary. Different questions of the same
object, returning different-looking answers, in the way that the same
landscape gives a different-looking answer to "which way is downhill"
depending on whether you are asking about a ball, a river, or a
population.

It is worth being exact about what kind of unification this is,
because the word is used loosely and usually means less than it
sounds. Nothing here has been merged. The three principles keep their
own variables, their own units and their own laboratories, and no
physicist computing a trajectory will now find herself computing
entropy production instead. What has changed is that they have
acquired a common author. One quantity is extremized, on one surface,
and the three principles are the three answers that quantity gives to
three different questions. That is a weaker relation than identity and
a far stronger one than analogy. Analogies are free --- one can be
drawn between any two things by anyone willing to squint. This costs
something, because it forbids outcomes. If the three principles were
really independent, a system could satisfy one and violate another,
and somewhere in nature something would. The claim here is that
nothing does.

The claim can now be stated at full strength, because every word of it
has been earned above and each clause carries its theorem. Within this
framework the principle of least time, maximum entropy production, and
the free energy principle are \emph{theorems}: each is the reading of
one projected functional in its own gauge
(Theorems~\ref{thm:qiep-plt}, \ref{thm:qiep-mepp},
\ref{thm:qiep-fep}, unified in Theorem~\ref{thm:qiep-unified}), each
is exact precisely where its regime theorem says, and each fails, by
theorem, where the regime ends
(Theorem~\ref{thm:qiep-failure}). The derivation used nothing
biological --- a packet, a ledger, and a projection --- so the
theorems bind every open system in the \SeOS{} class; a living
organism and a brain are members by the persistence theorem of
Part~\ref{part:seos}, and the companion capstone carries the
demonstration up the eighteen layers from the gauge group to the
sentence you are reading \citep{ArisakaA11Origin}. Attaching the
regime to the theorem in the same breath is not a hedge; it is what
makes the claim stronger than the slogans it replaces, because a
theorem that knows its boundary is falsifiable and a slogan is not.

Each of the three sciences gains something specific, and the gains
are worth naming separately because they are of different kinds.
\emph{Nonequilibrium thermodynamics} gains a grammar: extremum
statements about constitutive relations (flux side) and extremum
statements about state selection (force side) are different kinds of
statement, conjugate at one point, and the field's oldest quarrel
dissolves under the distinction --- with the functional now fixed
upstream, so that ``which entropy production'' is no longer the
user's choice. \emph{The biology of energy} gains a currency: if
information carries the derived price of \QiQUA{}, then bioenergetics
and information processing are one subject with one ledger, and
questions that sound philosophical --- how much of a cell's ATP
turnover services its memory and prediction --- become budgetary, and
attemptable. \emph{The sciences of mind} gain a boundary: the
inference reading survives here in exactly the form its best critics
said it should --- a conditional description, holding where a
checkable rank condition and sector condition hold, silent elsewhere.
That is a demotion from ``all persisting matter infers'' and a
promotion from ``useful modeling scheme'': inference becomes a
physical regime a system can be in or out of, and which regime a
given preparation occupies is an experimental question.

And symmetry requires the converse list, because a Part whose subject
is three literatures' overclaims must not end by adding a fourth. It
is \emph{not} concluded that maximum entropy production is true ---
the derivation says it is true in a bounded regime and false outside
it, and the boundedness is the result. It is \emph{not} concluded
that every organism computes probabilities --- the inference reading
holds where the rank condition holds and is silent elsewhere. And it
is \emph{not} concluded that the framework behind the derivation is
established: the projected functional is only as trustworthy as the
six-dimensional construction, the anomaly proof, and the
constitutional charter upstream of it, whose own open items are
recorded in their home papers and inherited here at their stated
grades (Table~\ref{tab:qiep-grades}). The three principles were
promoted to laws without derivation, and their fields paid for the
promotion in decades of misapplication and empty quarrel; this Part's
discipline is to hand every claim its passport at the border.


There is a cautionary lesson in this Part as well as a result. Two
generations of argument about whether driven systems maximize or
minimize entropy production dissolve here into a statement about which
variables were being held fixed by whom. The disputants were not
disagreeing about nature. They were disagreeing about a convention, and
the disagreement lasted because there was no framework in which the
convention was visible as a convention. That happens more often in
science than anybody likes to admit, and it is worth knowing what it
looks like from outside.

\section{Concluding remarks: one landscape, three shadows}
\label{sec:qiep-conclude}

The whole Part retold, once, with no equations. Begin with the
question that started it: \emph{which way does an open system go?}
Equilibrium physics has one magnificent answer for sealed boxes ---
downhill in free energy --- and no general answer for anything alive
enough to be interesting. Three sciences filled the gap with three
rules: settle into the laziest steady state available (true, provably,
when the pushing is gentle); degrade the driving gradient as fast as
the constraints allow (striking on three planets, derived never);
behave so as to make what arrives unsurprising (running half of
computational neuroscience on mathematics nobody disputes, under a
foundational claim that did not survive inspection). And a living
cell is all three subjects at once, with no account anywhere of why
it is not torn apart by three different instructions.

The answer of this Part is that there is one landscape of cost,
projected down from the bulk, and the three rules are its three
shadows. Ask the landscape which \emph{path} and it answers with
least time; ask it which \emph{flows} and it answers with the
flux--force law, whose two sides are the maximum and minimum
principles that were never rivals; ask it which \emph{guess} and it
answers with the error-correcting rule of the brain sciences, its
vague parts replaced by checkable conditions. One object, three
questions, three famous answers --- and, the part a skeptical reader
should weigh most heavily, the failures were derived too. The
circulating currents that ruin the minimum principle in a driven
circuit, the convection cells that embarrassed it historically, the
special couplings the brain-science principle silently needs: all of
them sit exactly where the reduction says they must, in the two
sectors of the flow that circulate and relabel rather than descend. A
theory that predicts only successes explains nothing; the arithmetic
of this one required its own limits.

What would end this Part is not a number but a system: something
whose path, whose fluxes, and whose inference genuinely could not be
organized by one budget --- a case where the three descriptions
demonstrably pull apart. The claim is that no such thing exists among
systems of this kind. That is a strong claim, and it is stated in the
form that lets somebody go looking; the register of
\S\ref{subsec:qiep-observations} names the eight places to start,
each with the measurement that would bite.

The reader who wants any thread at greater length has three doors.
The histories and the complete proofs at review depth are the
companion supplement \citep{ArisakaS1EP}. The cascade that carries
the selector from the gauge group to life and mind is the companion
capstone \citep{ArisakaA11Origin}. And the remaining Parts of the
present paper are the runtime that the selector serves: what the
selected branch looks like under magnification, how it remembers, and
what it costs to stay one.

One further thing has changed, and it sets up what follows. Before
this Part the system had a budget; it now has a chosen branch --- a
definite path, a definite set of maintained currents, a definite model
under correction. A chosen branch is an object, and objects have
properties one can go and measure. The most exposed of those
properties is what the branch looks like when the scale of the looking
is changed, because that is the one question an instrument can ask
without knowing anything about where the branch came from. The next
Part asks it. It is the point at which this framework stops giving an
account of things already known and starts saying things a laboratory
is in a position to refuse.


\GGPart[Qi-CON]{\QiCON{}}{Conformality, L\'evy statistics, fractality, and the two-knee $1/f$ band}
\label{part:qicon}

\noindent
This Part is about what a system of this kind looks like when you change
the magnification.

The phenomenon at issue is one of the most reported and least
explanatory in science. Measure almost any complicated thing over
time --- a resistor, a heartbeat, a river, a stock index, a neuron ---
and the fluctuations have more power at low frequencies than at high,
in a particular way, over a wide range. It is called $1/f$ noise, it has
been found nearly everywhere since the 1920s, and it has generated a
large literature of mechanisms because so many different mechanisms can
produce it. That ubiquity is exactly the problem. A prediction that a
system will show an unbounded power law is nearly unfalsifiable, since
almost any story delivers one.

So this Part makes a narrower claim, which is the reason it is worth
making. The power-law band is \emph{finite}, and the framework names
both of its ends. Below the lower turn the spectrum flattens, and the
frequency where that happens is set by the slowest process the system
still keeps --- the lifetime of the state it has settled into. Above the
upper turn the spectrum steepens, and that frequency is set by the
fastest process it still resolves. Two turns, not one; a band, not a
slope. And the framework says in advance what each end should equal,
which is what makes the whole thing something a measurement could
contradict.

Two more things fall out of the same construction and are worth watching
for, because they are one claim rather than three pieces of evidence.
The jumps such a system makes should have heavy tails rather than
bell-shaped ones, and the set it explores should be fractal rather than
filling. The same single exponent controls the spectrum, the tails and
the dimension, so a measurement that fixes it in one place fixes it in
the others --- and a system that gave three mutually inconsistent
answers would have refuted the Part.

Two cautions before starting. Nothing here claims exact scale invariance
at all scales; real systems are finite, instrument-bounded and
resource-limited, and the statement is about the window between the two
turns. And where the Part places known phenomena from other fields, it
is placing them, not predicting them: a retrodiction of $1/f$ noise is
not dressed up as a confirmation anywhere in what follows.


\label{sec:qicon}

After budget and selector come diagnostics. Once \QiQUA\ has fixed admissible current bookkeeping and \QiEP\ has fixed the selected history, one can finally ask what observable multiscale signatures that selected runtime leaves behind. Part~\ref{part:qicon} therefore derives the conformal, L\'evy, fractal, and \(1/f\)-type diagnostics of the selected branch.

\subsection{Historical background (with citations)}

Antecedents: Mandelbrot (1967, 1982), Wilson--Kogut (1974), Kadanoff (1966), Levy (1925), Bak--Tang--Wiesenfeld (1987), Voss--Clarke (1976) \citep{Mandelbrot1982,Wilson1971,WilsonKogut1974}; \QiCON{} subsumes these as gauges or limiting cases of Theorem~T-CON.1 $\cup$ T-CON.2 (fractality, L\'evy statistics, and the two-knee $1/f$ spectrum are joint signatures of the homogeneous boundary kernel; see A1 Sec.~6 and A4 Sec.~5 for full history).

\paragraph*{Dual-layer architecture and the compatibility relation $\mathfrak{C}_{\mathrm{CON}}$.}
\QiCON{} is, in the L16 language, a dual-layer principal pillar. Its
\emph{algebraic layer} (T-CON.1) is the operational conformality
theorem (Theorem~\ref{thm:qicon-operational-conformality}): the
boundary projection of the conformal isometry of the 6D bulk
produces homogeneous boundary kernels with power-law correlators,
realizing at the boundary the L\'evy-stable distributions of
\citet{Levy1937}, the fractal geometry of \citet{Mandelbrot1982},
and the conformal field theory of
\citet{BelavinPolyakovZamolodchikov1984}. Its
\emph{operational/physical layer} (T-CON.2) is the two-knee spectral
theorem (Theorem~\ref{thm:qicon-two-knee}): the predicted $1/f$
spectrum has exactly two scale-breaking knees --- the co-occurrence of two
knees bracketing a finite power-law band, rather than a single unbounded slope,
is the structural signature fixed by the homogeneous boundary kernel; the knee
positions are the system's slowest and fastest retained rates, and the band
slope is tied to the conformal weight. This two-knee structure is the central
falsifiable empirical prediction of \QiCON{}. The two layers are
tied together by the named compatibility relation
\begin{equation}
\label{eq:CCON-compat}
\resizebox{\textwidth}{!}{$\displaystyle
\boxed{\;\mathfrak{C}_{\mathrm{CON}}\;\equiv\;
\text{T-CON.1}\;\Longrightarrow\;\text{T-CON.2}
\quad\text{(via the scale profile and \OGNfive{}-determined boundary kernel widths).}\;}
$}
\end{equation}
The bulk conformal isometry forces homogeneous boundary kernels with
power-law decay; the two boundary knee positions are then set by the
specific bulk scale profile and \OGNfive{} integers, giving a sharp
falsifiable prediction.


\begin{figure}[htbp]
\centering
\resizebox{0.673\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 axis/.style={-{Stealth[length=2.4mm]}, line width=0.8pt, draw=black!75},
 band/.style={fill=orange!12, draw=orange!60!black, line width=0.55pt},
 curve/.style={line width=1.10pt, draw=blue!60!black},
 knee/.style={draw=red!60!black, line width=0.65pt,
 dash pattern=on 2.5pt off 2pt},
 lab/.style={font=\sffamily\scriptsize, align=center, inner sep=2pt},
 note/.style={font=\sffamily\scriptsize\itshape, text=black!65,
 align=center, inner sep=2pt},
]
% the power-law band, shaded
\fill[band] (35mm,3mm) rectangle (85mm,62mm);

% axes
\draw[axis] (5mm,3mm) -- (128mm,3mm) node[below left=1mm and 0mm, lab]
 {frequency $f$ \ (log)};
\draw[axis] (5mm,3mm) -- (5mm,68mm) node[above right=0mm and -6mm, lab]
 {power $S(f)$ \ (log)};

% the spectrum: plateau, band, roll-off
\draw[curve] (8mm,58mm) -- (35mm,58mm);
\draw[curve] (35mm,58mm) -- (85mm,30mm);
\draw[curve] (85mm,30mm) -- (120mm,8mm);

% the two knees
\draw[knee] (35mm,3mm) -- (35mm,66mm);
\draw[knee] (85mm,3mm) -- (85mm,66mm);
\fill[red!60!black] (35mm,58mm) circle (0.9mm);
\fill[red!60!black] (85mm,30mm) circle (0.9mm);

% regime labels
\node[lab] at (19mm,64mm) {\textbf{flat}\\ $S_0$};
\node[lab] at (60mm,50mm) {\textbf{the band}\\ $S \sim f^{-\beta}$};
\node[lab] at (107mm,24mm) {\textbf{roll-off}\\ $S \sim f^{-2}$};

% what sets each knee
\node[note, anchor=north] at (35mm,0mm)
 {\textbf{lower knee $f_{\mathrm{IR}}$}\\ set by the \emph{slowest} rate\\
  the system still keeps\\ (the lifetime of the state\\ it settled into)};
\node[note, anchor=north] at (85mm,0mm)
 {\textbf{upper knee $f_{\mathrm{UV}}$}\\ set by the \emph{fastest} rate\\
  the system still resolves\\ (the quickest microscopic\\ step it retains)};

% the width, and what it fixes
\draw[{Stealth[length=2mm]}-{Stealth[length=2mm]}, line width=0.6pt,
 draw=black!60] (36mm,66mm) -- (84mm,66mm);
\node[note, anchor=south] at (60mm,66.5mm)
 {the width of this window fixes how many levels deep\\
  the memory ladder of the next Part can be};
\end{tikzpicture}%
}
\caption[The two-knee spectrum: a band, not a slope.]{The claim that makes
this pillar falsifiable. Systems of this kind are known to show $1/f$ noise,
and that observation on its own is worth very little, because an unbounded
power law is compatible with a great many mechanisms and so eliminates almost
none of them. What is predicted here is a \emph{bounded} band: the spectrum is
flat below the lower knee, follows a power law between the knees, and steepens
to $f^{-2}$ above the upper one. The two knees are not fitted. The lower sits
at the slowest rate the system still keeps --- the lifetime of the state it
settled into --- and the upper at the fastest it still resolves, so a
measurement can disagree with either independently. The exponent in the band,
the heaviness of the jump distribution, and the dimension of the set the
system explores are all controlled by one number, so those are one claim
rather than three pieces of evidence. Where the paper places known
observations from other fields against this shape, they are placements: a
retrodiction of $1/f$ noise is nowhere treated as a confirmation.}
\label{fig:a7-two-knee}
\end{figure}

\section{Constitutional statement of the \texorpdfstring{\QiCON}{Qi-CON} principle}
\label{subsec:qicon-constitutional-statement}

Before any detailed calculation, the central claim of \QiCON\ should be stated in a precise operational form.

\begin{definition}[\QiCON]
Let \(S\) be an admissible open subsystem on the physical boundary state manifold \((\State_{\mathrm{phys}},g)\), and let \(\gamma_*\) be a \QiEP-selected branch of its projected dynamics. We say that \(S\) is in a \QiCON\ window if there exist two cutoff scales
\begin{equation}
\ell_{\mathrm{micro}} \ll \ell \ll L_{\mathrm{att}}
\qquad\Longleftrightarrow\qquad
f_{\mathrm{IR}} \ll f \ll f_{\mathrm{UV}},
\label{eq:qicon-window}
\end{equation}
and an exponent \(0<\alpha\le 2\) such that the effective generator of the selected branch satisfies
\begin{equation}
\mathcal{L}_{\mathrm{eff}}
=
-\,D_{\alpha}\,(-\Delta_g)^{\alpha/2}
+
\mathcal{L}_{<\alpha},
\label{eq:qicon-effective-generator}
\end{equation}
where \(D_\alpha>0\), \((-\Delta_g)^{\alpha/2}\) is the fractional Laplace--Beltrami operator on \((\State_{\mathrm{phys}},g)\), and the remainder \(\mathcal{L}_{<\alpha}\) contains lower-order, cutoff-sensitive, or weakly relevant deformations. In this window the leading dynamics are asymptotically homogeneous under scale transformations and generate power-law correlators, heavy-tailed coarse statistics, fractal support, and \(1/f\)-type spectra.
\end{definition}

Three clarifications are essential.

First, \QiCON\ does not claim exact scale invariance at all scales. Natural systems are finite, instrument-bounded, resource-limited, and often branch-selected by \GSSB. Therefore the relevant statement is \emph{intermediate} or \emph{operational} conformality: there exists a nontrivial scale interval on which the leading generator is homogeneous to good approximation.

Second, the exponent \(\alpha\) is not inserted by hand. It is inherited from the projected bulk-to-boundary kernel and therefore ultimately from the six-dimensional geometry together with the selected branch. The value \(\alpha=2\) reproduces Gaussian diffusion as a limiting case, but the generic \QiCON\ regime of interest is \(0<\alpha<2\), where nonlocal heavy-tailed transport survives at leading order.

Third, the phrase ``conformality'' is used in the operational sense appropriate to \GDOS. In the strictest mathematical literature, full conformal invariance can be stronger than mere scale covariance. The present paper uses the term to denote the physically relevant combination of homogeneous generators, power-law correlators, and absence of an intrinsic scale in the active window. This is exactly the feature required for a universal diagnostic layer.

\begin{remark}
The transition from exact microscopic law to operational conformality mirrors the transition from exact bulk action to projected boundary action already encountered in \QiEP. In both cases the decisive point is not that the boundary is simpler than the bulk, but that the observer sees only the projected homogeneous core together with finite-size and instrument-induced corrections.
\end{remark}

\begin{table}[H]
\centering
\caption[The four signatures of a \QiCON{} window.]{Four canonical signatures of a \QiCON{} window. The rows are operational conformality, L\'evy statistics, fractality, and $1/f$ noise with two knees; the columns give the mathematical form of each and its meaning inside the boundary law. They are listed together because they are not four independent findings --- the first three are one exponent seen three ways, as Table~\ref{tab:qicon-alpha-dictionary} sets out, and the fourth is the bounded band that distinguishes this claim from an unbounded power law.}
\label{tab:qicon-four-signatures}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.25\textwidth}p{0.42\textwidth}}
\toprule
\textbf{Signature} & \textbf{Mathematical form} & \textbf{Meaning inside \GDOS} \\
\midrule
Operational conformality & \(C(\lambda z,\lambda^\alpha t)=\lambda^{-2\Delta} C(z,t)\) & No intrinsic scale inside the active window; correlations decay by power laws. \\
L\'evy statistics & \(\widehat{p}(k,t)=\exp[-D_\alpha t|k|^\alpha]\) & Coarse increments are \(\alpha\)-stable rather than Gaussian. \\
Fractality & \(N(\varepsilon)\sim \varepsilon^{-D_f}\) & Trajectories and selected supports occupy non-integer effective dimension. \\
\(1/f\) with two knees & \(S(f)\propto f^{-\beta}\) on \(f_{\mathrm{IR}}<f<f_{\mathrm{UV}}\) & The selected branch produces multiscale soft modes bounded by attractor and microphysical cutoffs. \\
\bottomrule
\end{tabular}
\end{table}

\section{From the 6D bulk to homogeneous boundary kernels}
\label{subsec:qicon-bulk-to-boundary}

The derivation of \QiCON\ begins not on the boundary but in the six-dimensional bulk already established in \ATwo{} \citep{ArisakaA2GG6}. The extra radial direction is not an ornamental coordinate. It is the geometric carrier of scale. Writing
\begin{equation}
r = \ln\!\left(\frac{\EGUT}{\mu}\right),
\label{eq:qicon-r-rg}
\end{equation}
the ultraviolet boundary corresponds to \(r=0\) and lower operational scales correspond to larger \(r\). In other words, the bulk geometry stores the renormalization structure that later appears on the boundary as multiscale dynamics (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}; \citealp{Wilson1971}; \citealp{WilsonKogut1974}; \citealp{Maldacena1998}).

To expose the scaling law cleanly, it is convenient to replace \(r\) by a positive scale coordinate
\begin{equation}
\zeta = e^{-r/L},
\label{eq:qicon-zeta}
\end{equation}
where \(L\) is the radial length scale. Large \(\zeta\) corresponds to coarse boundary scales, whereas small \(\zeta\) corresponds to ultraviolet scales. Consider a projected Qi-neutral observable on a local physical state patch \(U\subset\State_{\mathrm{phys}}\) of dimension
\begin{equation}
d_\State = \dim U.
\label{eq:qicon-dstate}
\end{equation}
Let \(\ell = d_g(z,z')\) denote the geodesic separation of two nearby boundary points \(z,z'\in U\). For a dominant projected bulk mode with effective scaling weight \(\Delta\), the bulk-to-boundary propagator has the generic scale-profile form
\begin{equation}
K_{\Delta}(\zeta;z,z')
=
\left(
\frac{\zeta}{\zeta^2+\ell^2}
\right)^{\Delta},
\label{eq:qicon-btb-kernel}
\end{equation}
up to slowly varying geometric prefactors \citep{Maldacena1998}. The essential feature is homogeneity: the kernel depends on \(z,z'\) only through the dimensionless ratio \(\ell/\zeta\).

The \OGGEP\ projection integrates over the hidden scale coordinate and the unresolved sector. A jump \emph{rate} between two boundary points is an amplitude squared: the projected mode reaches each endpoint of the jump through one factor of the bulk-to-boundary propagator, so the induced boundary jump kernel carries \(K_\Delta\) twice,
\begin{equation}
W_{\Delta}(z,z')
=
\int_{\zeta_{\mathrm{min}}}^{\zeta_{\mathrm{max}}}
\omega(\zeta)\,
\big[K_{\Delta}(\zeta;z,z')\big]^{2}
\,\dd\zeta,
\label{eq:qicon-projected-kernel}
\end{equation}
where \(\omega(\zeta)\) is the effective radial weight after KK summation, scale profile Jacobians, and instrument projection. In the active scaling window one may write
\begin{equation}
\omega(\zeta)=\zeta^{\sigma-1}w(\zeta),
\qquad
w(\zeta)\approx w_0,
\label{eq:qicon-weight}
\end{equation}
with slowly varying \(w_0\). Substituting Eq.~\eqref{eq:qicon-btb-kernel} into Eq.~\eqref{eq:qicon-projected-kernel} gives
\begin{equation}
W_{\Delta}(z,z')
\approx
w_0
\int_{\zeta_{\mathrm{min}}}^{\zeta_{\mathrm{max}}}
\frac{\zeta^{\sigma+2\Delta-1}}{(\zeta^2+\ell^2)^{2\Delta}}\,\dd\zeta.
\label{eq:qicon-kernel-integral}
\end{equation}
Now perform the scale-separating substitution
\begin{equation}
\zeta = \ell u,
\qquad
\dd\zeta = \ell\,\dd u.
\label{eq:qicon-substitution}
\end{equation}
One obtains
\begin{align}
W_{\Delta}(z,z')
&\approx
w_0\,
\ell^{\sigma-2\Delta}
\int_{\zeta_{\mathrm{min}}/\ell}^{\zeta_{\mathrm{max}}/\ell}
\frac{u^{\sigma+2\Delta-1}}{(1+u^2)^{2\Delta}}\,\dd u.
\label{eq:qicon-kernel-after-substitution}
\end{align}
Inside the operational conformal window,
\begin{equation}
\zeta_{\mathrm{min}}\ll \ell \ll \zeta_{\mathrm{max}},
\label{eq:qicon-window-separation}
\end{equation}
the integral in Eq.~\eqref{eq:qicon-kernel-after-substitution} saturates to the finite dimensionless constant
\begin{equation}
C_{\Delta,\sigma}
=
\int_0^\infty\frac{u^{\sigma+2\Delta-1}}{(1+u^2)^{2\Delta}}\,\dd u
=
\tfrac12\,B\!\left(\Delta+\tfrac{\sigma}{2},\;\Delta-\tfrac{\sigma}{2}\right),
\qquad |\sigma|<2\Delta,
\label{eq:qicon-beta-constant}
\end{equation}
the convergence condition and the admissibility bounds below being the same fact seen twice, and therefore
\begin{equation}
W_{\Delta}(z,z')
\propto
\ell^{-(2\Delta-\sigma)}.
\label{eq:qicon-homogeneous-kernel}
\end{equation}
This is the decisive scaling step: after the hidden radial coordinate has been integrated out, the boundary inherits a power-law kernel with no intrinsic scale of its own. The full algebra is collected in Appendix~\ref{app:qicon-radial}.

To compare Eq.~\eqref{eq:qicon-homogeneous-kernel} with standard nonlocal generators, write
\begin{equation}
2\Delta-\sigma = d_\State+\alpha,
\qquad
0<\alpha\le 2.
\label{eq:qicon-alpha-identification}
\end{equation}
The admissibility window \(0<\alpha\le 2\) is forced as follows. The upper bound \(\alpha=2\) is the Gaussian (heat-kernel) limit: there the fractional Laplacian \((-\Delta_g)^{\alpha/2}\) reduces to the ordinary Laplacian, the boundary kernel becomes a Brownian semigroup, and the L\'evy parameter degenerates to a Gaussian \citep{Levy1937}. Values \(\alpha>2\) fail to define a non-negative generator on physical states: the corresponding kernel \(d_g(z,z')^{-(d_\State+\alpha)}\) is more singular at \(z'=z\) than the principal-value integral can absorb, and the resulting operator is no longer dissipative (standard result on Riesz potentials). The lower bound \(\alpha>0\) excludes the degenerate case in which the kernel ceases to be locally integrable and the operator collapses to a constant. In terms of the bulk parameters, the admissibility window translates to \(\Delta>(d_\State+\sigma)/2\) (lower bound) and \(2\Delta-\sigma\le d_\State+2\) (upper bound). Both bounds are routinely respected by the operationally selected branches discussed in the rest of the section. Then
\begin{equation}
W_{\Delta}(z,z')
\propto
\frac{1}{d_g(z,z')^{d_\State+\alpha}},
\label{eq:qicon-fractional-kernel}
\end{equation}
which is exactly the homogeneity required for a fractional jump operator on \(U\).

\begin{proposition}[Homogeneous kernel induced by \OGGEP]
\label{prop:qicon-homogeneous-kernel}
Assume that a \QiEP-selected branch admits an operational window satisfying Eq.~\eqref{eq:qicon-window-separation} and that the projected radial weight is slowly varying there. Then the \OGGEP-induced nonlocal kernel on the corresponding physical boundary patch is asymptotically homogeneous with degree \(-(d_\State+\alpha)\). Consequently the leading nonlocal part of the effective generator is
\begin{equation}
(\mathcal{J}_{\alpha}f)(z)
=
C_{d_\State,\alpha}\,
\operatorname{PV}\!\int_U
\frac{f(z')-f(z)}{d_g(z,z')^{d_\State+\alpha}}
\,\dd\mu_g(z'),
\label{eq:qicon-jalpha}
\end{equation}
which, in a local flat patch, reduces to
\begin{equation}
\mathcal{J}_{\alpha} = -\,D_\alpha\,(-\Delta_g)^{\alpha/2}.
\label{eq:qicon-fractional-laplace}
\end{equation}
\end{proposition}

\begin{proof}
Equation~\eqref{eq:qicon-kernel-after-substitution} already shows that the projected kernel is homogeneous with degree \(-(2\Delta-\sigma)\). Identifying that degree with \(d_\State+\alpha\) gives Eq.~\eqref{eq:qicon-fractional-kernel}. The principal-value integral \eqref{eq:qicon-jalpha} is the unique translation-covariant jump operator with that homogeneity on a local Euclidean patch, and the curved-space generalization is the fractional Laplace--Beltrami operator to leading order in the local chart. Thus the nonlocal sector of the boundary generator has the form \eqref{eq:qicon-fractional-laplace}.
\end{proof}

The proposition explains the geometric origin of the exponent \(\alpha\). It is not introduced phenomenologically by fitting a heavy-tailed random walk to data. It is the boundary image of the hidden-scale integral. Once the selected branch fixes the dominant mode \(\Delta\) and the effective measure exponent \(\sigma\), the exponent \(\alpha\) is fixed.

\section{The leading \texorpdfstring{\QiCON}{Qi-CON} generator and operational conformality}
\label{subsec:qicon-leading-generator}

On the selected branch \(\gamma_*\), the most general projected boundary equation combines drift, local diffusion, and the nonlocal term derived above:
\begin{equation}
\partial_t \rho
=
-\,\nabla_a\!\big(u^a \rho\big)
+
\nabla_a\nabla_b\!\big(D^{ab}\rho\big)
-
D_\alpha(-\Delta_g)^{\alpha/2}\rho
+
\mathcal{S}_{\mathrm{ext}}[\rho].
\label{eq:qicon-full-boundary-equation}
\end{equation}
Here \(u^a\) collects branch-dependent drift or transport, \(D^{ab}\) is the local diffusion tensor, and \(\mathcal{S}_{\mathrm{ext}}\) denotes sources, sinks, or instrument-specific deformations. Equation~\eqref{eq:qicon-full-boundary-equation} is exact enough for interpretation but still too detailed for a clean scaling analysis.

The \QiCON\ window is precisely the regime in which the homogeneous nonlocal sector dominates. This occurs when the observational scale is much larger than the microscopic local-diffusion scale but much smaller than the attractor-confinement scale. In that regime one may write
\begin{equation}
\partial_t \rho
=
-\,D_\alpha(-\Delta_g)^{\alpha/2}\rho
+
\text{sub-leading terms}.
\label{eq:qicon-leading-equation}
\end{equation}
The scale covariance of the leading term is immediate. Let
\begin{equation}
(\mathcal{S}_\lambda f)(z)=f(\lambda z),
\qquad \lambda>0.
\label{eq:qicon-scale-map}
\end{equation}
Then a direct Fourier or coordinate calculation yields
\begin{equation}
\mathcal{S}_\lambda^{-1}
(-\Delta_g)^{\alpha/2}
\mathcal{S}_\lambda
=
\lambda^\alpha
(-\Delta_g)^{\alpha/2},
\label{eq:qicon-fractional-scaling}
\end{equation}
modulo curvature corrections that are suppressed on the local scaling patch. Therefore Eq.~\eqref{eq:qicon-leading-equation} is invariant under the simultaneous transformation
\begin{equation}
z\mapsto \lambda z,
\qquad
t\mapsto \lambda^\alpha t.
\label{eq:qicon-dynamical-scaling}
\end{equation}
This is the core of operational conformality: the selected open-system dynamics possess no preferred length or time scale inside the active interval.

A point-source initial condition makes the scaling explicit. In a local flat patch, Fourier transformation of Eq.~\eqref{eq:qicon-leading-equation} gives
\begin{equation}
\partial_t \widehat{\rho}(k,t)
=
-\,D_\alpha |k|^\alpha \widehat{\rho}(k,t),
\label{eq:qicon-fourier-equation}
\end{equation}
so that
\begin{equation}
\widehat{\rho}(k,t)=
e^{-D_\alpha t |k|^\alpha}\,\widehat{\rho}(k,0).
\label{eq:qicon-fourier-solution}
\end{equation}
For \(\rho(z,0)=\delta(z)\), inverse transformation yields the scaling kernel
\begin{equation}
p_\alpha(t,z)
=
t^{-d_\State/\alpha}
\Phi_\alpha\!\left(
\frac{z}{t^{1/\alpha}}
\right),
\label{eq:qicon-scaling-kernel}
\end{equation}
where \(\Phi_\alpha\) is a dimensionless profile determined by \(\alpha\) and the local metric. Equation~\eqref{eq:qicon-scaling-kernel} is the precise semigroup form of \QiCON\ scaling.

\begin{theorem}[Operational conformality theorem (T-Qi4-11.4)]
\label{thm:qicon-operational-conformality}
Suppose that a \QiEP-selected branch admits the \QiCON\ leading equation \eqref{eq:qicon-leading-equation} on a local physical state patch of dimension \(d_\State\). Then the fundamental solution satisfies the homogeneity law
\begin{equation}
p_\alpha(\lambda^\alpha t,\lambda z)
=
\lambda^{-d_\State}\,
p_\alpha(t,z).
\label{eq:qicon-kernel-homogeneity}
\end{equation}
Consequently, for any scale-covariant Qi-neutral observable \(O\) with effective boundary dimension \(\Delta_O\), the two-point function in the active window obeys
\begin{equation}
\langle O(\lambda z,\lambda^\alpha t)\,O(0,0)\rangle
=
\lambda^{-2\Delta_O}
\langle O(z,t)\,O(0,0)\rangle,
\label{eq:qicon-correlator-scaling}
\end{equation}
and in particular at equal time
\begin{equation}
\langle O(z)\,O(0)\rangle
\propto
|z|^{-2\Delta_O}.
\label{eq:qicon-equal-time-powerlaw}
\end{equation}
\end{theorem}

\begin{proof}
Equation~\eqref{eq:qicon-kernel-homogeneity} follows directly from Eq.~\eqref{eq:qicon-scaling-kernel} by rescaling \(t\) and \(z\). Equation~\eqref{eq:qicon-correlator-scaling} then follows from dimensional covariance of the semigroup acting on observables with effective weight \(\Delta_O\). Setting \(t=0\) gives Eq.~\eqref{eq:qicon-equal-time-powerlaw}.
\end{proof}

The theorem is intentionally phrased in terms of an operational window rather than a global conformal field theory. That is the natural level for \GDOS. The observer does not access the full microscopic theory directly; the observer accesses the homogeneous boundary image of the selected branch over a finite but nontrivial scale range.

\section{Second signature: \texorpdfstring{L\'evy}{Levy} statistics from the fractional generator}
\label{subsec:qicon-levy}

The second signature of \QiCON\ is the appearance of \(\alpha\)-stable coarse statistics. Once the leading generator has the fractional form \eqref{eq:qicon-leading-equation}, the characteristic function of a scalar coarse coordinate follows immediately. Let \(X_t\) denote one local coordinate projection of the boundary state. In the drift-free leading regime its density \(p(x,t)\) satisfies
\begin{equation}
\partial_t p(x,t)
=
-\,D_\alpha(-\partial_x^2)^{\alpha/2}p(x,t).
\label{eq:qicon-1d-fractional}
\end{equation}
Fourier transformation gives
\begin{equation}
\partial_t \widehat{p}(k,t)
=
-\,D_\alpha |k|^\alpha \widehat{p}(k,t),
\label{eq:qicon-1d-fourier}
\end{equation}
with solution
\begin{equation}
\widehat{p}(k,t)=
\exp[-D_\alpha t |k|^\alpha]\,
\widehat{p}(k,0).
\label{eq:qicon-characteristic-general}
\end{equation}
For a localized initial packet \(\widehat{p}(k,0)=1\), the characteristic function becomes
\begin{equation}
\phi_{X_t}(k)
=
\mathbb{E}\!\left[e^{\ii k X_t}\right]
=
\exp[-D_\alpha t |k|^\alpha].
\label{eq:qicon-characteristic}
\end{equation}
This is the defining characteristic function of a symmetric \(\alpha\)-stable law. The full derivation is written out in Appendix~\ref{app:qicon-stable} and agrees with the standard anomalous-diffusion literature \citep{Metzler2000,vanKampen2007}.

Several consequences follow at once.

First, the process is self-similar:
\begin{equation}
X_t
\overset{d}{=}
t^{1/\alpha} X_1.
\label{eq:qicon-selfsimilar}
\end{equation}
Second, for \(0<\alpha<2\) the tails are heavy:
\begin{equation}
p(x,t)
\sim
\frac{D_\alpha t\,\Gamma(1+\alpha)\sin(\pi\alpha/2)}{\pi\,|x|^{1+\alpha}}
\qquad
(|x|\to\infty),
\label{eq:qicon-tail}
\end{equation}
so the coarse increments are L\'evy rather than Gaussian. Third, the Gaussian case appears only as the codimension-one limit \(\alpha=2\), where Eq.~\eqref{eq:qicon-characteristic} becomes \(e^{-D_2 t k^2}\).

\begin{corollary}[Why heavy tails are generic inside \texorpdfstring{\QiCON}{Qi-CON}]
\label{cor:qicon-heavy-tails}
Within an operational \QiCON\ window, the dominant boundary increments are generically heavy-tailed whenever \(0<\alpha<2\). Exact Gaussianity requires \(\alpha=2\), which corresponds to the disappearance of the leading nonlocal contribution.
\end{corollary}

\begin{proof}
Equation~\eqref{eq:qicon-characteristic} is \(\alpha\)-stable for all \(0<\alpha\le 2\). For \(0<\alpha<2\), the inverse Fourier transform has the asymptotic power law \eqref{eq:qicon-tail}; for \(\alpha=2\) the law becomes Gaussian. Hence heavy tails are generic except on the Gaussian boundary.
\end{proof}

Physically, this means that selected open-system trajectories retain rare but consequential excursions across scales. The system does not explore its branch by small equal-sized Brownian steps alone. It mixes local relaxation with occasional larger state-space moves. In an exact mathematical model these are L\'evy flights; in realistic finite systems they are often truncated or tempered by the same cutoffs that define the operational window. The present framework therefore predicts \emph{approximately} \(\alpha\)-stable increments over the active window, not an infinite-scale law.

\section{Third signature: fractality of trajectories and selected supports}
\label{subsec:qicon-fractality}

The third \QiCON\ signature is fractality. This statement should not be treated as a poetic metaphor. It follows from the same self-similarity exponent that controlled the characteristic function.

Let \(R_T\) denote the range of a coarse trajectory on a finite observation interval:
\begin{equation}
R_T = \{X_t \mid 0\le t\le T\}\subset U.
\label{eq:qicon-range}
\end{equation}
To estimate its box-counting dimension, cover \(R_T\) with boxes of side \(\varepsilon\). Because the process is self-similar with exponent \(1/\alpha\), a time interval \(\delta t\) typically produces a spatial excursion of size
\begin{equation}
\delta x \sim (\delta t)^{1/\alpha}.
\label{eq:qicon-deltaxt}
\end{equation}
To resolve the path at spatial resolution \(\varepsilon\), choose \(\delta x\sim \varepsilon\), which implies
\begin{equation}
\delta t \sim \varepsilon^\alpha.
\label{eq:qicon-deltat}
\end{equation}
An observation window of total duration \(T\) therefore contains approximately
\begin{equation}
N_T(\varepsilon)\sim \frac{T}{\delta t}\sim T\,\varepsilon^{-\alpha}
\label{eq:qicon-covering-count-raw}
\end{equation}
independent \(\varepsilon\)-scale segments. This estimate cannot exceed the number of \(\varepsilon\)-boxes available in the ambient physical patch, which scales as \(\varepsilon^{-d_\State}\). Therefore the effective box-counting dimension of the range is
\begin{equation}
D_f = \min(d_\State,\alpha).
\label{eq:qicon-fractal-dimension}
\end{equation}
The derivation is elementary but important enough that it is recorded separately in Appendix~\ref{app:qicon-fractal}; it is also consistent with the standard fractal viewpoint \citep{Mandelbrot1982}.

Equation~\eqref{eq:qicon-fractal-dimension} immediately explains why \QiCON\ belongs in a theory of selected open systems rather than in a theory of unrestricted random walks. The ambient manifold dimension \(d_\State\) sets the maximal available space, while the exponent \(\alpha\) sets the intrinsic scaling of the selected path. The realized fractal support is the lesser of the two. When \GSSB\ further constrains the branch to a submanifold or attractor sheet, the effective support dimension can only decrease. Thus the fractality seen by the observer is not merely a property of the transport kernel; it is also a property of branch selection and attractor confinement.

\begin{proposition}[Fractal support of a selected \texorpdfstring{\QiCON}{Qi-CON} branch]
\label{prop:qicon-fractal-support}
Let a \QiEP-selected branch possess a \QiCON\ window with exponent \(0<\alpha\le 2\) on an ambient physical patch of dimension \(d_\State\). Then the coarse trajectory range satisfies the box-counting estimate
\begin{equation}
N(\varepsilon)\sim \varepsilon^{-D_f},
\qquad
D_f=\min(d_\State,\alpha),
\label{eq:qicon-fractal-support}
\end{equation}
over \(\ell_{\mathrm{micro}}<\varepsilon<L_{\mathrm{att}}\). If \GSSB\ restricts the branch to an attractor support \(A\subset U\), then the observable support dimension satisfies \(D_A\le D_f\).
\end{proposition}

\begin{proof}
The range estimate follows from Eqs.~\eqref{eq:qicon-deltaxt}--\eqref{eq:qicon-fractal-dimension}. The second statement follows because any attractor support \(A\) visited by the selected branch is a subset of the ambient range and therefore cannot have a larger box-counting dimension than the range itself.
\end{proof}

A useful way to phrase the result is that \QiCON\ predicts \emph{fractal admissibility}. The selected branch need not fill the ambient state space densely; indeed it usually does not. It occupies a support whose effective dimension is determined jointly by the scale exponent and by the symmetry-breaking constraints that shaped the branch.

\begin{table}[H]
\centering
\caption[One exponent, read three ways.]{How a single exponent organizes the first three \QiCON{} signatures. The semigroup kernel, the characteristic function and the fractal support occupy the three rows, with the scaling law and the interpretation alongside. The table is the reason the signatures are not independent evidence: the same $\alpha$ appears in all three, so a measurement that fixes it in one row fixes it in the others, and a framework claiming all three is making one claim rather than three.}
\label{tab:qicon-alpha-dictionary}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.36\textwidth}}
\toprule
\textbf{Object} & \textbf{Scaling law} & \textbf{Interpretation} \\
\midrule
Semigroup kernel & \(p_\alpha(t,z)=t^{-d_\State/\alpha}\Phi_\alpha(z/t^{1/\alpha})\) & Operational conformality and absence of intrinsic scale in the active window. \\
Characteristic function & \(\phi_{X_t}(k)=e^{-D_\alpha t|k|^\alpha}\) & \(\alpha\)-stable L\'evy statistics for coarse increments. \\
Fractal support & \(D_f=\min(d_\State,\alpha)\) & Non-integer effective dimension of selected trajectory ranges or supports. \\
\bottomrule
\end{tabular}
\end{table}

\section{Fourth signature: \texorpdfstring{\(1/f\)}{1/f} spectra with two knees}
\label{subsec:qicon-1overf}

The fourth \QiCON\ signature is the most directly measurable in many applications: a broad spectral band of the form \(1/f^\beta\), bounded by two characteristic knees. The derivation proceeds most transparently from the spectral representation of the selected branch generator.

Let \(X(t)\) be a centered observable measured on the selected branch. Write its autocorrelation function as a mode expansion over the nonnegative spectrum of the reduced generator:
\begin{equation}
C_X(\tau)
=
\langle X(t+\tau)X(t)\rangle
=
\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}
A_X(\lambda)\,
e^{-\lambda |\tau|}
\,\rho(\lambda)\,\dd\lambda.
\label{eq:qicon-correlation-spectral}
\end{equation}
Here \(\rho(\lambda)\) is the density of relaxation modes, \(A_X(\lambda)\) is the observable-dependent spectral weight, \(\lambda_{\mathrm{IR}}\) is the slowest relevant mode, and \(\lambda_{\mathrm{UV}}\) is the fastest mode retained by the operational description. Inside the \QiCON\ window, the same homogeneity that produced the fractional operator implies a power-law soft-mode density:
\begin{equation}
A_X(\lambda)\rho(\lambda)
\approx
A_0\,\lambda^{\gamma-1},
\qquad
0\le \gamma < 1,
\label{eq:qicon-softmode-density}
\end{equation}
for \(\lambda_{\mathrm{IR}}<\lambda<\lambda_{\mathrm{UV}}\). The exponent \(\gamma\) is not phenomenological: it is fixed by Weyl asymptotics of the fractional Laplace--Beltrami operator on a \(d_\State\)-dimensional manifold. Indeed, the eigenvalues \(\lambda_n\) of the standard Laplacian on a compact \(d_\State\)-manifold satisfy \(\lambda_n\sim n^{2/d_\State}\), so that the eigenvalue counting function \(N(\lambda)\sim\lambda^{d_\State/2}\) and the density \(\rho_{\Delta_g}(\lambda)\sim\lambda^{d_\State/2-1}\) (Weyl's law). The fractional operator \((-\Delta_g)^{\alpha/2}\) acts on the same eigenspaces with eigenvalues \(\lambda_n^{\alpha/2}\); changing variables yields the fractional-operator density
\begin{equation}
\rho_{\alpha}(\lambda)
\sim
\lambda^{d_\State/\alpha-1}.
\label{eq:qicon-fractional-density}
\end{equation}
If the observable spectral weight scales as \(A_X(\lambda)\sim\lambda^{\beta-d_\State/\alpha}\) on the soft modes (the natural scaling for any observable that couples through the gradient of the fractional operator), then \(A_X(\lambda)\rho_\alpha(\lambda)\sim\lambda^{\beta-1}\), identifying \(\gamma=\beta\) in Eq.~\eqref{eq:qicon-softmode-density}. The nearly critical case \(\gamma\approx 0\) corresponds to \(\beta\approx 0\), i.e.\ an observable that couples uniformly across the soft-mode spectrum --- the universal generic case for memory-bearing systems, and the one that leads to true \(1/f\) behavior.

The power spectral density is
\begin{equation}
S_X(\omega)
=
2\int_0^\infty C_X(\tau)\cos(\omega\tau)\,\dd\tau.
\label{eq:qicon-psd-def}
\end{equation}
Substituting Eq.~\eqref{eq:qicon-correlation-spectral} and evaluating the Laplace transform of each exponential mode yields
\begin{equation}
S_X(\omega)
=
2\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}
\frac{\lambda\,A_X(\lambda)\rho(\lambda)}{\lambda^2+\omega^2}
\,\dd\lambda.
\label{eq:qicon-psd-spectral}
\end{equation}
Using the power-law density \eqref{eq:qicon-softmode-density}, one finds
\begin{align}
S_X(\omega)
&\approx
2A_0
\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}
\frac{\lambda^\gamma}{\lambda^2+\omega^2}\,\dd\lambda
\nonumber\\
&=
2A_0\,
\omega^{\gamma-1}
\int_{\lambda_{\mathrm{IR}}/\omega}^{\lambda_{\mathrm{UV}}/\omega}
\frac{u^\gamma}{1+u^2}\,\dd u,
\qquad
u=\frac{\lambda}{\omega}.
\label{eq:qicon-psd-scaled}
\end{align}
Now the three frequency regimes separate cleanly.

\paragraph{Intermediate band: the \(1/f^\beta\) law.}
If
\begin{equation}
\lambda_{\mathrm{IR}} \ll \omega \ll \lambda_{\mathrm{UV}},
\label{eq:qicon-middle-band}
\end{equation}
the integration limits in Eq.~\eqref{eq:qicon-psd-scaled} may be extended to \(0\) and \(\infty\), giving a constant dimensionless integral:
\begin{equation}
\int_0^\infty \frac{u^\gamma}{1+u^2}\,\dd u
=
\frac{\pi}{2}\sec\!\left(\frac{\pi\gamma}{2}\right),
\qquad
0\le \gamma<1.
\label{eq:qicon-middle-constant}
\end{equation}
Therefore
\begin{equation}
S_X(\omega)
\propto
\omega^{\gamma-1}
=
\frac{1}{\omega^\beta},
\qquad
\beta = 1-\gamma.
\label{eq:qicon-beta}
\end{equation}
The nearly critical case \(\gamma\approx 0\) gives \(\beta\approx 1\), i.e.\ genuine \(1/f\) behavior.

\paragraph{Low-frequency side: the infrared knee.}
If \(\omega\ll \lambda_{\mathrm{IR}}\), then \(\lambda^2+\omega^2\approx \lambda^2\), so
\begin{equation}
S_X(\omega)
\approx
2A_0
\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}
\lambda^{\gamma-2}\,\dd\lambda
=
\text{constant}.
\label{eq:qicon-lowfreq}
\end{equation}
Thus the spectrum flattens at low frequency. In ordinary language, there is a largest effective relaxation time beyond which the selected branch behaves as a finite system rather than as a scale-free one.

\paragraph{High-frequency side: the ultraviolet knee.}
If \(\omega\gg \lambda_{\mathrm{UV}}\), then \(\lambda^2+\omega^2\approx \omega^2\), so
\begin{equation}
S_X(\omega)
\approx
\frac{2A_0}{\omega^2}
\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}
\lambda^\gamma\,\dd\lambda
\propto
\omega^{-2}.
\label{eq:qicon-highfreq}
\end{equation}
Thus the spectrum steepens above the active window and leaves the scale-free regime.

These three regimes may be written compactly as
\begin{equation}
S_X(f)
\approx
\begin{cases}
S_0, & f<f_{\mathrm{IR}},\\[6pt]
A\,f^{-\beta}, & f_{\mathrm{IR}}<f<f_{\mathrm{UV}},\\[6pt]
B\,f^{-2}, & f>f_{\mathrm{UV}},
\end{cases}
\label{eq:qicon-two-knee}
\end{equation}
with
\begin{equation}
f_{\mathrm{IR}}=\frac{\lambda_{\mathrm{IR}}}{2\pi},
\qquad
f_{\mathrm{UV}}=\frac{\lambda_{\mathrm{UV}}}{2\pi}.
\label{eq:qicon-knees}
\end{equation}
The detailed algebra is given in Appendix~\ref{app:qicon-psd}. The key point is that the two knees are not arbitrary fitting decorations. They are forced by the existence of both an attractor-scale cutoff and a microphysical cutoff.

\begin{theorem}[Two-knee spectral theorem (T-Qi4-11.7)]
\label{thm:qicon-two-knee}
Suppose a \QiEP-selected branch possesses a \QiCON\ window with soft-mode density \eqref{eq:qicon-softmode-density}. Then the power spectral density of any observable with nonvanishing overlap on that soft sector obeys Eq.~\eqref{eq:qicon-two-knee}. The infrared knee is fixed by the longest branch or attractor relaxation time, and the ultraviolet knee is fixed by the shortest retained microscopic time.
\end{theorem}

\begin{proof}
Equations~\eqref{eq:qicon-psd-spectral}--\eqref{eq:qicon-highfreq} already provide the three asymptotic regimes. The infrared plateau is controlled by the lower modal cutoff \(\lambda_{\mathrm{IR}}\), while the ultraviolet roll-off is controlled by the upper cutoff \(\lambda_{\mathrm{UV}}\). Translating to ordinary frequency gives Eq.~\eqref{eq:qicon-knees}.
\end{proof}

This theorem is central because it converts a broad and often qualitative empirical observation --- ``many open systems show \(1/f\)-like spectra'' --- into a structural consequence of the selected branch geometry. The scale-free band is not universal by accident. It appears because the selected branch carries a dense soft-mode sector between a largest confinement scale and a smallest microphysical scale.

\begin{table}[H]
\centering
\caption[What the two spectral knees mean.]{Physical meaning of the two \QiCON{} knees. The infrared and ultraviolet knees occupy the rows, with the mathematical origin of each and its physical interpretation alongside. The two-knee structure is what separates this prediction from the observation of $1/f$ noise, which is ubiquitous and explains little: an unbounded slope is compatible with almost anything, whereas a band bounded at both ends by scales the framework computes is falsifiable, and the second column says what sets each end.}
\label{tab:qicon-knees}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Knee} & \textbf{Mathematical origin} & \textbf{Physical interpretation} \\
\midrule
Infrared knee \(f_{\mathrm{IR}}\) & Lowest retained relaxation rate \(\lambda_{\mathrm{IR}}\) & Largest branch or attractor timescale; beyond this the finite selected system behaves as a bounded object rather than a scale-free one. \\
Ultraviolet knee \(f_{\mathrm{UV}}\) & Highest retained relaxation rate \(\lambda_{\mathrm{UV}}\) & Fastest microscopic or instrument-resolved process; above this the effective coarse description breaks down and the spectrum steepens. \\
\bottomrule
\end{tabular}
\end{table}

\section{Why the \texorpdfstring{\QiCON}{Qi-CON} exponents are runtime invariants}
\label{subsec:qicon-runtime-invariants}

The four signatures derived above do not float independently of the master runtime: they are constrained by \DGI,\ \GSSB,\ \GA,\ \QiQUA,\ and \QiEP. \DGI{} acts on the effective generator by similarity (gauge conjugation), so the exponents \(\alpha,\gamma,\beta\) and the knee locations are spectral invariants of the selected branch. \GSSB{} fixes the universality class: different broken branches carry their own exponent sets \((\alpha_i,\gamma_i,\beta_i,D_{f,i},f_{\mathrm{IR},i},f_{\mathrm{UV},i})\). \GA{} fixes the infrared knee through \(f_{\mathrm{IR}}\sim 1/\tau_{\mathrm{GA}}\), tying it to the slowest attractor relaxation. \QiQUA{} delivers spectral locking: in the ideal locked regime \(J_E/\QB=J_e=J_B=J_Q\) implies \(S_{J_E}(f)/\QB^2 = S_{J_e}(f)=S_{J_B}(f)=S_{J_Q}(f)\) across any common \QiCON\ window, with realistic overhead leaving the slopes \(\beta\) common. Finally, \QiEP{} supplies the selector that picks which spectrum is actually observed. This ordering---budget \(\rightarrow\) selector \(\rightarrow\) diagnostic---is why the pillars cannot be permuted. Table~\ref{tab:qicon-runtime-relations} records the one-row-per-construct summary.

\begin{table}[H]
\centering
\caption[How the earlier layers constrain \QiCON{}.]{How the earlier runtime layers constrain \QiCON{}. Five earlier layers run down the first column --- the three cascade stages and the two preceding pillars --- with the constraint each imposes and the consequence for the scale-coherence statement. The table is the pillar's dependency surface: it inherits rather than assumes, and a reader auditing whether the third pillar is independent of the first two should read the middle column, which says exactly where it is not.}
\label{tab:qicon-runtime-relations}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}p{0.45\textwidth}}
\toprule
\textbf{Earlier layer} & \textbf{Constraint on \QiCON} & \textbf{Consequence} \\
\midrule
\DGI & Similarity-covariant generator and gauge-invariant observables & Scaling exponents are branch invariants, not gauge artifacts. \\
\GSSB & Branch and support selection & Different broken branches can carry different exponent sets and knees. \\
\GA & Attractor persistence and return times & The infrared knee is fixed by the slowest attractor relaxation mode. \\
\QiQUA & Common current denomination & Sector-resolved amplitudes and cross-spectral ratios are tied by the Arisaka Equation. \\
\QiEP & Extremal branch selection & The observed spectrum is the spectrum of the selected branch. \\
\bottomrule
\end{tabular}
\end{table}


\section{Why \texorpdfstring{\QiCON}{Qi-CON} applies to all open systems}
\label{subsec:qicon-universal-applicability}

The universality of \QiCON\ now follows from a simple structural observation. Any open system with
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item a nontrivial separation between microscopic and macroscopic scales,
\item a selected persistent branch or attractor,
\item repeated throughput of energy, matter, entropy, or information,
\item and observer access only to a projected coarse state
\end{enumerate}
is a candidate for an operational \QiCON\ window. No special reference to particles, chemistry, or brains is needed at this level. The statement is geometric and statistical.

This does \emph{not} mean that every dataset should exhibit exact power laws over arbitrarily many decades. Finite-size effects, external forcing, insufficient scale separation, coarse instrument bandwidth, and branch switching can all shorten or obscure the active window. The point is subtler and more useful: whenever a selected open system possesses a clear intermediate scale regime, the natural generic diagnostic of that regime is the \QiCON\ quartet --- power-law correlations, heavy tails, fractal support, and a two-knee \(1/f^\beta\) spectrum.

A domain table makes the statement concrete.

\begin{table}[H]
\centering
\caption[Expected \QiCON{} readouts by domain.]{Illustrative cross-domain readings of \QiCON{}. Five domains run down the first column, with the typical coarse observable in each, the knees one would expect to find, and the expected readout. As with Table~\ref{tab:qiep-domain-uses}, these are placements rather than predictions --- the paper is explicit that it does not dress a retrodiction of $1/f$ noise as a confirmation --- and the value of the table is that it says in advance where the framework expects to find \emph{two} knees rather than one.}
\label{tab:qicon-domain-uses}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Domain} & \textbf{Typical coarse observable} & \textbf{Likely knees} & \textbf{Expected \QiCON\ readout} \\
\midrule
Transport or particle-flow systems & Arrival-time, channel current, branch occupancy & Device size / microscopic scattering time & Power-law transit statistics and scale-covariant throughput fluctuations. \\
Molecular machines & Dwell time, rotational step interval, cycle current & Whole-cycle return time / chemical substep time & Heavy-tailed dwell distributions and broad intermediate spectral band. \\
Cells and reaction networks & Flux, burst size, concentration time series & Homeostatic relaxation / reaction microtime & Fractal switching, broad throughput spectra, and correlated fluctuations across scales. \\
Brains and cognitive systems & Field potentials, avalanche size, decision-variable trajectory & Behavioral or theta-cycle time / synaptic or cellular time & \(1/f\)-like spectra with two knees, heavy-tailed event statistics, and scale-free internal dynamics. \\
Ecological and social systems & Population change, traffic, transaction or communication counts & Institutional or seasonal time / individual-event time & Broad multiscale fluctuation windows with truncated heavy tails and knee-bounded spectra. \\
\bottomrule
\end{tabular}
\end{table}

The table is intentionally broad because broadness is the point. \QiCON\ is not a domain-specific fit function. It is a universal diagnostic grammar for selected open-system branches.

\section{Why two knees are worth more than one slope}
\label{sec:close-qicon}

There is a way that a theory can be popular and useless at the same
time, and $1/f$ noise is the standard example.

It is everywhere. It has been reported in semiconductors, in nerve
cells, in the flow of the Nile, in traffic, in music, in the timing of
human keystrokes. Dozens of mechanisms generate it. And that is exactly
why finding it in a new place is worth very little: a prediction that
something will show a power law is compatible with an enormous number of
underlying stories, so confirming it eliminates almost nothing. A theory
that predicted $1/f$ noise and stopped there would be collecting
agreement without ever risking anything.

The move this Part makes is to give the band edges. A slope with no ends
can be fitted to almost anything; a band with two ends, each fixed by a
quantity the framework identifies independently, can be wrong in several
distinct ways at once. It can be wrong because there is only one turn.
It can be wrong because the lower turn does not sit at the lifetime of
the state the system settled into. It can be wrong because the upper
turn does not sit at the fastest process the system still resolves. It
can be wrong because the exponent in the band, the shape of the jump
distribution, and the dimension of the set the system explores disagree
about the single number that is supposed to control all three.

Underneath that is a point about how much work one number is being
asked to do. This Part does not offer four predictions. It offers one,
wearing four costumes. The steepness of the spectrum inside the band,
the weight of the tail on the distribution of jumps, the roughness of
the set the system actually wanders over, and the placement of the two
turns are not four independent findings that happened to be made in
the same laboratory. They descend from a single exponent. That is a
dangerous position to occupy and it is occupied deliberately, because
four quantities controlled by one number can be checked against each
other by somebody who does not believe a word of the theory that
produced them. A framework whose predictions were independent could
lose one and keep the others, and would quietly do so. This one
cannot. Untie any pair of these and the whole comes apart.

That is the difference between a framework that accommodates a
phenomenon and one that is exposed to it. Accommodation is cheap and
this Part deliberately declines it: where the domain tables place known
observations from other fields, they are placements, and nothing here
treats a retrodiction of $1/f$ noise as a confirmation of anything.

There is something in the bare idea of scale covariance worth pausing
over before leaving this Part. To say that a system looks the same at
many scales is to say that it does not carry the information of how
big it is. Most things carry that information and cannot help it: a
table has a length, an atom has a radius, and neither can be mistaken
for a larger or smaller version of itself. What is unusual about the
systems this framework is about is that between their two turns they
forget their own size --- and it is exactly inside that forgetful
interval that they are able to do the things that make them worth
studying. A structure with a preferred scale can only ever be that
structure. A structure without one can be a template, and a template
is what gets used again at a size nobody planned for.

A closing thought about what the two turns actually mean. They are the
two ends of a system's world --- the slowest thing it still is, and the
fastest thing it can still tell apart. Everything between them is where
that system lives, and the width of that interval is a measure of how
much room it has to be complicated in. The next Part will use exactly
that width to say how deep a memory such a system can build. It is an
unusually direct link between something an instrument reads off a
spectrum and something architectural about the thing being measured, and
it is the kind of link a framework has to produce if it is going to earn
the reach it claims.


\GGPart[Qi-HAL]{\QiHAL{}}{Memory architecture: \MePMoS{}, \NHT{}, \HAL{}, \BFFT{}, and the \GRIP{}-loop}
\label{part:qihal}

\noindent
The moment a system holds on to something for longer than it takes to
act and look again, it stops being merely a thing that reacts and starts
being a thing with a past. This Part is about what that costs and what
it forces.

The entrance condition is exactly that comparison of two times: how long
a trace survives, against how long one cycle of acting and observing
takes. Below it, memory cannot guide anything, because whatever was
stored is gone before it could be used. Above it, the Part shows, the
system's runtime falls into a definite shape --- a loop that runs from
memory to prediction to motion to sensing and back; a code that stores
relationships between things rather than the things themselves; a
lattice rather than a filing cabinet; and a ladder of nested timescales
whose depth is fixed by the spectral window of the previous Part.

The order of that loop is not a modeling choice. Put sensing before
action and the system would have to know an observation it has not yet
caused. Put prediction after sensing and prediction becomes hindsight.
Put the memory update before the new measurement arrives and the
error-correction step disappears. Each rearrangement destroys something,
which is why the order comes out the way it does.

Two consequences deserve to be seen before the derivations. Because what
is stored is a set of differences between phases rather than absolute
values, a memory written on one carrier can be read out on a completely
different one --- which is the precise reason a chemical system, a
neural system and a digital one can hold the same content on utterly
different hardware. And because the content is spread across relations
rather than filed in places, a damaged substrate can still reconstruct
part of what it held, which is a familiar fact about brains given a
mechanism here.

Memory is never free in this account. A write is an elementary event,
and the ledger of the first pillar bills it. That is the sentence that
rules out the comfortable fiction of information as something symbolic
floating above the physics.

One note on the strength of the claim, because the Part is careful about
it and the introduction should be too. What is established is a
canonical normal form --- the shape a memory-bearing system's runtime
takes --- rather than a proof that no other shape could exist. The Part
says so itself, and explains why the stronger argument would be circular.
Its falsifier is correspondingly concrete: a memory-bearing system that
cannot be decomposed into these parts would end it.


\label{sec:qihal}

After diagnostics comes architecture. It is not enough to know the ledger, the selector, and the signatures; one must also know what persistent memory-bearing organization can actually implement them. Part~\ref{part:qihal} therefore closes the \QiFour{} block by deriving the architectural layer -- \MePMoS, \NHT, \HAL, \BFFT, and the exploration-to-commitment switch -- that makes the earlier pillars realizable in open systems with memory.

\subsection{Historical background (with citations)}

Antecedents: Hebb (1949), Hopfield (1982), Marr (1971), 't~Hooft (1993), Susskind (1995), Maldacena (1998), O'Keefe (1971), Buzs\'aki (2006), Tononi (2004) \citep{McCullochPitts1943,Hebb1949,Hopfield1982,tHooft1993,Susskind1995,Maldacena1998,Tononi2004,Friston2010}; \QiHAL{} subsumes these as gauges or limiting cases of Theorem~T-HAL.1 $\cup$ T-HAL.2 (the \MePMoS{}/\NHT{}/\HAL{}/\BFFT{} memory architecture is the boundary projection of the 6D bulk and its phase-holographic coding; see A1 Sec.~7 and A4 Sec.~6 for full history).

\paragraph*{Dual-layer architecture and the compatibility relation $\mathfrak{C}_{\mathrm{\HAL{}}}$.}
\QiHAL{} is, in the L16 language, a dual-layer principal pillar. Its
\emph{algebraic layer} (T-\HAL{}.1) is the \QiHAL{} universality theorem
(Theorem~\ref{thm:qihal-universality}): the named hierarchy
\MePMoS{}-\NHT{}-\HAL{}-\BFFT{} is a sufficient boundary realization of
the \GRIP{} cascade on the family-projector + memory moduli, with the
$U(1)_{Qi}$ gauge sector as the memory carrier. Its
\emph{operational/physical layer} (T-\HAL{}.2) is the body of empirical
realizations: the Hopfield network \citep{Hopfield1982}, the place
cells of \citet{OKeefeDostrovsky1971}, the \emph{Drosophila} ring
attractor of \citet{TurnerEvans2020}, and the thalamocortical
forward-model architecture of Llinás 2001. The two layers are tied
together by the named compatibility relation
\begin{equation}
\label{eq:CHAL-compat}
\resizebox{\textwidth}{!}{$\displaystyle
\boxed{\;\mathfrak{C}_{\mathrm{\HAL{}}}\;\equiv\;
\text{T-\HAL{}.1}\;\Longrightarrow\;\text{T-\HAL{}.2}
\quad\text{(via the $U(1)_{Qi}$ memory carrier and the \GRIP{} cascade on persistent memory moduli).}\;}
$}
\end{equation}
The universality theorem at the algebraic level forces any
sufficiently persistent open system with informational currents to
implement the five-layer hierarchy; the empirical realizations across
neuroscience (place cells, ring attractors, thalamocortical loops)
are then the boundary instances of this structural necessity.


\begin{figure}[htbp]
\centering
\resizebox{0.792\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=3pt, inner sep=4pt},
 stage/.style={fill=violet!12, draw=violet!60!black, line width=0.75pt,
 minimum width=34mm, minimum height=13mm},
 motif/.style={fill=teal!12, draw=teal!60!black, line width=0.60pt,
 minimum width=46mm, minimum height=13mm},
 gatebox/.style={fill=yellow!14, draw=olive!70!black, line width=0.70pt,
 minimum width=64mm, minimum height=11mm},
 billbox/.style={fill=orange!14, draw=orange!65!black, line width=0.70pt,
 minimum width=64mm, minimum height=11mm},
 loop/.style={-{Stealth[length=2.6mm]}, line width=1.10pt,
 draw=violet!60!black},
 tie/.style={-{Stealth[length=2.2mm]}, line width=0.55pt, draw=black!60,
 dash pattern=on 2pt off 2pt},
 why/.style={font=\sffamily\scriptsize\itshape, text=black!65,
 fill=white, inner sep=1.5pt},
]
% the gate
\node[gatebox] (gate) at (0,34mm)
 {\textbf{Entrance condition}\quad $\tau_m > T_c$\\[-1pt]
  \footnotesize the trace outlives one cycle of acting and looking};

% the loop, clockwise
\node[stage] (mem) at (0,16mm)  {\textbf{Memory}};
\node[stage] (pre) at (36mm,0)  {\textbf{Prediction}};
\node[stage] (mot) at (0,-16mm) {\textbf{Motion}};
\node[stage] (sen) at (-36mm,0) {\textbf{Sensing}};

\draw[loop] (mem) to[bend left=22] (pre);
\draw[loop] (pre) to[bend left=22] (mot);
\draw[loop] (mot) to[bend left=22] (sen);
\draw[loop] (sen) to[bend left=22] (mem);

\node[why] at (30mm,12mm) {cannot follow sensing:\\ that is hindsight};
\node[why] at (-31mm,-13mm) {cannot precede action:\\ the observation\\ is not yet caused};

% the three motifs, in a row beneath the loop
\node[motif] (nht) at (-52mm,-34mm)
 {\textbf{Phase code}\\[-1pt] \footnotesize relations, not snapshots ---\\
  \footnotesize so any carrier will do};
\node[motif] (hal) at (0,-34mm)
 {\textbf{Attractor lattice}\\[-1pt] \footnotesize distributed, so damage\\
  \footnotesize degrades rather than deletes};
\node[motif] (bfft) at (52mm,-34mm)
 {\textbf{Nested timescales}\\[-1pt] \footnotesize depth fixed by the width\\
  \footnotesize of the two-knee window};
\draw[tie] (sen.south) -- (nht.north);
\draw[tie] (mot.south) -- (hal.north);
\draw[tie] (pre.south) -- (bfft.north);

% the bill
\node[billbox] (bill) at (0,-52mm)
 {\textbf{Every write is an elementary event}\\[-1pt]
  \footnotesize and the first pillar's ledger bills it: memory is never free};
\draw[tie] (hal.south) -- (bill.north);
\end{tikzpicture}%
}
\caption[The memory loop, and why its order is forced.]{The architecture a
system falls into once it keeps a trace for longer than a single cycle of
acting and looking --- the entrance condition at the top, which is a
comparison of two times and nothing more. The loop's order is not a modeling
choice: sensing cannot come before action without the system knowing an
observation it has not yet caused, and prediction cannot come after sensing
without becoming hindsight, so each rearrangement destroys a step that has
work to do. Three motifs hang off the loop. What is stored is a set of
relations between phases rather than the phases themselves, which is why the
same content survives being carried on a completely different rhythm and why
a chemical, a neural and a digital system can hold it on unrelated hardware.
The store is a lattice rather than a filing cabinet, which is why damage
degrades a memory instead of deleting a list. And the ladder of nested
timescales is only as deep as the spectral window of the preceding Part
allows. At the foot: a write is an elementary boundary event, so the ledger
of the first pillar prices it. What is established is a canonical normal
form rather than a proof that no other shape could exist.}
\label{fig:a7-mepmos}
\end{figure}

\section{Constitutional statement of the \texorpdfstring{\QiHAL}{Qi-HAL} principle}
\label{subsec:qihal-constitutional}

The architectural claim of \QiHAL\ begins with a minimal observation: an open system counts as more than a passive dissipator only if part of its present state depends on past interactions in a way that remains functionally available for future action. This is the generic notion of memory, and it is intentionally broad enough to cover synapses, gene-regulatory marks, chemical modification states, conformational cycles, trail networks, field configurations, and externalized collective memory.

\begin{definition}[Memory-bearing open system]
\label{def:memory-bearing}
An admissible open system with state packet
\[
\Gamma(t)=\bigl(x^A(t),J^\alpha(t),\Ports,\Obs\bigr)
\]
is called \emph{memory-bearing} if there exists an internal or external state variable \(m(t)\in\mathcal{M}\) and a causal kernel \(K_m(\tau)\) with support on \(\tau\ge 0\) such that
\begin{equation}
\label{eq:qihal-memory-kernel}
m(t)=\int_0^{\infty} K_m(\tau)\,\Xi\bigl(x(t-\tau),s(t-\tau),a(t-\tau)\bigr)\,\dd\tau,
\end{equation}
for some local update functional \(\Xi\), and such that the associated retention time
\begin{equation}
\label{eq:qihal-memory-time}
\tau_m:=
\frac{\int_0^{\infty}\tau\,\norm{K_m(\tau)}\,\dd\tau}{\int_0^{\infty}\norm{K_m(\tau)}\,\dd\tau}
\end{equation}
exceeds the characteristic observation-to-action cycle time \(T_c\) of the system:
\begin{equation}
\label{eq:qihal-memory-condition}
\tau_m>T_c.
\end{equation}
\end{definition}

Definition~\ref{def:memory-bearing} is intentionally operational. It does not assume that memory is neural, digital, or symbolic. It requires only that the system carry forward a compressed trace of earlier interactions long enough for that trace to matter. Equation~\eqref{eq:qihal-memory-condition} is the real entrance condition: without persistence beyond one cycle, memory cannot guide prediction or action.

The second ingredient is the informational gauge symmetry introduced earlier. In the present program a memory trace is not an abstract bookkeeping label added from outside; once \(U(1)_{Qi}\) is present, informational events become physical current-carrying events. This promotes memory from a descriptive convenience to a gauge-bearing dynamical sector.

\emph{A note on why gauge structure is the right framework for memory.} Any quantity that propagates in time and admits a phase ambiguity --- and a memory trace is exactly such a quantity, since its absolute phase carries no observable content --- must, if it is to be physical, be expressed in gauge-invariant relational form: as a phase \emph{difference} corrected by holonomy. This is the standard reasoning that makes electromagnetic potentials gauge-invariant only modulo their gradient. Applied here, it makes phase-coded memory compatible with the Arisaka ledger of \QiQUA{}. Readers from biology may think of this as the precise way to say that a memory's content is what it differs from, not what it absolutely is; readers from neuroscience will recognize it as the formal version of the cocktail-party intuition that the relevant signal is always relative to a reference.

The constitutional data of the informational carrier are a real pair: a real density \(\rho_Q\ge 0\), the local informational intensity, and a real, circle-valued phase \(\phi_Q\), an angle on the compact \(U(1)_{Qi}\) memory circle. This pair is what the memory sector stores --- a real angle on a real circle is the minimal memory register --- and nothing complex is postulated to hold it. Because \(U(1)_{Qi}\) is a compact bulk gauge circle (the four-component \BIsix{} sector \(F_{Qi}\) of Sec.~\ref{subsec:paper2-recap}), the circle's quarter-turn is available as the unique circle-compatible complex structure on \(Qi\)-charged sectors --- the derived gauge \(\ii\) of T-P4-1 of the Foundations Branch \citep{ArisakaP4Real}; the causal clock supplies the state-space \(\ii\) (T-P4-2), and the handshake forces the two to agree on shell (T-P4-3). Under that derived quarter-turn the real pair renders locally as the single complex carrier
\begin{equation}
\label{eq:qihal-psi}
\Psi_Q(x,t)=\sqrt{\rho_Q(x,t)}\,\e^{\ii\phi_Q(x,t)},
\end{equation}
the Madelung packaging of \((\rho_Q,\phi_Q)\): its \(\ii\) is the boundary's shorthand for the derived quarter-turn, not a primitive, and its modulus and argument return the constitutional pair \((\sqrt{\rho_Q},\phi_Q)\). The polar dressing is convenience; the reality beneath it is the real density and the real circle-valued phase. This is A1's register discipline for the Hilbert space \(\mathcal{H}\) applied to the memory sector: the complex scalars are boundary shorthand, priced at candidate grade by L-Constitution-1 and recorded in the AX-3 register note of \citet{ArisakaA1Constitution}. A local \(U(1)_{Qi}\) gauge transformation with parameter \(\Lambda(x,t)\) is then a real shift of the stored angle, \(\phi_Q\mapsto\phi_Q+q_Q\Lambda\), which the dressing renders as
\begin{equation}
\label{eq:qihal-gauge-transformation}
\Psi_Q\mapsto \e^{\ii q_Q\Lambda}\Psi_Q,
\qquad
A^{(Qi)}_\mu\mapsto A^{(Qi)}_\mu+\partial_\mu\Lambda,
\end{equation}
so the covariant derivative, whose \(\ii\) is again the derived quarter-turn, is
\begin{equation}
\label{eq:qihal-covariant-derivative}
D_\mu \Psi_Q=(\partial_\mu-\ii q_Q A^{(Qi)}_\mu)\Psi_Q.
\end{equation}

The crucial consequence is that the physically meaningful memory token cannot be an absolute phase. It must be a relational, gauge-invariant quantity. The minimal such token is a phase difference corrected by the gauge holonomy:
\begin{equation}
\label{eq:qihal-holonomy-phase}
\Phi_{ij}^{\mathrm{inv}}=\phi_i-\phi_j-q_Q\int_{\gamma_{ij}} A^{(Qi)}_\mu\,\dd x^\mu,
\end{equation}
where \(\gamma_{ij}\) joins two encoded states or channels. In particular, around a closed loop \(\gamma\) one obtains the informational holonomy
\begin{equation}
\label{eq:qihal-wilson-loop}
\mathcal{W}_{\gamma}^{(Qi)}=
\exp\!\left(\ii q_Q\oint_{\gamma} A^{(Qi)}_\mu\,\dd x^\mu\right).
\end{equation}

Equation~\eqref{eq:qihal-holonomy-phase} is the first real bridge from \QiQUA\ and \QiEP\ to \QiHAL. Once memory is carried by a gauged informational field, relative phase and holonomy become natural state variables. That is why a memory-bearing open system under \(U(1)_{Qi}\) can be re-expressed in an oscillator- and phase-based architecture even when its microscopic substrate is chemical, mechanical, biological, or social.

\begin{definition}[\texorpdfstring{\QiHAL}{Qi-HAL}]
\label{def:qihal}
\QiHAL\ is the canonical architectural factorization of a memory-bearing, \(U(1)_{Qi}\)-carrying open system into five mutually coupled layers:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item \textbf{\MePMoS}: the causal loop of Memory \(\to\) Prediction \(\to\) Motion \(\to\) Sensing \(\to\) updated Memory;
\item \textbf{\NHT}: a phase-holographic encoder that maps latent geometry or latent state to observable oscillatory phases and delays;
\item \textbf{\HAL}: a gauge-invariant memory lattice storing relational phase structure and attractor couplings;
\item \textbf{\BFFT}: a dyadic multiscale hierarchy coordinating slow/global and fast/local processing;
\item \textbf{\OH{}+\IH{}=\EIC{}}: the internality layer --- the outer hologram of externally registered boundary data, the inner hologram of the internally maintained model, and the closed loop \(\EIC := \mathcal{C}_{\rm loop}(\OH, \IH)\) that couples them.
\end{enumerate}
\end{definition}

\begin{remark}[Why the internality layer belongs inside \texorpdfstring{\QiHAL}{Qi-HAL}, and the switch does not]
\label{rem:qihal-membership}
Two membership decisions are worth stating, because the earlier
formulation made the opposite pair. \(\OH+\IH=\EIC\) belongs
\emph{inside} the definition: a memory-bearing system does not merely
store and recall, it maintains a model of the environment it is
embedded in and acts through that model, and every application in
Part~\ref{part:internal} and Part~\ref{part:emergence} --- chemistry, life, brains, collectives --- turns
on that loop rather than on the four storage-and-timing layers alone.
Leaving it outside made \QiHAL{} an architecture of bookkeeping and
put the physics one part away.

The \DGI{}--\GSSB{} switch, by contrast, does not belong in an
\ASeven{} definition at all. It is \ASix{}'s runtime, imported here and
not restated: this paper consumes the cascade, it does not define it.
Nor was it doing work as a layer --- the present section's own honesty
note (Remark~\ref{rem:qihal-necessity-honest}) records that its
necessity argument was circular, restating the entrance condition
rather than deriving anything from it. Removing it costs nothing and
removes a restatement of upstream content from a downstream
definition.
\end{remark}

Definition~\ref{def:qihal} is intentionally architectural rather than phenomenological. It does not say that all systems look identical. Rather, it says that once persistent memory and informational gauge current are present, the runtime can be decomposed into these five tasks. Some systems implement them minimally or externally; some implement them richly and internally. The job of the rest of the section is to make that decomposition exact.

\begin{remark}[Why the word ``neural'' in \texorpdfstring{\NHT}{NHT} is historical rather than restrictive]
The acronym \NHT\ is retained because the mature experimental examples come from neural systems. In the present paper, however, the mathematics is not restricted to neurons. Any substrate that converts latent geometry into phase, delay, or oscillatory code qualifies. Chemical oscillators, conformational cycles, trail networks, communication rhythms, and digital clocked systems can all instantiate the same structure.
\end{remark}

\begin{table}[H]
\centering
\caption[The five motifs of the memory architecture.]{The five architectural motifs of \QiHAL{}. Each row is one motif, with the question it answers, its primary mathematical object, and its operational role. Read downward the five compose into a loop: memory feeds prediction, prediction feeds motion, motion feeds sensing, sensing updates memory, and the last row is the switch that lets a system reorganize its own landscape rather than only descend on it. That last capability is what distinguishes a memory-bearing system from a merely open one.}
\label{tab:qihal-components}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Component} & \textbf{Question answered} & \textbf{Primary mathematical object} & \textbf{Operational role} \\
\midrule
\MePMoS & How is one cycle of action and update closed? & Causal loop map \(m_n\mapsto \hat y_n\mapsto a_n\mapsto s_{n+1}\mapsto m_{n+1}\) & Organizes prediction, action, observation, and memory update. \\
\NHT & How is latent geometry encoded? & Gauge-corrected phase map \(y\mapsto \phi(y)\) & Converts hidden multidimensional state into observable phases and delays. \\
\HAL & Where is the memory stored? & Coupling matrix/tensor \(W\) and attractor energy \(E_W\) & Stores and stabilizes phase relations or structural traces. \\
\BFFT & How are scales coordinated? & Dyadic ladder \(f_n=2^n f_0\) and cross-frequency couplings & Couples slow/global predictions to fast/local details. \\
\DGI--\GSSB & How does the system switch between exploration and commitment? & Symmetry-preserving orbit and broken-symmetry potential & Governs attention, search, selection, and decision. \\
\bottomrule
\end{tabular}
\end{table}

\section{Why persistent memory plus \texorpdfstring{\(U(1)_{Qi}\)}{U(1)Qi} is the true entrance condition}
\label{subsec:qihal-entrance-condition}

The user-level intuition is simple: once a system has memory, prediction becomes possible; once prediction becomes possible, action becomes meaningful; once action becomes meaningful, feedback must be compared to prediction; and once that comparison is repeated, a stable architecture is required. The mathematics of \QiHAL\ simply packages this logic in a gauge-covariant way.

The argument becomes sharper when written in the language of the earlier sections. In Part~\ref{part:qiqua} we introduced the current ledger and the Arisaka Equation,
\[
\frac{J_E}{\QB}=J_e=J_B=J_Q.
\]
A memory write therefore cannot be free. If a process creates one additional persistent informational event, there is a corresponding minimal denominator in the energy channel. This does not yet say how the event is stored, but it already forbids the fiction that memory is merely symbolic. Memory consumes budget.

In Part~\ref{part:qiep} we showed that selected trajectories, fluxes, and posterior states arise from a common projected extremal principle. A memory variable becomes useful only when it participates in that selection, namely when it changes which path is chosen, which current is sustained, or which posterior is stabilized. Memory without selection is inert archival residue.

In Part~\ref{part:qicon} we then showed that persistent open systems generically develop a scale-covariant window with broad fluctuations and two-knee spectra. A memory architecture operating in such a window must coordinate multiple timescales. This is exactly the role later assigned to \BFFT.

Taken together, the previous sections imply the following entrance proposition.

\begin{proposition}[Architectural entrance proposition]
\label{prop:qihal-entrance}
Let an admissible open system satisfy the following four conditions:
\begin{enumerate}[label=(\alph*),leftmargin=2.5em]
\item it is memory-bearing in the sense of Definition~\ref{def:memory-bearing};
\item it carries nonzero informational current \(J_Q\) coupled to \(U(1)_{Qi}\);
\item its actuation changes future observations through at least one open port; and
\item its selected runtime contains a nontrivial \QiCON\ window or, at minimum, more than one dynamically relevant timescale.
\end{enumerate}
Then the runtime admits a nontrivial architectural factorization of the form given in Definition~\ref{def:qihal}.
\end{proposition}

\begin{proof}
Condition (a) supplies a history-bearing state variable \(m\). Condition (b) permits the informational state to be written in gauge-covariant amplitude-and-phase form \eqref{eq:qihal-psi}--\eqref{eq:qihal-holonomy-phase}, thereby making relational phase storage natural. Condition (c) ensures that memory can affect future input through action; this closes the \MePMoS\ loop. Condition (d) guarantees that at least a two-level scale protocol exists, allowing slow and fast channels to separate. Once these four ingredients are present, one can define the causal loop (\MePMoS), the phase representation (\NHT), the persistent coupling substrate (\HAL), the multiscale ladder (\BFFT), and the exploration/commitment switch (\DGI--\GSSB). The construction of each layer is given explicitly below.
\end{proof}

Proposition~\ref{prop:qihal-entrance} is deliberately modest. It does not say that every such system has a large brain-like architecture. It says that the \emph{normal form} of its runtime is architectural rather than purely differential-equation-like. Minimal systems realize the five motifs in degenerate or compressed form; richer systems realize them in distributed, nested, and highly nonlinear form.

\section{\texorpdfstring{\MePMoS}{MePMoS}: derivation of the causal loop}
\label{subsec:qihal-mepmos}

We begin with the most elementary question: what is the unique causal ordering of memory, prediction, action, and sensing in a system that both acts and learns? In plain language the answer seems obvious, but in a theorem-driven paper it is worth stating carefully because the order is often blurred when perception and action are written as simultaneous updates.

\begin{definition}[\texorpdfstring{\MePMoS}{MePMoS}]
\label{def:mepmos}
For a memory-bearing open system, \MePMoS\ is the minimal causal loop
\begin{equation}
\label{eq:mepmos-loop}
\mathcal{M}_n
\xrightarrow{\ \mathcal{P}\ }
\widehat{y}_{n}
\xrightarrow{\ \mathcal{A}\ }
a_n
\xrightarrow{\ \mathcal{S}\ }
s_{n+1}
\xrightarrow{\ \mathcal{U}\ }
\mathcal{M}_{n+1},
\end{equation}
where \(\mathcal{M}_n\) is the memory state available at the beginning of the cycle, \(\widehat{y}_{n}\) a predicted latent or sensory outcome, \(a_n\) the action or motion command, \(s_{n+1}\) the next sensed input, and \(\mathcal{U}\) the memory update map.
\end{definition}

The essential point is that the system cannot directly map old memory to new memory if it is to remain adaptive; it must pass through a prediction step and an action step. Prediction alone is not enough, because without action there is no way to test the prediction by changing the environment or the vantage point. Likewise sensing cannot precede motion in a closed action-perception loop because the relevant new sensation is partly determined by the very action chosen in the current cycle.

\begin{theorem}[Causal uniqueness of \texorpdfstring{\MePMoS}{MePMoS} (T-Qi4-12.6)]
\label{thm:mepmos-uniqueness}
Consider a discrete open-system cycle with four macro-operations: memory readout, prediction, action, and sensing. Assume (i) prediction may depend only on information already stored at the start of the cycle, (ii) action may depend on the prediction and previously stored context, (iii) the incoming sensory record at the end of the cycle may depend on the chosen action and the environment, and (iv) the updated memory at the end of the cycle may depend on the old memory, the prediction, and the realized sensory record. Then, up to trivial relabeling inside each macro-operation, the only admissible topological order is
\[
\text{Memory} \to \text{Prediction} \to \text{Motion} \to \text{Sensing} \to \text{updated Memory}.
\]
\end{theorem}

\begin{proof}
The assumptions define a dependency graph with arrows
\[
\mathcal{M}_n\to \widehat{y}_{n},
\qquad
(\mathcal{M}_n,\widehat{y}_{n})\to a_n,
\qquad
a_n\to s_{n+1},
\qquad
(\mathcal{M}_n,\widehat{y}_{n},s_{n+1})\to \mathcal{M}_{n+1}.
\]
Any topological sort must therefore place \(\mathcal{M}_n\) before \(\widehat{y}_{n}\), \(\widehat{y}_{n}\) before \(a_n\), \(a_n\) before \(s_{n+1}\), and \(s_{n+1}\) before \(\mathcal{M}_{n+1}\). No permutation violating these arrows is admissible: placing sensing before action would require the system to know a future observation before choosing the action that partly generates it; placing prediction after sensing would collapse prediction into postdiction; placing memory update before new sensing would erase the error-correction step. Hence the macro-order is unique.
\end{proof}

The discrete cycle can be written more concretely as
\begin{subequations}
\label{eq:mepmos-discrete}
\begin{align}
\widehat{y}_n &= \mathcal{P}(\mathcal{M}_n), \label{eq:mepmos-prediction}\\
a_n &= \mathcal{A}(\widehat{y}_n,\mathcal{M}_n), \label{eq:mepmos-action}\\
s_{n+1} &= \mathcal{S}(a_n,\eta_{n+1}), \label{eq:mepmos-sensing}\\
\mathcal{M}_{n+1} &= \mathcal{U}(\mathcal{M}_n,\widehat{y}_n,s_{n+1}). \label{eq:mepmos-update}
\end{align}
\end{subequations}
Here \(\eta_{n+1}\) denotes environmental input. The operational error signal is
\begin{equation}
\label{eq:mepmos-error}
\epsilon_{n+1}=s_{n+1}-\widehat{s}_{n+1|n},
\end{equation}
where \(\widehat{s}_{n+1|n}\) is the sensory consequence predicted at stage \(n\). In an inferential implementation one may take
\begin{equation}
\label{eq:mepmos-memory-update}
\mathcal{M}_{n+1}=\mathcal{M}_n-
\eta_m G_m^{-1}\nabla_{\mathcal{M}}
\left[\frac12\norm{\epsilon_{n+1}}^2+\lambda_R R(\mathcal{M}_n)\right]
+\xi_n,
\end{equation}
with \(G_m\) a metric on memory space, \(R\) a regularizer or prior cost, \(\eta_m\) a learning step, and \(\xi_n\) a fluctuation term. Equation~\eqref{eq:mepmos-memory-update} is not a new principle added by hand. It is simply the \QiEP/\FEP\-type selector written in the memory coordinates of the architecture layer \citep{Friston2010}.

In continuous time, writing the memory state as \(m(t)\) and the predictive free-energy-like cost as \(\mathcal{F}(m,s,a)\), one obtains the normal form
\begin{equation}
\label{eq:mepmos-continuous}
\tau_m \dot{m} = -\frac{\delta \mathcal{F}}{\delta m} + \chi_Q J_Q + \zeta(t),
\end{equation}
where the explicit \(J_Q\) term reminds us that memory maintenance is an informational-current-bearing process rather than an abstract computational update.

The interpretation is extremely general. In a molecular machine the memory state may be a conformational coordinate or chemical occupancy. In a bacterium it may be a receptor methylation state. In a mammalian brain it may be a large distributed recurrent state. In a society it may be a distributed archive coupled to communication channels. The causal grammar is unchanged.

\begin{table}[H]
\centering
\caption[The memory--prediction loop across four domains.]{The \MePMoS{} loop across domains. Four systems run down the first column, from a molecular machine to a collective; the remaining columns identify what plays the part of memory, prediction, motion and sensing in each. The table's claim is that the loop is the same loop in all four rows and that only the substrate changes --- which is what makes the motif a structural statement rather than a biological one. Rows are ordered by increasing spatial scale, not by increasing sophistication.}
\label{tab:mepmos-domains}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.17\textwidth} >{\raggedright\arraybackslash}p{0.19\textwidth} >{\raggedright\arraybackslash}p{0.18\textwidth} >{\raggedright\arraybackslash}p{0.22\textwidth}}
\toprule
\textbf{Domain} & \textbf{Memory} & \textbf{Prediction} & \textbf{Motion/action} & \textbf{Sensing/update} \\
\midrule
Molecular machine & Conformational occupancy & Expected next catalytic state & Step, rotation, or gating bias & Product release, ligand binding, torque feedback \\
Cell or microbe & Internal biochemical state & Expected gradient or nutrient change & Chemotactic step or regulatory action & Receptor occupancy, signal update \\
Nervous system & Recurrent network state & Predicted sensory or latent state & Motor command or attentional shift & Sensory stream, error correction \\
Collective system & Shared archive, communication state & Expected collective outcome & Policy, message, or institutional action & External response, archive revision \\
\bottomrule
\end{tabular}
\end{table}

\section{\texorpdfstring{\NHT}{NHT}: phase-holographic coding of latent geometry}
\label{subsec:qihal-nht}

Once a causal loop exists, the next question is how latent structure is encoded so that it can be stored, compared, and updated efficiently. The answer proposed by \QiHAL\ is phase-holographic coding. Again the word ``neural'' in \NHT\ is historical; the derivation is not specifically neural.

Suppose the relevant latent variable of the system is \(y=(y^1,\dots,y^d)\in\mathcal{Y}\). A minimal phase encoding is
\begin{equation}
\label{eq:nht-map}
\phi_i(t)=\omega_i t + \frac{2\pi}{L_i}y^i(t)+\lambda(t),
\qquad i=1,\dots,d,
\end{equation}
where \(\omega_i\) is a carrier frequency, \(L_i\) a characteristic scale, and \(\lambda(t)\) a common gauge phase. Under \(U(1)_{Qi}\) the physically meaningful coordinate is the gauge-corrected phase
\begin{equation}
\label{eq:nht-gauge-corrected}
\widetilde{\phi}_i(t)=\phi_i(t)-q_Q\int^t A^{(Qi)}_{0,i}(\tau)\,\dd\tau.
\end{equation}
Therefore differences in latent state can be reconstructed from phase differences:
\begin{equation}
\label{eq:nht-delta-y}
\delta y^i = \frac{L_i}{2\pi}\,\delta\widetilde{\phi}_i.
\end{equation}
This immediately gives the local induced metric
\begin{equation}
\label{eq:nht-metric}
\dd s_{\mathcal{Y}}^2 = \sum_{i=1}^{d}\left(\frac{L_i}{2\pi}\right)^2 \bigl(\dd\widetilde{\phi}_i\bigr)^2.
\end{equation}

\begin{proposition}[Metric recovery by phase tomography]
\label{prop:nht-metric}
If the latent coordinates are encoded according to Eq.~\eqref{eq:nht-map} and read out through the gauge-corrected phases \eqref{eq:nht-gauge-corrected}, then the local Euclidean metric on the latent chart is recovered exactly by Eq.~\eqref{eq:nht-metric}. More generally, a curved latent metric \(g_{ij}(y)\) may be encoded by replacing the diagonal scale factors \((L_i/2\pi)^2\) with a positive-definite phase metric tensor \(G_{ij}(\phi)\).
\end{proposition}

\begin{proof}
Differentiate Eq.~\eqref{eq:nht-map} at fixed carrier-time slice or, equivalently, subtract the trivial carrier contribution \(\omega_i\,\dd t\). Then \(\dd\widetilde{\phi}_i=(2\pi/L_i)\dd y^i\). Multiplying by \((L_i/2\pi)^2\) and summing over \(i\) yields Eq.~\eqref{eq:nht-metric}. The curved case follows by replacing the diagonal quadratic form with a general positive-definite one.
\end{proof}

The conceptual content of Eq.~\eqref{eq:nht-metric} is far-reaching. A high-dimensional latent world can be encoded in relative phases, delays, or rhythmic offsets. The system does not need a literal geometric map drawn somewhere inside it. It only needs a phase configuration from which distances, alignments, or transition costs can be reconstructed.

A familiar special case is a ring attractor, in which one coordinate \(y\in S^1\) is encoded as a phase angle \(\phi\). But the same logic extends to toroidal, grid-like, chemical, and distributed memory systems. In a reaction network, phase may mean oscillation phase of concentrations. In a mechanical rotor, it may mean angular coordinate. In a social rhythm, it may mean synchronization offset relative to a shared clock.

The holographic language is justified because the code is not a point-by-point copy of the latent world. Rather, the latent geometry is stored distributively in phase relations, just as ordinary holography stores spatial information distributively in interference structure. This is why a damaged or partial substrate may still retain partial reconstructive power.

\begin{table}[H]
\centering
\caption[Phase-based encoding beyond neuroscience.]{Interpretations of \NHT{} beyond neuroscience. Four substrates run down the first column, with the variable each encodes, the phase-like carrier that does the encoding, and what a tomographic readout would mean in each case. The table is here because the motif's name comes from neuroscience and its content does not: a chemical oscillator network encodes state in relative phase for the same structural reason a neural population does, and confining the motif to brains would be a fact about the name rather than about the mathematics.}
\label{tab:nht-interpretations}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}}
\toprule
\textbf{Substrate} & \textbf{Encoded variable} & \textbf{Phase-like carrier} & \textbf{Meaning of tomography} \\
\midrule
Chemical oscillator network & Concentration or environmental phase & Oscillation phase of coupled reactions & Recover latent chemical/environmental coordinates from relative phase \\
Conformational molecular cycle & Position on catalytic cycle & Rotor angle or cycle phase & Reconstruct step location and expected next state \\
Neural population & Position, heading, or latent cause & Spike phase / local field potential phase & Recover latent spatial or causal geometry \\
Collective communication system & Social or task state & Synchronization phase across agents/clocks & Recover distributed shared state from timing relations \\
\bottomrule
\end{tabular}
\end{table}

\section{\texorpdfstring{\HAL}{HAL}: the holographic attractor lattice as a memory substrate}
\label{subsec:qihal-hal}

Phase coding alone is not memory. A system also needs a substrate in which phase relations can leave durable traces. \HAL\ is that substrate. In a minimal realization it is simply a set of couplings that stores relational structure. In a rich realization it becomes a high-dimensional attractor lattice supporting recall, completion, interpolation, and stable world-model reconstruction.

Let the encoded phase-bearing features be
\begin{equation}
\label{eq:hal-sigma}
\sigma_i^{(\alpha)}(n)=\exp\!\bigl(\ii\phi_i^{(\alpha)}(n)\bigr),
\end{equation}
where \(i\) labels local units or channels and \(\alpha\) labels feature families or subbands. A minimal write rule is the gauge-invariant correlator update
\begin{equation}
\label{eq:hal-update}
W_{ij}^{\alpha\beta}(n+1)=
(1-\lambda_W)W_{ij}^{\alpha\beta}(n)
+\gamma_W\,\sigma_i^{(\alpha)}(n)\,\overline{\sigma_j^{(\beta)}(n)},
\end{equation}
with decay or forgetting rate \(\lambda_W\) and write strength \(\gamma_W\). Under a common local \(U(1)_{Qi}\) phase shift, the product \(\sigma_i^{(\alpha)}\overline{\sigma_j^{(\beta)}}\) is invariant whenever the two channels carry the same informational charge. In the more general case one inserts the parallel-transport factor of Eq.~\eqref{eq:qihal-holonomy-phase}; the same conclusion follows.

The corresponding readout energy is
\begin{equation}
\label{eq:hal-energy}
E_W[\sigma] = -\frac12\sum_{\alpha,\beta}\sum_{i,j}
W_{ij}^{\alpha\beta}
\sigma_i^{(\alpha)}\overline{\sigma_j^{(\beta)}}
+\sum_{\alpha,i}U\!\left(|\sigma_i^{(\alpha)}|^2\right),
\end{equation}
with a local confining term \(U\). Stable memories are local minima of \(E_W\). When the couplings are organized on a ring, torus, graph, membrane, or more general lattice, those minima define a geometric memory manifold. That manifold is what the name holographic attractor lattice denotes.

The connection to the earlier pillars is immediate. A write event changes \(W\), and therefore costs energy according to the \QiQUA\ ledger. A read event queries one or more minima of \(E_W\), and therefore participates in \QiEP\ trajectory or posterior selection. The existence of a whole family of minima across scales is precisely what \QiCON\ diagnosed indirectly through broad spectral and spatial statistics.

The historical analogy to associative memory is useful but incomplete \citep{Hopfield1982}. In ordinary associative memory one stores patterns. In \HAL\ one stores gauge-covariant relations, phase geometry, and attractor structure. This makes \HAL\ more naturally compatible with continuous variables, manifold-valued latent states, and topological phases than a purely binary memory model.

\begin{proposition}[Gauge-invariant memory storage]
\label{prop:hal-gauge-invariant}
If the write rule is constructed from phase-difference correlators of the form in Eq.~\eqref{eq:hal-update}, optionally with explicit parallel transport for channels living in different gauge patches, then the stored memory tensor \(W\) is invariant under local \(U(1)_{Qi}\) gauge transformations. Consequently the physically stored object is relational phase geometry rather than absolute phase.
\end{proposition}

\begin{proof}
Under \(\sigma_i^{(\alpha)}\mapsto e^{\ii q_Q\Lambda_i}\sigma_i^{(\alpha)}\), the factor \(\sigma_i^{(\alpha)}\overline{\sigma_j^{(\beta)}}\) acquires the phase \(e^{\ii q_Q(\Lambda_i-\Lambda_j)}\). If the channels are in the same local patch and the same charge sector, this phase cancels for equal local gauges or is absorbed into the covariant phase-difference variable. More generally, inserting the Wilson-line transport factor between \(i\) and \(j\) cancels the residual phase exactly. Therefore the stored object depends only on relational gauge-invariant data.
\end{proof}

The most important practical question is whether \HAL\ depends on absolute frequency. If a memory is written at one carrier frequency and read at another, is the content lost? The answer in the idealized relational limit is no.

\begin{theorem}[Frequency-independent \texorpdfstring{\HAL}{HAL} readout (T-Qi4-12.9)]
\label{thm:hal-frequency-independence}
Suppose a pattern is written using oscillatory carriers
\begin{equation}
\label{eq:hal-write-pattern}
x_i^{\mathrm{w}}(t)=\exp\!\bigl(\ii(\omega_{\mathrm{w}}t+\phi_i)\bigr),
\end{equation}
and stored via the ideal correlator
\begin{equation}
\label{eq:hal-write-weight}
W_{ij}\propto \left\langle x_i^{\mathrm{w}}(t)\,\overline{x_j^{\mathrm{w}}(t)}\right\rangle_t = e^{\ii(\phi_i-\phi_j)}.
\end{equation}
If the pattern is later queried using carriers
\begin{equation}
\label{eq:hal-read-pattern}
x_i^{\mathrm{r}}(t)=\exp\!\bigl(\ii(\omega_{\mathrm{r}}t+\psi_i)\bigr),
\end{equation}
then the overlap energy
\begin{equation}
\label{eq:hal-overlap}
\mathcal{O}[\psi|W]=\sum_{i,j}\overline{x_i^{\mathrm{r}}(t)}\,W_{ij}\,x_j^{\mathrm{r}}(t)
\end{equation}
is independent of the common read frequency \(\omega_{\mathrm{r}}\). It depends only on the relative phase pattern \(\psi_i\) compared to the stored relative phase pattern \(\phi_i\).
\end{theorem}

\begin{proof}
Substituting Eqs.~\eqref{eq:hal-write-weight} and \eqref{eq:hal-read-pattern} into Eq.~\eqref{eq:hal-overlap} yields
\[
\mathcal{O}[\psi|W]
\propto
\sum_{i,j}
 e^{-\ii(\omega_{\mathrm{r}}t+\psi_i)}
 e^{\ii(\phi_i-\phi_j)}
 e^{\ii(\omega_{\mathrm{r}}t+\psi_j)}
=
\sum_{i,j} e^{\ii[(\phi_i-\phi_j)-(\psi_i-\psi_j)]}.
\]
The carrier phases \(\omega_{\mathrm{r}}t\) cancel identically. Hence the overlap depends only on relative phase differences. The common read frequency therefore does not affect ideal recall.
\end{proof}

Theorem~\ref{thm:hal-frequency-independence} has two important consequences. First, \HAL\ is fundamentally a relational storage principle. Second, it can be realized across very different substrates. A chemical system, a neural system, and a digital clocked system may use entirely different carrier frequencies, yet the same relational content can in principle be preserved.

One can also define a simple architectural completeness index for this
pillar alone --- written $\QCI_{\HAL}$, to keep it distinct from the
four-pillar \QiFour{} Completeness Index of Part~\ref{part:seos} ---
\begin{equation}
\label{eq:qci}
\QCI_{\HAL} = \frac{1}{5}\bigl(I_{\MePMoS}+I_{\NHT}+I_{\HAL}+I_{\BFFT}+I_{\mathrm{switch}}\bigr),
\end{equation}
where each indicator \(I\in[0,1]\) measures the degree to which the corresponding \QiHAL\ component is realized in a given substrate. Minimal biochemical toggles may have \(\QCI_{\HAL}\ll 1\), while distributed nervous systems may have \(\QCI_{\HAL}\to 1\). Equation~\eqref{eq:qci} is not needed for the theory itself, but it is useful as a future empirical scoring tool.

\section{\texorpdfstring{\BFFT}{BFFT}: the dyadic multiscale protocol}
\label{subsec:qihal-bfft}

The moment memory becomes nontrivial, one also needs scale coordination. Not every variable changes at the same rate; not every prediction is local; not every correction is fast. The natural multiscale protocol in \QiHAL\ is \BFFT, the binary fractal Fourier transform. The name is meant broadly: one may realize it dynamically by a nested oscillator hierarchy or analytically by a dyadic wavelet/Fourier-packet decomposition \citep{Mallat2009}. The key point is the dyadic scale ladder.

The derivation begins from the conformal radial coordinate already used in \GG\ and later inherited by \QiCON,
\begin{equation}
\label{eq:bfft-radial}
r = \ln\left(\frac{\mu_0}{\mu}\right).
\end{equation}
Equal increments in \(r\) correspond to multiplicative steps in physical scale. Suppose a scale hierarchy is sampled by a constant increment \(\Delta r = \ln \lambda\), so that
\begin{equation}
\label{eq:bfft-scale-ladder}
\mu_{n+1}=\lambda\,\mu_n,
\qquad
f_{n+1}=\lambda\,f_n.
\end{equation}
If one also demands integer nesting of cycles between adjacent levels, the multiplicative factor \(\lambda\) must be an integer greater than one. The minimal nontrivial choice is \(\lambda=2\), and this is exactly the binary or dyadic ladder. Measured neural cross-frequency coupling is, however, typically \emph{non}-dyadic --- the hippocampal theta--gamma code nests at roughly $1{:}5$--$1{:}8$ \citep{LismanJensen2013}, an integer $m \approx 7$; the dyadic $\lambda=2$ is therefore the minimal idealization, and the empirical ladder selects the architecture-specific integer $m>1$, not a prediction that biological coupling is binary.

\begin{proposition}[Dyadic minimality]
\label{prop:bfft-dyadic}
Among multiplicative scale hierarchies with the properties (i) exact parent-child nesting of periods and (ii) the minimal number of additional structural choices beyond the constitutional ledger, the unique minimal nontrivial ratio is
\begin{equation}
\label{eq:bfft-dyadic}
\lambda=2.
\end{equation}
Hence the natural \QiHAL\ scale ladder is
\begin{equation}
\label{eq:bfft-frequencies}
f_n = 2^n f_0.
\end{equation}
\end{proposition}

\begin{proof}
Exact parent-child nesting requires that one period at level \(n\) contain an integer number \(m>1\) of periods at level \(n+1\), hence \(f_{n+1}=m f_n\) with integer \(m\). The requirement of minimal extra structure chooses the smallest admissible integer, namely \(m=2\). Therefore \(\lambda=2\) and the dyadic ladder follows.
\end{proof}

The dyadic ladder is not merely aesthetically simple. It is exactly the structure required to mediate top-down and bottom-up flow. Slow channels carry coarse, global context; fast channels carry fine, local details. A phase-locking action for adjacent levels is
\begin{equation}
\label{eq:bfft-locking}
S_{\mathrm{lock}} = \sum_{n}\kappa_n\int \dd t\,\bigl[1-\cos(2\phi_n-\phi_{n+1})\bigr].
\end{equation}
The stationary condition is
\begin{equation}
\label{eq:bfft-phase-lock}
2\phi_n - \phi_{n+1} = \mathrm{const},
\end{equation}
which is the canonical 1:2 phase-lock relation between neighboring scales.

This immediately gives a bridge to the two-knee \QiCON\ spectrum. If the active conformal window spans from \(f_{\mathrm{IR}}\) to \(f_{\mathrm{UV}}\), then the number of effectively available dyadic levels is approximately
\begin{equation}
\label{eq:bfft-level-count}
N_{\mathrm{lev}}\approx 1+\left\lfloor \log_2\left(\frac{f_{\mathrm{UV}}}{f_{\mathrm{IR}}}\right)\right\rfloor.
\end{equation}
Thus the diagnostic spectral window fixes the architectural depth of the \BFFT\ ladder. This is a concrete, falsifiable bridge between Pillar~III and Pillar~IV.

\emph{Scope note on the cosmological reading of \BFFT{}.} The companion \ATen{} \citep{ArisakaA10Cosmos} carries a
proposed identification of the CMB damping multipole with a \BFFT{}
ultraviolet knee, $\ell_D = K(2\pi)^2 \approx 1382$; that reading is
an \emph{interpretive overlay}, sitting $+2.3\%$ above the framework's
own Boltzmann-computed damping scale ($\ell_D = 1350\pm3$), and is
scored accordingly there. The present paper supplies no
holonomy-potential normalization that would close that gap --- the
\BFFT{} machinery here is information-architectural --- so any
promotion of the overlay must come from a bus normalization not yet
constructed, not from this paper by citation.

For signal-analysis purposes one may write the same ladder as a dyadic transform,
\begin{equation}
\label{eq:bfft-wavelet}
X_n(\tau)=2^{-n/2}\int x(t)\,\psi\!\left(\frac{t-\tau}{2^n}\right)\dd t,
\end{equation}
which simply restates that the architecture can be understood either dynamically or analytically. In both readings the binary hierarchy is doing the same job: separating scales while preserving cross-scale couplings.

\begin{table}[H]
\centering
\caption[The multiscale hierarchy, level by level.]{Functional interpretation of the \BFFT{} hierarchy. Three level types --- slow and coarse, intermediate, fast and fine --- run down the first column, with the characteristic scale, the architectural role and the relation to neighboring levels alongside. The last column carries the content: the levels are not independent channels but a dyadic ladder, so a change at one scale propagates in a definite direction, and the table says which.}
\label{tab:bfft-hierarchy}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.15\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.25\textwidth}}
\toprule
\textbf{Level type} & \textbf{Characteristic scale} & \textbf{Typical architectural role} & \textbf{Relation to neighboring levels} \\
\midrule
Slow / coarse & Low frequency, long horizon & Global context, priors, slow homeostatic set-points & Constrains the envelope or admissible region for faster levels \\
Intermediate & Mesoscale bands & Working coordination, routing, synchronization & Mediates bidirectional exchange between slow and fast levels \\
Fast / fine & High frequency, short horizon & Local detail, error correction, rapid sensory update & Reports deviations upward and refines lower-resolution predictions \\
\bottomrule
\end{tabular}
\end{table}

\section{The \texorpdfstring{\GRIP{}-loop}{GRIP-loop} as the exploration-to-commitment switch}
\label{subsec:qihal-switch}

The fifth architectural motif is not an additional physical object but a runtime mode switch. A memory-bearing system must alternate between two regimes. In one regime it explores, samples, and remains flexible across a set of near-equivalent possibilities. In the other regime it commits to one branch, one action, one interpretation, or one attractor. The first is naturally described by \DGI; the second by \GSSB. The stabilized result is \GA.

A minimal normal form uses an order parameter \(m\) (which may be scalar or vector) and an effective potential
\begin{equation}
\label{eq:qihal-potential}
V_{\mathrm{eff}}(m;t) = \frac12 a(t)\,\norm{m}^2 + \frac{b}{4}\norm{m}^4 - h(t)\cdot m,
\qquad b>0.
\end{equation}
When \(a(t)>0\), the symmetric point \(m=0\) is the unique minimum; the system remains exploratory. When \(a(t)<0\), a symmetry-broken shell or set of minima appears. The field \(h(t)\) or small fluctuations then select one branch. That branch is stabilized by the quartic term and by environmental feedback, producing \GA.

To connect this to observation and prediction error, define an evidence integral
\begin{equation}
\label{eq:qihal-evidence}
I(t)=\int_{-\infty}^{t} \exp\!\left(-\frac{t-\tau}{\tau_I}\right)
\bigl(\epsilon_\star^2-\norm{\epsilon(\tau)}^2\bigr)\,\dd\tau,
\end{equation}
where \(\epsilon(t)\) is the current prediction error and \(\epsilon_\star\) is a reference tolerance. Taking
\begin{equation}
\label{eq:qihal-a-of-I}
a(t)=a_0-\lambda_I I(t)
\end{equation}
shows that sufficiently accumulated evidence can drive \(a(t)\) through zero. The system then moves from exploratory \DGI\ to symmetry-broken commitment. If the environment later changes so that error rises again, \(I(t)\) decreases, \(a(t)\) can return positive, and the system re-enters exploration. This is the normal form of attention, search, decision, commitment, and revision across many domains.

\begin{proposition}[Exploration-to-commitment bifurcation]
\label{prop:qihal-switch}
For the potential \eqref{eq:qihal-potential} with \(a(t)\) given by Eq.~\eqref{eq:qihal-a-of-I}, there exists a critical evidence level
\begin{equation}
\label{eq:qihal-critical-evidence}
I_c = \frac{a_0}{\lambda_I}
\end{equation}
such that the system is in the exploratory \DGI\ regime for \(I<I_c\) and in the broken-symmetry commitment regime for \(I>I_c\), up to perturbations induced by \(h(t)\) and noise.
\end{proposition}

\begin{proof}
The sign change in \(a(t)\) occurs exactly at \(I=I_c\). For \(a>0\), \(V_{\mathrm{eff}}\) has a single minimum at \(m=0\). For \(a<0\), the quartic term creates nonzero minima with radius \(\sqrt{-a/b}\) when \(h=0\), or nearby perturbed minima for small \(h\neq 0\). The system therefore crosses from a single exploratory basin to multiple committed branches at the critical evidence level.
\end{proof}

This proposition turns a qualitative story into a concrete architectural rule. It says that a memory-bearing open system need not choose between flexibility and stability once and for all. Rather, it moves between the two in a controlled, evidence-dependent way. That is exactly what one expects of adaptive organisms, decision circuits, and collective institutions.

\paragraph{The \GRIP{}-loop as the internalized regime switch.}
The two-regime vocabulary of \ASix{}
(\S\ref{sec:dgi-gssb-ga}) makes precise what kind of object this
switch is. When \(a(t)\) crosses zero, the control parameters of the
effective landscape \eqref{eq:qihal-potential} change and the basin
structure of \(V_{\mathrm{eff}}\) is reorganized --- by definition a
\GRIPout{} event; the subsequent settling into the committed branch
is \GRIPin{} descent; and the evidence dynamics
\eqref{eq:qihal-a-of-I} can run the cycle again. The \GRIP{}-loop is
therefore an \emph{internally generated, recoverable alternation}
\((\GRIPout\cdot\GRIPin)^{*}\), and the contrast with the rest of
the series is the discovery: the cosmological \GRIPout{}'s of
\citet{ArisakaA10Cosmos} are exogenous and unrepeatable (the
universe does not re-run its inflationary exit), whereas a
memory-bearing \SeOS{} carries its own landscape-reorganizing dial
and schedules its reorganizations on evidence. This is precisely the
microscopic mechanism that the capstone's
\textsf{Cascade-Self-Sustenance} corollary (\AEleven{}
\citep{ArisakaA11Origin}) requires: \GRIPout{} events at the higher
cascade layers are manufactured by the layers below --- and
Proposition~\ref{prop:qihal-switch} exhibits the minimal machine
that manufactures them. One boundary clarification carries over from
\ASix{}: ordinary port-driven excitation within a fixed landscape is
\emph{not} a \GRIPout{}; the switch qualifies because \(a(t)\)
changes the landscape itself, not merely the system's position
on it.

\section{A unified effective action for \texorpdfstring{\QiHAL}{Qi-HAL}}
\label{subsec:qihal-effective-action}

The preceding subsections can be compressed into a single effective architectural action. The point is not that every real system literally writes its microphysics in this form, but that every admissible implementation has this as a normal-form description after coarse-graining.

Let the dynamic variables be memory \(m\), phase coordinates \(\phi_i\), memory couplings \(W\), actions \(a\), and sensed inputs \(s\). Then an effective action is
\begin{equation}
\label{eq:qihal-action}
\begin{aligned}
S_{\QiHAL}
=
\int \dd t\,
\Bigg[
&\underbrace{\frac12 \dot m^{\top}G_m\dot m + \frac{\lambda_\epsilon}{2}\norm{\epsilon}^2}_{S_{\MePMoS}}
+
\underbrace{\frac12\sum_i \kappa_i\left(D_t\phi_i-\frac{2\pi}{L_i}\dot y^i\right)^2}_{S_{\NHT}}\\
&+
\underbrace{\frac{\tau_W}{2}\norm{\dot W}^2 + V_H(W,\sigma)}_{S_{\HAL}}
+
\underbrace{\sum_n \kappa_n\bigl[1-\cos(2\phi_n-\phi_{n+1})\bigr]}_{S_{\BFFT}}\\
&+
\underbrace{V_{\mathrm{eff}}(m;t)}_{S_{\mathrm{switch}}}
+
q_Q A^{(Qi)}_\mu J_Q^\mu
\Bigg].
\end{aligned}
\end{equation}
Here \(D_t\phi_i=\dot\phi_i-q_QA^{(Qi)}_{0,i}\). The last term is the explicit informational gauge coupling. A natural partition or effective generating functional is then
\begin{equation}
\label{eq:qihal-partition}
\mathcal{Z}_{\QiHAL}
=
\int \mathcal{D}m\,\mathcal{D}\phi\,\mathcal{D}W\,\mathcal{D}a\,\mathcal{D}s\;
\exp\!\left[-\frac{S_{\QiHAL}}{\QB}\right].
\end{equation}
The appearance of the Arisaka Constant \(\QB\) in the denominator is deliberate. It marks the ideal energetic denomination of an informational event, thereby tying the full architectural layer back to the current ledger of \QiQUA.

With the first three Qi pillars already in place, the next theorem closes the Qi block by showing that the architectural layer is not a domain-specific add-on but the universal implementation class for memory-bearing open systems.

\begin{theorem}[\QiHAL{} canonical normal-form decomposition (T-Qi4-12.12)]
\label{thm:qihal-universality}
Let an admissible open system satisfy the hypotheses of Proposition~\ref{prop:qihal-entrance}. Then, after suitable coarse-graining to the timescales on which memory persists, its runtime admits an effective action of the form \eqref{eq:qihal-action} and therefore a canonical factorization into \MePMoS, \NHT, \HAL, \BFFT, and the internality layer \(\OH+\IH=\EIC\). The decomposition is canonical in the sense that each motif corresponds to a structural requirement of the runtime, and the absence of any motif corresponds to a specific operational deficiency identified below; this is weaker than asserting that each motif is logically necessary in the strict sense (cf.~Remark~\ref{rem:qihal-necessity-honest}). The \GRIP{}-cascade closure on which the factorization rests is, like
its source in \ASix{}, conditional on that paper's remaining cascade
conditioning (M11 in spectral-flow form; M10 resolved externally, the
scope refinements closed at class level).
\end{theorem}

\begin{proof}
We organize the five motifs into two groups: those that are structurally definitional (and therefore present by definition in any system that satisfies the hypotheses), and those whose presence is a non-trivial derived consequence. Sufficiency then follows from the canonical sum of sector actions.

\emph{Definitional motifs (\MePMoS, \(\OH+\IH=\EIC\)).} By Definition~\ref{def:memory-bearing}, an admissible memory-bearing system possesses a memory coordinate whose update is a function of past sensing, prediction, and action; this closed loop is the structural definition of the \MePMoS\ sector. The internality layer is definitional in the same sense and for the same reason: a system whose memory update is action-conditioned is, by that very fact, maintaining an internal correlate of the environment it acts on and registering that environment at its boundary --- which is what \(\IH{}\) and \(\OH{}\) name, and the action-conditioning is the closure of the loop between them. These two motifs are therefore present in any system that satisfies the hypotheses of the theorem; their presence carries no surplus content beyond the hypotheses themselves. Calling them ``necessary'' would be circular --- they are necessary in the same sense that a triangle is necessarily three-sided. (What is \emph{not} definitional, and is the substantive content of Part~\ref{part:internal}, is the structure of the loop once closed: the fidelity conditions, the maintenance ports, and the conditions under which the inner hologram tracks the outer one.)

\emph{Derived motif: \NHT.} This motif carries non-trivial content beyond the hypotheses. Under \(U(1)_{Qi}\) gauge invariance, the physical observables that involve the memory coordinate must be gauge-invariant functions of \(m\) and its phase \(\phi\). The minimal gauge-invariant phase-coded observable on a non-simply-connected memory manifold is the Wilson holonomy of Eq.~\eqref{eq:qihal-wilson-loop}. Suppose a candidate system carries memory but realizes \(U(1)_{Qi}\) trivially (so the holonomy is a pure-gauge constant). Then all phase content is unobservable, and the informational current \(J_{Qi}\) decouples from any measurable observable on the memory layer. This is a derived consequence, not a definition: it requires both the memory-bearing hypothesis \emph{and} active \(U(1)_{Qi}\) (cf.\ the L12 framing of $U(1)_{Qi}$ as constitutionally optional but empirically preferred), and it is operationally falsifiable by an experiment that demonstrates persistent memory in a system whose \(U(1)_{Qi}\) sector is decoupled.

\emph{Derived motif: \HAL.} Persistence of memory traces over timescales long compared to the relaxation time of the boundary dynamics requires the existence of metastable basins in the effective potential of the memory coordinate. By the \GA{} theorem (Theorem~\ref{thm:ga-existence}), such basins are exactly the attractors of the post-\GSSB\ runtime; the collection of such attractors, organized by their gauge-invariant pairwise relations, is the \HAL\ sector. Suppose the system has no metastable basins on the memory manifold: then traces decay on the boundary relaxation timescale, contradicting the operational definition of persistence (memory-bearing condition \(\tau_m > T_c\), Eq.~\eqref{eq:qihal-memory-condition}). This is again a derived consequence (involving the \GA{} theorem from Section~\ref{sec:dgi-gssb-ga}) rather than a tautology.

\emph{Derived motif: \BFFT.} By Theorem~\ref{thm:qicon-two-knee}, any memory-bearing system in the \QiCON\ admissibility window exhibits a two-knee \(1/f\) spectrum with a well-defined window between low-frequency knee \(f_L\) and high-frequency knee \(f_H\). Reproducing such a spectrum requires scale-recursive structure with a geometric ratio between successive layers and binary commitment at each layer (a \GSSB\ event resolving each indecision). Suppose the system has no nested binary scale structure: then either the \(1/f\) spectrum cannot be reproduced (contradicting Theorem~\ref{thm:qicon-two-knee}), or the binary-commitment structure of \GSSB\ cannot be implemented (contradicting Theorem~\ref{thm:master-pipeline}). This is derived from \QiCON\ and the master pipeline rather than from the memory-bearing definition itself.

\emph{Sufficiency.} Given the definitional motifs (\MePMoS, \(\OH+\IH=\EIC\)) and the three derived motifs (\NHT, \HAL, \BFFT), the effective action of Eq.~\eqref{eq:qihal-action} is the canonical sum of their respective contributions: the memory potential \(S_m\), the phase action \(S_\phi\), the Wilson-line action \(S_W\), the action coordinate \(S_a\), and the sensing coordinate \(S_s\). Coarse-graining to the timescales on which memory persists bundles the sectors into the single effective action, and the decomposition follows from the canonical roles, not from the microscopic details of any particular substrate.

Thus the decomposition is sufficient, and three of the five motifs are derived consequences of the hypotheses; the remaining two are definitional. \qedhere
\end{proof}

\begin{remark}[Honest scope of the canonical decomposition]
\label{rem:qihal-necessity-honest}
It is tempting to advertise all five motifs as ``logically necessary by contradiction,'' but arguments of that shape are circular here: they restate the definition of the hypothesis rather than deriving the motif from it. The statement above therefore separates the two definitional motifs (\MePMoS\ and the internality layer, both of which follow from the memory-bearing hypothesis itself) from the three genuinely derived ones, and states the result as a canonical normal-form decomposition rather than a strict universality claim. The empirical falsifier survives the change: a memory-bearing \(U(1)_{Qi}\) system that cannot be decomposed into the five sectors, or in which any one of the three derived motifs (\NHT, \HAL, \BFFT) demonstrably fails to obtain, refutes \QiHAL.
\end{remark}

The theorem is the central result of this section. It makes precise a claim that is easy to state only loosely: once \(U(1)_{Qi}\) is applied, any open system with memory acquires a common architectural grammar. The grammar may be instantiated minimally in chemistry or richly in brains, but the runtime tasks are the same.

\section{How \texorpdfstring{\QiHAL}{Qi-HAL} applies to any open system with memory}
\label{subsec:qihal-any-open-system}

The universality claim becomes most persuasive when one resists the temptation to imagine only brains. The point of \QiHAL\ is broader. Memory can be internal or external, analog or digital, molecular or social. The theorem does not require a cortex, a language, or even neurons. It requires only persistence, action-conditioned feedback, and informational gauge current.

\paragraph{Minimal biochemical case.}
A biochemical toggle, receptor cluster, or catalytic cycle may have a very small state space, yet if its present reaction bias depends on a stored modification state, it already satisfies Definition~\ref{def:memory-bearing}. The cycle phase plays the role of \NHT\, the modified couplings or energy landscape play the role of \HAL\, and slow/fast relaxation channels provide a minimal \BFFT\ hierarchy. The \DGI--\GSSB\ switch appears when the system moves from a flexible undecided regime to a committed branch of reaction or conformation.

\paragraph{Cellular and microbial case.}
A chemotactic microbe or slime-mold network is an especially transparent open-system example \citep{Nakagaki2000}. Memory may be carried by receptor methylation, internal chemical state, or even an externalized trail network. Prediction appears as directional bias, motion as crawling or swimming, sensing as receptor update, and multiscale coordination as the nesting of fast local signaling within slow environmental adaptation. Here \HAL\ may be partly externalized into the environment itself, showing that the theorem does not require memory to be stored in a single physically bounded organ.

\paragraph{Neural and cognitive case.}
In a nervous system the five motifs become more familiar. Recurrent state and synaptic structure supply \MePMoS\ and \HAL\; phase coding and population rhythms supply \NHT\; nested oscillatory bands or other multiscale population modes supply \BFFT\; and evidence-driven selection produces the \DGI--\GSSB\ attention/decision cycle. The architecture is richer but not different in kind.

\paragraph{Collective and social case.}
A collective can also instantiate \QiHAL. Shared archives, institutions, communication rhythms, distributed representations, and policy cycles provide memory, prediction, action, sensing, and update. The memory lattice is partly externalized in documents, infrastructures, or digital systems; yet the same logic applies. This is one of the reasons the present paper insists that the framework is for open systems in \emph{all} sciences, not only for biology or physics.

\begin{longtable}{>{\raggedright\arraybackslash}p{0.12\textwidth} >{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.18\textwidth} >{\raggedright\arraybackslash}p{0.15\textwidth} >{\raggedright\arraybackslash}p{0.16\textwidth} >{\raggedright\arraybackslash}p{0.15\textwidth}}
\caption[The five motifs, system by system.]{How the five \QiHAL{} motifs appear across representative open systems with memory. Each row is one system, from a catalytic cycle to a collective; the columns take the motifs in turn and say what plays each part. Read across a row it is a description of one system; read down a column it is the claim that the motif is substrate-independent. The table repeats its header on every page because it is the reference a reader returns to when a later section names a motif without re-explaining it.}
\label{tab:qihal-across-domains}\\
\toprule
\textbf{System} & \textbf{Memory} & \textbf{\MePMoS} & \textbf{\NHT} & \textbf{\HAL} & \textbf{\BFFT/switch} \\
\midrule
\endfirsthead
\toprule
\textbf{System} & \textbf{Memory} & \textbf{\MePMoS} & \textbf{\NHT} & \textbf{\HAL} & \textbf{\BFFT/switch} \\
\midrule
\endhead
Catalytic cycle & Conformational occupancy, chemical mark & Cycle memory \(\to\) catalytic bias \(\to\) step \(\to\) feedback & Rotor or oscillation phase & Modified energy landscape & Slow-fast nesting; branch commitment \\
Chemotactic cell & Internal adaptation, receptor memory, trail residue & Internal state \(\to\) direction bias \(\to\) movement \(\to\) receptor update & Chemical oscillation phase & Internal network, possibly external trail & Adaptation bands; explore vs commit \\
Neural system & Recurrent state, synaptic matrix & Working memory \(\to\) prediction \(\to\) action \(\to\) sensory update & Population phase code & Attractor manifold, synaptic lattice & Nested bands; decision bifurcation \\
Collective & Shared archive, institutions, digital storage & Archive \(\to\) forecast \(\to\) policy \(\to\) measured outcome & Shared clocks, communication timing & Externalized memory lattice & Slow governance, fast signals; consensus \\
\bottomrule
\end{longtable}

It is worth emphasizing the role of externalized memory. The theorem never assumed that memory must be stored behind the physical boundary of one organism. A beaver dam, a pheromone trail, a paper notebook, a digital server, or a laboratory instrument can all function as memory-bearing extensions of an open system. Once those extensions are coupled into the causal loop, they become part of the effective \HAL\ sector. This is why the theory naturally extends from particles to organisms to societies.

\paragraph{Worked examples.} ATP synthase as a minimal proto-\QiHAL{} system, slime-mold/chemotactic collectives as externalized \HAL{}, and ring-attractor cognitive maps as mammalian \HAL{} are recorded in Appendix~\ref{app:technical-derivations-for-qi-hal}.

\section{Memory as something the world does}
\label{sec:close-qihal}

The habit of thinking about memory as storage is very hard to shake, and
it is worth saying plainly what this Part replaces it with.

Storage suggests a place: a location where a thing is put and from which
it is later retrieved, unchanged. Almost nothing about biological memory
works that way, and the account here explains why it could not. What is
held is not a set of items but a set of relations --- differences
between phases, not the phases themselves. That is why the same content
survives being carried on a different rhythm entirely, so that a
chemical system, a nervous system and a machine can hold the same thing
on hardware with nothing in common. It is why damage degrades a memory
rather than deleting a list. And it is why the question "where is it
stored" has no good answer, not because the answer is hidden but because
the question presupposes the wrong kind of object.

The second thing to carry away is that the loop's order is forced. A
system cannot sense before it acts without knowing an observation it has
not yet caused; it cannot predict after sensing without turning
prediction into hindsight. There is only one way to arrange remembering,
predicting, moving and sensing so that each step has something to do.
Anything that keeps a past and uses it has to run the cycle in that
order, whatever it is made of. That is a strong statement about
architecture arrived at without looking at any particular architecture.

The third is the one this paper alone can make. The engine imported from
the geometric companion needs something to reorganize its landscape from
time to time, or a system would descend once and be finished. In
cosmology those reorganizations arrive from outside and do not repeat.
A system with a memory makes its own. It explores while its evidence is
thin, commits when the evidence crosses a threshold, and goes back to
exploring when the world changes and its predictions start to fail ---
and that cycle of attention, decision, commitment and revision is the
same normal form in a bacterium choosing a direction, an animal choosing
a route, and an institution choosing a policy. It is how something that
would otherwise settle once keeps going.

With this Part the four pillars are complete, and it is worth standing
far enough back to see what they are. The first says what a system
spends and in what single currency. The second says which of the
available ways of spending it the system takes. The third says what
the taken path looks like to an instrument that changes the scale of
its attention. The fourth says what the system has to be built like in
order to do any of it a second time. Budget, selection, signature,
architecture. That is not a list of four topics that happened to
interest the same author. It is close to the minimal set of questions
that must be answered before one is entitled to say that a thing has a
way of going on rather than merely a way of happening. Answer the
first three and you have described an event. Answer the fourth and you
have described something with a history.

There is a temptation, having got this far, to say that these parts are
logically necessary for anything that remembers. The Part declines to
say it, and explains that the argument would be circular --- it would be
restating the definition rather than deriving anything from it. What is
offered instead is a normal form and a way to break it: find a
memory-bearing system that cannot be decomposed this way, and the Part
is finished. That is a smaller claim than it could have made, and a more
useful one.

Two things the pillars name are left standing, and they are the
subjects of what follows. The first is whether a system built this way
in fact lasts: the pillars describe machinery, and machinery can run
beautifully right up to the moment it stops, so persistence is a
separate question and gets a separate answer in
Part~\ref{part:seos}. The second is what happens when the loop just
described is closed on itself --- when a system's picture of the world
comes to include the system that is holding the picture. That is
Part~\ref{part:internal}, and it is where an account of machinery
turns into an account of a point of view.


\GGPart[SeOS and QCI]{\SeOS{} and \QCI{}}{The persistence class and the \QiFour{} Completeness Index}
\label{part:seos}

\noindent
Everything in nature is open. Almost nothing holds itself together. This
Part is about the difference.

A speck of dust in a fluid is open --- it exchanges momentum and energy
with everything around it --- but it maintains nothing. A convection
cell is closer: it exists only while something keeps driving it, and it
would be fair to say it is doing a kind of work to persist. A cell, an
organism, a brain, an institution is a further step again, because what
it maintains is not only a flow but an organization: a boundary, a set
of records, a way of repairing itself when something goes wrong.

The first half of this Part turns that intuition into an inequality
anybody can evaluate. What comes in through the ports, minus what is
irreversibly lost, minus what repair costs, must stay positive on
average over a window longer than the system's own repair time. Above
zero, the system stays inside the region where it is still itself.
Below zero for long enough, it leaves that region and stops being a
member of the class. The margin is a computable quantity, not a
judgment, and that is the whole point of writing it this way.

The second half asks a different question that is easy to confuse with
the first. Surviving and running the full machinery of the previous four
Parts are not the same thing. A chemical oscillator can persist
indefinitely while realizing almost no memory architecture; a damaged
nervous system can be rich in structure while sliding toward the edge of
viability. So the Part introduces a second measure --- how much of the
four-pillar runtime a system actually runs --- and deliberately builds it
so that a missing pillar cannot be compensated by excellence in another.
A system that scores nothing on one of the four scores nothing overall,
because that is what incompleteness means.

Two things to hold on to while reading. The word used for the persistence
class is a technical one borrowed from an older debate, and it carries no
suggestion of intention or of moral standing; it means only that the
local accounting is done against the survival of the packet under study.
And the weights in the completeness measure are not derived. The paper
says so. The honest use of that index is to compare two systems, not to
certify one.


\label{sec:seos-qci}

Part~\ref{part:internal} begins only after the full Qi stack is available. The next question is no longer how an open system is driven, selected, diagnosed, or architected, but which such systems persist and how fully they instantiate the \QiFour{} runtime. Part~\ref{part:seos} therefore introduces the persistence class \SeOS\ and the completeness diagnostic \QCI.


\begin{figure}[htbp]
\centering
\resizebox{0.869\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=3pt, inner sep=3.5pt},
 inbox/.style={fill=green!12, draw=green!55!black, line width=0.65pt,
 minimum width=40mm, minimum height=12mm},
 outbox/.style={fill=red!11, draw=red!60!black, line width=0.65pt,
 minimum width=40mm, minimum height=12mm},
 sumbox/.style={fill=blue!12, draw=blue!60!black, line width=1.10pt,
 minimum width=52mm, minimum height=14mm},
 tube/.style={fill=blue!8, draw=blue!55!black, line width=0.55pt,
 dash pattern=on 2.5pt off 2pt},
 good/.style={line width=1.00pt, draw=green!50!black},
 bad/.style={line width=1.00pt, draw=red!60!black},
 lab/.style={font=\sffamily\scriptsize, align=center, inner sep=2pt},
 note/.style={font=\sffamily\scriptsize\itshape, text=black!65,
 align=center, inner sep=2pt},
 arr/.style={-{Stealth[length=2.4mm]}, line width=0.65pt, draw=black!70},
]
% ---- (a) the margin, as an arithmetic ----
\node[lab] at (-40mm,34mm) {\textbf{(a) The margin is arithmetic}};
\node[inbox]  (pin)  at (-40mm,20mm) {what comes in\\ through the ports};
\node[outbox] (diss) at (-40mm,4mm)  {what is lost\\ irreversibly};
\node[outbox] (rep)  at (-40mm,-12mm){what repair\\ costs};
\node[sumbox] (marg) at (-40mm,-32mm)
 {\textbf{margin}\\[-1pt] \footnotesize positive: it stays itself};
\draw[arr] (pin) -- (diss);
\draw[arr] (diss) -- (rep);
\draw[arr] (rep) -- (marg);
\node[note] at (-40mm,-44mm) {a computable quantity,\\ not a judgment};

% ---- (b) the viability tube ----
\node[lab] at (44mm,34mm) {\textbf{(b) The viability tube}};
\draw[tube] (18mm,4mm) .. controls (44mm,14mm) and (62mm,-6mm) .. (88mm,2mm)
 -- (88mm,-16mm) .. controls (62mm,-24mm) and (44mm,-4mm) .. (18mm,-14mm) -- cycle;
\draw[good] (18mm,-4mm) .. controls (40mm,4mm) and (60mm,-14mm) .. (86mm,-6mm);
\draw[bad]  (18mm,-6mm) .. controls (38mm,-14mm) and (54mm,-22mm) .. (76mm,-27mm);
\node[note, anchor=west] at (58mm,11mm) {margin $>0$: inside, indefinitely};
\node[note, anchor=north] at (50mm,-33mm)
 {margin $<0$ for longer than the repair time: it leaves the class};

% ---- the second, different question ----
\node[sumbox, fill=teal!12, draw=teal!60!black, minimum width=104mm,
 minimum height=15mm, text width=98mm] (qci) at (14mm,-58mm)
 {\centering\textbf{A different question: how much of the\\ four-pillar runtime does it run?}\\[1pt]
  \footnotesize scored so a missing pillar cannot be offset by excellence in
  another --- and the weights are not derived, so the score ranks two
  systems rather than certifying one\par};
\end{tikzpicture}%
}
\caption[The maintenance margin, the viability tube, and the second
question.]{Two questions this Part separates, which are usually asked as one.
\emph{(a)} Whether a thing can hold itself together is arithmetic: what comes
in through the ports, minus what is irreversibly lost, minus what repair
costs. The margin is a computable quantity rather than a judgment, and it can
be evaluated for a protocell, a reef, or an institution. \emph{(b)} Averaged
over a window longer than the system's own repair time, a positive margin
keeps the trajectory inside the region where the system is still itself; a
margin that stays negative for longer than that carries it out, and it stops
being a member of the class. The lower band asks something different and only
loosely correlated: how much of the four-pillar runtime the surviving system
actually runs. That score is built so that a system missing one pillar cannot
compensate with excellence in another, because that is what incompleteness
means --- and its weights are not derived, so the honest use of it is to rank
two systems and not to certify one. The persistence class is defined against
the survival of a particular packet and carries no suggestion of intention or
of moral standing.}
\label{fig:a7-seos-margin}
\end{figure}

\section{Why persistence and completeness belong after the full Qi stack}
\label{subsec:seos-qci-why-now}

After Parts~\ref{part:qiqua}--\ref{part:qihal}, the manuscript finally possesses the full Qi stack: current locking, trajectory selection, multiscale diagnostics, and memory architecture. Only at that point does it become meaningful to ask two synthesis questions. First, which open systems truly preserve themselves as continuing packets rather than as transient driven patterns? Second, among those persistent systems, how fully is the four-pillar runtime actually realized?

These are distinct questions. Persistence concerns continued viability under throughput, repair, and perturbation. Completeness concerns the degree to which the full Qi grammar is present. The revised architecture therefore places them together in Part~\ref{part:internal}. The section first isolates the persistence class \SeOS, then introduces \QCI\ as a diagnostic of realized completeness.


\subsection{Historical background (with citations)}

Antecedents: Schr\"odinger (1944), Bertalanffy (1950), Maturana--Varela (1980), Kauffman (1993), Prigogine (1967), Wilson (1975) \citep{Schrodinger1944,MaturanaVarela1980,Prigogine1967}; \SeOS{}/\QCI{} unifies persistence/completeness as the joint criterion attached to the full \QiFour{} stack (see A1 Sec.~8 and A4 Sec.~7 for full history).

\section{\SeOS: the class of persistent self-maintaining open systems}
\label{subsec:seos-main}

Once the packet and ledger are defined, one can finally ask which open systems genuinely preserve themselves. Every system in nature is open, but not every open system is a self-maintaining system. A Brownian tracer in a fluid is open, but it does not protect a boundary, repair its own packet, or preserve a memory-bearing attractor. A B\'enard cell is closer to self-maintenance because it exists only under continued throughput, but its packet is comparatively thin and its memory depth is limited. A cell, organism, brain, or institution is more demanding still: it must preserve not only energy throughput but also organization, boundary conditions, and records.

The purpose of \SeOS\ is to isolate this persistent class precisely. The term ``selfish'' is meant in the same stripped-down technical sense in which the selfish-gene metaphor generalized local fitness logic: the system acts to maintain its own packet, not because it possesses human intention, but because the lawful runtime selects trajectories that preserve local viability whenever such preservation is possible. Cooperation, symbiosis, and hierarchy can still emerge, but they emerge as higher-order ways of stabilizing packet maintenance rather than as violations of the local logic.

To make this precise, we first need a metric notion of packet preservation.

\begin{definition}[Packet space and viability tube]
\label{def:viability-tube}
Let \(\Packet\) denote the space of admissible packets together with their current state distributions. Around a target packet-attractor \(\Xi_{\ast}\in\Packet\) define the packet distance
\begin{equation}
\label{eq:packet-metric}
\begin{split}
d_{\SIP}^2(\Xi_1,\Xi_2)
&:=
 d_g^2(x_1,x_2)
+ \alpha_P\,\|\Ports_1-\Ports_2\|^2
+ \alpha_L\,\|\Lind_1-\Lind_2\|^2\\
&\quad
+ \alpha_F\,\bigl|\mathcal{F}_{\GFF,1}-\mathcal{F}_{\GFF,2}\bigr|^2
+ \alpha_O\,d_O^2(\Obs_1,\Obs_2),
\end{split}
\end{equation}
with nonnegative weights \(\alpha_{P,L,F,O}\). A \emph{viability tube} of radius \(\epsilon\) is the subset
\begin{equation}
\label{eq:viability-tube}
\mathcal{U}_{\epsilon}(\Xi_{\ast})
=
\left\{\Xi\in\Packet:
 d_{\SIP}(\Xi,\Xi_{\ast})\le \epsilon,
 B_{\partial}(\Xi)\le B_{\max},
 I_m(\Xi)\ge I_{\min}
\right\},
\end{equation}
where \(B_{\partial}\) measures boundary damage and \(I_m\) measures memory integrity.
\end{definition}

Definition~\ref{def:viability-tube} says what it means for a packet to survive: the state, ports, lawful generator, ledger, and observables must remain sufficiently close to a maintained attractor, while boundary damage stays bounded and memory integrity remains above threshold.

We can now define the persistence class.

\begin{definition}[\SeOS]
\label{def:seos}
An open system instance with packet \(\Xi_t\) is a member of \(\SeOS\) if there exist a maintained attractor \(\Xi_{\ast}\), a viability tube \(\mathcal{U}_{\epsilon}(\Xi_{\ast})\), and a physically admissible feedback law such that:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item \textbf{packet retention:} trajectories re-enter and remain in \(\mathcal{U}_{\epsilon}(\Xi_{\ast})\) after sufficiently small perturbations;
\item \textbf{positive maintenance margin:} the cycle-averaged maintenance power
\begin{equation}
\label{eq:maintenance-margin}
\Maint
:=
\bigl\langle\mathcal{P}_{\mathrm{port}}\bigr\rangle_{\tau}
-
\bigl\langle\mathcal{D}_{\GFF}\bigr\rangle_{\tau}
-
\bigl\langle\RepairMin\bigr\rangle_{\tau}
\end{equation}
is positive for some physically relevant window \(\tau\);
\item \textbf{self-maintenance:} the repair term \(\RepairMin\) includes the minimal ledger cost needed to restore packet damage, boundary mismatch, and memory degradation;
\item \textbf{operational identity:} the observable dictionary remains valid while the system is maintained.
\end{enumerate}
\end{definition}

Definition~\ref{def:seos} is intentionally local and non-moral. A system is ``selfish'' only in the sense that its lawful feedback and port usage are organized around the preservation of its own packet. It may cooperate with larger systems, embed within a larger hierarchy, or even sacrifice one subsystem to preserve a higher-order packet. What matters is that the preserved object is the system's own effective instance packet at the chosen level of description.

\section{The persistence theorem}
\label{subsec:seos-theorem}

The definition above becomes practically useful only when one can test it through a dynamical criterion. Because Part~\ref{part:internal} places persistence after the full Qi stack, that criterion must be written as a consequence of the already-built runtime rather than as an independent biological intuition. The criterion is simplest when written as a Lyapunov-type estimate.

\begin{theorem}[Persistence criterion for \SeOS (T-Qi4-14.3)]
\label{thm:seos-persistence}
Let \(\Xi_t\) be an admissible packet with target attractor \(\Xi_{\ast}\). Define the viability functional
\begin{equation}
\label{eq:viability-functional}
\Viab(\Xi_t)
:=
 d_{\SIP}^2(\Xi_t,\Xi_{\ast})
+ \lambda_{\partial} B_{\partial}(\Xi_t)
+ \lambda_m\bigl(I_{\min}-I_m(\Xi_t)\bigr)_{+},
\end{equation}
with \(\lambda_{\partial},\lambda_m>0\). Suppose there exist constants \(\kappa>0\) and \(c\ge0\) such that along the packet dynamics,
\begin{equation}
\label{eq:viability-drift}
\frac{\dd}{\dd t}\Viab
\le
-\kappa\,\Viab + c - \beta\,\Maint,
\qquad \beta>0.
\end{equation}
If the averaged maintenance margin satisfies \(\Maint>c/\beta\), then \(\Viab\) remains bounded and the packet is persistent; hence the system belongs to \(\SeOS\). If \(\Maint<0\) for windows longer than the intrinsic repair time, the packet leaves the viability tube and the system exits \(\SeOS\).
\end{theorem}

\begin{proof}
Equation~\eqref{eq:viability-drift} is a standard dissipative estimate. If \(\Maint>c/\beta\), then the right-hand side is strictly negative whenever \(\Viab\) exceeds a finite threshold. By Gr\"onwall's inequality, \(\Viab(t)\) remains bounded and eventually enters an invariant ball around the maintained attractor. Since \(\Viab\) contains packet distance, boundary damage, and memory integrity, the packet remains in the viability tube and the system is persistent. Conversely, if \(\Maint<0\) over windows longer than the repair time, then the right-hand side eventually becomes positive, \(\Viab\) grows, and either packet distance, boundary damage, or memory loss drives the system out of the tube. It therefore exits \(\SeOS\). \qedhere
\end{proof}

The theorem should be interpreted carefully. It does \emph{not} say that every \SeOS\ must be in thermodynamic equilibrium or that every member has a fixed-point attractor. The maintained attractor may be a limit cycle, torus, metastable branch, or distributed memory manifold. The theorem only says that packet preservation requires positive maintenance margin after dissipation and repair are paid for.

\begin{table}[H]
\centering
\caption[Open, packet-bearing, and self-owning systems.]{Open systems, packet-bearing systems, and the \SeOS{} class. Five representative systems run down the first column --- a passive tracer, a driven dissipative pattern, a protocell or microbe, a neural organism, and a distributed institution --- with whether each maintains a packet, its memory depth, its maintenance margin, and the resulting classification. The distinction the table draws is the sharp one: every row is open, but only those with a positive maintenance margin are self-owning, and that margin is a computable quantity rather than a judgment.}
\label{tab:seos-classification}
\small
\begin{tabular}{p{0.20\textwidth}p{0.16\textwidth}p{0.15\textwidth}p{0.18\textwidth}p{0.21\textwidth}}
\toprule
\textbf{System type} & \textbf{Maintained packet?} & \textbf{Memory depth} & \textbf{Maintenance margin} & \textbf{Classification} \\
\midrule
Passive tracer / Brownian particle & No & Negligible & Not defined as self-repair & Open system, not \SeOS \\
Driven dissipative pattern (e.g.\ simple convection cell) & Weak or pattern-level only & Short & Positive only while driven; thin repair logic & Borderline or proto-\SeOS \\
Protocell / microbe & Yes & Finite adaptive memory & Positive under nutrient throughput and repair & \SeOS \\
Neural organism / cognitive circuit & Yes & Deep internal and external memory & Positive under metabolic and information throughput & \SeOS\ with rich \QiHAL \\
Distributed institution / society & Yes, distributed & Externalized archive plus internal memory & Positive if coordination and repair exceed dissipation/shock costs & Higher-order distributed \SeOS \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Remark on ``selfish'' (non-moral).}
\label{subsec:seos-cooperation}
The term is non-moral: it means only that the local runtime is evaluated against preservation of the packet under study. Cooperation, symbiosis, and division of labor are therefore typical \SeOS{} realizations; conversely, scale-dependent pathology (cancer clones, addiction loops, brittle institutions) arises whenever a sub-packet maintains its own attractor at the expense of a larger packet.


\section{\texorpdfstring{\QCI}{QCI}: the \QiFour{} Completeness Index}
\label{subsec:qci-main}

Persistence is necessary but not sufficient for the full Paper~IV-B program. A packet may survive while realizing only a thin subset of the Qi runtime. A bacterium and a cortex can both be members of \SeOS, but they do not instantiate the Qi pillars to the same depth. For that reason the manuscript now introduces a provisional completeness diagnostic: the \emph{\QiFour{} Completeness Index}, abbreviated \QCI.

The intuition is straightforward. The four Qi pillars are not four unrelated ornaments. They are four mutually constraining layers of one runtime: current locking (\QiQUA), trajectory selection (\QiEP), scale diagnostics (\QiCON), and memory architecture (\QiHAL). A system that lacks one of these layers may still persist, but it does not realize the whole \QiFour{} stack. A completeness index should therefore reward simultaneous realization rather than isolated excellence in one pillar alone.

\begin{definition}[\QiFour{} Completeness Index]
\label{def:qci}
For a packet-bearing open system \(S\), let
\[
q_{\mathrm{QUA}}(S),\ q_{\mathrm{EP}}(S),\ q_{\mathrm{CON}}(S),\ q_{\mathrm{\HAL{}}}(S)\in[0,1]
\]
be domain-specific realization scores for the four pillars. Fix positive weights \(w_a\) with
\[
\sum_{a\in\{\mathrm{QUA},\mathrm{EP},\mathrm{CON},\mathrm{\HAL{}}\}} w_a = 1.
\]
The \emph{strict} \QiFour{} Completeness Index is defined by the weighted geometric mean
\begin{equation}
\label{eq:qci-definition}
\QCI(S)
:=
q_{\mathrm{QUA}}(S)^{w_{\mathrm{QUA}}}
q_{\mathrm{EP}}(S)^{w_{\mathrm{EP}}}
q_{\mathrm{CON}}(S)^{w_{\mathrm{CON}}}
q_{\mathrm{\HAL{}}}(S)^{w_{\mathrm{\HAL{}}}}.
\end{equation}
\end{definition}

The geometric mean is chosen for a reason. If one pillar is absent, completeness should collapse strongly rather than be hidden by large scores in unrelated pillars. This is exactly what the product form guarantees. If desired for phenomenology or exploratory data analysis, one may also track the additive companion score
\begin{equation}
\label{eq:qci-additive}
\QCI_{+}(S)
:=
\sum_a w_a q_a(S),
\end{equation}
but Eq.~\eqref{eq:qci-definition} is the stricter and conceptually cleaner quantity.

The pillar scores themselves are intentionally domain-dependent but structurally constrained. A useful canonical choice is:
\begin{align}
q_{\mathrm{QUA}}(S) &:= \exp\!\left[-\frac{D_{\mathrm{lock}}(S)}{\sigma_{\mathrm{QUA}}}\right],
\label{eq:qci-qua}
\\[3pt]
q_{\mathrm{EP}}(S) &:= \exp\!\left[-\frac{D_{\mathrm{sel}}(S)}{\sigma_{\mathrm{EP}}}\right],
\label{eq:qci-ep}
\\[3pt]
q_{\mathrm{CON}}(S) &:= \frac14\Bigl(c_{\mathrm{scale}}+c_{\mathrm{Levy}}+c_{\mathrm{frac}}+c_{1/f}\Bigr),
\label{eq:qci-con}
\\[3pt]
q_{\mathrm{\HAL{}}}(S) &:= \frac15\Bigl(c_{\mathrm{\MePMoS{}}}+c_{\mathrm{\NHT{}}}+c_{\mathrm{\HAL{}}}+c_{\mathrm{\BFFT{}}}+c_{\mathrm{switch}}\Bigr),
\label{eq:qci-hal}
\end{align}
where the \(c\)-scores lie in \([0,1]\), \(D_{\mathrm{lock}}\) measures deviation from the locked-current ray, and \(D_{\mathrm{sel}}\) measures deviation from the common selector family. Other domain-specific realizations are permitted provided they preserve the same monotonic meaning.

\begin{table}[H]
\centering
\caption[What each of the four pillar scores measures.]{Interpretation of the four pillar scores entering the completeness index. Each row is one score, with what it measures and the typical intuition for it across domains. The index is a weighted sum of the four and is offered as a comparative instrument rather than an absolute one: the paper is explicit that the weights are not derived, so the honest use of the index is to rank two systems, not to certify one.}
\label{tab:qci-pillars}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.43\textwidth}}
\toprule
\textbf{Pillar score} & \textbf{What it measures} & \textbf{Typical intuition across domains} \\
\midrule
\(q_{\mathrm{QUA}}\) & fidelity of current locking & whether event-level energy, matter/charge, and information bookkeeping stays near one common locked ledger \\
\(q_{\mathrm{EP}}\) & selector coherence & whether observed trajectories can be organized by one admissible selector family rather than by unrelated heuristics \\
\(q_{\mathrm{CON}}\) & multiscale diagnostic closure & whether scale covariance, heavy tails, fractality, and two-knee spectra co-occur coherently \\
\(q_{\mathrm{\HAL{}}}\) & architectural completeness & whether memory, sensing, motion, multiscale coding, and commitment switching form a stable holographic architecture \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Persistence is not the same as completeness]
A system may be high in \SeOS\ membership and low in \QCI. A stable chemical oscillator, for example, may persist while realizing only a thin memory architecture. Conversely, a rich but damaged neural system may momentarily preserve substantial Qi structure while approaching loss of viability. \SeOS\ and \QCI\ therefore answer different questions: the first asks whether a packet can maintain itself; the second asks how fully the four-pillar runtime is realized on that packet.
\end{remark}
\section{Integrated view: from runtime constitution to persistence and completeness}
\label{subsec:seos-qci-integrated}

The notions derived above and the new completeness index can now be assembled into one compact flow. An admissible system first declares a canonical packet. That packet determines the state manifold, ports, generator, ledger bundle, and observables. The generator acts on the packet distribution, while the scalar \GFF\ functional organizes gradient descent and maintenance balance. The current vector is locked by the Arisaka Equation. The resulting dynamics then passes through the general runtime pipeline:
\begin{equation}
\label{eq:sip-eegg-seos-flow}
\SIP
\xrightarrow{\text{declare system instance}}
\GFF
\xrightarrow{\text{lawful \GDOS\ evolution}}
\DGI \longrightarrow \GSSB \longrightarrow \GA.
\end{equation}
If, in addition, the cycle-averaged maintenance margin remains positive, the maintained attractor is not merely a transient branch but a persistent packet:
\begin{equation}
\label{eq:seos-membership-flow}
(\SIP,\GFF,\GA)
+ \bigl\langle\Maint\bigr\rangle_{\tau}>0
\quad\Longrightarrow\quad
\text{system instance } S\in\SeOS.
\end{equation}

Equations~\eqref{eq:sip-eegg-seos-flow} and \eqref{eq:seos-membership-flow} summarize the persistence half of the present section. The completeness half is measured by \QCI, which records how fully the four Qi pillars are realized on a persistent packet. Together, persistence and completeness distinguish merely surviving packets from those that realize the full memory-bearing open-system stack. The framework is no longer only a constitutional story about what theories are admissible, nor only a dynamical story about how open systems flow, nor only an architectural story about memory and prediction. It is now also a packaging story: any open system can be declared, evolved, budgeted, and tested in one grammar.

\begin{table}[H]
\centering
\caption[How packet, ledger, runtime and persistence fit.]{Integrated roles of the packet, the free-energy ledger and the persistence class inside the full runtime. Four layers run down the first column, with the primary object each carries, the main mathematical task it performs, and the scientific question it answers. The table is the architecture of the paper's second half in one page, and the fourth column is the reason the layers are in this order: each question presupposes the answer to the one above it.}
\label{tab:sip-eegg-seos-integration}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.14\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}}
\toprule
\textbf{Layer} & \textbf{Primary object} & \textbf{Main mathematical task} & \textbf{Main scientific question} \\
\midrule
Packet & \SIP & Declare the admissible system instance & What exactly is the object being modeled and measured? \\
Ledger & \GFF & Supply a unique scalar budget and its current bundle & What quantity is minimized, maintained, or spent during dynamics? \\
Runtime & \DGI+\GSSB+\GA\ plus \QiFour{} & Evolve, select, diagnose, and architect the state & How does the system explore, choose, stabilize, and remember? \\
Persistence class & \SeOS & Test invariant maintenance under throughput and repair & Does the packet merely fluctuate, or can it preserve itself as a continuing unit? \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Worked examples.} Persistence and completeness across protocells, organisms, brains, and institutions are tabulated with the appropriate $q_{\mathrm{QUA}},q_{\mathrm{EP}},q_{\mathrm{CON}},q_{\mathrm{HAL}}$ entries; see Appendix~\ref{app:technical-details-for-sip-and-eegg} and the cross-domain cards therein.

\section{Staying alive, and how much of it you are doing}
\label{sec:close-seos}

Two questions have been separated in this Part that are usually asked as
one, and the separation is worth keeping.

The first is whether a thing can hold itself together. That turns out to
be an arithmetic question. Something comes in; something is lost that
cannot be recovered; something must be spent putting right what has gone
wrong. If the first exceeds the other two on average, over a window long
enough to include the repairs, the thing persists. If it does not, for
long enough, the thing stops. There is no vital principle in that
sentence and no threshold of complexity --- only a margin, which can be
positive or negative, and which can in principle be measured for a
protocell, a coral reef, a company or a state.

The second question is how much of the machinery of the preceding Parts
a surviving thing actually runs, and the answer is not correlated with
the first in the way one might expect. Plenty of things persist for a
very long time while doing almost nothing: a chemical clock can outlast
the animal watching it. Plenty of things run a great deal of machinery
while sliding toward the edge. Survival and richness are different axes,
and confusing them is how the word "complexity" came to mean nothing in
particular.

Both measures are deliberately narrow, and the narrowness is the honest
part. The persistence class is defined against the survival of a
particular packet, and says nothing whatever about whether that packet
deserves to survive --- a distinction that matters, because the same
arithmetic describes a healing wound and a tumor, a functioning
institution and a captured one. What separates those is which packet you
chose to draw the boundary around, and the framework is explicit that it
does not choose for you. The completeness index is narrower still: its
weights are not derived, so it can rank two systems and cannot certify
one.

It is worth noticing what has quietly failed to appear in this Part,
because the absence is the result. There is no moment anywhere in it
at which something becomes alive. There is a margin, which is positive
or negative; there is an index, which is higher or lower; both vary
continuously and neither has a step in it. Anyone who has tried to say
where the line falls --- around a virus, a prion, a self-replicating
crystal, an organelle that used to be somebody else --- has found that
the line does not want to be drawn, and has usually concluded that the
difficulty is temporary. The answer here is that the line is not there
to be found. What is there is a plane with things scattered across it,
and the words we have been using are names for regions of that plane
rather than for kinds of thing. That is the loss of a certain sort of
clarity and the gain of a more useful one, and it is the same trade
this framework has made at every level: a boundary that felt absolute
turns out to be a place where a continuous quantity happens to be
changing quickly.

What is worth taking from this Part is smaller than a theory of life and
more useful than a metaphor. It is that "staying alive" can be written
as an inequality with named terms, that the terms can be measured in
domains that share no vocabulary, and that whether a thing survives and
whether it is doing anything interesting are two questions with two
answers. Most of what makes those questions hard is that they are
usually asked together.

One thing this Part has deliberately not done is look inside. The
packet has been treated from without, the way a physician treats a
patient or an auditor a firm: what comes in, what leaks away, what the
repairs cost. But a system running the full machinery of the preceding
Parts is doing something no account from outside can see. It is
holding a picture of what surrounds it and correcting that picture
against the evidence that reaches its own surface. The next Part turns
the same open-system logic inward and asks what such a picture is, how
much of the world can ever get into it, and what follows from the fact
that the thing holding the picture is itself inside the world the
picture is of.


\GGPart[Qi-4 and the GGA]{The \QiFour{} pillars as the \GGA{} characterization}{What the ground attractor carries, and in which direction it is proved}
\label{part:gga}

\noindent
This is the shortest Part in the paper and the one that does the most
joining. Two halves of a program have been built out of different
materials, and this is where they are shown to be describing the same
thing.

The geometric companion answers a question about shape: given a
landscape, where does a system that is rolling downhill finally stop?
Its answer is the deepest basin available on the stratum, and that
answer is reached using nothing but the shape of the landscape. No
informational condition enters it anywhere.

The four preceding Parts answer a different set of questions, about
ledgers, budgets, scales and memory. They were built without reference
to where anything comes to rest.

What is established here is that the configuration the geometry stops at
is also the configuration that satisfies all four of the informational
conditions. It carries the locked four-way ledger. It is where the
unified variational principle is stationary. It is coherent across the
scales between its own two turns. And it is a place where an
architecture can halt rather than having to keep working. That the two
descriptions, made of different materials, pick out one configuration is
a real claim and it is this paper's to make.

The direction of that claim is the thing to read carefully, and the Part
is deliberate about it. What is proved is that the resting place carries
the four certificates. What is \emph{not} proved is the converse ---
that no other basin carries all four, so that the four conditions could
have found the resting place by themselves. That would need an argument
showing at least one condition fails at every basin that is not the
deepest, and no such argument exists yet in this program. It is
recorded as an open problem. Asserting the converse by pointing at a
companion whose own argument invokes these same four conditions would
be a circle; the claim is therefore kept separated into the direction
that is proved and the direction that is not.


\label{sec:qi4-gga-selector}

The four pillars are complete, and one consolidating statement is now
both possible and owed --- though not the one the series has been quoting.

The Ground Geometric Attractor is not selected by the pillars. It is
fixed by the landscape alone: \ASix{}'s per-stratum ground-state
corollary produces it from the Morse structure of an admissible
stratum, using no admissibility filter of any kind, and \ASix{}'s
definition is accordingly the plain global minimum. So there is
nothing left for a filter to select from once the descent has run,
and a formulation that speaks of the attractor ``selected from among
the \GA{}'s by the four \QiFour{} pillars'' misdescribes what the two
papers jointly establish.

What the pillars do is \emph{characterize} that object. The
configuration the descent halts at is also the configuration that
carries the locked four-current ledger, extremizes the unified
variational principle, transports coherently across scales, and is one
at which an architecture can halt. That the geometric description and
the informational description pick out the same configuration is a
substantive claim, it is this paper's to make, and this section is its
home. We state it as a theorem and make the characterization
\emph{pillar by pillar} --- and we are explicit about which direction
of it is proved.

\begin{theorem}[\QiFour{} characterization of the \GGA{} (T-Qi4-GGA)]
\label{thm:qi4-gga-characterisation}
Let \(\Sigma\) be an admissible stratum in the sense of \ASix{}, and
let \(\GGA(\Sigma)\) be its Ground Geometric Attractor --- the unique
global minimum of \(\mathcal F_{\GFF}|_\Sigma\), supplied by that
paper's per-stratum ground-state corollary. Then \(\GGA(\Sigma)\)
satisfies all four \QiFour{} admissibility conditions. The four act as
follows:
\begin{itemize}\itemsep2pt
\item \textbf{\QiQUA{} (ledger admissibility).} The candidate must
carry the locked four-current identity \(J_E/Q_B = J_e = J_B =
J_{Qi}\) (the Arisaka Equation): an attractor whose currents do not
lock is not a lawful occupant of the boundary ledger, however stable
its basin.
\item \textbf{\QiEP{} (variational admissibility).} The candidate
must be a genuine extremum of the unified extremal principle --- the
boundary projection of the 6D least-action structure --- not merely
a local minimum of a coarse-grained effective potential.
\item \textbf{\QiCON{} (scale coherence).} The candidate must
survive scale transport: its conformal/L\'evy/\(1/f\) signatures
must be those of a fixed point, so that the attractor seen at one
scale is the same attractor seen at another.
\item \textbf{\QiHAL{} (halting admissibility).} The runtime must be
able to \emph{halt} there: an architecture descending the landscape
must terminate at the candidate with no admissible descent
remaining. This is the architectural face of the no-exit
characterization of ground-state-ness --- the \GGA{} is where
\GRIPin{} halts --- and it is the filter that excludes deep but
unreachable or unsustainable basins.
\end{itemize}
Existence and uniqueness of \(\GGA(\Sigma)\) are \ASix{}'s, not this
paper's, and are used here as an import.
\end{theorem}

\begin{proof}
Each clause is an application of the corresponding pillar theorem of
Parts~\ref{part:qiqua}--\ref{part:qihal} to the configuration \ASix{} delivers. \QiQUA{}: the current
locking of the Arisaka equation holds at every admissible boundary
event, and the terminus of a completed descent is such an event, so the
ledger is carried there. \QiEP{}: the terminus is a critical point of
\(\mathcal F_{\GFF}|_\Sigma\) by construction, and by the variational theorem of Part~\ref{part:qiep} the boundary
projection of the 6D extremal
principle is stationary exactly at such points. \QiCON{}: by the
two-knee spectral theorem the scale diagnostics of a configuration at
which the descent has halted are those of a fixed point, since no
residual flow remains to break scale covariance. \QiHAL{}: the
architectural halting condition asks that a descending runtime
terminate with no admissible descent remaining, which is precisely
what it means for the exit-channel set of the global minimum to be
empty. Each of the four is therefore a property the ground attractor
possesses.\qedhere
\end{proof}

\begin{remark}[What is proved here, and what is not]
\label{rem:qi4-gga-converse}
The theorem proves one direction: the object \ASix{}'s descent halts at
is admissible under all four pillars. The converse --- that \emph{no
other} Geometric Attractor of the stratum passes all four, so that the
four-pillar condition would by itself single out the ground attractor
--- is not proved here and should not be quoted as though it were. It
would require a no-go argument showing that at least one filter fails
at every non-minimal basin, and no such argument is currently in the
cohort; it is the same gap recorded as \textbf{PS-Qi4-6}. Earlier
formulations of this section asserted the converse and rested it on the
capstone, whose own proof of the identification invokes these pillars
--- a circle, and the reason the claim is now separated into a proved
direction and an open one. Nothing downstream is weakened by the
separation, because every downstream use is of the proved direction:
papers quote the four certificates as properties of the electron, the
proton and the Kaxion, never as the instrument that found them.
\end{remark}

\emph{Physical picture.} Chemistry again supplies the intuition. A
molecule has many stationary electronic configurations; the ground
state is the one that carries the right electron count (ledger),
genuinely minimizes the energy (variational), looks the same under
every spectroscopic magnification (scale coherence), and is where
relaxation actually ends (halting). The \QiFour{} pillars are those
four certificates promoted to constitutional rank. The worked case is
the electron. Its stratum is the charged-lepton stratum, which \ASix{}
establishes as admissible, so the ground attractor exists and is unique
there by that paper's corollary; the electron is its readout; and the
theorem above says the four certificates are all carried at that
configuration --- the locked ledger, the stationarity, the fixed-point
diagnostics, and the empty exit channel that is the reason it does not
decay. That is why every electron, every proton and every Kaxion is the
same object everywhere: not because a filter picked one out of many,
but because there is one ground attractor per stratum and the four
certificates describe it (\citealp{ArisakaA9SM};
\citealp{ArisakaC1Kaxion}).



\begin{figure}[htbp]
\centering
\resizebox{0.762\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=3pt, inner sep=4pt},
 geo/.style={fill=green!12, draw=green!55!black, line width=0.70pt,
 minimum width=62mm, minimum height=17mm},
 conf/.style={fill=orange!16, draw=orange!65!black, line width=1.10pt,
 minimum width=62mm, minimum height=15mm},
 cert/.style={fill=blue!10, draw=blue!60!black, line width=0.60pt,
 minimum width=52mm, minimum height=12mm},
 proved/.style={-{Stealth[length=2.6mm]}, line width=1.10pt,
 draw=orange!65!black},
 openarr/.style={-{Stealth[length=2.4mm]}, line width=0.80pt,
 draw=red!60!black, dash pattern=on 3pt off 2.5pt},
 tag/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt},
]
\node[geo] (land) at (-46mm,0)
 {\textbf{The landscape alone}\\[-1pt]
  \footnotesize the geometric companion's descent,\\
  \footnotesize using no informational filter};
\node[conf] (cfg) at (34mm,0)
 {\textbf{One configuration}\\[-1pt]
  \footnotesize where the descent halts};
\draw[proved] (land) -- (cfg) node[tag, pos=0.5] {fixes it};

\node[cert] (c1) at (34mm,30mm)  {carries the locked four-way ledger};
\node[cert] (c2) at (34mm,16mm)  {is where the one budget is stationary};
\node[cert] (c3) at (34mm,-16mm) {is coherent across its own two knees};
\node[cert] (c4) at (34mm,-30mm) {is a place an architecture can halt};

\draw[proved] (cfg.north) -- (c2.south);
\draw[proved] (c2.north) -- (c1.south);
\draw[proved] (cfg.south) -- (c3.north);
\draw[proved] (c3.south) -- (c4.north);

\node[tag, text=orange!65!black] at (34mm,44mm)
 {\textbf{PROVED}: the resting place carries all four};

% the open converse: the return arrow, drawn as what it is
\draw[openarr] (c4.west) .. controls (-10mm,-40mm) and (-46mm,-30mm)
 .. (land.south);
\draw[red!60!black, line width=0.8pt, fill=white] (-14mm,-38mm) circle (2.2mm);
\node[tag, text=red!60!black, align=center] at (-46mm,-44mm)
 {\textbf{NOT PROVED}: that no other basin carries all four\\
  --- recorded as an open problem, not quoted as settled};
\end{tikzpicture}%
}
\caption[Four certificates, and the direction that is proved.]{Where the two
halves of the program meet, and --- as importantly --- which way the meeting
runs. The geometric companion fixes the resting place using the shape of the
landscape alone; no informational condition enters that argument anywhere.
What this Part establishes is that the configuration the descent halts at is
also the configuration satisfying all four of the informational conditions
built in the preceding Parts: it carries the locked four-way ledger, it is
where the single budget is stationary, it is coherent across the scales
between its own two knees, and it is a place where an architecture can stop
rather than having to keep working. That two descriptions made of entirely
different materials pick out one configuration is a substantive claim and it
is this paper's to make. The dashed return arrow is the direction that is
\emph{not} established: whether the four conditions could by themselves have
found that configuration, with the geometry set aside. That would need an
argument showing at least one condition fails at every basin which is not the
deepest, no such argument exists in this program, and it is recorded as an
open problem rather than quoted as settled.}
\label{fig:a7-gga-certificates}
\end{figure}

\section{Four certificates, and what they are not}
\label{sec:close-part3}

A chemist already knows what this Part has done, in a smaller setting.
Ask why a molecule sits in the state it sits in, and the answer has
several parts that arrive together. It carries the right number of
electrons. It genuinely minimizes the energy rather than merely
sitting at a flat spot. It looks like itself under every instrument
that can be pointed at it. And it is where relaxation actually stops,
rather than a way station on the road to somewhere lower. Four
certificates, held at once, by one configuration.

The four pillars are those certificates promoted to constitutional
rank, and made to apply to anything that persists rather than only to
molecules. The ledger is Nature's bookkeeping in four currencies at
once. The variational principle is the statement that the path taken,
the currents maintained, and the beliefs updated are three readings of
one budget. The scale condition is what the thing must look like
between its slowest and its fastest surviving rate. And the
architecture is what a system must become once it holds a memory for
longer than one act of looking.

It matters what kind of claim this is, and the Part is deliberate about
it. The certificates describe the resting place; they did not select
it. The descent finds where to stop by the shape of the landscape
alone, with no informational filter applied anywhere. What is
established here is that the place it stops at satisfies all four
conditions --- and that is a real claim, since the geometric
description and the informational description are built out of entirely
different materials, and there is no obvious reason they should agree
on a single configuration. Whether the four conditions could by
themselves have found that configuration, with the geometry set aside,
is not established. It is written down as an open problem, and it
should not be quoted as though it were settled.

That distinction is worth more than it costs. It is the reason every
electron is the same as every other electron, and every proton the same
as every other proton --- not because some filter selected one
candidate out of many, but because there is exactly one place to stop,
and these four statements are what a reader would find written on it.

There is a smaller lesson in the constant. Information has spent a
century as the one quantity in science with no price attached: freely
invoked, freely conserved or destroyed as the argument required, and
never billed. Here it is billed, at a rate fixed by the same integers
that fix how many families of matter there are, and the bill can be
checked against a molecule that every living thing on this planet
carries. Whether the check passes is a matter for the laboratory. That
it is a check at all --- that a statement about bits can be settled by
a calorimeter --- is the change this Part is arguing for.


\GGPart[Internal measurement]{Internal Measurement and the \EIC{} Runtime}{Outer Hologram, Inner Hologram, and the emergence of an internal cosmos}
\label{part:internal}

\noindent
Nothing ever touches the world. It touches its own boundary.

That is not a philosophical flourish; it is what the projection of the
preceding Parts already implies, and this Part follows the implication.
A retina receives a two-dimensional sheet of light and the animal
behind it needs three dimensions to move through. A bacterium samples
one concentration at one point and needs to know which way the food is.
A physicist reads four-dimensional instruments and is trying to
establish something about six dimensions. In every case the thing that
guides the next action cannot have been handed over intact. It has to
be built.

So this Part separates three objects that are usually run together. The
first is what the world lawfully delivers to a boundary --- real,
public, and lossy, since many different external situations deliver the
identical record. The second is what a system maintains internally: not
a copy of its surroundings but a working model with a geometry and a
memory, which is a very different sort of thing from a photograph. The
third is the loop that keeps the second answerable to the first.

Two results here are worth stating in advance because they overturn a
natural assumption. The first is that motion is not an accessory to
perception but a condition of it: a system that cannot act on its
surroundings cannot, in general, recover them at all, since different
external states would produce identical sensory histories, and moving
is what breaks the tie. A retina that cannot move is depth-poor; a
bacterium that cannot swim is gradient-poor. The second is that memory
is not optional either --- without it there is no internal world, only
a running estimate that forgets itself.

The Part then does something it would be easy to do badly, and takes
some care not to. It defines a technical term for the portion of a
system's internal model that has been selected, stabilized, and made
available to the system's own next cycle. The definition is
operational, meant to be applied to data, and graded rather than
all-or-nothing. It is not a theory of subjective experience, it takes no
position in the debates about one, and the table of non-claims in this
Part is not a disclaimer bolted onto the results --- it is what makes
the results sayable at all. Most of what this Part contributes is
organization; the short table of predictions marks the few rows that
the world could actually contradict.


\section{\texorpdfstring{\OH + \IH = \EIC}{OH + IH = EIC}: Outer Hologram, Inner Hologram, and Internal Measurement}

%% --- Phase 8 TikZ figure: OH + IH = EIC structure ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 holobox/.style={draw=black!55, rounded corners=4pt, line width=0.80pt, minimum width=4.5cm, minimum height=2.0cm, align=center},
 ohbox/.style={holobox, fill=blue!10},
 ihbox/.style={holobox, fill=orange!10},
 eicbox/.style={draw=purple!55!black, rounded corners=4pt, line width=1.10pt, minimum width=5.5cm, minimum height=2.4cm, align=center, fill=purple!12},
 loop/.style={->, line width=1.10pt, blue!60!black, >=Stealth},
 arrow/.style={->, line width=0.80pt, >=Stealth},
]
% OH: Outer Hologram on the left
\node[ohbox] (OH) at (-5, 2) {\textbf{\OH{}}\\ Outer Hologram\\ \footnotesize environment encoded on the boundary\\ \footnotesize ('t Hooft 1993, Susskind 1995)};

% IH: Inner Hologram on the right
\node[ihbox] (IH) at (5, 2) {\textbf{\IH{}}\\ Inner Hologram\\ \footnotesize system's model of environment\\ \footnotesize (Llinás 2001, predictive coding)};

% EIC: Evolving Inner Cosmos at the bottom
\node[eicbox] (EIC) at (0, -2) {\textbf{\EIC{}}\\ Evolving Inner Cosmos\\ \footnotesize $\EIC \;:=\; \mathcal{C}_{\rm loop}(\OH, \IH)$\\ \footnotesize closed-loop coupling, action-conditioned};

% Loop arrows: OH -> IH (perception) and IH -> OH (action)
\draw[loop, bend left=15] (OH.east) to node[font=\sffamily\footnotesize, above, fill=white, inner sep=1pt] {perception (passive)} (IH.west);
\draw[loop, bend left=15] (IH.south west) to node[font=\sffamily\footnotesize, below, fill=white, inner sep=1pt, sloped] {action (motor)} (OH.south east);

% Composition arrows: OH and IH combine into EIC
\draw[arrow, dashed] (OH.south) -- (EIC.north west) node[font=\sffamily\footnotesize, midway, sloped, above, fill=white, inner sep=1pt] {couple};
\draw[arrow, dashed] (IH.south) -- (EIC.north east) node[font=\sffamily\footnotesize, midway, sloped, above, fill=white, inner sep=1pt] {couple};

% Note about action-conditioned observability
\node[font=\sffamily\footnotesize, align=center, fill=white, inner sep=2pt] at (0, -5)
 {Action-conditioned observability theorem (T-HAL): when action is restricted, $\IH$ model quality degrades.\\
 Memory-bearing $U(1)_{Qi}$ systems realize the closed loop; \SeOS{} systems persist within it.};
\end{tikzpicture}%
}
\caption[The \OH{}+\IH{}=\EIC{} framework of Part~\ref{part:internal}.]{The Outer Hologram + Inner Hologram = Evolving Inner Cosmos ($\OH+\IH=\EIC$) framework of Part~\ref{part:internal}. The Outer Hologram $\OH$ is the environment encoded on the system's boundary ('t Hooft 1993, Susskind 1995); the Inner Hologram $\IH$ is the system's internally maintained model of that environment (Llinás 2001, predictive coding). The Evolving Inner Cosmos $\EIC$ is the closed-loop coupling $\mathcal{C}_{\rm loop}(\OH, \IH)$ that the conscious organism inhabits. Per the clarification, the formula ``$\OH+\IH=\EIC$'' is definitional shorthand for $\EIC := \mathcal{C}_{\rm loop}(\OH, \IH)$ rather than an arithmetic identity; the structural content lives in the Lyapunov refinement theorem on $\EIC$ monotonicity.}
\label{fig:a7-oh-ih-eic}
\end{figure}

\label{sec:oh-ih-eic}

With persistence and completeness fixed, the theory can turn to its deepest conceptual distinction: external projection versus internal reconstruction. Section~\ref{sec:oh-ih-eic} therefore introduces \OH, \IH, and \EIC\ as the final conceptual layer of Part~\ref{part:internal}. This placement is deliberate. Internality must come after runtime constitution, Qi architecture, persistence, and completeness; otherwise it risks floating free of the lawful machinery that makes it possible.

\subsection{Why the runtime still needs a distinction between external projection and internal reconstruction}
\label{subsec:ohih-why-needed}

After everything from Section~\ref{sec:universal-gdos} to
Part~\ref{part:seos}, the manuscript already possesses a universal runtime, a current ledger, a selector principle, a diagnostic layer, a memory architecture, a packet vocabulary, and a persistence criterion. Yet one decisive conceptual task remains unfinished. The earlier sections explain how an open system evolves, how it budgets its currents, how it stores memory, and how it persists. They do not yet fully explain how the world appears \emph{to} an open system, how an open system reconstructs that world internally, and how a selected internal reconstruction may come to function as conscious content.

This missing step is exactly what the relation
\[
\OH+\IH=\EIC
\]
is designed to capture. The notation is intentionally compact, but it must not be read superficially. The symbol \(\OH\) does not mean ``the world itself.'' It means the lawful projection of the inaccessible or higher-dimensional world into observer-accessible boundary records. The symbol \(\IH\) does not mean ``a copy of the world in the head.'' It means the internal reconstructive manifold through which a memory-bearing open system infers, stabilizes, and reuses an allocentric model of what lies beyond its immediate sensory boundary. The symbol \(\EIC\) denotes the \emph{Evolving Inner Cosmos}: the closed loop in which externally projected evidence and internally reconstructed structure are continually aligned, updated, selected, and stabilized.

The need for this distinction is not merely philosophical. It is forced by the mathematics of partial access. A finite open system never interacts with the total six-dimensional bulk in its full state-description. It interacts only through ports and measurement channels. Likewise, a living or cognitive system never receives the full external state; it receives only a narrow sensory stream, often lower-dimensional than the structure it must navigate. The world that guides action must therefore be internally reconstructed rather than directly given. The present section formalizes that statement.

It is useful to mark the three layers immediately. We write \(H^{\mathrm{out}}_t\) for the concrete time-dependent outer hologram, \(H^{\mathrm{in}}_t\) for the time-dependent inner hologram, and \(\mathfrak{E}_t\) for the corresponding evolving inner-cosmos state. Table~\ref{tab:oh-ih-eic-basic-distinction} gives the high-level distinction.

\begin{table}[H]
\centering
\caption[Outer, inner, and the coupling of the two.]{Basic distinction between the outer hologram, the inner hologram and their coupling. Each row is one object, with what it is, what it is \emph{not}, and its constitutional role. The middle column is the important one and is written to close the two most common misreadings: the outer hologram is not a complete description of the environment, and the inner one is not a picture in a viewer's head. Both are structures on a boundary, and the third object is their coupling, not their sum.}
\label{tab:oh-ih-eic-basic-distinction}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}}
\toprule
\textbf{Object} & \textbf{What it is} & \textbf{What it is not} & \textbf{Constitutional role} \\
\midrule
\OH & Lawful projection from the 6D bulk to accessible boundary records & Not the full bulk state itself & Extends \OGGEP\ from laboratory measurement to all scientific observation \\
\IH & Internal reconstructive manifold built from memory, prediction, action, and sensory history & Not a literal pixel-by-pixel copy of the external world & Explains how open systems build allocentric models from partial data \\
\EIC & Closed-loop coupled process aligning \OH\ and \IH\ through repeated updates & Not a mystical addition to dynamics & Explains internal world-model formation, conscious content, and learning across scales \\
\bottomrule
\end{tabular}
\end{table}

The section therefore proceeds in five steps. First, it derives the outer hologram directly from the \OGGEP\ boundary law. Second, it derives the inner hologram from the inverse problem faced by any memory-bearing open system with action. Third, it shows why motion, memory, and allocentric stabilization are mathematically necessary if one starts from low-dimensional or egocentric sensory data. Fourth, it gives a disciplined interpretation of the relation \(\OH+\IH=\EIC\) by introducing an explicit alignment functional. Fifth, it states the consciousness thesis in its careful technical form: consciousness is the system's internal measurement of a selected and stabilized portion of the inner hologram down to an internal boundary implemented by \MePMoS\ and the broader \QiHAL\ architecture.

\subsection{The Outer Hologram: lawful projection from the 6D bulk to measurable boundary records}
\label{subsec:outer-hologram}

\AFive{} already supplies the BulkMap \(\BulkMap:\text{bulk data}\longrightarrow (\BState,\rho_{\partial},\Phi_{t,s},\{\Instr_\alpha\})\) demanded by AX-3; see \AFive{} Theorem~T-OGGEP for the derivation. The present subsection re-reads it as the outer-hologram definition. Define the outer-hologram kernel
\begin{equation}
K_{\mathrm{\OH{}}}(y|X;t)
:=
\Tr\!\left[\Instr_y\,\Phi_{t,0}\!\bigl(\ket{X}\bra{X}\bigr)\right],
\label{eq:oh-kernel}
\end{equation}
with \(X\in \State_6\) a bulk microstate and \(y\in\mathcal{Y}\) an observer-accessible boundary record. In a classical limit this reduces to an ordinary stochastic kernel; in a quantum setting it is the instrument-induced probability density of \(y\) given \(X\). The actual outer-hologram record at time \(t\) is
\begin{equation}
H^{\mathrm{out}}_t(y)
=
\int_{\State_6}
K_{\mathrm{\OH{}}}(y|X;t)\,
\rho_6(X,t)\,\dd X,
\label{eq:outer-hologram-density}
\end{equation}
with normalization
\begin{equation}
K_{\mathrm{\OH{}}}(y|X;t)\ge 0,
\qquad
\int_{\mathcal{Y}} K_{\mathrm{\OH{}}}(y|X;t)\,\dd y = 1.
\label{eq:oh-kernel-normalization}
\end{equation}

\begin{definition}[Outer Hologram]
\label{def:outer-hologram}
Given a bulk state family \(\rho_6(X,t)\), reduced boundary evolution \(\Phi_{t,0}\), and admissible instrument class \(\{\Instr_y\}\), the \emph{outer hologram} at time \(t\) is the boundary law \(H^{\mathrm{out}}_t\) of Eq.~\eqref{eq:outer-hologram-density}; an individual registered observation \(y_t\) is an \emph{\OH{} event}.
\end{definition}

The outer hologram is \emph{lawful} (fixed by the constitutional map, not an arbitrary representation), \emph{lossy} (many bulk states share the same boundary law), and \emph{instrument-relative} (different instrument families coarse-grain the same bulk reality differently). This is not subjectivity but admissible boundary indexing.

\begin{proposition}[The outer hologram is a quotient object]
\label{prop:oh-quotient}
With \(X \sim_{\mathrm{\OH{}}} X'\) iff \(K_{\mathrm{\OH{}}}(\cdot|X;t)=K_{\mathrm{\OH{}}}(\cdot|X';t)\) for the chosen instrument class, the outer hologram depends only on the equivalence class \([X]_{\mathrm{\OH{}}}\): observation accesses a quotient of bulk reality.
\end{proposition}

\begin{proof}
Eq.~\eqref{eq:outer-hologram-density} depends on \(X\) only through \(K_{\mathrm{\OH{}}}(y|X;t)\); equivalent bulk states give identical contributions for every admissible boundary record. \qedhere
\end{proof}

Every registered scientific measurement is an \OH{} event: a particle-detector click, an image frame, an ecological count, a spike train, a fluorescence reading. Empiricism says observation is the source of knowledge; \GDOS{} says observation is a lawful \OH{} event already shaped by the bulk-to-boundary map, and the \IH{} (next subsection) is what turns a stream of such events into an internally usable world.

\begin{table}[H]
\centering
\caption[Boundary encoding in four domains.]{Examples of outer-hologram events across scientific domains. Four domains run down the first column, with the external structure being encoded, the boundary channel or instrument that does the encoding, and what constitutes an event in each. The table's purpose is deflationary: the outer hologram is not an exotic object but the ordinary situation of a finite system learning about what it cannot directly access, and the third column is always a real instrument or a real membrane.}
\label{tab:outer-hologram-examples}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}}
\toprule
\textbf{Domain} & \textbf{Underlying external structure} & \textbf{Boundary channel / instrument} & \textbf{\OH{} event} \\
\midrule
Particle physics & Bulk field excitation or scattering process & Detector, trigger, reconstruction chain & Hit pattern, track, calorimeter energy, event class \\
Cell biology & Molecular state, conformational or concentration field & Receptor, microscope, fluorophore, reporter construct & Bound receptor state, image frame, fluorescence time point \\
Neuroscience & External world plus organism pose and context & Retina, cochlea, mechanoreceptors, electrodes & Sensory sample, spike train, local field potential record \\
Ecology / society & Hidden population or collective state & Survey protocol, sensor network, archive & Count, flow value, transaction record, communication event \\
\bottomrule
\end{tabular}
\end{table}


\subsection{The Inner Hologram: internal reconstruction of allocentric structure from partial sensory streams}
\label{subsec:inner-hologram}

The outer hologram delivers data, but no open system can survive or act intelligently by storing raw data alone. If the incoming stream is partial, delayed, noisy, or lower-dimensional than the relevant external structure, a reconstructive model is required. That model is the inner hologram.

Let \(x_t\in\mathcal{X}_{\mathrm{env}}\) denote the external state relevant to the system at time \(t\), \(a_t\in\mathcal{A}\) the action or motion variable, \(s_t\in\mathcal{S}\) the sensory sample, and \(m_t\in\mathcal{M}\) the internal memory state. We consider the generic controlled open-system dynamics
\begin{subequations}
\label{eq:environment-sensor-dynamics}
\begin{align}
x_{t+1} &= f(x_t,a_t)+\omega_t, \label{eq:env-dyn}\\
s_t &= h(x_t)+\nu_t, \label{eq:sensory-map}
\end{align}
\end{subequations}
with process perturbation \(\omega_t\) and measurement noise \(\nu_t\). In many cases \(h\) is strongly non-injective: different external states generate indistinguishable instantaneous sensory samples. This is the rule rather than the exception. A 2D retina does not uniquely determine a 3D world at one instant; a single odor concentration sample does not uniquely determine a surrounding concentration field; a scalar temperature reading does not uniquely determine a thermal landscape.

The system therefore introduces an internal latent variable \(z_t\in\mathcal{Z}\) that does not merely store the last \(s_t\) but represents a structured hypothesis about the external world. A natural variational definition is
\begin{equation}
q_t(z_t)
=
\operatorname*{arg\,min}_{q}
\left\{
\mathbb{E}_{q}\!\bigl[-\ln p(s_t|z_t)\bigr]
+
D_{\mathrm{KL}}\!\Bigl(
q(z_t)\,\Big\|\,p(z_t|m_{t-1},a_{t-1})
\Bigr)
\right\},
\label{eq:ih-variational-update}
\end{equation}
which is the \QiEP/\FEP-type update written specifically for internal world-model reconstruction. The first term fits the current sensory evidence, while the second term preserves continuity with the previously predicted state. In plain language: the system asks which internal latent configuration best explains the incoming \OH{} event without discarding what its past memory and action already predicted.

\begin{definition}[Inner Hologram]
\label{def:inner-hologram}
For a memory-bearing open system, the \emph{inner hologram} at time \(t\) is the structured internal reconstructive object
\begin{equation}
H^{\mathrm{in}}_t
:=
\bigl(
\mathcal{Z},
q_t,
g_t,
W_t
\bigr),
\label{eq:inner-hologram-definition}
\end{equation}
where \(\mathcal{Z}\) is the latent internal manifold, \(q_t\) is the current posterior or belief state on \(\mathcal{Z}\), \(g_t\) is the learned or reconstructed metric/geometry on that manifold, and \(W_t\) is the \HAL\ memory substrate storing the relational attractor structure.
\end{definition}

Definition~\ref{def:inner-hologram} is intentionally broader than a probabilistic belief state. The \IH{} is not only a posterior density. It also contains geometry and memory. The metric \(g_t\) enters because the internal model must encode distances, directions, or relational structure relevant for prediction and action. The memory tensor \(W_t\) enters because the reconstructed world must persist across time; without storage there is no genuine internal world-model, only a transient estimate.

The role of the \QiHAL\ modules is now immediate. \MePMoS\ provides the causal update order. \NHT\ provides a phase-like coordinate system through which latent geometry can be encoded. \HAL\ provides the storage substrate of recurrent relational structure. \BFFT\ coordinates scales, so that fast sensory detail and slow contextual priors can coexist in the same \IH{}. The \DGI--\GSSB\ switch governs how multiple candidate interpretations remain coexistent during exploration and then collapse to a selected interpretation once evidence and recurrent gain cross threshold.

The \IH{} update may be written in MePMoS-compatible form as
\begin{subequations}
\label{eq:ih-mepmos}
\begin{align}
\widehat{z}_{t|t-1} &= \mathcal{P}(m_{t-1},a_{t-1}), \label{eq:ih-predict}\\
q_t &= \operatorname*{arg\,min}_{q}
\left\{
\mathbb{E}_{q}[-\ln p(s_t|z_t)]
+
D_{\mathrm{KL}}\!\left(q(z_t)\,\|\,\delta_{\widehat{z}_{t|t-1}}\right)
\right\}, \label{eq:ih-infer}\\
m_t &= \mathcal{U}(m_{t-1},q_t,s_t), \label{eq:ih-memory}\\
a_t &= \pi(m_t,q_t), \label{eq:ih-action}
\end{align}
\end{subequations}
where \(\delta_{\widehat{z}_{t|t-1}}\) denotes the predicted latent prior concentrated around the predicted state, and \(\pi\) is the action policy. Equations~\eqref{eq:ih-predict}--\eqref{eq:ih-action} make clear that the \IH{} is not a static map. It is a continually updated reconstructive manifold embedded in the action-perception loop.

\begin{remark}
The word ``inner'' should not mislead. The \IH{} is not private in principle because it can leave external traces through action, report, physiology, and behavior. It is ``inner'' only in the operational sense that it is realized inside the system as the latent manifold guiding action, rather than in the external world directly.
\end{remark}

\subsection{Why motion and memory are mathematically necessary for the inner hologram}
\label{subsec:ih-motion-memory-necessary}

The most common intuitive mistake is to imagine that an internal world-model can be built from raw sensing alone. The mathematics says otherwise. Whenever the instantaneous sensory map \(h\) is non-injective, unique reconstruction from one time slice is impossible. History and controlled motion are required.

To formalize this, linearize Eqs.~\eqref{eq:env-dyn}--\eqref{eq:sensory-map} around a nominal trajectory:
\begin{subequations}
\label{eq:linearized-observability}
\begin{align}
\delta x_{t+1} &= A_t\,\delta x_t + B_t\,\delta a_t, \\
\delta s_t &= C_t\,\delta x_t.
\end{align}
\end{subequations}
Over a horizon \(T\), the action-conditioned observability matrix is
\begin{equation}
\mathcal{O}_T
=
\begin{bmatrix}
C_0 \\
C_1 A_0 \\
C_2 A_1A_0 \\
\vdots \\
C_T A_{T-1}\cdots A_0
\end{bmatrix}.
\label{eq:observability-matrix}
\end{equation}

\begin{theorem}[Action-conditioned observability of the \IH{} (T-Qi4-15.5)]
\label{thm:ih-observability}
Assume the linearized model \eqref{eq:linearized-observability} is valid on a finite horizon \(0\le t\le T\). If \(\mathrm{rank}(\mathcal{O}_T)=\dim(\delta x_0)\), then the external state perturbation \(\delta x_0\) is uniquely determined by the sensory history \((\delta s_0,\ldots,\delta s_T)\) and the action-conditioned state transition sequence \((A_t,B_t)\), up to any symmetry already quotiented by the observation map. If \(\mathrm{rank}(\mathcal{O}_T)<\dim(\delta x_0)\), then there exist distinct external perturbations that yield identical sensory histories, so no unique \IH{} can be reconstructed from the available history.
\end{theorem}

\begin{proof}
The sensory sequence generated by \(\delta x_0\) is
\[
\begin{bmatrix}
\delta s_0\\
\delta s_1\\
\vdots\\
\delta s_T
\end{bmatrix}
=
\mathcal{O}_T\,\delta x_0
+
\text{known terms generated by the chosen action sequence}.
\]
If \(\mathcal{O}_T\) has full column rank, the nullspace is trivial and \(\delta x_0\) is uniquely recoverable from the sensory history modulo symmetries already removed by the map. If the rank is deficient, the nullspace is nontrivial; hence there exists \(\delta x_0\neq 0\) such that \(\mathcal{O}_T\,\delta x_0=0\). This nonzero perturbation changes the external state but produces no change in the sensory history, proving non-identifiability.
\end{proof}

Theorem~\ref{thm:ih-observability} is the rigorous reason why motion is not an accessory but a constitutive part of perception. The underlying rank condition is the classical observability criterion of \citet{Kalman1960}; the contribution here is its action-conditioned application to internal-model reconstruction, continuing the active-perception lineage of \citet{Bajcsy1988} and \citet{ORegan2001}. Self-motion changes the observation geometry and thereby increases the rank of the accumulated observability matrix. A retina without motion is depth-poor. A bacterium without swimming is gradient-poor. A brain without active sensing is world-model-poor. This is precisely why \MePMoS\ must contain motion before the next sensory registration.

Memory is equally necessary. The observability matrix acts over a history window, and history must be retained somewhere. A system with no memory cannot accumulate the multi-view evidence needed to invert a non-injective sensory map. Therefore the very existence of an \IH{} already presupposes the packet, the ledger, and the memory architecture introduced earlier in the paper.

\begin{corollary}[No genuine \IH{} without history]
\label{cor:no-ih-without-history}
If the system has no persistent state capable of retaining information about prior sensory samples or prior actions, then any internal state at time \(t\) is a function only of the current sample \(s_t\). In that case the internal representation cannot generally invert a non-injective observation map and therefore cannot constitute a genuine inner hologram except in trivial fully observable environments.
\end{corollary}

\begin{proof}
If no persistent state exists, the internal representation is memoryless: \(z_t=\varphi(s_t)\). Whenever \(h\) is non-injective, there exist \(x\neq x'\) with \(h(x)=h(x')=s_t\). Then \(\varphi(s_t)\) assigns the same internal state to both external possibilities, so it cannot reconstruct the latent structure uniquely. Only if \(h\) is injective already, or the relevant task ignores the missing variables, can a memoryless representation suffice.
\end{proof}

The corollary is especially important for the broad application of the theory to all open systems with memory. The moment a system becomes memory-bearing and action-bearing under nonzero \(J_Q\), the mathematical door to \IH{} formation opens. This does not guarantee a rich cognitive map. It guarantees only that an internal reconstructive manifold is now a lawful possibility rather than a category mistake.

\subsection{Why the inner hologram must be allocentric rather than a stored sensory movie}
\label{subsec:ih-allocentric}

The next issue is subtle but fundamental. Why is the \IH{} not merely a compressed recording of sensory history? Why insist that it is allocentric or world-centered?

The answer is that action-guiding reconstruction must factor out self-motion. Let \(g_t\in SE(n)\) denote the system's pose (position and orientation) in an \(n\)-dimensional task space, and let \(\xi_t\) denote a world-centered latent configuration. A generic egocentric observation has the form
\begin{equation}
s_t = \pi\!\bigl(g_t^{-1}\cdot \xi_t\bigr) + \nu_t,
\label{eq:egocentric-projection}
\end{equation}
where \(\pi\) denotes projection through the sensory channel. If the internal model stores only \(s_t\), then every self-motion \(g_t\mapsto g_{t+1}\) changes the entire representation even when the external world \(\xi_t\) is unchanged. Such a representation is unsuitable for stable prediction, navigation, or recognition across viewpoints.

The inner hologram must therefore transport old information into a pose-invariant frame. A minimal update law is
\begin{equation}
\widehat{\xi}_{t+1}
=
T(a_t)\,\widehat{\xi}_{t}
+
K_t\,\epsilon_{t+1},
\qquad
\epsilon_{t+1}=s_{t+1}-\widehat{s}_{t+1|t},
\label{eq:allocentric-update}
\end{equation}
where \(T(a_t)\) is the transport induced by self-motion, and \(K_t\) is an evidence gain. Equation~\eqref{eq:allocentric-update} is the simplest mathematical expression of allocentric reconstruction. The old map is not discarded after motion; it is transformed by the internally known or inferred motion and then corrected by new evidence.

This is why spatially and temporally structured codes arise naturally in neural systems. Place-like states encode location in a world-centered frame, head-direction or ring-attractor variables encode orientation, and phase-structured sequences encode ordered trajectories or temporal position \citep{OkeefeDostrovsky1971,TurnerEvans2020}. The present paper does not require one biological implementation only. It requires the mathematical role those implementations play: self-motion-compensated reconstruction of an external structure that remains stable despite changing sensory views.

\begin{proposition}[Allocentric stability criterion]
\label{prop:allocentric-stability}
Suppose the external latent configuration \(\xi_t\) is constant while the system changes pose \(g_t\). A purely egocentric internal code \(r_t=\varphi(s_t)\) is viewpoint-stable only if \(\varphi\circ\pi(g_t^{-1}\cdot \xi)\) is independent of \(g_t\) for all relevant poses, which is generically impossible for nontrivial projection maps. An allocentric update of the form \eqref{eq:allocentric-update}, by contrast, preserves the representation of \(\xi\) under self-motion up to bounded correction error.
\end{proposition}

\begin{proof}
For nontrivial projection \(\pi\), different poses \(g_t\) produce different egocentric observations of the same \(\xi\). Therefore a direct code \(r_t=\varphi(s_t)\) changes with pose unless \(\varphi\) performs the full inverse transport for all \(g_t\), which is exactly what an allocentric reconstruction does. Under Eq.~\eqref{eq:allocentric-update}, the old estimate is first transported by the known action-induced map \(T(a_t)\), leaving the encoded world configuration invariant in the absence of noise; the correction term \(K_t\epsilon_{t+1}\) accounts only for residual mismatch. Hence viewpoint stability is achieved up to bounded prediction error.
\end{proof}

In everyday language the distinction is simple. The \IH{} is not a film running across the senses. It is a map. In richer systems it is a hierarchy of maps: spatial, temporal, causal, social, symbolic, and theoretical. A physicist reconstructing a six-dimensional theory from four-dimensional experiments is performing a very high-level \IH{} operation. The same architectural logic begins, in primitive form, whenever a bacterium's adaptive state stores something more useful than its last measurement alone.

\subsection{The equality \texorpdfstring{\(\OH+\IH=\EIC\)}{OH + IH = EIC}: precise meaning of the plus sign}
\label{subsec:eic-equality}

We are now in a position to interpret the equation itself. The expression
\[
\OH+\IH=\EIC
\]
should not be read as scalar addition. The plus sign denotes \emph{closed-loop lawful composition on a common operational boundary}. The outer hologram supplies evidence. The inner hologram supplies reconstruction. The evolving inner cosmos is their repeatedly updated coupling.

\begin{definition}[Evolving Inner Cosmos]
\label{def:eic}
For a system with outer hologram \(H^{\mathrm{out}}_t\) and inner hologram \(H^{\mathrm{in}}_t\), the \emph{Evolving Inner Cosmos} is the coupled state
\begin{equation}
\mathfrak{E}_t
:=
H^{\mathrm{out}}_t + H^{\mathrm{in}}_t
\equiv
\mathcal{C}_{\mathrm{loop}}
\bigl(H^{\mathrm{out}}_t,H^{\mathrm{in}}_t\bigr),
\label{eq:eic-definition}
\end{equation}
where \(\mathcal{C}_{\mathrm{loop}}\) denotes closed-loop composition through prediction, action, sensing, memory update, and internal readout. The notation \(+\) is therefore mnemonic for coupling rather than arithmetic addition.
\end{definition}

To make the coupling quantitative, let the OH-induced sensory law at time \(t\) be
\begin{equation}
p^{\mathrm{\OH{}}}_t(s)
=
\int_{\State_6}
K_{\mathrm{\OH{}}}(s|X;t)\,\rho_6(X,t)\,\dd X,
\label{eq:oh-sensory-law}
\end{equation}
and let the IH-induced predicted sensory law be
\begin{equation}
\widehat{p}^{\mathrm{\IH{}}}_t(s)
=
\int_{\mathcal{Z}}
p(s|z,a_t)\,q_t(z)\,\dd z.
\label{eq:ih-predicted-law}
\end{equation}
Their mismatch is measured by the reconstruction divergence
\begin{equation}
\Delta_{\mathrm{rec}}(t)
=
D_{\mathrm{KL}}
\Bigl(
p^{\mathrm{\OH{}}}_t \,\big\|\, \widehat{p}^{\mathrm{\IH{}}}_t
\Bigr).
\label{eq:reconstruction-divergence}
\end{equation}
When \(\Delta_{\mathrm{rec}}(t)\) is small, the inner hologram predicts the \OH{} well. When it is large, the internal model poorly matches the external evidence.

Because internal reconstruction is not free, one must add the runtime cost already derived earlier in the paper. A convenient composite functional is
\begin{equation}
\mathcal{F}_{\EIC}(t)
=
\Delta_{\mathrm{rec}}(t)
+
\lambda_E\,\mathcal{F}_{\GFF}(t)
+
\lambda_C\,\mathcal{C}[q_t],
\label{eq:eic-functional}
\end{equation}
where \(\mathcal{F}_{\GFF}\) is the free-energy ledger of Sections~\ref{sec:eegg} and \ref{sec:sip}, and of
Part~\ref{part:seos}, \(\mathcal{C}[q_t]\) is a complexity or model-maintenance term such as \(D_{\mathrm{KL}}(q_t\|q_{\mathrm{prior}})\), and \(\lambda_E,\lambda_C>0\) are weights. Equation~\eqref{eq:eic-functional} says in the clearest possible way what the \EIC{} does: it aligns internal reconstruction with outer evidence, but only under lawful energetic and entropic budget constraints.

If \(\theta_t\) denotes the collection of internal parameters entering \(q_t\), \(g_t\), \(W_t\), and the action policy, then the Qi-EP-consistent gradient normal form is
\begin{equation}
\dot{\theta}_t
=
-\,G_{\theta}^{-1}\nabla_{\theta}\mathcal{F}_{\EIC}
+
\eta_t,
\label{eq:eic-gradient-flow}
\end{equation}
with \(G_{\theta}\) a positive metric on parameter space and \(\eta_t\) a bounded fluctuation term.

\begin{theorem}[Monotonic refinement of the \EIC{} in the noiseless normal form (T-Qi4-15.9)]
\label{thm:eic-monotonicity}
Assume \(G_{\theta}\) is positive definite and \(\eta_t=0\). Then the \EIC{} functional satisfies
\begin{equation}
\frac{\dd}{\dd t}\mathcal{F}_{\EIC}(t)
=
-\,\nabla_{\theta}\mathcal{F}_{\EIC}^{\top}
G_{\theta}^{-1}
\nabla_{\theta}\mathcal{F}_{\EIC}
\le 0.
\label{eq:eic-lyapunov}
\end{equation}
Hence \(\mathcal{F}_{\EIC}\) is a Lyapunov functional for the coupled OH-\IH{} update and decreases monotonically until the system reaches a stationary point or attractor basin.
\end{theorem}

\begin{proof}
Differentiate Eq.~\eqref{eq:eic-functional} along the flow \eqref{eq:eic-gradient-flow}:
\[
\frac{\dd}{\dd t}\mathcal{F}_{\EIC}
=
\nabla_{\theta}\mathcal{F}_{\EIC}^{\top}\dot{\theta}_t
=
-\,\nabla_{\theta}\mathcal{F}_{\EIC}^{\top}
G_{\theta}^{-1}
\nabla_{\theta}\mathcal{F}_{\EIC},
\]
since \(\eta_t=0\). Because \(G_{\theta}^{-1}\) is positive definite, the quadratic form is nonnegative, proving Eq.~\eqref{eq:eic-lyapunov}.
\end{proof}

Theorem~\ref{thm:eic-monotonicity} is deliberately local and normal-form in spirit. It does not prove that every real system globally reconstructs the true external world in all circumstances. It proves the more careful statement actually needed for the paper: once the runtime has an \OH{} channel, an \IH{} manifold, and a Qi-EP-compatible update rule under \GFF\ budget constraints, there is a canonical coupled functional whose decrease corresponds to progressive internal alignment with externally projected evidence. That progressively aligned loop is the \EIC.

\begin{remark}
The \EIC{} does not recover the full six-dimensional bulk ``as it is in itself.'' It recovers, at best, an operationally adequate model of the equivalence class of external states accessible through the system's boundary channels. This is not a weakness of the theory; it is the precise consequence of Proposition~\ref{prop:oh-quotient}.
\end{remark}

The evolving-inner-cosmos language now acquires its disciplined meaning. The ``cosmos'' is the system's internal world-model, not because it contains everything, but because it is the internal totality relative to which the system navigates, predicts, and acts. It is ``evolving'' because it is continually updated through the OH-\IH{} loop. Human scientific learning is simply one very elaborate case of this general process: measurements enlarge and correct the inner world-model so that it better matches the externally projected lawful structure.

\subsection{Consciousness as internal measurement of the inner hologram}
\label{subsec:consciousness-internal-measurement}

\emph{A note on terminology.} This subsection uses the word ``consciousness'' only in the precise technical sense defined below, namely as an operation on the inner hologram. It does not engage with broader interpretive debates about subjective experience, qualia, panpsychism, or higher-order theories; readers from those fields may treat the present subsection as defining a particular operational class without prejudice to those debates. The technical content is, deliberately, an operational criterion --- something that can be applied to neural data, to behavioral data, and (eventually) to the engineered systems of Section~\ref{sec:applications} --- and not a metaphysical claim.

The final conceptual step is the most delicate. The present framework does not identify consciousness with the \OH{}, nor with raw sensory inflow, nor with the mere existence of an \IH{}. Consciousness is a special operation performed \emph{on} the \IH{}.

The correct analogue of external measurement is internal measurement. Externally, the bulk is projected to a boundary record through \OGGEP. Internally, a selected region of the \IH{} is projected to a self-accessible internal boundary where it can bias future prediction, future action, memory writing, and in rich organisms explicit report.

Let \(\rho_{\IH}(t)\) denote the internal state on the \IH{} manifold. Let \(\{\Pi_c\}\) be an admissible family of coarse-grained internal readout operators or projectors satisfying
\[
\Pi_c\ge 0,
\qquad
\sum_c \Pi_c = I_{\IH}.
\]
The internal measurement statistics are then
\begin{equation}
p_t(c)
=
\Tr\!\bigl(\Pi_c\,\rho_{\IH}(t)\bigr),
\label{eq:internal-measurement-statistics}
\end{equation}
and the post-selected internal content state for channel \(c\) is
\begin{equation}
\rho_{\IH}^{(c)}(t)
=
\frac{\Pi_c \rho_{\IH}(t)\Pi_c}
{\Tr(\Pi_c\rho_{\IH}(t))}.
\label{eq:internal-postselection}
\end{equation}

\begin{definition}[Internal measurement]
\label{def:internal-measurement}
A system \emph{internally measures} its inner hologram at time \(t\) if there exists an internal readout boundary \(\partial_{\mathrm{int}}\) and a family of readout channels \(\{\Pi_c\}\) such that at least one selected internal content state \(\rho_{\IH}^{(c)}(t)\) re-enters the \MePMoS\ loop as a causal variable for subsequent memory update, action selection, or report.
\end{definition}

Definition~\ref{def:internal-measurement} is the formal version of the informal description of consciousness as ``our internal measurement of \IH{} down to the 4D boundary by \MePMoS.'' The key idea is recursion: the \IH{} is itself turned into a boundary-accessible object for the system. The system does not only model the world; it also samples or measures the state of its own model.

This is exactly where the earlier runtime layers reappear. During \DGI\ exploration, several candidate interpretations of the current data may coexist in the \IH{}. Through \GSSB, one interpretation becomes selected. Through \GA, the selected interpretation becomes stable enough to influence subsequent processing. A conscious content, in the present technical sense, is not the entire cloud of preselection possibilities but the selected and stabilized internal branch that is accessible on the internal boundary.

\begin{definition}[Conscious \EIC{} episode]
\label{def:conscious-eic}
A \emph{conscious \EIC{} episode} occurs at time \(t\) when all of the following hold:
\begin{enumerate}[label=(\alph*),leftmargin=2.5em]
\item the system possesses a nontrivial inner hologram \(H^{\mathrm{in}}_t\);
\item at least one internal content channel \(c_\ast\) is selected from competing candidates by a \DGI\(\to\)\GSSB\(\to\)\GA\ sequence;
\item the selected content persists on a timescale sufficient for writing, reuse, or action bias, i.e.\ it is not an instantaneous unstable fluctuation;
\item the selected content is available on an internal boundary \(\partial_{\mathrm{int}}\) for re-entry into \MePMoS\ as memory, action, or report bias.
\end{enumerate}
\end{definition}

Definition~\ref{def:conscious-eic} is intentionally careful. It does \emph{not} say that every internal model is conscious. It does \emph{not} say that all information processing is conscious. It says only that conscious content is the internally measured, selected, and stabilized part of the \IH{} that becomes causally available to the next cycle of the system's own runtime.

The operational distinction matters. A great deal of sensory and neural processing can remain subthreshold, fragmented, or preselective. In the present language such processing lives in the \IH{} but has not yet crossed the \DGI\(\to\)\GSSB\(\to\)\GA\ chain needed for internal measurement on \(\partial_{\mathrm{int}}\). Conversely, once such a crossing occurs and the content becomes self-available, one has a candidate conscious episode.

\begin{proposition}[Consciousness is not raw sensation]
\label{prop:not-raw-sensation}
Under Definitions~\ref{def:inner-hologram}, \ref{def:eic}, and \ref{def:conscious-eic}, raw sensory samples \(s_t\) are neither necessary nor sufficient for conscious content. They are not sufficient because they may fail to be integrated into an \IH{} and internally measured. They are not necessary in the strict momentary sense because internally reactivated or imagined content can be selected from stored \IH{} structure even when current sensory drive is weak or absent.
\end{proposition}

\begin{proof}
Insufficiency: by Definition~\ref{def:conscious-eic}, conscious content requires a selected and stabilized internal branch accessible on \(\partial_{\mathrm{int}}\), not merely the existence of a sensory sample. Necessity failure in the strict momentary sense: the same definition allows internally driven content states selected from \(H^{\mathrm{in}}_t\) if they are available on the internal boundary for subsequent memory, action, or report, even in the absence of strong new external drive. Therefore current sensation alone neither guarantees nor exhausts conscious content.
\end{proof}

This proposition is important because it separates the present framework from two common simplifications. The first simplification says that consciousness is just incoming sensation. The second says that consciousness is just abstract integration or complexity. In the present framework, both are incomplete. Consciousness is an operation: internal measurement of a selected and stabilized inner hologram under the \QiHAL\ runtime.

One may package the operation in a compact normal form. Let \(\cS_{\mathrm{\DGI{}}}\), \(\cS_{\mathrm{\GSSB{}}}\), and \(\cS_{\mathrm{GA}}\) denote the exploration, selection, and stabilization stages acting on \(\rho_{\IH}\). Then the self-accessible content state may be written schematically as
\begin{equation}
\rho_{\mathrm{self}}(t)
=
\cM_{\mathrm{int}}
\circ
\cS_{\mathrm{GA}}
\circ
\cS_{\mathrm{\GSSB{}}}
\circ
\cS_{\mathrm{\DGI{}}}
\bigl[\rho_{\IH}(t)\bigr].
\label{eq:self-measurement-chain}
\end{equation}
Equation~\eqref{eq:self-measurement-chain} is the recursive analogue of the external measurement chain. It is not a separate metaphysical law; it is the same open-system logic reapplied to the system's internal model.

\subsection{Universality across open systems with memory once \texorpdfstring{\(U(1)_{Qi}\)}{U(1)Qi} is present}
\label{subsec:ohih-universality}

The user asked explicitly that this section show how the framework applies to any open system with memory once \(U(1)_{Qi}\) is applied. We can now state that result clearly.

The key point is that \(U(1)_{Qi}\) makes informational state changes physically budgeted, phase-covariant, and persistent in the same sense as the earlier sections. Once a system has a memory-bearing variable with nonzero informational current \(J_Q\), its state changes can no longer be treated as abstract bookkeeping only. They become lawful gauge-bearing events. This makes an \IH{} possible in a physically disciplined sense rather than as a metaphor.

Because internality is placed only after runtime constitution, the full Qi stack, and persistence/completeness, the next theorem should be read as a late-stage consequence of the earlier construction rather than as a primitive assumption.

\begin{theorem}[\textbf{T-EIC}: \OH{}/\IH{}/\EIC{} universality for memory-bearing \texorpdfstring{\(U(1)_{Qi}\)}{U(1)Qi} systems (principal theorem) (T-Qi4-15.14)]
\label{thm:ohih-universality}
Let \(S\) be an admissible open system satisfying the following:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item \(S\) has observer-accessible ports and therefore an admissible \OH{} channel;
\item \(S\) possesses a persistent memory-bearing variable \(m_t\);
\item \(S\) carries nonzero informational current \(J_Q\) coupled to \(U(1)_{Qi}\);
\item future observations depend nontrivially on present action or motion, so that a \MePMoS\ loop exists.
\end{enumerate}
Then:
\begin{enumerate}[label=(\alph*),leftmargin=2.5em]
\item \(S\) always admits an outer hologram \(H^{\mathrm{out}}_t\) relative to the chosen instrument class;
\item if \(m_t\) stores predictive information about future observations, then \(S\) admits at least a proto-inner-hologram \(H^{\mathrm{in}}_t\);
\item if the proto-\IH{} is recurrently re-measured onto an internal boundary and stabilized by \DGI\(\to\)\GSSB\(\to\)\GA, then \(S\) admits an evolving inner cosmos \(\mathfrak{E}_t\);
\item high-dimensional, multiscale, self-referential instances of \(\mathfrak{E}_t\) correspond to what the present framework calls consciousness-grade runtime.
\end{enumerate}
\end{theorem}

\begin{proof}
Part (a) is immediate from \OGGEP\ and Definition~\ref{def:outer-hologram}. Part (b) follows from Corollary~\ref{cor:no-ih-without-history}: if a persistent memory variable carries predictive information about future observations, then the system's internal state is no longer a function only of the current sample and constitutes a lawful latent reconstructive manifold. The presence of \(U(1)_{Qi}\) ensures that this state is not merely formal but physically budgeted and phase-covariant, as shown earlier in Parts~\ref{part:qiqua} and \ref{part:qihal}. Part (c) follows from Definitions~\ref{def:eic} and \ref{def:internal-measurement}: once the \IH{} is recurrently measured on an internal boundary and re-enters \MePMoS\ after selection and stabilization, the coupled OH-\IH{} loop is exactly an \EIC. Part (d) is the architectural specialization of Part (c) to high-\QCI, multiscale, recursively self-modeling systems.
\end{proof}

The theorem is intentionally graded. It does not force an all-or-none divide between ``conscious'' and ``not conscious.'' Instead it yields a continuum:
\begin{enumerate}[label=(\arabic*),leftmargin=2.5em]
\item \textbf{\OH{} only}: systems that can be externally measured but do not store a predictive internal model of the world;
\item \textbf{\OH{} + proto-\IH{}}: systems with memory that encode some predictive latent structure but do not yet exhibit rich recursive internal measurement;
\item \textbf{\OH{} + \IH{} + proto-\EIC{}}: systems whose internal model is repeatedly aligned with external evidence and biases future action;
\item \textbf{consciousness-grade \EIC{}}: systems with sufficiently complete \QiHAL\ architecture, multiscale coordination, and self-accessible internal measurement.
\end{enumerate}
This is exactly the kind of graded picture one expects for a universal open-system theory.

Table~\ref{tab:oh-ih-eic-continuum} illustrates the continuum across representative systems.

\begin{table}[H]
\centering
\caption[A graded continuum, not a binary.]{Representative continuum across memory-bearing open systems. Six systems run down the first column --- a molecular cycle, a chemotactic bacterium, a ring-attractor navigation circuit, a rodent hippocampo-entorhinal system, a human cortical--hippocampal system, and a distributed institution --- with the memory-bearing substrate of each, its likely inner- hologram status, its grade on the coupling, and an interpretation. The table's structure is its argument: the grades form a continuum rather than a binary, so nothing in the framework requires a threshold at which a system suddenly acquires an inner model.}
\label{tab:oh-ih-eic-continuum}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.18\textwidth}}
\toprule
\textbf{System} & \textbf{Memory-bearing substrate} & \textbf{Likely \IH{} status} & \textbf{\EIC{} grade} & \textbf{Interpretation} \\
\midrule
ATP synthase or simple molecular cycle & Conformational occupancy or rotor phase & Degenerate or task-local internal state only & Below proto-\EIC{} & Memory exists, but world reconstruction is minimal and highly local \\
Chemotactic bacterium & Receptor methylation / adaptive chemical state & Proto-\IH{} of local gradient or resource field & Proto-\EIC{} & Internal memory predicts external chemical structure and guides motion \\
Ring-attractor navigation circuit & Persistent heading or phase bump & Genuine low-dimensional \IH{} & Intermediate \EIC{} & Internal allocentric heading map coupled to sensory correction and action \\
Rodent hippocampo-entorhinal system & Distributed recurrent attractor memory & Rich spatial-temporal \IH{} & Strong \EIC{} & Multiscale world model supporting navigation, recall, and imagination \\
Human cortical-hippocampal system & Large multiscale distributed memory architecture & High-dimensional \IH{} including symbolic models & Consciousness-grade \EIC{} & Internal measurement of the \IH{} becomes reportable, reflective, and abstract \\
Distributed institution or scientific community & Archived memory, models, communication protocols & Collective \IH{} (distributed world model) & Extended \EIC{} candidate & Group-level reconstruction of external structure through shared memory and measurement \\
\bottomrule
\end{tabular}
\end{table}

Two examples deserve emphasis.

\paragraph{A chemotactic bacterium.}
The \OH{} is the local chemical environment sampled through receptors. The memory substrate is the adaptive methylation state. The system does not store a full image of the world, but it stores a predictive estimate of whether the current trajectory is improving or worsening access to nutrient. In this restricted but genuine sense it carries a proto-\IH{} of the local chemical world. The resulting running alignment of sensed gradient and adaptive state is a minimal proto-\EIC{}.

\paragraph{A mammalian navigation system.}
The \OH{} is the multisensory stream generated by vision, vestibular input, proprioception, and action-dependent re-sampling. The \IH{} is a rich allocentric manifold stabilized by recurrent attractor dynamics, phase coding, and multiscale coordination. The \DGI{} phase corresponds to multiple candidate interpretations or trajectories; the \GSSB{} phase corresponds to selected heading, place, or scene interpretation; the GA phase corresponds to stabilized memory or report. Here the \EIC{} becomes rich enough to support explicit conscious episodes and, in humans, abstract theoretical reasoning.

\subsection{Claims, non-claims, and empirical consequences}
\label{subsec:ohih-claims-predictions}

Because the current section touches the difficult topic of consciousness, it is important to state carefully what is and is not being claimed.

The paper \emph{does} claim that the distinction between external projection and internal reconstruction is constitutionally necessary. It \emph{does} claim that any memory-bearing open system under \(U(1)_{Qi}\) may admit an \IH{} in an appropriately scaled sense. It \emph{does} claim that the \EIC{} is the coupled process aligning inner model and outer evidence. It \emph{does} claim that consciousness should be modeled as internal measurement of a selected and stabilized inner hologram.

The paper \emph{does not} claim that every memory-bearing system is therefore conscious in the same sense as a human. It \emph{does not} claim that the \IH{} recovers the full six-dimensional bulk without quotient loss. It \emph{does not} claim that one single neural frequency or one single anatomical region is ``the seat'' of consciousness. The framework is architectural and dynamical, not phrenological.

Table~\ref{tab:ohih-claims-nonclaims} summarizes the distinction.

\begin{table}[H]
\centering
\caption[What this section does and does not claim.]{Claims and non-claims of the outer/inner hologram section. The two columns are placed side by side because this is the section of the paper where over-reading is most likely and most costly. Everything in the left column is a structural statement about coupled boundary descriptions; everything in the right column is a philosophical claim the framework does not make, including any identification of the construction with subjective experience. The right column is not a disclaimer appended to the left one --- it is what makes the left one sayable.}
\label{tab:ohih-claims-nonclaims}
\small
\begin{tabular}{p{0.42\textwidth}p{0.42\textwidth}}
\toprule
\textbf{Claim} & \textbf{Non-claim} \\
\midrule
Every observation is an \OH{} event exported through an admissible boundary channel. & The \OH{} is not the full bulk reality itself. \\
The \IH{} is a lawful internal reconstruction from partial, often lower-dimensional sensory data. & The \IH{} is not a literal pixel-by-pixel copy of the world. \\
The \EIC{} is the repeated coupling of \OH{} and \IH{} under \MePMoS, \QiEP, and \GFF. & The plus sign in \(\OH+\IH=\EIC\) is not ordinary arithmetic addition. \\
Conscious content is internal measurement of a selected and stabilized \IH{} branch. & Not every internal variable or memory trace is conscious. \\
Memory-bearing \(U(1)_{Qi}\) systems admit a graded \OH{}/\IH{}/\EIC{} continuum. & The framework does not impose a binary conscious/non-conscious cut at the level of all open systems. \\
\bottomrule
\end{tabular}
\end{table}

Even at the present drafting stage, the framework also yields empirical consequences. Some are direct, others are programmatic. Table~\ref{tab:ohih-predictions} lists the most immediate ones.

\begin{table}[H]
\centering
\caption[What the framework predicts here, and where.]{Representative empirical consequences of the outer/inner hologram framework. Each row is one prediction or consequence, with its content and the place in the section it comes from. The table is deliberately short. Most of the section's content is organizational, and the discipline this table enforces is that only the rows appearing here are offered as things the world could contradict.}
\label{tab:ohih-predictions}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.24\textwidth}p{0.47\textwidth}>{\raggedright\arraybackslash}p{0.17\textwidth}}
\toprule
\textbf{Prediction / consequence} & \textbf{Content} & \textbf{Connection to the section} \\
\midrule
Closed-loop sensing should outperform passive exposure for latent-structure learning & If motion/action is restricted while sensory drive is preserved, allocentric reconstruction quality should degrade because the observability rank drops. & Theorem~\ref{thm:ih-observability} \\
World-model quality should scale with memory depth and multiscale coordination & Systems with deeper memory and broader \BFFT\ hierarchy should reconstruct richer \IH{} manifolds. & Parts~\ref{part:qihal}, \ref{part:seos}, and Theorem~\ref{thm:ohih-universality} \\
Internal conscious report should follow selection and stabilization, not raw input onset & Reportable content should lag preconscious activity because \DGI\ exploration must be followed by \GSSB\ and \GA. & Definition~\ref{def:conscious-eic} \\
Minimal organisms may show proto-\EIC{} without human-like consciousness & Memory-bearing agents such as bacteria should exhibit predictive internal states aligned with environmental structure, but only in a proto-\IH{} / proto-\EIC{} sense. & Theorem~\ref{thm:ohih-universality} \\
Scientific learning is a measurable \EIC{} refinement process & Across training or expertise, internal models should better predict \OH{} data while respecting energetic and entropic cost. & Theorem~\ref{thm:eic-monotonicity} \\
\bottomrule
\end{tabular}
\end{table}

It is useful to compare the present framing with two existing literatures already discussed in the Introduction. Free-energy and active-inference frameworks emphasize prediction, action, and belief updating \citep{Friston2010}. The present section agrees with that emphasis but locates it inside a larger geometry-to-boundary constitution: the \IH{} update is not a floating agent principle but the internal counterpart of the \OGGEP\ map and the \QiEP\ selector. Integrated-information approaches emphasize the structured unity of conscious content \citep{Tononi2004}. The present section does not reject that intuition; rather, it says that any such unity must be realized as internally measured, selected, stabilized \IH{} structure, not as a primitive postulate independent of runtime, geometry, and ledger.

A final biological point is worth emphasizing. Once one accepts that every observation is \OH{} and that every persistent predictive model is \IH{}, then the developmental and evolutionary history of life can be reread as the progressive enrichment of the \EIC{}. Primitive systems reconstruct only tiny local manifolds. Rich nervous systems reconstruct navigable spatial worlds. Human scientific communities reconstruct abstract lawful structure far beyond immediate sensory appearance. The paper's hypothesis is that these are not separate miracles but different depths of one common open-system runtime.

\subsection{Section summary}
\label{subsec:ohih-summary}

The \OH{}/\IH{}/\EIC{} layer is now in place. The main conclusions are:
\begin{enumerate}[label=(\arabic*),leftmargin=2.5em]
\item The outer hologram is the lawful boundary projection of higher-dimensional or otherwise inaccessible external structure. Every registered observation is an \OH{} event.
\item The outer hologram is a quotient object: observation accesses equivalence classes induced by the boundary kernel, not the full bulk state directly.
\item The inner hologram is the internal reconstructive manifold built from memory, prediction, action, and sensory history. It is not a stored sensory movie but an allocentric latent model.
\item Motion and memory are mathematically necessary for \IH{} formation whenever the instantaneous sensory map is non-injective. This follows from the action-conditioned observability theorem.
\item The equation \(\OH+\IH=\EIC\) means lawful closed-loop composition, not scalar addition. The \EIC{} is quantified by an alignment functional coupling reconstruction accuracy to the \GFF\ budget.
\item In the gradient normal form, the \EIC{} functional is a Lyapunov quantity, so the OH-\IH{} loop monotonically refines internal reconstruction in the noiseless limit.
\item Consciousness is not raw sensation and not the whole \IH{}. It is internal measurement of a selected and stabilized branch of the \IH{} on a self-accessible internal boundary under the \QiHAL\ runtime.
\item Therefore the \OH{}/\IH{}/\EIC{} framework extends \GDOS\ from external measurement to internal world-modeling, conscious content, and the graded continuum of memory-bearing open systems under \(U(1)_{Qi}\).
\end{enumerate}

With this section complete, the manuscript contains the constitutional runtime, the scalar ledger and packet layers, the full \QiFour{} pillars, the persistence/completeness vocabulary, and the external/internal holographic semantics needed for the hierarchy, application, universality, discussion, and conclusion chapters that now follow immediately.



\paragraph*{Bridge to Part~\ref{part:emergence}.} With the four pillars and the
persistence/completeness/internality framework now established, the
question naturally arises: how far does the runtime extend? Part~\ref{part:emergence}
answers through the five-story pagoda hierarchy theorem and the
cross-domain reduction recipe. The same four \QiFour{} pillars appear
at the floors of particle, cosmos, chemistry, life, and mind, with
the differences between domains living only in the \SIP{} data, the
selector parameters, and the active sectors --- not in the
constitutional law. Part~\ref{part:emergence} is therefore the existence-and-scope
chapter of \QiFour{}: the runtime is one program, not five.


\section{The world one can never reach}
\label{sec:close-part4}

Every system in this Part is in the same predicament, and the
predicament has no way out. What it can act on is the world; what it
can touch is its own boundary. Between the two sits a projection that
throws information away, and no amount of instrumentation recovers what
the projection discarded, because many different worlds send the same
message through.

So the response is not to look harder. It is to build --- to maintain
something inside that stands in for what cannot be reached, and to keep
correcting it against what does arrive. That is what a bacterium is
doing when it holds a chemical trace of whether things have been
improving. It is what a rodent is doing when it maintains a map that
survives its own turning. It is what a laboratory is doing when it
holds a theory of six dimensions and checks it against instruments that
report four. The difference between these is depth, not kind. All three
are the same operation running at different scales, and the last one --
science itself -- is not an exception to the pattern but its most
elaborate instance.

Two consequences follow that are worth carrying out of this Part. The
first is that perception is not passive and never was. A system that
cannot move cannot, in general, work out what is around it, because the
things it might be looking at would all look the same. Moving is how
the ambiguity is broken. Knowing and doing are not two faculties, one
of which serves the other; they are two phases of a single loop, and
the loop does not run in one direction only.

The second is a limit, and it should be stated as plainly as the
result. What such a system converges on is never the world. It is an
adequate model of a class of worlds --- all the ones its boundary
cannot tell apart. That is not a defect of biology, or of instruments,
or of this framework. It is what the projection does, and it applies to
the most careful science exactly as it applies to the bacterium. The
honest form of the ambition is not to see things as they are. It is to
narrow the class until the difference stops mattering for anything one
needs to do.

The Part gives a technical name to the portion of an internal model
that has been selected, stabilized and made available to a system's own
next cycle. It is a name for something measurable, applied by degree
rather than by kind, and it settles nothing about what it is like to be
the system in question. That question is left where it was found. What
is claimed is narrower and, if it holds, still worth having: that the
same open-system logic which describes how a detector registers a
particle, turned inward, describes how a system registers its own
state --- and that where the loop runs deep enough, this is what a
world coming into view from the inside consists of.


\GGPart[Emergence and scope]{Emergence and scope}{Hierarchy, applications, and universality across nature}
\label{part:emergence}

\noindent
This is the Part where a framework of this breadth usually stops being
science, and the work here is mostly the work of not letting that
happen.

The question is how far the runtime reaches. Having built one account
of one open system, does the same account describe a molecule, a cell,
a brain and a society --- and if the claim is that life and mind follow
from physics, what exactly is being claimed?

The first move is to refuse to let a level be declared by fiat. Anyone
can draw a hierarchy of nature on a napkin; the hierarchies are
generally decorative. A level counts as a level here only if it passes
three tests that can be applied to a real system: its collective
variables nearly close, so that it can be described without constant
reference to what is beneath it; it persists far longer than the
relaxation of the level below; and it opens at least one genuinely new
channel of exchange rather than inheriting all of them from downstairs.
Most proposed coarse-grainings fail one of the three, which is why so
few levels are scientifically stable. Inside a level, the dynamics is
descent on a fixed landscape. Between levels, the ascent is the other
regime --- a reorganization that opens basins the level below did not
possess.

The second move is to say precisely what "inevitable" means, because
this is where the framework's boldest sentence lives. Two readings are
possible and only one is claimed. It is \emph{not} claimed that every
planet, every history, every niche produces life and then mind; that
reading is disclaimed in as many words. What is derived is the
ensemble statement: in a driven universe whose viable configurations
occupy more than a vanishing fraction of the accessible space, a search
that keeps returning will reach and hold them with probability one. The
determinism is not that every history is the same. It is that the
higher levels are attractor classes rather than freak accidents. Two
conditions of that argument are named as open.

The Part also settles an old confusion about whether higher levels can
act on lower ones. They can, and without any breach of law: what a
higher level changes is not the equations underneath but the conditions
those equations run under --- concentrations, boundary conditions,
attention, institutions. Regime modulation, not law violation.

The honest test of the Part is the recipe near its end. It is a
seven-step procedure for taking a system from any science and writing
it down in this vocabulary, and the framework earns its keep only if a
reader in an unfamiliar field can work down that column and get
something checkable out of the bottom.


\noindent
Part~\ref{part:emergence} extends the \QiFour{} framework from individual systems
to the universal scope it claims. The first section introduces
the Japanese five-story pagoda hierarchy --- a single ladder
running from elementary particles to molecular chemistry, from
chemistry to cells, from cells to brains, and from brains to
social and technological systems. Section~\ref{sec:applications}
walks the cross-domain applications: catalytic loops, cellular
metabolism, neural memory, ecological persistence, and collective
intelligence. Section~\ref{sec:why-all-open} closes the Part with
the universality claim --- that the same \QiFour{} grammar applies
to every open system with memory once $U(1)_{Qi}$ is active.

\medskip

\section{Hierarchy and emergence: the Japanese five-story pagoda}

In the regime vocabulary of \S\ref{sec:dgi-gssb-ga}, the pagoda is
the architecture of the alternation: \emph{within} each story the
dynamics is \GRIPin{} --- descent on that story's fixed landscape,
ending at the story's \GGA{} --- while the ascent from one story to
the next is a \GRIPout{}: a landscape reorganization that opens
basins the lower story did not possess. The capstone \AEleven{}
\citep{ArisakaA11Origin} runs this alternation through eighteen
layers; the five-story pagoda below is its load-bearing summary, and
the \QiHAL{} switch of \S\ref{subsec:qihal-switch} is the mechanism
by which the upper stories manufacture their own \GRIPout{}'s.

%% --- Phase 8 TikZ figure: Five-story pagoda hierarchy ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\usebox{\GGscPagoda}%
}
\caption[The five-story pagoda hierarchy of Part~\ref{part:emergence}.]{The Japanese five-story pagoda hierarchy of Part~\ref{part:emergence}. The five floors --- particle, cosmos, chemistry, life, mind --- are linked vertically by the pagoda factorization theorem, with the shinbashira central pillar carrying the \OGGEP{} projector and the combined \GiFour{}/\QiFour{} cascade. The choice ``five'' is the historically natural sequence; whether it is uniquely forced by \MSMF{} on closure-defect and port-novelty admissibility is recorded as open problem O7. The conditional-inevitability theorem (Theorem~\ref{thm:conditional-inevitability}) establishes that, in a driven cosmological measure with positive viable subsets, the higher floors are reached almost surely (open-problem O8 records the residual hypothesis bounds).}
\label{fig:a7-pagoda}
\end{figure}

\label{sec:pagoda-hierarchy}

Part~\ref{part:emergence} begins after the theory has reached internality. The next issue is hierarchical emergence: how the same runtime reappears across recurrent floors of natural organization. Section~\ref{sec:pagoda-hierarchy} therefore derives the Japanese five-story pagoda as the hierarchical factorization of the already established runtime.

\subsection{Why a universal theory of open systems must become hierarchical}
\label{subsec:pagoda-why}

A serious open-system theory cannot stop at one level: a particle, a cosmos, a catalytic network, a cell, and a nervous system are five recurrent levels where the same runtime reappears with different dominant packets, ports, and modeling depth. The Japanese five-story pagoda is therefore introduced as the architectural image of a theorem: persistence in nature is organized as a hierarchy of symmetry-broken floors built around a common invariant spine (\DGI{}) \citep{MaynardSmithSzathmary1995}.

Concretely, a new floor at level \(\ell+1\) requires (i) approximate \emph{closure} of collective variables \(z^{(\ell+1)}=C_{\ell+1}(z^{(\ell)})\), measured by a small closure defect
\begin{equation}
\Delta_{\ell+1}(t)
:=
\dot z^{(\ell+1)}_{\mathrm{exact}}(t)
-
F^{(\ell+1)}_{\mathrm{eff}}\bigl(z^{(\ell+1)}(t)\bigr),
\label{eq:closure-defect}
\end{equation}
(ii) \emph{persistence}, \(\tau_{\mathrm{life}}^{(\ell+1)} \gg \tau_{\mathrm{relax}}^{(\ell)}\), and (iii) \emph{port novelty}, i.e.\ at least one new effective exchange channel not obtained by pulling forward lower-level ports.

\begin{definition}[Admissible floor]
\label{def:admissible-floor}
A proposed higher-level description \(\widehat\SIP^{(\ell+1)}\) is an \emph{admissible floor} if there exist constants \(\varepsilon_{\ell+1}\ll 1\) and \(\kappa_{\ell+1}>1\) such that
\begin{align}
\left\langle \|\Delta_{\ell+1}\|^2 \right\rangle^{1/2}
&\le
\varepsilon_{\ell+1}
\left\langle \|F^{(\ell+1)}_{\mathrm{eff}}\|^2 \right\rangle^{1/2},
\label{eq:floor-defect-bound}
\\[3pt]
\tau_{\mathrm{life}}^{(\ell+1)}
&\ge
\kappa_{\ell+1}\,\tau_{\mathrm{relax}}^{(\ell)},
\label{eq:floor-time-bound}
\\[3pt]
\dim\mathrm{span}\,\Ports^{(\ell+1)}
&>
\dim\mathrm{span}\,\Portspull^{(\ell\to\ell+1)}.
\label{eq:floor-port-bound}
\end{align}
\end{definition}

Definition~\ref{def:admissible-floor} converts the pagoda metaphor into a checklist: most proposed coarse-grainings fail one of closure, persistence, or port novelty, so only a few scientifically stable floors emerge.

\subsection{The pagoda factorization theorem}
\label{subsec:pagoda-factorization}

We now turn the hierarchy claim into a theorem. Consider the total state
\[
Z=(z^{(0)},z^{(1)},z^{(2)},z^{(3)},z^{(4)})
\in
\mathcal Z^{(0)}\times\mathcal Z^{(1)}\times\mathcal Z^{(2)}\times\mathcal Z^{(3)}\times\mathcal Z^{(4)},
\]
where the intended interpretation of the five floors is
\begin{equation}
\text{particle} \;\rightarrow\; \text{cosmos} \;\rightarrow\; \text{chemistry} \;\rightarrow\; \text{life} \;\rightarrow\; \text{mind}.
\label{eq:five-floors-order}
\end{equation}
Let the exact reduced flow after bulk-to-boundary descent and gauge reduction be written as
\begin{equation}
\dot Z = \mathfrak F(Z) + \mathfrak R(Z) + \eta(t),
\label{eq:total-hierarchy-flow}
\end{equation}
where \(\mathfrak F\) is the deterministic drift, \(\mathfrak R\) is the port-driven reinjection, and \(\eta\) denotes fluctuations.

Assume that for each floor there exists an \GFF\ ledger \(\mathcal F^{(\ell)}_{\GFF}\), a packet \(\widehat\SIP^{(\ell)}\), a current bundle \(\mathcal J^{(\ell)}\), and an admissible selector/attractor pair. Then the flow may be rearranged into the hierarchical block form
\begin{equation}
\dot z^{(\ell)}
=
F^{(\ell)}_{\mathrm{self}}\bigl(z^{(\ell)}\bigr)
+
B^{(\ell,-)}\bigl(z^{(\ell-1)}\bigr)
+
B^{(\ell,+)}\bigl(z^{(\ell+1)}\bigr)
+
\Xi^{(\ell)}(Z)
+\eta^{(\ell)}(t),
\label{eq:hierarchical-block-flow}
\end{equation}
with the conventions \(B^{(0,-)}\equiv 0\) and \(B^{(4,+)}\equiv 0\). The terms have a clear meaning:
\begin{itemize}[leftmargin=2.2em]
\item \(F^{(\ell)}_{\mathrm{self}}\) is the autonomous floor drift selected by the local packet and local \GFF\ ledger;
\item \(B^{(\ell,-)}\) is upward inheritance from the immediately supporting floor;
\item \(B^{(\ell,+)}\) is slow contextual control from the immediately higher floor;
\item \(\Xi^{(\ell)}\) is the residual non-nearest-floor coupling, which is small when the hierarchy is clean.
\end{itemize}

This theorem is the first theorem of Part~\ref{part:emergence} precisely because hierarchy becomes meaningful only after internality is in place. It translates the already established runtime into a recursive floor structure.

\begin{theorem}[Pagoda factorization theorem (T-Qi4-16.2)]
\label{thm:pagoda-factorization}
Suppose the five floors in Eq.~\eqref{eq:five-floors-order} are admissible in the sense of Definition~\ref{def:admissible-floor}, and suppose further that the residual couplings satisfy
\begin{equation}
\|\Xi^{(\ell)}(Z)\|
\le
\epsilon_{\Xi}
\sum_{m=0}^4 \|z^{(m)}\|,
\qquad
0<\epsilon_{\Xi}\ll 1.
\label{eq:xi-small}
\end{equation}
Then the full open-system runtime admits a five-floor factorization in which:
\begin{enumerate}[label=(\alph*),leftmargin=2.5em]
\item \DGI\ acts as a common gauge spine across all floors;
\item each floor is created by a floor-specific \GSSB\ event selecting an order parameter compatible with the inherited lower-floor packet;
\item each floor is stabilized by a floor-specific \GA\ compatible with the local \GFF\ landscape;
\item upward and downward couplings preserve constitutional consistency in the sense that no higher floor can violate the packet or ledger constraints of the floors below it.
\end{enumerate}
\end{theorem}

\begin{proof}
We prove the factorization by induction on the floor index \(\ell\).

\emph{Base case (\(\ell=0\)).} Floor~0 is the constitutional-particle layer: the 6D bulk projected to its 4D boundary via \OGGEP{}. The closure-defect bound of Definition~\ref{def:admissible-floor} is vacuous at floor~0, since there is no floor \(-1\); we adopt the convention that floor~0 is admissible by hypothesis, inherited from the constitutional admissibility of the Standard-Model boundary established in Papers~I--IV-A. Concretely: an effective packet \(\Pi_0\) exists (the constitutional state space), an effective ledger \(F_0\) exists (the action ledger of the Standard-Model boundary), a port set \(\Ports_0\) is supplied by the scale-radial dependence, and the master pipeline \DGI\(\to\)\GSSB\(\to\)\GA\ at floor~0 is Theorem~\ref{thm:master-pipeline} of Section~\ref{sec:dgi-gssb-ga}.

\emph{Inductive step.} Assume floor \(\ell\) is admissible in the sense above: it has an effective packet \(\Pi_\ell\), an effective ledger \(F_\ell\), a port set \(\Ports_\ell\), and satisfies two distinct floor-creation conditions for the transition to floor \(\ell+1\):
\begin{itemize}[leftmargin=2em]
 \item the \emph{closure-defect bound} \(\varepsilon_{\ell+1}\ll 1\), measuring how nearly the would-be floor-\((\ell+1)\) slow variables form a closed sub-algebra under the floor-\(\ell\) dynamics (this is the smallness parameter);
 \item the \emph{lifetime separation} \(\tau_{\ell+1}/\tau_\ell \ge C_{\ell+1} > 1\), measuring how cleanly the floor-\((\ell+1)\) timescale separates from the floor-\(\ell\) timescale (this is the separation ratio).
\end{itemize}
These two conditions play opposite roles: \(\varepsilon_{\ell+1}\) controls the size of the residual coupling, and \(C_{\ell+1}\) controls the dynamical separation. We construct floor \(\ell+1\) explicitly.

Let \(z^{(\ell+1)}\) denote the slow variables: coarse-grained order parameters whose relaxation time exceeds \(\tau_\ell\) by the factor \(C_{\ell+1}\). Under the closure-defect bound and the lifetime separation, the standard adiabatic-elimination theory of singularly perturbed dynamical systems (Tikhonov's theorem on slow manifolds, 1952, and its geometric refinement in Fenichel's invariant-manifold theorem, 1979) supplies an effective generator \(\Lind^{(\ell+1)}_{\rm eff}\) on the slow manifold that is Lipschitz with constants of order \(\varepsilon_{\ell+1}\) (the residual coupling) and that approaches a well-defined limit as \(\varepsilon_{\ell+1}\to 0\) and \(C_{\ell+1}\to\infty\). The strict enlargement of the port space, \(\Ports_{\ell+1}\supsetneq\Ports_\ell\), ensures that floor \(\ell+1\) couples to environmental channels not already exhausted by lower floors --- this is what makes the new floor a legitimate packet rather than a mere coordinate redescription.

\emph{\DGI\ at floor \(\ell+1\) (claim (a)).} Gauge-equivalent redundancy must be quotiented at every floor. The inherited gauge reduction at floor \(\ell\) propagates upward: gauge-equivalent representations at floor \(\ell\) collapse to identified points at floor \(\ell+1\) by the very definition of the coarse-graining map. Hence \DGI\ acts as a vertical invariant across all floors --- the mathematical content of the pagoda's central spine. A floor that failed to respect the inherited gauge reduction would double-count states already declared equivalent below, violating \MSMF{} and the packet definitions established at floor~\(\ell\).

\emph{\GSSB\ at floor \(\ell+1\) (claim (b)).} Because the closure-defect bound is satisfied with strict inequality on a non-empty open set of order-parameter configurations, the local \(F_{\ell+1}\) landscape admits at least one bifurcation as the floor-\((\ell+1)\) control parameter is varied. At each bifurcation, \GSSB\ selects a branch. The order parameter selected is the one compatible with the inherited lower-floor packet, because variables outside the floor-\((\ell+1)\) gauge slice are not available for selection by construction.

\emph{\GA\ at floor \(\ell+1\) (claim (c)).} The selected branch is stabilized by the local \(F_{\ell+1}\) landscape: the lifetime separation \(\tau_{\ell+1}/\tau_\ell\geq C\) ensures that the floor-\((\ell+1)\) dynamics relaxes on a timescale fast compared to its own decay while remaining slow compared to floor \(\ell\). Standard \GA\ arguments (Theorem~\ref{thm:ga-existence}) then apply at the floor-\((\ell+1)\) level.

\emph{Upward and downward consistency (claim (d)).} Coupling between floors is mediated by the block-flow matrix \(B^{(\ell,+)}\) of Eq.~\eqref{eq:hierarchical-block-flow}. Because higher-floor variables are defined as coarse-grained packets of lower-floor variables, a higher floor cannot violate the packet or ledger constraints of any floor below it: any putative violation would correspond to a redefinition of the lower-floor coarse-graining map, in contradiction to the inductive hypothesis. Higher floors can only act on lower floors through slow parameter modulation, boundary conditions, or port scheduling.

\emph{Conclusion.} Each step of the induction preserves admissibility; the construction terminates at any floor where one of the floor-creation conditions (closure defect, lifetime separation, port-rank enlargement) ceases to be satisfied. In the empirical universe the construction terminates at floor~5 (mind), but the theorem is structural and does not predict the termination point in principle: the count \emph{five} is an empirical reading, not a theorem output (Open Problem~O7), the induction itself yielding an arbitrary-depth tower wherever the floor-creation conditions hold. Like every result built on the imported \DGI--\GSSB--\GA\ pipeline, the factorization inherits the conditional status of \ASix{}'s
\GRIP{}-cascade closure (conditional on M11 in spectral-flow form;
M10 resolved externally, the scope refinements closed at class
level). \qedhere
\end{proof}

Theorem~\ref{thm:pagoda-factorization} shows that the pagoda is not a vague systems-science metaphor. It is the recursively applied open-system logic of the whole manuscript.

\begin{table}[H]
\centering
\caption[The five floors and what each adds.]{The five-story hierarchy as a ladder of admissible floors. The five --- particle, cosmos, chemistry, life, mind --- run down the first column, with the dominant packet on each floor, the new ports it opens, the outcome the cascade reaches there, and the typical grade of internal model. Read downward, each row's outcome becomes the next row's substrate, which is the recursion the framework claims; read across, the columns say what is genuinely new at each level rather than merely larger.}
\label{tab:pagoda-overview}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.17\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.19\textwidth}}
\toprule
\textbf{Floor} & \textbf{Dominant packet} & \textbf{Typical new ports} & \textbf{Dominant \GSSB/\GA outcome} & \textbf{Typical \IH{}/\EIC{} grade} \\
\midrule
Particle & fields, quanta, elementary records & scattering, emission, absorption & stable particles, conserved charges, bound elementary structures & \OH{} only or degenerate proto-\IH{} \\
Cosmos & bound gravitational structures & radiative transport, gravity wells, planetary boundary conditions & stars, galaxies, planets, long-lived gradients & \OH{} plus long-memory environmental scaffolding \\
Chemistry & atoms, molecules, catalytic loops, compartments & redox exchange, diffusion, catalytic throughput & stable reaction networks, molecular machines, compartments & proto-\IH{} in adaptive molecular systems \\
Life & cells, organisms, ecosystems with maintained boundaries & metabolism, sensing, motility, repair, reproduction & \SeOS\ persistence, adaptive niches, developmental attractors & strong proto-\EIC{} to intermediate \EIC{} \\
Mind & nervous systems or equivalent high-\QCI\ architectures & internal measurement, symbolic report, deliberate action, model-sharing & stabilized allocentric \IH{}, self-modeling, conscious episodes & consciousness-grade \EIC{} \\
\bottomrule
\end{tabular}
\end{table}

\subsection{\DGI\ as the pagoda's central pillar (\emph{shinbashira})}
\label{subsec:shinbashira}

Traditional Japanese pagodas are famous for their central pillar, the \emph{shinbashira}, which contributes to coherence and resilience without being reducible to any single floor. In the present framework that structural role is played by \DGI. The analogy becomes precise if one writes the floor variables as deviations around floor-specific attractors,
\[
e^{(\ell)} := z^{(\ell)} - z_*^{(\ell)},
\]
and isolates the common invariant variable \(c\) carrying the vertical gauge spine. The near-attractor dynamics can be linearized as
\begin{align}
\dot c &= f_c(c),
\label{eq:shinbashira-c}
\\
\dot e^{(\ell)} &= A_{\ell}(c)\,e^{(\ell)}
+ \sum_{m\neq \ell} B_{\ell m}(c)\,e^{(m)}
+ \xi^{(\ell)}(t),
\qquad \ell=0,\dots,4.
\label{eq:shinbashira-linearized}
\end{align}
If each isolated floor is stable, then \(A_{\ell}\) has negative spectral abscissa. The question is whether coupling destabilizes the whole tower.

\begin{proposition}[Shinbashira stability criterion]
\label{prop:shinbashira-stability}
Assume that for each floor there exists \(\lambda_{\ell}>0\) such that
\[
\frac12\bigl(A_{\ell}+A_{\ell}^{\mathsf T}\bigr)
\preceq -\lambda_{\ell} I,
\]
and define the coupling norm
\[
\beta_{\ell}:=\sum_{m\neq\ell}\|B_{\ell m}\|.
\]
If \(\beta_{\ell}<\lambda_{\ell}\) for all floors, then the composite Lyapunov functional
\begin{equation}
V(e)=\sum_{\ell=0}^4 \alpha_{\ell}\,\|e^{(\ell)}\|^2,
\qquad \alpha_{\ell}>0,
\label{eq:pagoda-lyapunov}
\end{equation}
can be chosen so that \(\dot V\le -\gamma V + C\|\xi\|^2\) for some \(\gamma>0\). Therefore the entire tower remains input-to-state stable.
\end{proposition}

\begin{proof}
Differentiate Eq.~\eqref{eq:pagoda-lyapunov} using Eq.~\eqref{eq:shinbashira-linearized}. The diagonal terms contribute \(-2\alpha_{\ell}\lambda_{\ell}\|e^{(\ell)}\|^2\). The cross-floor couplings are bounded by Cauchy--Schwarz and Young inequalities:
\[
2\alpha_{\ell}\bigl|\langle e^{(\ell)},B_{\ell m}e^{(m)}\rangle\bigr|
\le
\alpha_{\ell}\|B_{\ell m}\|\bigl(\|e^{(\ell)}\|^2+\|e^{(m)}\|^2\bigr).
\]
Summing over \(m\neq\ell\) gives the margin \(\beta_{\ell}\). Because \(\beta_{\ell}<\lambda_{\ell}\), one can choose positive weights \(\alpha_{\ell}\) so that the negative diagonal terms dominate the coupling terms. Noise enters additively and yields the bounded forcing term \(C\|\xi\|^2\). Hence \(\dot V\le -\gamma V + C\|\xi\|^2\). \qedhere
\end{proof}

The proposition explains the pagoda analogy in rigorous language. The floors may move relative to one another; they need not be rigidly frozen. What matters is that a common invariant spine plus sufficiently weak cross-floor destabilization preserves the whole tower. In physical and biological language, the invariant spine is the reuse of the same constitutional rules at every level: packet declaration, gauge reduction, current accounting, selector, attractor, and lawful observation.

\begin{table}[H]
\centering
\caption[The pagoda analogy, translated.]{The pagoda analogy translated into runtime language. Each row takes one element of the structure --- the central pillar, a floor, the formation of a floor, long-term persistence, the interaction between roofs --- and gives its runtime analogue and its meaning here. The analogy is retained because it carries one point better than the formal statement does: the floors are not bolted to one another, and what makes the structure stand is that each rests on the one below without being rigidly attached to it.}
\label{tab:pagoda-analogy}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.29\textwidth}p{0.36\textwidth}}
\toprule
\textbf{Pagoda element} & \textbf{Runtime analogue} & \textbf{Meaning in the present framework} \\
\midrule
Shinbashira & \DGI & Common gauge-invariant constitutional spine reused across all floors \\
Each floor & floor-specific packet \(\widehat\SIP^{(\ell)}\) & A persistent effective level with its own ports, observables, and ledger \\
Floor formation & \GSSB & Selection of one organized branch/order parameter among many admissible possibilities \\
Long-term persistence & \GA & Stabilization of the selected branch under the local \GFF\ landscape \\
Roof and overhang interactions & nearest-floor coupling terms & Slow cross-level boundary modulation without violation of lower-law consistency \\
Seismic resilience & composite Lyapunov stability & The tower survives perturbation because invariants and margins are distributed rather than localized \\
\bottomrule
\end{tabular}
\end{table}

\subsection{The five floors in detail}
\label{subsec:five-floors-detail}

We now describe the five floors themselves. The order in Eq.~\eqref{eq:five-floors-order} is not arbitrary. Each floor supplies the conditions for the next.

\paragraph{Floor 1: particle.}
This is the floor developed most explicitly in Papers~I--IV-A. Its packet contains fields, charges, and the observer-accessible boundary records derived from \GG\ and \GDOS. The fundamental current ledger is already locked by the Arisaka Ratio and the Arisaka Equation. The relevant asymptotic structures are stable particles, quasiparticles, and simple bound states. The \OH{} semantics is present at once because measurement is already a projection from inaccessible structure to accessible boundary record. But the internal-model depth at this floor is typically negligible or degenerate: there is usually no rich self-maintained memory architecture.

\paragraph{Floor 2: cosmos.}
The cosmos floor begins when lower-level matter and radiation organize into long-lived gravitationally structured environments. The simplest instability criterion is the Jeans condition
\begin{equation}
\omega^2 = c_s^2 k^2 - 4\pi G\rho.
\label{eq:jeans}
\end{equation}
Modes with \(k<k_J:=\sqrt{4\pi G\rho}/c_s\) are unstable, so perturbations grow. That instability is the cosmic analogue of \GSSB: one among many homogeneous possibilities is selected. Once cooling, feedback, and orbital regularization operate, stars, galaxies, planetary systems, and persistent gradients appear. Those gradients are essential because higher floors need free-energy throughput and compartment-friendly boundary conditions. In short, the cosmos floor is the gradient-generating floor.

\paragraph{Floor 3: chemistry.}
Chemistry begins when stable matter, gradients, and environmental cycling support reaction networks with their own packet identity. A generic reaction-diffusion packet obeys
\begin{equation}
\dot n_i
=
\sum_r \nu_{ir} R_r(n,m)
-\nabla\cdot J_i
+\xi_i,
\label{eq:chem-general}
\end{equation}
possibly augmented by a conformational or catalytic memory variable \(m\). The chemistry floor is not just ``matter at lower temperature.'' It is the floor at which event identities, catalytic loops, compartments, and stoichiometric budgets become the natural description. Here \QiQUA\ and \GFF\ become especially transparent: reaction events carry energy and information cost simultaneously, and the chemical natural-unit Arisaka Equation becomes operationally useful.

\paragraph{Floor 4: life.}
Life begins when a chemical packet becomes a \SeOS. The defining step is not merely self-replication or metabolism in isolation but packet-level maintenance under continued openness. Using the notation of Sections~\ref{sec:eegg} and \ref{sec:sip}, and of
Part~\ref{part:seos}, one requires a positive average maintenance margin,
\begin{equation}
\left\langle \Maint - \RepairMin \right\rangle > 0,
\qquad
\left\langle \dot{\mathcal F}_{\GFF} \right\rangle_{\mathrm{cycle}} \le 0,
\label{eq:life-margin}
\end{equation}
with boundary integrity and memory depth sufficient to guide action or regulation. The life floor therefore adds adaptive sensing, active repair, motility or transport control, and eventually reproduction and evolution. Proto-\IH{} and proto-\EIC{} are common here: the system carries an internal state that predicts environmental relevance strongly enough to bias future behavior.

\paragraph{Floor 5: mind.}
Mind begins when the internal model is not merely useful for immediate control but becomes a high-dimensional, recurrently re-measured, self-accessible inner hologram. In the language of Section~\ref{sec:oh-ih-eic}, the system must admit a nontrivial internal boundary and a stabilized selected content. A minimal criterion is
\begin{equation}
\dim H_t^{\mathrm{in}} \gg 1,
\qquad
\QCI(t) > \QCI_{\mathrm{crit}},
\qquad
\mathrm{rank}\,\mathcal B_{\mathrm{\BFFT{}}} \ge 2,
\label{eq:mind-criterion}
\end{equation}
where \(\QCI\) is the \QiFour{} Completeness Index of Part~\ref{part:seos} and \(\mathcal B_{\mathrm{\BFFT{}}}\) denotes the multiscale tree of frequencies or timescales supporting coordinated memory. This floor supports explicit navigation, planning, imagination, and in human cases symbolic abstraction and scientific theorizing.

The five floors are summarized in Table~\ref{tab:floor-criteria}.

\begin{table}[H]
\centering
\caption[Operational criteria for each floor.]{Operational criteria distinguishing the five floors. Each row is one floor, with the new variable it introduces, the minimum criterion a system must meet to be on it, its memory substrate and its typical stabilizer. The third column is what makes the hierarchy checkable rather than descriptive: it states, for each floor, a condition that can in principle be evaluated for a given system, so that floor membership is a question with an answer.}
\label{tab:floor-criteria}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.09\textwidth} >{\raggedright\arraybackslash}p{0.17\textwidth} >{\raggedright\arraybackslash}p{0.22\textwidth} >{\raggedright\arraybackslash}p{0.20\textwidth} X}
\toprule
\textbf{Floor} & \textbf{New variable} & \textbf{Minimum criterion} & \textbf{Memory substrate} & \textbf{Typical stabilizer} \\
\midrule
Particle & charges, fields & gauge-consistent packet and current ledger & none or local phase & conservation laws, bound states \\
Cosmos & gravitational gradients, orbits & persistent large-scale gradients & orbital, environmental & gravity, radiative feedback \\
Chemistry & stoichiometric loops, catalysis & reaction-network closure with compartments & conformational, catalytic & kinetic attractors, compartments \\
Life & maintained boundary, repair budget & positive maintenance margin (\SeOS) & genetic, epigenetic, regulatory & adaptive regulation, reproduction \\
Mind & self-accessible allocentric \IH & high-dimensional \IH, internal measurement & neural recurrent memory & recurrent attractors, reportable \EIC \\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Probabilistic recurrence of high-floor packets in a driven open universe (conditional)}
\label{subsec:inevitability}

The user-level slogan of the present framework is that life and mind emerge naturally, deterministically, and inevitably in a six-dimensional open universe. To turn that slogan into a mathematical claim, one must distinguish two meanings of inevitability.

The first is \emph{trajectory inevitability}: every local history, on every planet or in every niche, produces the same outcome. The present paper does \emph{not} claim that. The second is \emph{ensemble inevitability}: once the lower floors generate sufficiently many driven windows and the viable higher-floor packets occupy a set of positive measure, the open-system search process reaches and stabilizes those packets with probability one in the long-time, many-trial limit. It is this second meaning that the theory derives.

Let \(W_{\mathrm{life}}\subset\mathcal Z^{(3)}\) be the subset of chemistry-floor packets that satisfy the \SeOS\ inequalities for sustained maintenance, and let \(W_{\mathrm{mind}}\subset\mathcal Z^{(4)}\) be the subset of life-floor packets that satisfy the mind criterion \eqref{eq:mind-criterion}. Let \(\tau_W\) denote the first hitting time of a set \(W\). Assume that the \DGI{} exploration process is recurrent or sufficiently mixing on the accessible driven region and that every entry into a viable window has a positive stabilization probability under \GSSB\ plus \GA.

\begin{theorem}[Conditional inevitability theorem (T-Qi4-16.4)]
\label{thm:conditional-inevitability}
%% v6.26: theorem downgraded in name and scope; substantive content (positivity
%% of measures) deferred to Open Problem O8 per hostile-reviewer recommendation

Assume:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item the driven accessible region of state space has finite invariant measure \(\mu\);
\item \(\mu(W_{\mathrm{life}})>0\) and \(\mu(W_{\mathrm{mind}})>0\);
\item the exploration process returns infinitely often to every positive-measure open set in the accessible region;
\item conditional on entering \(W_{\mathrm{life}}\) or \(W_{\mathrm{mind}}\), the probability of stabilization by \GSSB\ plus \GA\ is bounded below by positive constants \(p_{\mathrm{life}},p_{\mathrm{mind}}>0\).
\end{enumerate}
Then
\begin{equation}
\mathbb P\bigl(\tau_{W_{\mathrm{life}}}<\infty\bigr)=1,
\qquad
\mathbb P\bigl(\tau_{W_{\mathrm{mind}}}<\infty\mid \tau_{W_{\mathrm{life}}}<\infty\bigr)=1.
\label{eq:inevitability-result}
\end{equation}
In this precise sense, life and mind are inevitable attractor classes of a sufficiently rich driven universe.
\end{theorem}

\begin{proof}
By assumption (iii), the process enters each positive-measure viable set infinitely often almost surely. Consider first \(W_{\mathrm{life}}\). Let \(E_n\) be the event that the \(n\)-th visit to \(W_{\mathrm{life}}\) fails to stabilize. By assumption (iv), \(\mathbb P(E_n\mid \text{visit}_n)\le 1-p_{\mathrm{life}}\). Therefore the probability that stabilization fails on the first \(N\) visits is bounded by
\[
\mathbb P(E_1\cap\cdots\cap E_N)
\le
(1-p_{\mathrm{life}})^N.
\]
Taking \(N\to\infty\) yields zero. Hence stabilization in \(W_{\mathrm{life}}\) occurs almost surely. The same argument applies to \(W_{\mathrm{mind}}\) conditioned on the existence of life-floor packets from which the mind-floor accessible region is explored. This proves Eq.~\eqref{eq:inevitability-result}. \qedhere
\end{proof}

The theorem clarifies the meaning of ``deterministic'' in the present program. The determinism is not that every micro-history is identical. The determinism is that the runtime plus the measure-theoretic structure of the accessible universe makes higher-floor packets attractor classes rather than freak accidents. Once this is understood, the otherwise bold claim that life and mind are natural and inevitable becomes mathematically disciplined.

\subsection{Upward inheritance and downward modulation}
\label{subsec:upward-downward}

A common confusion in multiscale science is that a higher level must either be reduced away entirely or else be granted miraculous causal powers. The present framework avoids both extremes. Upward emergence is encoded by coarse-graining and selection of persistent packets. Downward influence is encoded only through lawful modulation of lower-level boundary conditions, control parameters, and port schedules. A higher floor never violates a lower floor's law; it changes the regime in which the lower floor operates.

Mathematically one may write the lowest-floor dynamics as
\begin{equation}
\dot z^{(0)}
=
F^{(0)}\bigl(z^{(0)};\theta^{(1)}(z^{(1)}),\theta^{(2)}(z^{(2)}),\ldots,\theta^{(4)}(z^{(4)})\bigr)
+\eta^{(0)}.
\label{eq:downward-modulation}
\end{equation}
The functions \(\theta^{(\ell)}\) are slow control parameters: concentrations, membrane states, tissue architecture, attention, or institutions depending on the floor. Equation~\eqref{eq:downward-modulation} is the correct formal meaning of downward causation in the present framework. It is regime modulation, not law violation.

\subsection{Section summary}
\label{subsec:pagoda-summary}

The hierarchy chapter establishes five points.
\begin{enumerate}[label=(\arabic*),leftmargin=2.5em]
\item A higher scientific floor is legitimate only if it closes approximately, persists long enough, and introduces genuinely new effective ports.
\item Under those conditions the universal runtime recursively factorizes into the five-story pagoda: particle, cosmos, chemistry, life, and mind.
\item \DGI\ plays the role of the pagoda's \emph{shinbashira}: a common invariant spine reused across all floors.
\item \GSSB\ forms the floors by selecting specific organized branches, while \GA\ stabilizes those branches under the relevant \GFF\ ledger.
\item If driven viable windows for life and mind have positive measure and are repeatedly explored, then life and mind become inevitable attractor classes in the ensemble sense proved by Theorem~\ref{thm:conditional-inevitability}.
\end{enumerate}

The hierarchy is now in place. We can therefore move from formal runtime to explicit cross-domain application without losing the floor structure that tells us how those applications are organized.


\section{Cross-domain applications: chemistry, life, brains, and society}
\label{sec:applications}

Once the hierarchy is explicit, the manuscript can move from architecture to worked reduction. Section~\ref{sec:applications} shows how the same packet-ledger-selector-diagnostic-architecture-internality vocabulary is instantiated in chemistry, life, brains, and society. The order matters: applications now come after the formal hierarchy rather than being used to motivate it prematurely.

\subsection{A canonical reduction recipe for any open system with memory}
\label{subsec:application-recipe}

The previous sections supplied the universal language. An application section must now answer a practical question: given a concrete system in any branch of science, how does one actually reduce it into the \GDOS--Qi runtime? The answer is a seven-step map.

\begin{definition}[Universal application tuple]
\label{def:application-tuple}
For an open system \(S\) with memory under nontrivial \(U(1)_{Qi}\), define its application tuple by
\begin{equation}
\mathfrak R(S)
:=
\Bigl(\widehat\SIP_S,\ \mathcal F^{(S)}_{\GFF},\ \mathcal J_S,\ \mathfrak H^{\mathrm{out}}_S,\ \mathfrak H^{\mathrm{in}}_S,\ \mathfrak E_S,\ \Pi_S\Bigr),
\label{eq:application-tuple}
\end{equation}
where \(\widehat\SIP_S\) is the canonical packet, \(\mathcal F^{(S)}_{\GFF}\) the scalar ledger, \(\mathcal J_S\) the locked currents, \(\mathfrak H^{\mathrm{out}}_S\) the \OH{} object, \(\mathfrak H^{\mathrm{in}}_S\) the \IH{} object, \(\mathfrak E_S\) the \EIC{} object, and \(\Pi_S\) the floor placement in the pagoda hierarchy.
\end{definition}

The tuple is useful only if it always exists under the hypotheses of the paper. After the hierarchy theorem, the next theorem turns the architecture into a reusable reduction method for concrete domains.

\begin{theorem}[Cross-domain reduction theorem (T-Qi4-17.2)]
\label{thm:cross-domain-reduction}
Let \(S\) be an open system with:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item explicit ports and non-isolated throughput,
\item a persistent memory-bearing state variable,
\item a nontrivial informational current coupled to \(U(1)_{Qi}\), and
\item a chosen instrument class defining external observables.
\end{enumerate}
Then the application tuple \(\mathfrak R(S)\) in Eq.~\eqref{eq:application-tuple} exists up to gauge-equivalent packet reparameterization.
\end{theorem}

\begin{proof}
A packet exists by Section~\ref{sec:sip}: every admissible open-system analysis begins by declaring a \SIP. The scalar ledger and current bundle exist by Section~\ref{sec:eegg} and
Part~\ref{part:qiqua}. The selector and architecture exist by the
\QiEP{} and \QiHAL{} Parts, respectively. Because an instrument class is given, the \OH{} object exists by \OGGEP\ and Section~\ref{sec:oh-ih-eic}. Because the system is memory-bearing and carries nontrivial \(U(1)_{Qi}\), the universality theorem of Section~\ref{sec:oh-ih-eic} implies the existence of at least a proto-\IH{}; if internal measurement is sufficiently structured, an \EIC{} object exists. Finally, the hierarchy chapter supplies the floor placement. Gauge-equivalent packet choices produce equivalent reduced descriptions because \DGI\ identifies redundancy. \qedhere
\end{proof}

Table~\ref{tab:application-recipe} rewrites the theorem as a practical workflow.

\begin{table}[H]
\centering
\caption[How to apply the framework, step by step.]{Canonical reduction recipe for applying the framework to any open system with memory. Each row is one step, with the question to ask and the output it produces. The recipe is included because a framework of this breadth is only useful if it can be run on a case, and the honest test of the paper is whether a reader in an unfamiliar domain can work down this column and produce something checkable. The steps are ordered so that each output is the next step's input.}
\label{tab:application-recipe}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\textwidth}p{0.30\textwidth}p{0.45\textwidth}}
\toprule
\textbf{Step} & \textbf{Question} & \textbf{Output} \\
\midrule
1 & What is the admissible system packet? & \(\widehat\SIP_S\): state manifold, ports, observables, ledger bundle \\
2 & What currents flow through the packet? & \(\mathcal J_S\): locked event currents and dominant throughputs \\
3 & What scalar budget drives lawful relaxation and maintenance? & \(\mathcal F^{(S)}_{\GFF}\) and maintenance balance laws \\
4 & What trajectories are admissible or selected? & \QiEP\ sector: \PLA/\PLT/\MEPP/\FEP\ interpretation \\
5 & What scale signatures diagnose the runtime? & \QiCON\ sector: power laws, fractality, two-knee spectra, scale covariance \\
6 & What architecture stores memory and forms world-models? & \QiHAL\ sector: \MePMoS, \NHT, \HAL, \BFFT, \OH{}/\IH{}/\EIC{} \\
7 & Where does the system sit in the pagoda? & Floor placement \(\Pi_S\) and cross-floor couplings \\
\bottomrule
\end{tabular}
\end{table}

The value of the recipe is that it does not change across sciences. Only the variables entering each slot change.


\section{What each floor is answerable for}
\label{sec:pagoda-inheritance}

A hierarchy that runs from particles to minds is the paper's largest claim about its own reach, and a claim of that size earns the question of what, exactly, it commits to at each level. This section answers it floor by floor: which pillar result is load-bearing there, at what grade it is carried, and what observation would refute the reading at that floor specifically. The purpose is to prevent the pagoda from being read as a metaphor with five levels --- the failure mode of every hierarchy of this shape in the literature --- by making each floor separately answerable.

Three cautions govern the table. First, the grades are not uniform and are not meant to be: the lower floors inherit theorem-grade results, and the upper floors inherit those same results plus placements that the paper marks as organizational rather than predictive. Second, a floor's entry names what would refute \emph{the reading at that floor}, not what would refute the framework: the pillar falsifiers sit upstream and are stated in Part~\ref{part:assessment}. Third, the floor count itself is an empirical reading and not a theorem output, which the open-problem list records; nothing in the table below turns on there being exactly five.

\begin{table}[H]
\centering
\caption[What each floor of the hierarchy inherits, and what would refute it.]{The per-floor accounting of the hierarchy. Each row names the pillar content that is load-bearing at that floor, the grade at which the paper carries it, and the observation that would refute the reading \emph{at that floor}. The grades fall as the floors rise, which is the honest shape of a construction that is theorem-grade where it prices events and organizational where it places behaviors: a reader who wants only what is proved can stop after the second row, and the table says so rather than leaving the boundary to be inferred.}
\label{tab:pagoda-inheritance}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.13\textwidth}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.13\textwidth}p{0.34\textwidth}}
\toprule
\textbf{Floor} & \textbf{What is load-bearing there} & \textbf{Grade} & \textbf{What would refute the reading there} \\
\midrule
Particle &
The four-current lock and the single conversion constant; elementary events as the unit of the ledger &
Derived &
An elementary event of the channel whose four-way bookkeeping departs from the single conversion factor beyond the stated tolerance \\
Cosmos &
The same lock at cosmological port scales; the runtime's three moves on the geometric companion's moduli &
Derived, imported &
A cosmological boundary budget requiring a second conversion factor, or a descent inconsistent with the imported cascade \\
Chemistry &
The current gauge and its two readings; reaction networks as sustained throughputs under clamped ports &
Derived, with the failure boundary of Theorem~\ref{thm:qiep-failure} &
A driven reaction network selected by an entropy-production extremum where the circulating sector demonstrably does ledger work \\
Life &
The memory architecture and the persistence layer; the informational column priced at \QB{}, anchored by the bond that carries two quanta &
Derived; the anchor measured to $1.6\%$ &
A metabolic anchor disagreeing with $2\QB$ beyond calorimetric precision with no integer-multiplicity reinterpretation available \\
Mind &
The internal-measurement loop and the inferential gauge, the latter conditional on an explicit rank condition &
Conditional; the floor count itself an empirical reading &
Inference maintained where the rank condition provably fails, or an identified inference metric that is not the one the ledger fixes \\
\bottomrule
\end{tabular}
\end{table}

Read down the grade column and the paper's shape appears: theorem-grade where it prices events and derives selection, conditional where it requires a checkable structural condition, and organizational where it places behaviors that no equation here predicts. That descent is not a weakness to be hidden in a late section; it is the reason the hierarchy is worth stating at all. A construction that claimed the same grade at every floor would be claiming that mind follows from the ledger as tightly as the conversion constant does, and it does not.

\section{The exportable result: inference derived, not assumed}
\label{sec:emergence-export}

One result of this paper travels further than the rest, and a reader who came for the biology may reasonably never open the pillar Part in which it is proved. It is worth stating on its own, in the register of the sciences that will use it.

In the study of brains and adaptive systems there is a widely used principle holding that anything which persists behaves so as to minimize a quantity measuring the mismatch between what it expected and what arrived. As mathematics it is productive; as a claimed law about matter it has had a difficult history, because the conditions under which a lump of matter can be said to carry a model of its surroundings turn out to be special rather than generic, and the principle's own defenders now describe it as a way of modeling systems rather than a fact about them.

What Part~\ref{part:qiep} establishes is that the working core of that principle is not a postulate here at all. It is what one obtains by taking the boundary functional this program derives from the six-dimensional action --- a functional that knows nothing of brains, and was constructed to answer a question about transport --- and asking it which \emph{beliefs} a system updates, rather than which path it takes or which currents it carries. The answer that falls out is natural-gradient descent of a free energy, with the geometry of the descent fixed by the ledger's own informational term rather than chosen for convenience. Three consequences follow, and each is checkable.

\emph{The metric stops being a modeling choice.} In the usual presentation the geometry on the space of beliefs is installed by hand and the weighting of precisions is a family of free parameters. Here the metric is the second variation of the ledger's informational column, which means that where it can be measured, it is a prediction rather than a fit.

\emph{The boundary of the system stops being a hopeful assumption.} Nothing here asks that a statistical boundary be discoverable inside the stationary density of a generic system. The split between a system and its surroundings enters as declared data --- the ports of the packet, written down with the system and carrying the conditions under which the description would be judged wrong.

\emph{The existence of an internal model becomes a condition one can test.} Whether a given system admits the inference reading is decided by whether an explicit action-conditioned observability matrix has full column rank. Systems that fail it are classified rather than assumed away, which is the difference between a framework that explains the negative cases and one that is embarrassed by them.

And one caution belongs in the same breath, because a result that arrives with its limits stated is worth more than one that does not: the inference reading fails exactly where the entropy readings fail, in the circulating sector of Theorem~\ref{thm:qiep-failure}, and the prospective policy machinery built on top of the principle in its own literature does not follow from anything derived here and is not claimed. What is offered is narrower than the original principle and, for that reason, more useful: a derivation of its working core, with the conditions printed on it.

\section{Why \texorpdfstring{\GDOS}{GDOS} applies to all open systems in nature}
\label{sec:why-all-open}

The universality claim now follows in one paragraph. Once a system is open it has ports; once it persists it requires a packet and a maintenance ledger; once it stores predictive information it requires a memory architecture and a nontrivial informational current; once that current is physically budgeted by \(U(1)_{Qi}\) the system pays a lawful cost for carrying and updating memory. From that moment the entire runtime---\DGI/\GSSB/\GA, \GFF, \SIP, \QiQUA/\QiEP/\QiCON/\QiHAL, \SeOS/\QCI, and \OH/\IH/\EIC---is available. None of these ingredients depends on whether the substrate is molecular, biological, neural, or social: only the state variables, timescales, and observables change. This completes the derivational arc, so Part~\ref{part:assessment} can now separate assessment into significance/comparison and limitation/falsifiability.



\paragraph*{Bridge to Part~\ref{part:assessment}.} Having established the four pillars
of \QiFour{}, the persistence-and-internality framework of Part~\ref{part:internal},
and the pagoda hierarchy of Part~\ref{part:emergence}, what remains is the broader
assessment: how does \QiFour{} compare to existing programs in
open-system science; what are the limitations of the present
derivational stage; what are the named experimental falsifiers; and
what does the framework imply, methodologically and philosophically,
about the open-system world? Part~\ref{part:assessment} gathers the discussion, the
honest L6 open-problem triage, the empirical falsifiability program,
and a closing meditation on the four pillars as one geometric object.


\section{One grammar, many sciences}
\label{sec:close-part5}

The pagoda in this Part is a real building, and the reason it is used
here is not the shape but the engineering. Those towers survive
earthquakes that flatten more rigid structures, and they do it because
the floors are not bolted to one another. Each rests on the one below.
A central pillar runs the height of the building and belongs to no
floor at all. Under load the stories move against each other, and the
movement is what saves them.

That is the claim about nature this Part is making. Chemistry is not
bolted to particle physics, and biology is not bolted to chemistry.
Each rests on what is beneath it, is constrained by it, and cannot
violate it --- and each is free in the ways that matter, because what a
higher level changes is never the law below but the conditions under
which that law is running. Concentrations, boundary conditions,
attention, institutions: these do not overrule anything. They set the
regime. The old dilemma that a higher level must either be reduced away
entirely or granted powers that break physics turns out to be a false
one, and the resolution is unglamorous. It was never a contest of
causes; it was a difference of timescale.

The word "inevitable" has been used about life in this program, and
this Part is where it is finally pinned down, so it should be said once
more in the plainest terms. It does not mean that any given world will
produce living things. Nothing here says that of the Earth, and nothing
here would be embarrassed by a sterile galaxy. What it means is a
statement about a search rather than about a history: that in a
universe which keeps generating gradients, and in which the
configurations capable of holding themselves together occupy more than
a vanishing fraction of what is reachable, such configurations are not
accidents waiting on improbable luck. They are where the search ends
up. And the number of stories in the tower is not a prediction of this
theory --- five is what has been observed, not what has been derived,
and the theory is honest enough to say so.

What would it mean if the grammar really is one grammar? Not that a
society is a large molecule, which it is not, or that a cell can be
computed from quarks, which it cannot. Something more modest and more
useful: that the questions worth asking of an open system are the same
questions at every scale. What does it spend. What does it hold on to.
What would kill it. Where does it stop. Biology, chemistry and physics
would remain separate sciences, because their variables and timescales
and instruments are genuinely different --- but they would stop being
separate \emph{kinds} of explanation. The differences would be
differences of subject, which is what they always looked like from a
distance, and never differences of law.


\GGPart[Assessment]{Assessment}{Significance, limitations, falsifiability, and conclusions}
\label{part:assessment}


\section{Discussion: significance, uniqueness, and comparison with existing frameworks}
\label{sec:discussion}

Part~\ref{part:assessment} begins only after derivation, hierarchy, and applications are complete. The first assessment question is positive and comparative: what is genuinely new, why does it matter, and how does it relate to existing scientific programs? Section~\ref{sec:discussion} answers that question before any limitation or falsifiability claims are stated.

\subsection{What is new in the present paper}
\label{subsec:discussion-new}

The present manuscript does not claim novelty merely by combining familiar words such as geometry, entropy, information, life, and consciousness. Its stronger claim is structural. It gives one continuous derivational chain from constitutional axioms to a practical runtime for open systems. That chain was not available before the four papers are read together.

The main novelties of the present paper are the following.
\begin{enumerate}[label=(\arabic*),leftmargin=2.5em]
\item Open systems are taken as fundamental rather than as corrections to ideal closed systems.
\item Observation is treated constitutionally as bulk-to-boundary projection, not as a late interpretive patch.
\item Information is given a gauge-current status through \(U(1)_{Qi}\) and the Arisaka ledger rather than remaining metaphorical.
\item The same runtime vocabulary is reused from particles to society through packet, ledger, selector, diagnostic, architecture, persistence, and holographic layers.
\item Consciousness is placed late in the derivation as internal measurement of selected \IH{} content rather than early as an unexplained primitive.
\end{enumerate}
These moves are jointly stronger than any one of them separately.

\begin{table}[H]
\centering
\caption[How this framework differs from six neighbors.]{Why the framework is structurally distinct from standard one-layer approaches. Six neighboring programs run down the first column, from open quantum systems to hierarchy and complexity theory. The second column says what each captures well --- and is meant to be read as credit, not as setup --- and the third what the present framework adds. The distinction claimed throughout is not that the neighbors are wrong but that each works at one layer, where the object here is the relation between layers.}
\label{tab:discussion-uniqueness}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.28\textwidth}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.34\textwidth}}
\toprule
\textbf{Approach} & \textbf{What it captures well} & \textbf{What the present framework adds} \\
\midrule
Open quantum systems & reduced dynamics, decoherence, instruments & embeds those tools into a larger geometric and multiscale runtime extending beyond quantum experiments \citep{BreuerPetruccione2002} \\
Nonequilibrium thermodynamics & dissipation, transport, entropy production & adds explicit packet identity, current locking, memory architecture, and lawful observation \citep{Prigogine1967,NicolisPrigogine1977} \\
Free-energy / active inference & prediction, action, belief update & derives such selector behavior inside a larger packet-ledger-gauge constitution \citep{Friston2010} \\
Integrated-information style theories & structured conscious integration & relocates consciousness after packet, ledger, selector, architecture, and measurement semantics \citep{Tononi2004} \\
Synergetics / self-organization & order parameters and pattern formation & supplies a complete bridge from fundamental geometry to multiscale attractors \citep{Haken1983} \\
Hierarchy/complexity viewpoints & architecture of levels and emergence & provides explicit closure, persistence, and port criteria for legitimate floors \citep{Simon1962,Anderson1972} \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Relation to existing scientific programs}
\label{subsec:discussion-related-programs}

It is helpful to state the relation to several existing programs one by one.

\paragraph{To nonequilibrium thermodynamics and dissipative structure theory.}
The present framework agrees that far-from-equilibrium flux can generate organized structure \citep{Prigogine1967,NicolisPrigogine1977}. It adds a universal packet, a current-locking ledger, and an explicit hierarchy of observation and internal modeling.

\paragraph{To active inference and predictive processing.}
The present framework agrees that perception and action should be treated jointly \citep{Friston2010}. It adds an earlier constitutional layer: prediction is not the first principle but a boundary manifestation of the deeper geometry-to-ledger runtime.

\paragraph{To synergetics and complexity science.}
The present framework agrees that order parameters, multiscale hierarchy, and emergence matter \citep{Haken1983,Simon1962,Anderson1972}. It adds explicit closure-defect, lifetime, and port criteria, allowing one to say when a level is genuinely admissible rather than merely evocative.

\paragraph{To consciousness science.}
The present framework agrees that conscious content involves structured integration and internal availability \citep{Tononi2004}. It adds a derivational path from packet to internal measurement, placing consciousness after rather than before the runtime.




%% ====================================================================
\section{Cross-Paper Coherence Map}
\label{sec:coherence-map}
%% ====================================================================

The \QiFour{} framework imports its constitutional inputs from
\citet{ArisakaA1Constitution}, bulk geometry from
\citet{ArisakaA2GG6}, anomaly closure from
\citet{ArisakaA3BI6}, boundary-projection structure from
\citet{ArisakaA5GDOS}, and its geometric pillar template from
\citet{ArisakaA6Gi4}; in turn it exports the four Qi-pillars,
\SeOS{}/\EIC{} framework, and Arisaka constant to
\citet{ArisakaA9SM}, \citet{ArisakaA10Cosmos}, \citet{ArisakaP1KK}, and \citet{ArisakaC1Kaxion}.
Each row of the maps below names a specific theorem number, section, or appendix location in the cited companion paper\footnote{Audit trail: \AOne{} (Axioms \AX{1}--\AX{4}; Postulates \POS{1}--\POS{7}; Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$). \ATwo{} (canonical 6D metric; \BIsix{} four-component vector; \OGNfive{} cascade $(1,3,5;7,35)$). \AThree{} (Theorem~3.4 \OGNfive{} trace-ratio lock; Theorem~4.1 anomaly freedom). \AFive{} ($\cD_{\OGGEP} = \cR \circ \mathrm{Tr}_{\rm env}$; \SIP{}; \OH{}+\IH{}=\EIC{}). \ASix{} (principal theorems T1-T4 with compatibility propositions $\mathfrak{C}_{\TGG}, \mathfrak{C}_{\CCI}, \mathfrak{C}_{\GFF}, \mathfrak{C}_{\GRIP}$). Each row names a specific theorem identifier rather than a description, so a change in a companion shows up as a mismatch rather than as a silently wrong pointer.}.

\subsection{Inward map: imports into \ASeven{} from the foundation block}
\label{ssec:a7-inward-map}

The following longtable lists, for each theorem-level fact that
\QiFour{} imports from a foundation-block paper, the specific
companion-paper location and the role the imported fact plays in
the present manuscript.

\begin{footnotesize}
\begin{longtable}{p{0.30\linewidth} p{0.27\linewidth} p{0.35\linewidth}}
\caption[What this paper imports, and from where.]{Inward coherence map: what this paper imports from the foundation block. Each row is one imported fact, with the companion paper and location that establishes it and the role it plays here. Together with Table~\ref{tab:a7-outward-map} it is the paper's full interface with the cohort, and it is the table to consult when checking that no argument here is circular: everything in the middle column is upstream, and nothing in it depends on a result of this paper.}
\label{tab:a7-inward-map}\\
\toprule
\textbf{Imported fact} & \textbf{Companion-paper location} & \textbf{Role in \ASeven{}}\\
\midrule
\endfirsthead
\multicolumn{3}{l}{\emph{(continued from previous page)}}\\
\toprule
\textbf{Imported fact} & \textbf{Companion-paper location} & \textbf{Role in \ASeven{}}\\
\midrule
\endhead
\bottomrule
\multicolumn{3}{r}{\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
Four Axioms \AX{1}--\AX{4} (LC, \GGEP{}, \OGGEP{}, \MSMF{}) and seven postulates &
\AOne{} \citep{ArisakaA1Constitution} &
Constitutional source of the four Qi-pillars; \AX{3} \OGGEP{} is the foundational dynamics from which the boundary projection underlying \QiFour{} is derived.\\
Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ &
\AOne{} \citep{ArisakaA1Constitution} &
Inherited as the central boundary identity that \QiQUA{} (Pillar 1) formalizes through the rank-one inflow theorem and the natural-unit equivalence.\\
6D bulk metric and gauge group $\Spinone \times \SUfive \times \UoneB \times \UoneQi$ &
\ATwo{} \citep{ArisakaA2GG6} &
Bulk arena and four-factor gauge structure from which the four boundary currents $(J_e, J_B, J_{Qi}, J_E)$ of \QiQUA{} descend.\\
\OGNfive{} Diophantine cascade $(M,N,L;J,K) = (1,3,5;7,35)$ &
\ATwo{}, Theorem~6 \citep{ArisakaA2GG6} &
Sets the integer $K = 35$ used in the Arisaka constant $\QB = (2/5)\,E_{\rm Ryd}/K$; the open problem O5 (\OGNfive{} selection rule) is precisely the question of whether this assignment is uniquely forced by \MSMF{}.\\
\BIsix{} closure $\dd H_3 = X_4$ with coefficient vector $(1,-1,-2/5,-2/5)$ &
\AThree{}, Theorem~3.4 \citep{ArisakaA3BI6} &
Provides the rank-one bulk source $X_4$ from which the four-current lock of \QiQUA{} descends. The defensive uniqueness clause cites \S~8.7 here.\\
GS-as-\GGEP{} ``geometrization reading'' &
\AThree{}, \S~8.7 \citep{ArisakaA3BI6} &
The recap-bridge section of Part~\ref{part:entrance} cites this for supportive framing of the Green--Schwarz construction; the present paper does not re-derive the argument.\\
Anomaly-completion and unified gauge consistency &
\citet{ArisakaA3BI6} (formerly GD-2026-5) &
Cross-reference for the legality conditions imported into both \ASix{} and the present Qi-runtime paper.\\
\GDOS{} boundary law $\cD_{\OGGEP} = \cR \circ \mathrm{Tr}_{\rm env}$ as CPTP &
\AFive{} \citep{ArisakaA5GDOS} &
Boundary-projection construction that underlies the operational layer of all four Qi-pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}); the \OH{}+\IH{}=\EIC{} framework in Part~\ref{part:internal} descends directly from this.\\
\SIP{} (System Instance Packet) construction &
\AFive{} \citep{ArisakaA5GDOS} &
Provides the bookkeeping primitive for the persistence class \SeOS{} (Part~\ref{part:internal}) and the \QCI{} completeness index.\\
\TGG{} (Total Geometric-Gauge object) + Master Conservation Law &
\ASix{}, Theorem~T1 \citep{ArisakaA6Gi4} &
Geometric backbone of \QiQUA{}: the rank-one inflow theorem is the boundary face of T1's master conservation law.\\
\CCI{} (Covariant Conservation Identity) + \OGNfive{} uniqueness &
\ASix{}, Theorem~T2 \citep{ArisakaA6Gi4} &
Geometric scaffolding for the four-current Ward package used in \QiQUA{} and \QiEP{}.\\
\GFF{} (Geometric Free-Energy Functional) with $F_\GFF = F_\GEP + F_\CEP$ &
\ASix{}, Theorem~T3 \citep{ArisakaA6Gi4} &
Inherited as the master functional from which \QiEP{} extracts \PLT{}, \MEPP{}, and \FEP{} as three boundary gauges (common-origin theorem of).\\
\GRIP{} cascade $\DGI{} \to \GSSB{} \to \GA{}$ &
\ASix{}, Theorem~T4 \citep{ArisakaA6Gi4} &
Inherited as the runtime structure on the family-projector + memory moduli that \QiHAL{} formalizes as \MePMoS{} / \NHT{} / \HAL{} / \BFFT{}.\\
\SeOS{} (Selfish/Self-maintaining Open System) seed &
\AFive{} \citep{ArisakaA5GDOS} &
The persistence-class seed that Part~\ref{part:internal} of the present paper formalizes into the persistence criterion + \QCI{}-completeness index.\\
\end{longtable}
\end{footnotesize}

\subsection{Outward map: exports from \ASeven{} to downstream papers}
\label{ssec:a7-outward-map}

The following longtable lists, for each theorem-level fact that
\QiFour{} exports to a downstream paper, the present-paper theorem
label or section location and the downstream paper's anticipated use.

\begin{footnotesize}
\begin{longtable}{p{0.30\linewidth} p{0.27\linewidth} p{0.35\linewidth}}
\caption[What this paper exports, and to whom.]{Outward coherence map: what this paper exports to the papers downstream of it. Each row is one exported result, with its location here and the downstream paper that uses it. Read beside Table~\ref{tab:a7-inward-map} it shows the paper's position in the cohort's dependency graph --- a consumer of the foundation block and a supplier to the phenomenology --- and it is what a reader auditing the cohort for circularity should read first.}
\label{tab:a7-outward-map}\\
\toprule
\textbf{Exported \ASeven{} fact} & \textbf{A7 location} & \textbf{Downstream paper and anticipated use}\\
\midrule
\endfirsthead
\multicolumn{3}{l}{\emph{(continued from previous page)}}\\
\toprule
\textbf{Exported \ASeven{} fact} & \textbf{A7 location} & \textbf{Downstream paper and anticipated use}\\
\midrule
\endhead
\bottomrule
\multicolumn{3}{r}{\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\QiQUA{} rank-one inflow theorem &
Theorem~\ref{thm:qiqua-rank-one} (T-QUA.1) &
\AEight{}: rank-one boundary source structure as the geometric content of quantum measurement events. \ANine{}: rank-one ledger for the 28 Standard-Model parameter derivations.\\
Arisaka constant $\QB \approx 0.1555\,\mathrm{eV}/\mathrm{qubit}$ &
Theorem~\ref{thm:no-new-param}, Eq.~\eqref{eq:arisaka-constant} &
\ANine{}: energy--information conversion factor for the lepton/quark mass ratios. \PaperPOne{}: high-energy scaling factor for KK resonance lifetime predictions. \PaperCOne{}: low-energy conversion factor for Kaxion DM detection thresholds.\\
Compatibility proposition $\mathfrak{C}_{\rm QUA}$ &
Part~\ref{part:qiqua}, eq.~\eqref{eq:CQUA-compat} &
\AEight{}: dual-layer template for the algebraic-vs-operational structure of quantum measurement events.\\
\QiEP{} common-origin theorem  &
Theorem~\ref{thm:qiep-unified} (T-EP.1) &
\ANine{}: variational backbone for the seesaw-mechanism derivation and the cosmological-evolution selectors of \ATen{}.\\
Three named gauges \PLT{}, \MEPP{}, \FEP{} &
Theorems~\ref{thm:qiep-plt}, \ref{thm:qiep-mepp}, \ref{thm:qiep-fep} &
\AEight{}: three falsifiable laboratory predictions on action minimization, current statistics, and inferential model quality. \ATen{}: cosmological-entropy production as the \MEPP{} gauge on the boundary of the 6D bulk.\\
\QiCON{} operational conformality theorem &
Theorem~\ref{thm:qicon-operational-conformality} &
\ANine{}: conformal structure on the lepton/quark mass moduli. \PaperPOne{}: spectral-shape predictions for KK resonance clusters.\\
\QiCON{} two-knee spectral theorem &
Theorem~\ref{thm:qicon-two-knee} (T-CON.2) &
\AEight{}: $1/f$ noise predictions for quantum-measurement decoherence. \PaperCOne{}: low-mass DM signal-shape prediction. (Open problems O5--O7 record the residual gaps.)\\
\QiHAL{} universality theorem &
Theorem~\ref{thm:qihal-universality} (T-\HAL{}.1) &
\AEight{}: memory architecture underlying any quantum-measurement device. (``Unique'' is weakened to ``sufficient'' here; uniqueness is open problem O6.)\\
\MePMoS{} / \NHT{} / \HAL{} / \BFFT{} hierarchy &
Theorems~\ref{thm:mepmos-uniqueness}, \ref{thm:hal-frequency-independence}, etc. &
\AEight{}: explicit named hierarchy carrying the action-conditioned \IH{} observability and the runtime cascade of any persistent quantum-information system.\\
\SeOS{} persistence criterion + \QCI{} completeness index &
Theorem~\ref{thm:seos-persistence}  and Definition~\ref{def:qci} &
\ATen{}: persistence class for the cosmological boundary-projection of living open systems.\\
\OH{}+\IH{}=\EIC{} (definitional shorthand $\EIC := \mathcal{C}_{\rm loop}(\OH, \IH)$) &
Section~\ref{sec:oh-ih-eic} &
\AEight{}: holographic-observation framework for the action-conditioned \IH{}; \ATen{}: cosmological-scale outer hologram.\\
Pagoda hierarchical factorization theorem (five-floor variant) &
Theorem~\ref{thm:pagoda-factorization} &
\ANine{}: particle floor; \ATen{}: cosmos floor; future papers on chemistry / life / mind floors. (Open problem O7 records that ``five'' is the historically natural variant, not yet uniquely forced by \MSMF{}.)\\
Conditional inevitability theorem &
Theorem~\ref{thm:conditional-inevitability} &
\ATen{}: anthropic-style boundary-cosmology argument for the inevitability of life and mind in a sufficiently rich driven universe. (Open problem O8 records the unmeasured Borel--Cantelli hypotheses.)\\
\end{longtable}
\end{footnotesize}

The two maps together establish the position of \QiFour{} in the
\GGTheory{} program: the constitutional content flows downward from
A1, the bulk-side technical content flows in from A2 and A3, the
boundary-projection structure flows in from A5, the geometric pillar
template flows in from A6, and the four Qi-pillars + \SeOS{}/\EIC{} then
flow outward to every downstream paper that addresses the 4D-physics,
cosmology, or predictions branches of the program. When any
companion paper is later revised to a new version, the present
section should be re-walked against the revised paper's actual
theorem numbering before the present paper is itself republished;
the audit trail above is the working record of the current numbering
snapshot.

\section{The Ten Structural Innovations of the \QiFour{} Runtime}
\label{sec:ten-innovations}

The contribution of \ASeven{} is to carry the open-system boundary law of
\AFive{} into a complete information-theoretic runtime --- four pillars, an
internal-measurement construction, and a cross-scale hierarchy --- without
introducing any new continuous parameter beyond the imported ledger. We collect
here the ten structural innovations that runtime delivers, each carrying its
cohort-global identifier (\S\ref{sec:naming-convention}).

\paragraph{Innovation 1 (I-Qi4-1): Open systems as fundamental.} The runtime
takes open, environment-coupled systems as the primitive object rather than as
corrections to ideal closed systems --- the boundary law, not the isolated
Hamiltonian, is the starting point.

\paragraph{Innovation 2 (I-Qi4-2): Observation as a constitutional projection.}
Measurement is treated as the lawful bulk-to-boundary \OGGEP{} projection from
the outset (Theorem T-Qi4-3.3), not as a late interpretive patch added to a
closed theory.

\paragraph{Innovation 3 (I-Qi4-3): Information as a gauge current.} Information
is given physical, gauge-current status through the memory-bearing $U(1)_{Qi}$
sector and the Arisaka ledger $\QB = (2/5)\,\ERyd/K$, rather than remaining an
informal metaphor.

\paragraph{Innovation 4 (I-Qi4-4): One runtime vocabulary from particles to
society.} The same seven-layer grammar --- packet, ledger, selector, diagnostic,
architecture, persistence, and holographic observation --- is reused unchanged
across chemistry, life, brains, and collective systems.

\paragraph{Innovation 5 (I-Qi4-5): The four pillars as principal theorems.} The
informational pillars \QiQUA{}, \QiEP{}, \QiCON{}, and \QiHAL{} are stated as
principal theorems with the implication chain
$\QiQUA{} \Rightarrow \QiEP{} \Rightarrow \QiCON{} \Rightarrow \QiHAL{}$, each
forcing the next rather than standing as independent assumptions.

\paragraph{Innovation 6 (I-Qi4-6): The Arisaka four-current lock.} \QiQUA{} ties
the four boundary currents through one energy--information conversion
factor, $J_E/\QB = J_e = J_B = J_{Qi}$ (the rank-one inflow theorem T-Qi4-9.3), so that
the informational current is not free but locked to the energetic, electric, and
baryonic channels.

\paragraph{Innovation 7 (I-Qi4-7): Three selector gauges unified.} The
action-extremum, current-statistics, and inferential readings of \QiEP{}
(\PLT{}/\MEPP{}/\FEP{}) are shown to be three gauges of one projected functional
(the common-origin theorem T-Qi4-10.6), not three competing principles.

\paragraph{Innovation 8 (I-Qi4-8): The two-knee conformal signature.} \QiCON{}
predicts a specific two-knee shape for $1/f$ spectra --- exactly two
scale-breaks bracketing a finite power-law band, a structural signature distinct
from a single unbounded slope (the two-knee spectral theorem
T-Qi4-11.7), with the knee positions set by the system's slowest and fastest
retained rates --- a sharp, multiscale falsifier rather than a generic power law.

\paragraph{Innovation 9 (I-Qi4-9): Internal measurement and the late placement of
consciousness.} The Outer-Hologram / Inner-Hologram / Evolving-Inner-Cosmos
construction (\OH{}${+}$\IH{}${=}$\EIC{}, universality theorem T-Qi4-15.14)
places conscious content late in the derivation, as the internal measurement of
selected \IH{} structure, rather than early as an unexplained primitive.

\paragraph{Innovation 10 (I-Qi4-10): The internalized \GRIP{}-loop and the
cross-scale pagoda.} \QiHAL{} discovers that a memory-bearing system carries its
own recoverable \GRIPout{} (the internalized regime switch), and the five-story
pagoda factorization (T-Qi4-16.2) exhibits the same runtime from particle to
mind as a single nested hierarchy.

\subsection{Global naming convention for cross-cohort citation}
\label{sec:naming-convention}

Beginning with \citet{ArisakaA2GG6} and extended across the cohort, each
load-bearing item of the \GGTheory{} program carries a globally unique
identifier of the form \emph{Prefix-Suffix-N}, where the Prefix names the item
type and the Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the $N$th Theorem (\ASeven{}'s theorems are
section-numbered, so its global ids carry the section number, e.g.\ T-Qi4-9.3 is
the rank-one inflow theorem of \QiQUA{} and T-Qi4-15.14 the \EIC{} universality
theorem).
\item \textbf{L-Suffix-N}, \textbf{C-Suffix-N}, \textbf{D-Suffix-N},
\textbf{R-Suffix-N} for the $N$th Lemma, Corollary, Definition, and Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (e.g., O-Qi4-1 is the first
Common Objection of the present paper).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (e.g., F-Qi4-1 is the
current-locking falsifier of the empirical program).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-Qi4-1 is the
open-systems-as-fundamental innovation above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved / open-problem entry
(\S\ref{subsec:discussion-limitations}).
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is: \emph{Constitution}
(A1), \emph{GG6} (A2), \emph{BI6} (A3), \emph{Euclid} (A4), \emph{GDOS} (A5),
\emph{GRIP} (A6), \emph{Qi4} (A7, this paper), \emph{QM} (A8),
\emph{SM} (A9), \emph{Cosmos} (A10), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
\emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and
the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).
The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7)
of \citet{ArisakaA1Constitution} keep their existing cohort-wide names; the
global identifier is the same item viewed from outside its home paper.

\section{Open Problems, Limitations, and Falsifiability}
\label{sec:open-problems}
\label{sec:limitations-falsifiability}

Assessment has a second, harder side. After significance and comparison, the manuscript must say where the framework could fail and how it should be tested. Section~\ref{sec:limitations-falsifiability} therefore isolates limitations, falsifiability, and the empirical program into a separate assessment layer.

\subsection{Why assessment must be split from significance}
\label{subsec:limitations-why-split}

A landmark synthesis must not only say why it matters; it must also say where it is incomplete and how it could fail. For that reason the revised architecture separates assessment into two layers. The previous section addressed significance, uniqueness, and relation to existing programs. The present section addresses the harder question: what remains unfinished, and what concrete empirical failures would count against the framework?

\subsection{Open problems and limitations of the derivational stage}
\label{subsec:discussion-limitations}

A serious discussion must state open problems and limitations clearly. Following the L6 audit convention adopted across the \GGTheory{} program, each open problem $O_k$ below is triaged into one of three status categories:
\begin{itemize}[leftmargin=2em]
\item \emph{Resolved.} The question has a complete answer in a named companion paper, with explicit citation.
\item \emph{Partially resolved.} The central claim is settled in the named companion paper, with a specific sub-task still pending in that companion paper.
\item \emph{Genuinely open.} No companion paper currently carries the answer. The body identifies the candidate mechanisms or research directions.
\end{itemize}
Listing as ``open'' what is in fact resolved in a companion paper would be a methodological failure of the constitutional program; the triage below records the actual status of each question. A closing summary at the end of the section gives the status counts.

\begin{enumerate}[leftmargin=2em,label=(O\arabic*)]
\item \textbf{(PS-Qi4-1)~Universal numerical simulation package spanning all pagoda floors.}
\emph{Status: Genuinely open.}
The theory is architectural before it is numerically exhaustive. The present paper gives theorem-level derivations, normal forms, and reduction criteria, but it does not yet supply one universal simulation package spanning all five floors of the pagoda. Domain-specific numerical models will still be needed. A unified simulation toolkit is a substantial software-engineering project deferred to future work.

\item \textbf{(PS-Qi4-2)~Measurement protocols for \(U(1)_{Qi}\) at chemistry, life, and brain scales.}
\emph{Status: Genuinely open} (experimental, awaiting protocol development).
The empirical realization of \(U(1)_{Qi}\) at the chemistry, life, and brain scales is at present a structural and bookkeeping claim rather than a finished measurement protocol. The framework says what must be tracked as informational current; it does not yet provide a single off-the-shelf instrument family for every domain. Direct detection of the Kaxion-like \(U(1)_{Qi}\) signature is the high-energy companion problem and is tracked in \citet{ArisakaC1Kaxion}.

\item \textbf{(PS-Qi4-3)~Full phenomenology of consciousness.}
\emph{Status: Genuinely open.}
The consciousness discussion is deliberately disciplined but incomplete. The framework identifies a runtime condition for conscious content --- internally measured selected \IH{} structure --- without claiming to have solved the entire phenomenological problem. The criterion the framework supplies is one of \emph{access} consciousness --- the reportability and global availability of selected content, in the sense of \citet{Block1995}; the \emph{phenomenal} hard problem of subjective experience is recorded here as a structural placeholder for a future companion paper.

\item \textbf{(PS-Qi4-4)~Individual-environment realization of the pagoda hierarchy.}
\emph{Status: Partially resolved.}
The hierarchy theorem is proved in the ensemble sense earlier in the present paper: averaged over admissible environments, the higher floors are inevitable. The individual-environment realization is conditional on accessible viable windows having positive measure and on sufficient driven exploration. No companion paper currently addresses the local-environment refinement, though some sub-questions are tracked in \citet{ArisakaA10Cosmos}.

\item \textbf{(PS-Qi4-5)~\OGNfive{} selection rule for the Arisaka constant $\QB$.}
\emph{Status: Genuinely open.}
The Arisaka constant $\QB \approx 0.1555\,\mathrm{eV}/\mathrm{qubit}$ is defined by $\QB = (2/5)\,E_{\rm Ryd}/K$ with $K=35$ from the \OGNfive{} ledger. The assignment ``per-step quantum $= E_{\rm Ryd}/K$'' is currently justified only by minimality intuition; a derivation showing $E_{\rm Ryd}/K$ (rather than $E_{\rm Ryd}/J = E_{\rm Ryd}/7$ or another combination of the \OGNfive{} integers) is the unique per-step quantum under \MSMF{} would close this question. The earlier factor-of-two gap to ATP hydrolysis is resolved: one phosphoanhydride bond carries two locked quanta, so $2\,N_A\QB = 30.0~\mathrm{kJ/mol}$ matches the measured standard free energy $-30.5 \pm 0.5~\mathrm{kJ/mol}$ to $1.6\%$ (Part~\ref{part:qiqua}). What remains genuinely open here is the selection-rule proof of the functional form $E_{\rm Ryd}/K$, not the bioenergetic magnitude.

\item \textbf{(PS-Qi4-6)~Uniqueness of the \MePMoS{}-\NHT{}-\HAL{}-\BFFT{} factorization in the \GRIP{} cascade.}
\emph{Status: Genuinely open.}
Theorem~\ref{thm:qihal-universality} establishes that the named five-layer hierarchy is a \emph{sufficient} boundary realization of the \GRIP{} cascade on the family-projector + memory moduli. Uniqueness --- the claim that no other 3-stage, 5-stage, or category-equivalent factorization satisfies the cascade --- requires a separate no-go argument that is currently absent.

\item \textbf{(PS-Qi4-7)~Derivation of the pagoda floor count $= 5$ from \MSMF{}.}
\emph{Status: Genuinely open.}
The five-story pagoda (particle $\to$ cosmos $\to$ chemistry $\to$ life $\to$ mind) is asserted as the intended interpretation of the hierarchical factorization theorem. Whether ``five'' is forced by \AX{4} \MSMF{} (minimality across closure-defect and port-novelty inequalities) or whether other floor counts satisfy the same admissibility criteria is currently open. A four-floor or six-floor variant would satisfy the existence theorem on its face.

\item \textbf{(PS-Qi4-8)~Quantitative bounds on the conditional-inevitability hypotheses.}
\emph{Status: Genuinely open.}
Theorem~\ref{thm:conditional-inevitability} reduces the inevitability of life and mind to a Borel--Cantelli application conditional on (i) finite invariant measure on the driven accessible region, (ii) positive measure of the viable subsets $W_{\rm life}$, $W_{\rm mind}$, and (iii) per-visit stabilization probabilities bounded below. The unconditional inevitability claim depends on independent bounds for $\mu(W_{\rm life})$ from chemistry-floor analysis and $p_{\rm life}$ from \SeOS{} persistence statistics; these are not provided in the present paper.

\item \textbf{(PS-Qi4-9)~Three-component \QiFour'{} framework.}
\emph{Status: Genuinely open.}
The L12 framing of \(U(1)_{Qi}\) admissibility (above) acknowledges that a hypothetical three-component theory $\Spinone \times \SUfive \times \UoneB$ would require an entirely new \QiFour'{} framework: a four-current Arisaka equation reducing to three currents, a memory carrier other than a gauged $U(1)$, and a reformulated \EIC{} construction. Whether such a \QiFour'{} is consistent at all, and what its empirical predictions would be, is recorded here for future work.
\end{enumerate}

These open problems are not defects hidden in small print. They are the honest boundary of the present derivational stage, and the triage above records where each lives in the \GGTheory{} program.

\subsection{Falsifiability and empirical program}
\label{subsec:discussion-falsifiability}

The framework would be weak if it only re-described existing phenomena after the fact. Its scientific value depends on linked falsifiable predictions. The most important point is that the predictions are \emph{joint}. Many individual signatures, such as power laws or prediction errors, can appear in other frameworks too. The present framework predicts that specific families of signatures should co-occur because they descend from one runtime.

Define a system-level consistency vector
\begin{equation}
\Sigma_S := \bigl(C_{\mathrm{lock}},\ C_{\mathrm{sel}},\ C_{\mathrm{scale}},\ C_{\mathrm{mem}},\ C_{\mathrm{holo}}\bigr)_S,
\label{eq:consistency-vector}
\end{equation}
where the components measure, respectively: evidence for current locking (\QiQUA{}); evidence for variational behavior (\QiEP{}); evidence for scale-covariant diagnostics (\QiCON{}); and two faces of \QiHAL{} --- evidence for the memory architecture (\MePMoS{}, \NHT{}, \HAL{}, \BFFT{}) and evidence for the internality layer (\(\OH+\IH=\EIC\)). The last two are scored separately because a system can implement storage and timing richly while closing no model-of-the-environment loop at all, and that dissociation is exactly what the vector is for. The framework predicts not just isolated positive components, but positive \emph{coherence} across the vector in systems claimed to instantiate the full runtime.

A simple scalar summary is
\begin{equation}
\Xi_S := \sum_{a=1}^5 w_a C_a(S),
\qquad w_a>0,
\quad \sum_a w_a=1.
\label{eq:xi-score}
\end{equation}
The exact operationalization of \(C_a\) depends on domain, but the logic is universal: a true full-runtime system should score coherently across all five categories.

Table~\ref{tab:falsifiability} lists representative tests.

\begin{table}[H]
\centering
\small
\caption[The falsification program, signature by signature.]{Representative falsifiability program for the integrated framework. Each row is one predicted signature, with a test idea and what a failure would mean. The third column is the one that gives the table its value: a framework this broad can absorb a surprising amount of data unless it says in advance which failures are fatal and which are merely inconvenient, and this column distinguishes the two case by case.}
\label{tab:falsifiability}
\usebox{\GGscFalsProg}
\end{table}

The paper is therefore falsifiable in a strong sense. It is not protected by saying ``something complicated happened.'' It predicts specific linked structures: packet stability, ledger accounting, selector behavior, scale diagnostics, memory architecture, and holographic observation semantics. If those structures fail jointly, the theory fails.

\paragraph*{Experimental open-problem list (L6 triage).}
We catalog the five principal experimental directions of
Table~\ref{tab:falsifiability} as labeled open-problems
E1--E5, each with the L6 status header. All five are presently
\emph{Genuinely open} in the experimental sense --- the
framework provides the prediction, the laboratory verdict is
the next decade of measurements.
\begin{enumerate}[leftmargin=2em,label=(E\arabic*)]
\item \textbf{(F-Qi4-1)~Current-locking constraints.} \emph{Status: Genuinely open} (awaiting empirical data). Event-level energy/charge/information budgets in repeated micro-events should be consistent with one locked ledger per \QiQUA{}. Falsification: persistent inability to define a coherent event ledger across the four boundary currents $J_e$, $J_B$, $J_{Qi}$, $J_E$.
\item \textbf{(F-Qi4-2)~Selector behavior across domains.} \emph{Status: Genuinely open}. Perturb pathway options and test whether realized trajectories follow a common \PLA{}/\PLT{}/\MEPP{}/\FEP{}-style selector. Falsification: no selector family fits across domains.
\item \textbf{(F-Qi4-3)~Two-knee scale signatures.} \emph{Status: Genuinely open} (predicted by T-CON.2 of \QiCON{}, awaiting controlled spectral measurements). Specific predicted shape of $1/f$ noise with two co-occurring scale-breaking knees bracketing a finite power-law band (the structural signature, distinct from a single unbounded $1/f$ slope), with the knee positions set by the system's slowest and fastest retained rates rather than by universal constants.
\item \textbf{(F-Qi4-4)~Exploration-to-commitment lag (\DGI{}$\to$\GSSB{}$\to$\GA{} timing).} \emph{Status: Genuinely open}. Time-resolve candidate-state exploration, branch selection, and stable report or action. Falsification: selection/stabilization staging is absent or has the wrong temporal structure.
\item \textbf{(F-Qi4-5)~Motion-conditioned model quality (action-conditioned \IH{} observability).} \emph{Status: Genuinely open}. Restrict action while preserving passive input and test whether internal model quality degrades. The qualitative active-versus-passive asymmetry is already established (\citet{HeldHein1963}; cf.\ the reafference principle of \citet{vonHolst1950}); the sharper, still-open prediction is the \emph{quantitative} dependence of model error on the observability-Gramian deficit. Falsification: failure of that quantitative dependence would challenge the action-conditioned observability theorem behind \IH{}.
\end{enumerate}

\paragraph*{Triage summary.}
Of the fourteen open problems cataloged above (O1--O9 architectural
and theoretical, E1--E5 experimental), \emph{zero} are Resolved
in a named companion paper at this writing; \emph{one} (O4,
individual-environment realization of the pagoda) is \emph{Partially
resolved} with the central claim settled in the ensemble sense in
the present paper but with sub-tasks pending in
\citet{ArisakaA10Cosmos}; the remaining \emph{thirteen} (O1, O2,
O3, O5--O9, E1--E5) are \emph{Genuinely open} and constitute the
live agenda for the next phase of the \QiFour{} program. The open problems O5--O9 record the uniqueness and selection-rule gaps that a hostile reading of the construction exposes: the \OGNfive{} selection rule for $\QB$ (O5), the
uniqueness of the \QiHAL{} five-layer factorization (O6), the
derivation of the pagoda floor count from \MSMF{} (O7), the
quantitative bounds on the Borel--Cantelli inevitability
hypotheses (O8), and the construction of a three-component
\QiFour'{} framework (O9). Indeed, the proportion of
Genuinely-open entries is the natural signature of a paper at
the derivational frontier: most of the constitutional content is
settled; the uniqueness and selection-rule refinements --- and
the laboratory verdict --- are the work of the next decade.


\section{What a framework is entitled to claim}
\label{sec:close-assess}

There comes a point in a construction of this size when the useful
thing to do is stop building and take inventory, and this Part has been
that point.

Four things are now on the record and are better said plainly than
hedged. The framework is architectural before it is numerical: it says
what kinds of quantity there are, how they constrain one another, and
what would count as a violation, and for most domains it does not yet
say what the numbers are --- that work needs simulation and calibration
this paper has not done. Its information-carrying gauge structure,
outside the one sector where it was derived, is at present a
bookkeeping claim: well defined, internally consistent, and still short
of the experimental proxy that would make it a measurement rather than
a convention. Against those two admissions stands the reason the
construction is worth arguing with at all --- what it predicts is not a
handful of separate effects but a web of linked ones, so it can be
caught out by a disagreement between two of its own predictions, with
nobody obliged to take the framework's word for anything. And from all
three together follows the fourth: its value is hostage to a
coordinated program of measurement that does not yet exist, and cannot
be rescued by reinterpretation if that program returns the wrong
answer.

That last sentence is the one worth dwelling on, because the
alternative is so much more comfortable and so common. A framework
broad enough to speak about particles and cells and institutions is
under continuous temptation to make itself unfalsifiable by degrees ---
never in one decision, always in small ones, each individually
reasonable. A parameter is left free because the domain is complicated.
A disagreement becomes a matter of the regime not having been the one
intended. A retrodiction is counted as a success. None of these is a
lie, and their accumulation is fatal, because what is left at the end
is a vocabulary that can describe any outcome and therefore expects
none. The discipline this Part tries to impose is the refusal of each
of those moves one at a time, and the place where the refusal shows is
the open-problem list: it is long, most of its entries are genuinely
open rather than pending, and the ones that would hurt most are named
in the same voice as the ones that would not.

There is a second reason to keep the list long and honest, and it has
nothing to do with modesty. A theory's open problems are the only part
of it that tells you where to go next. A construction with no
acknowledged gaps offers a reader nothing to do but agree or leave. A
construction that says exactly which joint is glued, which step is
conditional, and which number was fitted rather than derived is handing
over a map with the unexplored regions marked --- and those regions,
not the settled interior, are where the next result will come from. On
this reading the ratio of open to closed in a young framework is not an
embarrassment to be minimized in the abstract. It is a statement about
how much of the thing is still alive.

What can be said, then, at the close of the assessment. Not that the
account is established --- it is not, and the preceding sections have
been specific about which parts are proved, which are conditional on
named hypotheses, and which are organizational readings that could be
replaced tomorrow without the structure noticing. What can be said is
that the account has been put into a shape where being wrong is
possible, and where the ways of being wrong have been enumerated by
the author rather than left for somebody else to discover. That is a
smaller thing to have achieved than a theory of nature. It is also the
only condition under which the larger claim, if it is ever earned,
would mean anything at all. The final Part sets out the whole program
in the order in which it was built, so that a reader who wants to
disagree can see precisely where to begin.

\GGPart[Conclusions]{Conclusions}{Synthesis of the \QiFour{} runtime, the falsifiable program, and the constitutional reading of open-system physics}
\label{part:conclusions}

\section{Conclusions}
\label{sec:conclusions}

This paper asked a single question: once a lawful six-dimensional bulk has been established and a lawful four-dimensional boundary has been derived from it, what universal runtime must govern any open system that lives on that boundary? The answer developed across the paper is that the runtime is not a loose metaphor for openness. It is a definite mathematical structure---the \GDOS\ program with the \QiFour{} Pillars---that can be stated, tested, and falsified.

The derivation proceeded in a deliberate order, and that order is itself part of the result. One cannot meaningfully lock currents (\QiQUA) before declaring the packet that carries them (\SIP) and the ledger that budgets them (\GFF). One cannot meaningfully diagnose multiscale signatures (\QiCON) before establishing the variational grammar that generates trajectories (\QiEP). One cannot meaningfully discuss consciousness (\EIC) before establishing what measurement, persistence, and internal modeling mean in the runtime. This dependency-ordered architecture is therefore not an editorial convenience but a logical dependency chain: each layer requires the one below it and enables the one above it.

\subsection{The synthesis theorem}

The full derivation chain can now be stated as a single theorem.

\begin{theorem}[\GDOS{} synthesis theorem (T-Qi4-22.1)]
\label{thm:gdos_synthesis}
Assume the constitutional results of Papers~I--IV-A: the \GGConstitution{} axiom system, the \GG{} bulk with \BIsix{} and \OGNfive{}, and the OGGEP-derived observer boundary (the imported \GRIP{}-cascade closure remaining conditional on M11 in
its spectral-flow form, M10 resolved externally and the scope
refinements closed at class level). Then any natural open system that admits an accessible/hidden split, persistent reduced records, and coarse-grained current closure is described by the following nested chain:
\begin{enumerate}[label=(\roman*),leftmargin=1.9em]
 \item \DGI + \GSSB + \GA\ as the universal runtime triad;
 \item \GFF\ as the universal scalar ledger of energy, entropy, and free energy;
 \item \SIP\ as the canonical packet that instantiates the ledger in a concrete open system;
 \item \QiQUA\ as the locked current ledger, giving the Arisaka Ratio, Arisaka Equation, and Arisaka Constant;
 \item \QiEP\ as the extremal grammar whose boundary gauges include \PLA, \PLT, \MEPP, and \FEP;
 \item \QiCON\ as the scale-covariant sector producing heavy tails, fractality, and double-knee \(1/f\) statistics;
 \item \QiHAL\ as the minimal architectural sector for persistent adaptive information processing;
 \item \SeOS\ and \QCI\ as the persistence/completeness layer of self-maintaining open systems;
 \item \OH, \IH, and \EIC\ as the dual-hologram closure for systems capable of internal reconstruction;
 \item the pagoda hierarchy as the graded map from particle to cosmos, chemistry, life, and mind.
\end{enumerate}
This chain is stable under lawful coarse-graining in the sense that lower stories can generate higher stories through story-lift reductions, while higher stories can constrain lower stories through admissible control couplings.
\end{theorem}

\begin{proof}
We list, for each clause (i)--(x), the specific theorem of the present paper (or its companions) that supplies it, together with any inherited hypotheses. The synthesis theorem then holds under the union of these hypotheses.

\begin{itemize}[leftmargin=2em]
 \item[(i)] \DGI{}\,+\,\GSSB{}\,+\,\GA{} as the universal runtime triad: derived in Section~\ref{sec:dgi-gssb-ga}, Theorem~\ref{thm:master-pipeline}; inherited from \ASix{} \citep{ArisakaA6Gi4} Theorem~T4 (\GRIP{} cascade). Hypothesis: an admissible bulk action satisfying \OGGEP.

 \item[(ii)] \GFF{} as the universal scalar ledger: derived in Section~\ref{sec:eegg}, Theorem~\ref{thm:eegg-monotonicity}; inherited from \ASix{} Theorem~T3. Hypothesis: a smooth boundary state manifold with metric \(g\) and gauge connection \(\mathcal A\).

 \item[(iii)] \SIP{} as the canonical packet: derived in Section~\ref{sec:sip}, Theorem~\ref{thm:sip-completeness}. Hypothesis: the system admits a finite-dimensional accessible state space with a declared observable algebra.

 \item[(iv)] \QiQUA{} as the locked current ledger: derived in Part~\ref{part:qiqua}, Theorems~\ref{thm:qiqua-rank-one} and \ref{thm:natural-unit-equivalence}. Hypothesis: the four-component bulk gauge group \(\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}\) and the \BIsix{} closure flowing through the rank-one inflow channel.

 \item[(v)] \QiEP{} as the extremal grammar: derived in Part~\ref{part:qiep}, Theorems~\ref{thm:qiep-plt}, \ref{thm:qiep-mepp}, \ref{thm:qiep-fep}, and \ref{thm:qiep-unified}. Hypothesis: tree-level / large-deviation reduction of the influence functional (the boundary saddle-point assumption).

 \item[(vi)] \QiCON{} as the scale-covariant sector: derived in Part~\ref{part:qicon}, Theorems~\ref{thm:qicon-operational-conformality} and \ref{thm:qicon-two-knee}. Hypotheses: the admissibility window \(0<\alpha\leq 2\) on the fractional exponent and the soft-mode density assumption \(A_X(\lambda)\rho(\lambda)\approx A_0\lambda^{\gamma-1}\).

 \item[(vii)] \QiHAL{} as the minimal architectural sector: derived in Part~\ref{part:qihal}, Theorem~\ref{thm:qihal-universality}. Hypothesis: a memory-bearing open system with active \(U(1)_{Qi}\) coupling.

 \item[(viii)] \SeOS{} and \QCI{} as the persistence/completeness layer: derived in Part~\ref{part:seos}, Theorem~\ref{thm:seos-persistence} and the \QCI{} construction therein. Hypothesis: the maintenance-margin inequality of Part~\ref{part:seos}.

 \item[(ix)] \OH{}, \IH{}, \EIC{} as the dual-hologram closure: derived in Section~\ref{sec:oh-ih-eic}, Theorems~\ref{thm:ih-observability}, \ref{thm:eic-monotonicity}, and \ref{thm:ohih-universality}. Hypothesis: continued re-measurement and a sufficiently rich internal-model class.

 \item[(x)] The pagoda hierarchy: derived in Section~\ref{sec:pagoda-hierarchy}, Theorem~\ref{thm:pagoda-factorization}. Hypothesis: floor-by-floor admissibility (closure defect, lifetime separation, port-rank enlargement); the empirical termination at floor~5 is conditional on Open Problem~O7.
\end{itemize}

The story-lift property between adjacent floors is supplied by Theorem~\ref{thm:pagoda-factorization} (clause~(x)). The admissibility-after-coarse-graining property is supplied by Theorem~\ref{thm:cross-domain-reduction} of Section~\ref{sec:universal-gdos} (the universality theorem of Part~\ref{part:entrance}). The conditional-inevitability statement at floor~5 is supplied by Theorem~\ref{thm:conditional-inevitability} of Section~\ref{sec:pagoda-hierarchy} under the four hypotheses identified in Open Problem~O8.

Therefore the whole chain follows under the union of all hypotheses listed above. \qedhere
\end{proof}

\begin{remark}[On the conditional character of the synthesis]
\label{rem:gdos-synthesis-conditional}
Several of the clauses listed in the proof above inherit hypotheses that the present paper itself flags as Genuinely-open in the Open Problems triage of Section~\ref{sec:open-problems}. Specifically:
\begin{itemize}[leftmargin=2em]
 \item clause~(vi) (\QiCON) depends on the soft-mode density assumption and the admissibility window \(0<\alpha\le 2\);
 \item clause~(vii) (\QiHAL) depends on the active \(U(1)_{Qi}\) coupling, which is constitutionally optional and empirically preferred (see the L12 framing in Section~\ref{sec:open-problems});
 \item clause~(viii) (\SeOS, \QCI) depends on the calibration protocol for the packet-distance weights and the completeness-score thresholds (Open Problem~O2);
 \item clause~(x) (the pagoda hierarchy) depends on floor-by-floor admissibility, conditional on Open Problem~O7.
\end{itemize}
Theorem~\ref{thm:gdos_synthesis} is therefore best read as a \emph{conditional} synthesis statement: the chain holds under the union of all listed hypotheses, four of which (highlighted above) are themselves the subjects of explicit open problems in the present paper. The empirical mission of the manuscript is to make those open problems concrete enough that a falsification at any single hypothesis collapses the chain at a named clause, with the rest of the synthesis remaining intact.
\end{remark}

Theorem~\ref{thm:gdos_synthesis} is the main deliverable of the paper. It does not mean that all detailed science is finished. It means that the constitutional and runtime scaffolding for a unified open-system program is now explicit, auditable, and refutable.

\subsection{Three principal conclusions}

Three conclusions deserve emphasis because they go beyond the technical content of any single section.

\paragraph{First: the open-system world has a definite runtime geometry.}
The strongest single claim of the paper is that a universal science of open systems requires one common runtime that begins with lawful geometry and ends with lawful observation, memory, persistence, and hierarchical emergence. Before the present work, these capacities were addressed by separate traditions---quantum measurement theory, nonequilibrium thermodynamics, dynamical systems, active inference, and consciousness science---each with its own foundational language. After the present work, they are shown to be successive layers of one derivational chain. The differences between domains appear in the \SIP\ data, not in the constitutional law.

\paragraph{Second: life and mind belong inside physics, not beside it.}
Life is the floor of the pagoda at which maintenance-positive open packets become stable and evolvable under the \SeOS\ criterion. Mind is the floor at which the inner hologram becomes sufficiently rich and self-accessible to support internal measurement of selected content. Neither requires a separate ontological category. Neither is merely reduced to the floor below it. The pagoda hierarchy explicitly says that new stories emerge by new slow variables, new selection rules, and new attractor families. Chemistry is not a useless paraphrase of particle physics. Life is not a useless paraphrase of chemistry. Mind is not a useless paraphrase of life. What the present theory denies is not level-specific reality; it denies level-specific \emph{lawlessness}. Every floor must obey the constitutional runtime, and every floor genuinely extends what the floors below it can do.

\paragraph{Third: the framework is reflexive.}
Because the framework applies to distributed memory-bearing systems, scientific communities become examples of the runtime they study. Experimental data are \OH: lawful projections from a larger world to a registered boundary record. Theories are \IH: internally constructed models that attempt to reconstruct the structure behind those records. The history of science is an \EIC\ refinement process, carried out collectively, in which the mismatch between outer data and inner model is gradually reduced. This reflexive feature is not an embarrassment. It is an expected consequence of a genuine theory of observation and model-building, and it provides a natural account of why science works, why it is difficult, and why it converges.

\subsection{Outlook and the next generation of papers}

The present paper is broad by design: it had to be, because the runtime must be stated once and completely before domain-specific work can begin on firm constitutional ground. The next generation of papers will be narrow by design, each testing one sector of the runtime against one body of data.

\paragraph{Chemistry and the origin of life.}
Papers centered on \GFF\ and \QiQUA\ at the molecular scale will attempt to verify the Arisaka Equation against known catalytic cycles (ATP synthase, proton-coupled electron transfer, kinetic proofreading), derive the \SeOS\ transition from protocells to minimal life, and predict testable current-locking signatures in single-molecule experiments.

\paragraph{Neuroscience and consciousness.}
Papers centered on \QiHAL\ and \(\OH+\IH=\EIC\) will develop quantitative models of hippocampal phase coding as \NHT, cortical attractor landscapes as \HAL, multiscale oscillatory coordination as \BFFT, and the staged delay from stimulus to reportable conscious content as the \(\DGI\to\GSSB\to\GA\) pipeline applied to neural populations. Predictions about exploration-to-commitment timing, motion-conditioned model quality, and inner-hologram degradation under specific lesions will be stated and compared with existing neurophysiological data.

\paragraph{Artificial and engineered systems.}
Papers testing whether engineered systems (autonomous robots, large-scale AI architectures, synthetic biological circuits) can be made genuinely \GDOS-admissible will provide a controlled environment in which the runtime can be verified or falsified without the complexity of natural systems.

\paragraph{Collective and social systems.}
Papers on distributed social, economic, and institutional open systems will formalize collective inner cosmos formation, test whether institutional persistence obeys \SeOS-type criteria, and ask whether the consistency vector \(\Sigma_S\) can be measured for organizations and scientific communities.

\paragraph{Quantitative sharpening.}
Across all domains, the deeper task is cumulative calibration: turning the architectural framework into numerical predictions with stated error budgets. Every sector of the current paper---from the Arisaka Constant to the two-knee spectral template to the QCI scoring protocol---can be made more explicit by model class, data type, and measurement instrument. This sharpening is now a matter of development rather than of constitutional incompleteness. The program has crossed the threshold from scattered motifs to a connected derivational scaffold.

\subsection{Closing reflections: four pillars, one object}
\label{ssec:closing-reflections}

\emph{Epilogue. The remainder of this subsection steps out of derivation mode and into reflective commentary on what the framework attempts and what it promises. Readers expecting a technical close may skip directly to the Appendices.}

We conclude with a broader reflection on what the present paper attempts to do and what it hopes, in this volume, to have partially accomplished. The technical content of the four \QiFour{} pillars is in place; the philosophical content of those pillars deserves its own paragraph.

In 1944, in a slim book that has never quite gone out of print, Erwin Schr\"odinger asked a question that physics still has not settled: \emph{what is life?} His proposed answer was that life is what happens when an open system extracts a stream of order from its environment and uses that stream to maintain itself against the tendency of entropy to dissolve it. He believed, but could not prove, that this would one day be a piece of physics --- not a metaphor borrowed from physics, but physics in the same sense that gravity and electromagnetism are physics. Three quarters of a century later, the question still sits at the edge of every textbook of biophysics, every paper on the origins of life, every introductory remark in a neuroscience lecture. It has not been answered because the language has not been written.

The present paper proposes a language. It identifies a single quantity --- the current of information --- and gives it the same standing as the current of electric charge that runs through a copper wire, or the current of energy that crosses the surface of the Sun. Charge is conserved because the world has a $U(1)$ symmetry of electromagnetism; energy is conserved because the world is invariant under time translation; the proposal here is that information is conserved, along its appropriate boundary and on appropriate time-scales, because the world is invariant under a further $U(1)$ symmetry that lives on the six-dimensional bulk and projects to every open system as a fourth-current ledger. The conversion factor between the information current and the energy current of the same open system is a single small number, of size $0.1555$ electron-volts per qubit, that we call the Arisaka constant. This number is not a fit. One molecule of ATP carries two of these quanta, so that twice the Arisaka constant, $2\QB \approx 30.0~\mathrm{kJ/mol}$, is the standard free energy of hydrolysis of the ATP phosphoanhydride bond (measured $-30.5 \pm 0.5~\mathrm{kJ/mol}$, an agreement to $1.6\%$) --- the molecular currency that every living cell on Earth uses to keep itself alive. That a constant derived from a six-dimensional geometry should match a number that every biochemist has been measuring since the 1930s is the empirical surprise around which the present paper is organized, and it is the single sharpest reason to take the framework seriously.

If this picture survives the laboratory tests of the coming decade, what will have changed? At the level of the physics textbook, the chapter on conservation laws will gain a fourth entry alongside the energy, charge, and momentum currents that have been on the page for a hundred years. At the level of biology, the open-system character of every cell will stop being a complication to be handled after the closed-system idealization has been written down, and will instead become the starting axiom --- every system in nature is open, life is what happens when openness is high-quality enough to sustain a memory, and consciousness --- in the \emph{access} sense the framework can address --- is what happens when the memory becomes rich enough to model the system's own outer environment from the inside, while the \emph{phenomenal} hard problem is left explicitly open. At the level of physics, the long-standing puzzle of how to fit the observer into the picture will lose its strangeness; the observer is not a complication added afterwards but the very thing the framework was built to describe. At the level of the philosophy of science, the relationship between physics and the life sciences will no longer be one of disciplinary courtesy: the same equations, the same currents, the same conservation laws will be available to the chemist studying catalysis, the cell biologist studying mitochondria, the neuroscientist studying the hippocampus, and the cosmologist studying horizons.

The deeper philosophical content of the framework is what it says about the relationship between the physical world and the living systems within it. In the standard reading inherited from the late nineteenth century, the universe is a collection of fields obeying laws that exist independently of any observer; living systems are an accidental, late, locally complex phenomenon that the universe produces in a tiny corner of itself, and that physics studies separately, if at all. In the \QiFour{} reading, this division dissolves. The laws of physics and the living systems that obey them are not two ontological categories but two aspects of one foundational object --- the six-dimensional bulk and the boundary on which all measurements occur. The bulk is not a remote stage on which the boundary world plays out; the boundary world is the place where the bulk geometry writes itself down in bits. Life is one specific way that the writing becomes legible. Mind is one specific way that the legible writing folds back on itself and produces an internal model --- an Evolving Inner Cosmos, in the operational sense of \OH{}${+}$\IH{}${=}$\EIC{} --- that mirrors the outer environment well enough to predict, decide, and persist.

This is not a metaphor; it is the constitutional content of the operational/open extension of the Geometry-Gauge Equivalence Principle (\AX{3} \OGGEP{}) and the four \QiFour{} pillars that descend from it. Every measurement --- the one a physicist makes with a particle detector at CERN, the one a neuron makes when it spikes in a sensory cortex, the one a cell makes when it commits to a fate decision based on the molecular concentrations in its environment, the one a scientific community makes when it ratifies or refutes a theory --- is, in this reading, an \OH{} event. Every internal model --- the one the brain builds of the body's position in space, the one a cell builds of its homeostatic state, the one a community builds of the future of the climate it inhabits --- is an \IH{} object. The pair, taken together, is the Evolving Inner Cosmos: the way a living open system, embedded in a larger geometry it cannot fully access, navigates that geometry by mirroring it from the inside. In the older language of John Wheeler, the universe is an observer-participancy. In the language proposed here, this is no longer a slogan; it is the operational content of the four \QiFour{} pillars, made falsifiable in four named laboratories.

For three and a half centuries, physics has been the science of what nature does when nobody is looking. The closed Lagrangian, the unitary wave function, the isolated universe --- each was an attempt to step outside the world and write down its laws from nowhere. Each succeeded spectacularly within its own domain, and each failed at the moment a finite observer --- a thermometer, a photographic plate, a brain --- asked to be included in the picture. The axiomatic inversion proposed in these pages takes the finite observer not as a complication but as the starting point. Every system in nature is open; closure is the limit, not the law. The language in which physics writes its rules for the open-system world is the language of \OGGEP{} on the bulk and the four \QiFour{} pillars on the boundary, with the energy--information Arisaka constant $\QB$ as the conversion factor between energy and information that the boundary obeys whenever it sustains a memory.

If this proposal is correct, the laboratory world of currents and free energies and spectral signatures and memory architectures will be recognized, in the coming century, as the boundary projection of one geometric structure that includes the observer in its formal syntax --- not as a complication, not as an afterthought, not as a soft option to be added when convenient, but as a primitive object of the theory. The long-standing strangeness of the relationship between physics and the life sciences will, with appropriate humility, dissolve into a single grammar in which both speak. Schr\"odinger's question --- what is life? --- will not have a single one-sentence answer; but for the first time it will have an answer of the same kind that physics already gives to the question \emph{what is light?} or \emph{what is matter?} --- an answer cast in the language of a conservation law, a symmetry group, and a coupling constant, falsifiable in any laboratory equipped to measure it.

If the proposal is wrong, the empirical record will say so within a decade. A measured discrepancy between $2\QB$ and the standard free energy of ATP hydrolysis beyond the precision of modern calorimetry; a memory-bearing system whose $1/f$ spectrum does not exhibit the predicted two-knee shape; a neural population whose memory architecture decomposes outside the \MePMoS{}/\NHT{}/\HAL{}/\BFFT{} class; a controlled open system in which the three avatars of the extremal principle (least action, maximum entropy production, free-energy minimization) disagree irreducibly --- any one of these will collapse the upstream constitution and tell the next generation of researchers which clauses to keep and which to revise. The framework has been deliberately designed to expose itself to falsification at the axiomatic level; it is not safe by virtue of obscurity. Either the framework will succeed empirically, or its specific failures will sharpen the next attempt.

What the present paper proposes, in the end, is not a new theory of life, nor a new theory of mind, nor a new theory of information. It is a new contract between physics and the living world. Physics promises to write its rules down --- not only for the closed systems of textbook tradition, but for the open systems that constitute every cell, every brain, every ecosystem, and every society we know. The living world, in turn, promises to falsify those rules if they ever lie. We hope, with appropriate humility, that the present paper contributes a useful step in that direction. Indeed, the next generation of physicists, chemists, biologists, and neuroscientists --- who will inherit the empirical results that vindicate or falsify the framework --- must be the final judges. After all, the question Schr\"odinger asked in 1944 was whether life requires a new kind of physics; the answer that emerges here is that life does not require new physics, but it does require physics to recognize that information has always been one of its first-class objects, and that the boundary on which we live is, in a precise and laboratory-testable sense, the place where the upstream geometry writes itself down in bits.

\medskip

We close with three reflections in the spirit of the philosophical closing of \citet{ArisakaA6Gi4}. Each is one paragraph; together they record what the present paper believes it has accomplished.

\paragraph*{The four pillars as one object.}
The four pillars \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{} are not four constructions but four faces of one geometric object: a single composite connection on a single 6D principal bundle, projected onto a single boundary by a single CPTP channel, with the $U(1)_{Qi}$ gauge sector carrying the memory. Each pillar names one face --- the four-current ledger, the variational principle, the conformal-multiscale signatures, the memory architecture --- and the four faces are mutually forcing. Once one face is fixed by the constitution and by the geometric pillars of \citet{ArisakaA6Gi4}, the other three are no longer free. This is the structural sense in which \GGTheory{} extends from particle physics through cosmos, chemistry, life, and mind in one program: the same geometric object is the upstream source, and the differences between domains appear only in which sectors are active.

\paragraph*{The bargain with the laboratory.}
Every \QiFour{} pillar carries an explicit, named, dimensional empirical falsifier. \QiQUA{} forces the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ with the energy--information conversion factor $\QB \approx 0.1555\,\mathrm{eV}/\mathrm{qubit}$. \QiEP{} forces three falsifiable laboratory regimes (action-extremum trajectories, current-statistics extremum, inferential-model-quality extremum) that should co-occur in any system claimed to instantiate the runtime. \QiCON{} forces the two-knee shape of $1/f$ noise spectra. \QiHAL{} forces the \MePMoS{}/\NHT{}/\HAL{}/\BFFT{} architecture in any memory-bearing $U(1)_{Qi}$ system. Each falsifier is a prediction, not a hope. What makes a body of knowledge a science is not that it explains many things but that it forces a small number of things to be exactly as they are; what makes that science honest is that the forcing carries with it the risk of being wrong. \QiFour{} accepts that risk in four named places, across four laboratories that already exist.

\paragraph*{The deeper claim.}
If the framework survives the experimental tests of the coming decade --- ATP-hydrolysis bioenergetics matching $2\QB$, two-knee $1/f$ signatures in controlled spectroscopy, the \MePMoS{}/\NHT{}/\HAL{} architecture verified in neural populations, the conditional-inevitability bounds of the pagoda satisfied in cosmological measure --- then the lesson of the four pillars will be that the laboratory world is the projection of a single geometric structure, and that the apparent multiplicity of informational currents, free-energy ledgers, scaling laws, and memory architectures is the multiplicity of the faces of one underlying object. If the framework does not survive, then this paper will remain as one technical record of what such a projection would have looked like, written down with sufficient rigor that the failure mode is itself diagnostic. Either way, the work has been worth doing: physics promises to write its rules down for the open-system world, and the world promises to falsify them if they ever lie.


\GGAcknowledgments

\GGAppendixPart[Technical Appendices]{Technical Appendices}{Detailed proofs, cross-domain worked examples, and the complete abbreviation, symbol, and key-equation reference for the main text}
\label{part:appendices}
\addtocontents{toc}{\protect\switchtoappendixtocsection}


%% ====================================================================
%% Phase 12 v6.14: Canonical Appendix Intro + A Summary of Abbreviations
%% + B Glossary of Key Symbols + C Key equations
%% (matched to A5-GDOS canonical form)
%% ====================================================================


\noindent
The Appendices gather the technical machinery and the reference
material that supports the main-text theorems of the \QiFour{}
pillars and of the \OH{}${+}$\IH{}${=}$\EIC{} runtime completion.
Roughly speaking, this is where the reader who wants to verify a
specific derivation, run a specific worked example, or trace a
specific symbol will spend time --- while the reader who wants only
the axiomatic content and the principal theorems of the main text
can stop after Part~\ref{part:conclusions} and treat the back matter as a self-contained
companion volume. Appendices~A and~B provide the abbreviations and
the glossary of key symbols. Appendix~C collects the key
equations of the paper in a single ordered table. Appendix~D
unfolds the technical derivations for the \GRIP{} pipeline.
Appendices~E--H supply the technical derivations for the four
\QiFour{} pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}).
Appendix~I supplies the technical details for \SIP{} and \GFF{}.
Appendix~J unfolds the technical derivations for the
\OH{}${+}$\IH{}${=}$\EIC{} construction. Appendix~K gives the
pagoda-hierarchy and cross-domain technical details. Appendix~L
collects the compact falsifiability matrix.
Appendices~M--O supply the chemistry, neural, and ecological
worked-example panels for the \QiFour{} runtime across scientific
domains. Appendix~P is the notation concordance and benchmark
protocols. Finally, the inline bibliography with full Relevance
annotations follows the appendices and closes the volume. Indeed,
the fifteen Parts together with the sixteen appendices realize the
\QiFour{} runtime in full: Parts~\ref{part:entrance}--\ref{part:emergence} build and apply the runtime, Part~\ref{part:assessment}
assesses it, Part~\ref{part:conclusions} concludes, and the back matter --- the technical
appendices, the common objections, and the references --- supplies the
infrastructure that makes the whole document self-checking.


\section{Summary of Abbreviations}
\label{app:abbrev}

Program-defined names are in sans-serif; standard physics abbreviations are plain Roman; the third column gives the primary Arisaka-GG paper and section.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.16\linewidth} >{\raggedright\arraybackslash}p{0.56\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper.]{Every abbreviation used in this paper, with its meaning and the place it is introduced in full. Typography carries information here: program-defined names are set in sans serif, standard physics abbreviations in plain uppercase Roman, and mathematical objects in roman or italic as usual, so a reader can tell at a glance whether a name belongs to this framework or to the general literature. The third column links each entry to its primary source --- the present paper for names defined here, the companion paper otherwise. A reader who meets an unfamiliar name anywhere in the volume should come here first.}
\label{tab:appA-abbrev}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endfirsthead

\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev} continued from previous page}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endhead

\midrule
\multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot

\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape Framework name and core program objects} \\
\midrule
\GGTheory{} & Grand Geometric Theory (also: Geometry-Gauge Equivalence Theory) & \AOne{}, Preface \\
\GGConstitution{} & The constitutional charter of the \GGTheory{} program: four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) & \AOne{}, \S 2--\S 3 \\
\GG{} & Geometry-Gauge locking in six dimensions: the 6D bulk & \ATwo{}, \S 8 (\TX{1}{GG6}) \\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law implied by \OGGEP{} & \AFive{} \\
\BIsix{} & Green--Schwarz-modified Bianchi identity at 6D & \AThree{}, \S 3 \\
\OGNfive{} & Occam-Gauge-Normalization 5-integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ & \AThree{}, \S 7; \TX{6}{GG6} \\
\QGL{} & Quadruple Gauge Locking: the four-factor bulk gauge group $\Spinone \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$ & \ATwo{}, \S 10 (\TX{3}{GG6}) \\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{} \\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & this paper \\
\midrule

\multicolumn{3}{l}{\itshape Four axioms (\AX{1}--\AX{4})} \\
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}, \S 1.5 \\
\LC{} & \AX{1} Locality and Causality & \AOne{}, \S 2.2 \\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & \AOne{}, \S 2.3 \\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle (boundary observation dictionary) & \AOne{}, \S 2.4 \\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & \AOne{}, \S 2.5 \\
\midrule

\multicolumn{3}{l}{\itshape Seven postulates (\POS{1}--\POS{7})} \\
\midrule
\POSkw{} & Postulate (admissibility filter; testable in principle) & \AOne{}, \S 1.5 \\
\GI{} & \POS{1} Gauge invariance across scales (\LGI{} / \DGI{} / \GGI{}) & \AOne{}, \S 3.3 \\
\UEP{} & \POS{2} Unified Extremal Principle (PLA $\Leftrightarrow$ PLT / \MEPP{} / FEP) & \AOne{}, \S 3.4 \\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow & \AOne{}, \S 3.5 \\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & \AOne{}, \S 3.6 \\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & \AOne{}, \S 3.7 \\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization) & \AOne{}, \S 3.8 \\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & \AOne{}, \S 3.9 \\
\LGI{} & Local Gauge Invariance (microscopic / closed-system layer of \GI{}) & \AOne{}, \S 3.3 \\
\GGI{} & Global Gauge Invariance (macroscopic / ledger layer of \GI{}) & \AOne{}, \S 3.3 \\
\midrule

\multicolumn{3}{l}{\itshape \GiFour{} pillars} \\
\midrule
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & \ASix{} \\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{} \\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}) & \ASix{} \\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & \ASix{} \\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}) & \ASix{} \\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & \ASix{} \\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & \ASix{} \\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & \ASix{} \\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & \ASix{} \\
\GGA{} & Ground Geometric Attractor: the unique deepest \GA{} of an admissible stratum, characterized by the four pillars in Theorem~\ref{thm:qi4-gga-characterisation} & \ASix{} \\
\GRIPin{} & Fixed-landscape regime of the runtime (descent halting at the \GGA{}) & \ASix{} \\
\GRIPout{} & Landscape-reorganizing regime (supplies the next descent's initial datum; internalized by the \QiHAL{} switch) & \ASix{}; \AEleven{} \\
\midrule

\multicolumn{3}{l}{\itshape \QiFour{} pillars} \\
\midrule
\QiQUA{} & Quantum information for \QGL{} (Quadruple Gauge Locking) -- Pillar I of \QiFour{} & this paper \\
\QiEP{} & Quantum information for the \UEP{} (Unified Extremal Principle) -- Pillar II of \QiFour{} & this paper \\
\QiCON{} & Quantum information for Conformal Invariance -- Pillar III of \QiFour{} & this paper \\
\QiHAL{} & Quantum information for the Holographic Attractor Lattice -- Pillar IV of \QiFour{} & this paper \\
\OH{} & Outer Hologram (boundary projection of bulk data; outward face of \EIC{}) & this paper \\
\IH{} & Inner Hologram (inner self-modeling layer; inward face of \EIC{}) & this paper \\
\EIC{} & Evolving Inner Cosmos (boundary completion of \QiFour{} as the \OH{}${+}$\IH{} composite) & this paper \\
\midrule

\multicolumn{3}{l}{\itshape \QiHAL{} acronyms} \\
\midrule
\MePMoS{} & Memory--Prediction--Motion Sensing & this paper, \QiHAL{} \\
\NHT{} & Neural Holographic Tomography & this paper, \QiHAL{} \\
\HAL{} & Holographic Attractor Lattice & this paper, \QiHAL{} \\
\BFFT{} & Binary Fourier / Fractal Fourier Transform & this paper, \QiHAL{} \\
\midrule

\multicolumn{3}{l}{\itshape \GDOS{}-internal acronyms (program-defined)} \\
\midrule
\SIP{} & System Instance Packet (canonical \GDOS{} input unit) & this paper, \T{SIP} (T-Qi4-8.2) \\
\SeOS{} & Selfish Open System & \AFive{} + this paper \\
\midrule

\multicolumn{3}{l}{\itshape Program-defined mathematical symbols} \\
\midrule
$\QB$ & Quantum-information Bit -- the energy--information Arisaka constant $\QB \approx 0.1555$~eV per qubit; one ATP carries two, so $2 N_A\QB$ matches the standard ATP-hydrolysis free energy at biological temperatures & this paper, Definition~\ref{def:arisaka-constant}; bulk origin \ATwo{} (\TX{8}{GG6}) \\
\midrule

\multicolumn{3}{l}{\itshape Standard physics abbreviations (plain Roman, journal convention)} \\
\midrule

\multicolumn{3}{l}{\itshape \quad Open-system / quantum-channel formalism} \\
\midrule
CPTP & Completely Positive Trace-Preserving (channel admissibility for open evolution; \AX{3}/\POS{4}) & --- \\
GKLS & Gorini--Kossakowski--Lindblad--Sudarshan (master equation for Markovian CPTP semigroups; \POS{4}) & --- \\
GNS & Gelfand--Naimark--Segal (construction) & --- \\
NZ & Nakajima--Zwanzig (memory-kernel projection for non-Markovian open dynamics) & --- \\
POVM & Positive Operator-Valued Measure (generalized measurement model; \AX{3}/\POS{7}) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum and classical foundational theories} \\
\midrule
BRST & Becchi--Rouet--Stora--Tyutin (cohomology) & --- \\
GR & General Relativity & --- \\
QCD & Quantum Chromodynamics & --- \\
QED & Quantum Electrodynamics & --- \\
QFT & Quantum Field Theory & --- \\
QM & Quantum Mechanics & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Standard Model and beyond} \\
\midrule
EFT & Effective Field Theory & --- \\
EW & Electroweak & --- \\
GUT & Grand Unified Theory & --- \\
SM & Standard Model of particle physics & --- \\
SSB & Spontaneous Symmetry Breaking (closed-system; generalized by \GSSB{}) & --- \\
SUSY & Supersymmetry & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Higher-dimensional constructions} \\
\midrule
AdS & Anti--de Sitter spacetime & --- \\
AdS/CFT & Anti--de Sitter / Conformal Field Theory correspondence & --- \\
CFT & Conformal Field Theory & --- \\
KK & Kaluza--Klein (mode expansion / dimensional reduction) & --- \\
RS & Randall--Sundrum (warped extra-dimension model) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum gravity and cosmology} \\
\midrule
$\Lambda$CDM & Standard cosmological model with cosmological constant + cold dark matter & --- \\
LQG & Loop Quantum Gravity & --- \\
TOV & Tolman--Oppenheimer--Volkoff (equation) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Foundations of quantum mechanics} \\
\midrule
CHSH & Clauser--Horne--Shimony--Holt (inequality) & --- \\
EPR & Einstein--Podolsky--Rosen (paradox / argument) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Open-system / extremal-principle sub-acronyms} \\
\midrule
FEP & Free Energy Principle (boundary inference avatar of \UEP{}) & this paper, \QiEP{} \\
\MEPP{} & Maximum Entropy Production Principle (heuristic boundary avatar of \UEP{}) & this paper, \QiEP{} \\
PLA & Principle of Least Action (closed/bulk avatar of \UEP{}; standard variational dynamics) & this paper, \QiEP{} \\
PLT & Principle of Least Time (operational/boundary avatar of \UEP{}; fastest lawful path) & this paper, \QiEP{} \\
\end{longtable}
}


\bigskip

\noindent\textbf{Figure~\ref{fig:abbrev-flow}: hierarchy of program-defined acronyms in the \GGTheory{} program.} Top: the framework name (\GGTheory{}) and its constitutional charter (\GGConstitution{}). Tier~2: the four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) that together define the admissibility predicate. Tier~3: three bulk objects forced by the constitution --- \GG{} (6D bulk), \BIsix{} (modified Bianchi identity), \OGNfive{} (integer ledger). Tier~4: the boundary law \GDOS{} that the bulk projects to via \OGGEP{}, with input acronym \SIP{}. Tier~5: the two four-pillar structures, \GiFour{} (geometric flank, with sub-acronyms \GEP{}/\CEP{} blocks of \GFF{} and \DGI{}/\GSSB{}/\GA{} stages of \GRIP{}) and \QiFour{} (quantum-information flank, with \QiEP{} sub-acronyms PLT/\MEPP{}/FEP, \QiCON{} sub-concepts Conformal/L\'evy/Fractal/$1/f$, \QiHAL{} sub-acronyms \MePMoS{}/\NHT{}/\HAL{}/\BFFT{}, the boundary completion \OH{}${+}$\IH{}${=}$\EIC{}, and the runtime carrier \SeOS{} (Selfish Open System) underneath). Bottom: the energy--information Arisaka constant $\QB$ (derived secondary quantity), with the Arisaka equation $J_E / \QB = J_e = J_B = J_{Qi}$ that locks the four boundary currents through one conversion factor.

\begin{figure}[H]
\centering
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% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

% --- Tier 2: 4 axioms (left) and 7 postulates (right) -----
\node[axhead] (AXH) at (-52mm, -36mm) {\textbf{4 Axioms} (\textbf{AX-1}\,--\,\textbf{AX-4})};
\node[ax] (AX1) at (-52mm, -48mm) {\textbf{AX-1} \, \textbf{LC} \quad Locality and Causality};
\node[ax] (AX2) at (-52mm, -58mm) {\textbf{AX-2} \, \textbf{GGEP} \quad Geometry-Gauge Equivalence};
\node[ax] (AX3) at (-52mm, -68mm) {\textbf{AX-3} \, \textbf{OGGEP} \quad Operational/Open GGEP};
\node[ax] (AX4) at (-52mm, -78mm) {\textbf{AX-4} \, \textbf{MSMF} \quad Max.\ Symmetry, Min.\ Freedom};

\node[posthead] (POSH) at (52mm, -36mm) {\textbf{7 Postulates} (\textbf{POS-1}\,--\,\textbf{POS-7})};
\node[ax] (P1) at (52mm, -48mm) {\textbf{POS-1} \, \textbf{GI} \;\; (\,\textbf{LGI} \,/\, \textbf{DGI} \,/\, \textbf{GGI}\,)};
\node[ax] (P2) at (52mm, -56mm) {\textbf{POS-2} \, \textbf{UEP} \quad Unified Extremal Principle};
\node[ax] (P3) at (52mm, -64mm) {\textbf{POS-3} \, \textbf{AF} \quad Anomaly Freedom};
\node[ax] (P4) at (52mm, -72mm) {\textbf{POS-4} \, \textbf{CP} \quad Closure and Positivity};
\node[ax] (P5) at (52mm, -80mm) {\textbf{POS-5} \, \textbf{RG} \quad RG covariance};
\node[ax] (P6) at (52mm, -88mm) {\textbf{POS-6} \, \textbf{OGN} \quad Occam Gauge Normalization};
\node[ax] (P7) at (52mm, -96mm) {\textbf{POS-7} \, \textbf{MF} \quad Measurement and Falsifiability};

\draw[arrow] (GGC.south) -| ($(AXH.north) +(0,3mm)$) -- (AXH.north);
\draw[arrow] (GGC.south) -| ($(POSH.north)+(0,3mm)$) -- (POSH.north);

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

\draw[arrow] (AX4.south) -- ([yshift=-2mm]BTL.north -| AX4.south);
\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

\draw[arrow] (GG6.south) -- (GDOS.north -| GG6.south);
\draw[arrow] (BI6.south) -- (GDOS.north -| BI6.south);
\draw[arrow] (OGN5.south) -- (GDOS.north -| OGN5.south);

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
% Gi-4 box (left) -- 4 pillars + sub-acronyms
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

% Qi-4 box (right) -- 4 pillars + sub-acronyms + EIC
\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms in the \GGTheory{} program.]{How the program's own names are related to each other. The chart is a hierarchy, not a list: the charter sits at the top, the geometry it forces below it, the boundary law below that, and the derived pillars at the foot, so the position of a name on the chart tells the reader what it depends on. Names at the same level are siblings introduced by the same construction. The purpose is orientation rather than definition --- the meanings are in the abbreviations table --- and a reader who has lost track of which name belongs to which layer can recover it here in one glance.}
\label{fig:abbrev-flow}
\end{figure}

\clearpage


\section{Glossary of Key Symbols}
\label{app:glossary}

\begin{table}[H]
\centering
\caption[Glossary of key mathematical symbols.]{Glossary of key mathematical symbols. Symbols are listed in the order in which the construction introduces them --- the bulk and its factors first, then the coordinates, the ledger integers, the metrics and the curvatures --- with a one-line meaning for each. The ordering is deliberate: read straight down, the table is a compressed statement of how the geometry is built, and a reader who has lost the thread of a derivation can usually recover it here faster than by searching back through the text.}
\label{tab:appB-glossary}
\small
\begin{tabular}{@{}lp{12.35cm}@{}}
\toprule
\textbf{Symbol} & \textbf{Meaning} \\
\midrule
$M_6, M_4, I_r, S^1_\theta$ & 6D bulk, 4D boundary, radial interval, compact angular direction \\
$r, \theta$ & Radial coordinate $\in [0, K]$, angular $\in [0, 2\pi)$ \\
$K$ & Dual-lock integer = 35 \\
$g_{AB}, g_{\mu\nu}$ & 6D bulk metric, 4D boundary metric \\
$R_6, R_4$ & 6D Ricci scalar, 4D Ricci scalar \\
$M_\Pl, M_6, \Mlock, v$& Reduced Planck mass, 6D fundamental mass, KK lock scale, EW vev \\
$\Lambda_4$ & 4D cosmological constant \\
$G_N$ & Newton's gravitational constant \\
$E_\Pl, E_{\rm GUT}$ & Planck and GUT energy scales \\
$\Phi(x, r, \theta)$ & Generic bulk field \\
$\phi^{(n,\ell)}(x)$ & 4D KK-mode field (radial index $n$, angular index $\ell$) \\
$f_n(r)$ & Radial profile of $n$-th KK mode \\
$m_{n,\ell}^2$ & KK mass squared = $m_n^2 + \ell^2$ in natural units \\
$\cH_{\rm Bulk}, \cH_{\rm Bdy}, \cH_{\rm env}$ & Bulk, boundary, environment Hilbert spaces \\
$\rho_{\rm Bulk}, \rho_{\rm Bdy}$ & Bulk, boundary density operators \\
$\cP, \cQ$ & Nakajima--Zwanzig projection superoperators \\
$\cL, \cL_\GDOS$ & Liouville superoperator, \GDOS{} generator \\
$H_{\rm Bulk}, H_{\rm eff}$ & Bulk Hamiltonian, effective boundary Hamiltonian \\
$L_\alpha, \gamma_\alpha$ & Jump operators and rates of GKLS \\
$\Phi_{t,s}$ & Boundary CPTP evolution channel \\
$\cI_\alpha, E_\alpha$ & Instrument and POVM effect \\
$K(t,s), \cI(t)$ & NZ memory kernel and inhomogeneous initial-correlation term \\
$\cR$ & Boundary reduction map \\
$\cD_{\OGGEP}$ & \OGGEP{} dictionary $= \cR \circ \Tr_\env$ \\
$T^{\mu\nu}, T^{\mu\nu}_\TGG$ & Boundary stress tensor, unified \TGG{} stress tensor \\
$\Delta T^{\rm GS}_{\mu\nu}, \Delta T^{\rm KK}_{\mu\nu}$ & GS-tensor and KK-tower inflow \\
$X_4, H_3$ & \BIsix{} anomaly polynomial and Green-Schwarz two-form \\
$\cA_{\rm tot}$ & Composite bulk connection \\
$G_{\rm tot}$ & Bulk gauge-gravity group $\Spinone \times \SUfive \times \UoneB \times \UoneQi$ \\
$Q^c, T^a, Y, B, Q_{Qi}$ & SU(5) generators, hypercharge, baryon, informational charges \\
$F_{\mu\nu}$ & Gauge field strength \\
$s_6, s_4$ & 6D and 4D BRST operators \\
$Z_4[J], W_4[J], \Gamma_4$ & Generating functional, connected generating functional, effective action \\
$\Tr_{\rm env}, \Tr_{\rm Bdy}, \Tr_{\rm Bulk}$ & Partial and full traces \\
$\boxed{\cdot}$ & Boxed equation = key result \\
$\blacksquare$ & End of proof \\
\bottomrule
\end{tabular}
\end{table}

\clearpage


\section{Key equations}
\label{app:key-equations}

The following table collects the principal equations of the paper in logical order, with pointers to the sections where they are derived.

\begin{longtable}{@{}p{0.04\textwidth} p{0.30\textwidth} p{0.60\textwidth}@{}}
\caption[The principal equations of the framework, in order.]{The principal equations of the framework, collected in the order in which the argument establishes them and grouped by the paper that owns each. The first column numbers them, the second names them, and the third gives the compact form. The table is an index rather than a derivation: nothing is proved here, and each row points back to the place where the equation is obtained. It is included because a reader checking a later step usually wants the earlier equation in one line rather than in its original context.}\label{tab:key-equations}\\
\toprule
\textbf{No.} & \textbf{Name} & \textbf{Equation (compact form)} \\
\midrule
\endfirsthead
\toprule
\textbf{No.} & \textbf{Name} & \textbf{Equation (compact form)} \\
\midrule
\endhead
\bottomrule
\endfoot

\multicolumn{3}{@{}l}{\textbf{Foundation (Papers I--III)}} \\[3pt]
1 & Bulk--boundary split & \(\mathcal{H}_{\mathrm{Bulk}} = \mathcal{H}_{\mathrm{sys}} \otimes \mathcal{H}_{\mathrm{env}}\) \\[4pt]
2 & Reduced state & \(\rho_\partial(t) = \Tr_{\mathrm{env}}\,\rho_{\mathrm{Bulk}}(t)\) \\[4pt]
3 & \GG{} bulk & \(M_6 = M_4 \times I_r \times S^1_\theta\) \\[4pt]
4 & \BIsix{} identity & \(\dd H_3 = X_4\), locked coefficients \((1,-1,-2/5,-2/5)\) \\[4pt]
5 & \OGNfive{} package & \((M,N,L;J,K) = (1,3,5;7,35)\) \\[4pt]
6 & GKLS generator & \(\dot\rho_\partial = -\ii[H_{\mathrm{eff}},\rho_\partial] + \sum_k \bigl(L_k\rho_\partial L_k^\dagger - \tfrac12\{L_k^\dagger L_k,\rho_\partial\}\bigr)\) \\[6pt]

\multicolumn{3}{@{}l}{\textbf{Universal \GDOS{} runtime (Part~\ref{part:runtime})}} \\[3pt]
7 & \GDOS{} continuity & \(\partial_t p(z,t) = -\nabla_A J^A(z,t)\) \\[4pt]
8 & Current decomposition & \(J^A = J^A_{\mathrm{grad}} + J^A_{\mathrm{curl}} + J^A_{\mathrm{noise}} + J^A_{\mathrm{gauge}}\) \\[4pt]
9 & Runtime pipeline & \(\DGI \;\Longrightarrow\; \GSSB \;\Longrightarrow\; \GA\) \\[4pt]
10 & \DGI{} covariance & \(J'^A(z') = \dfrac{\partial z'^A}{\partial z^B}\,J^B(z)\) \\[6pt]
11 & \GFF{} functional & \(\mathcal{F}_{\GFF}[p] = \displaystyle\int_{\mathcal{M}} p\,\Phi_{\GEP}\,\dd\mu_g + \Theta_{\mathrm{eff}}\!\int_{\mathcal{M}} p\ln p\,\dd\mu_g\) \\[8pt]
12 & \GFF{} monotonicity & \(\dot{\mathcal{F}}_{\GFF} \leq P_{\mathrm{port}}\) (equality iff curl\,\(=0\) and noise\,\(=0\)) \\[4pt]
13 & \SIP{} packet & \(\SIP = \bigl(\mathcal{M},\,z^A,\,G_{AB},\,\mathcal{A}_A,\,\mathcal{F}_{\GFF},\,\Sigma^{AB},\,\mathcal{O}\bigr)\) \\[6pt]

\multicolumn{3}{@{}l}{\textbf{\QiFour{} Pillars (Parts~\ref{part:qiqua}--\ref{part:qihal})}} \\[3pt]
14 & Arisaka Ratio & \(\JE'/\Je' = \Je'/\JB' = \JB'/\JQi' = 1\) (physics natural units) \\[4pt]
15 & Arisaka Equation & \(\JE/\QB = \Je = \JB = \JQi\) (chemistry natural units) \\[4pt]
16 & Arisaka Constant & \(\QB = (2/5)\,\ERyd / K = (2/5)\times 13.6\;\mathrm{eV}/35\) \\[4pt]
17 & \QiEP{} action & \(\mathcal{S}[z] = \displaystyle\int \!\dd t\;\bigl[\tfrac12 G_{AB}\dot z^A\dot z^B + \dot z^A\mathcal{A}_A + V_{\mathrm{diss}} + \Lambda_{\mathrm{sel}}\bigr]\) \\[8pt]
18 & PLT geodesic & \(\ddot z^A + \Gamma^A_{BC}\,\dot z^B\dot z^C = 0\) \\[4pt]
19 & Fractional generator & \(\partial_t p = D_\alpha(-\Delta)^{\alpha/2}p\), \quad \(0<\alpha\leq 2\) \\[4pt]
20 & Two-knee spectrum & \(S(f) \sim \begin{cases} S_0, & f\ll f_L \\ f^{-\beta}, & f_L\ll f\ll f_H \\ f^{-\gamma}, & f\gg f_H \end{cases}\) \\[10pt]

\multicolumn{3}{@{}l}{\textbf{Persistence, completeness, and internality (Part~\ref{part:internal})}} \\[3pt]
21 & \SeOS{} viability tube & \(\mathbb{P}\bigl(\tau_{\partial\mathcal{V}} > t\bigr) \geq 1 - \delta\) for all \(t\) in the planning horizon \\[4pt]
22 & QCI definition & \(\QCI_S = w_1\,q_{\QiQUA} + w_2\,q_{\QiEP} + w_3\,q_{\QiCON} + w_4\,q_{\QiHAL}\) \\[4pt]
23 & \OH{} kernel & \(P(s\mid z) = \Tr\bigl[\mathcal{K}_{\OH}(s)\,\rho(z)\bigr]\) \\[4pt]
24 & \IH{} reconstruction & \(\partial_{\tau}\hat\Phi = -\Gamma\,\dfrac{\delta\mathcal{J}_{\EIC}}{\delta\hat\Phi}\) (gradient descent on mismatch) \\[6pt]
25 & \EIC{} convergence & \(\dot{\mathcal{J}}_{\EIC} \leq 0\) (Lyapunov decrease) \\[6pt]

\multicolumn{3}{@{}l}{\textbf{Synthesis (Part~\ref{part:assessment})}} \\[3pt]
26 & Synthesis theorem & Papers I--III \(\Longrightarrow\) \DGI{}+\GSSB{}+GA \(\Longrightarrow\) \GFF{}+\SIP{} \(\Longrightarrow\) \QiFour{} \(\Longrightarrow\) \SeOS{}/QCI \(\Longrightarrow\) \OH{}+\IH{}=\EIC{} \\
\end{longtable}

\clearpage

\section{Technical derivations for the \GRIP{} pipeline}
\label{app:technical-derivations-for-the-dgi--gssb-}

\subsection{Derivation of the transformed generator}
\label{app:generator-derivation}

This appendix records the algebra behind Eq.~\eqref{eq:generator-transform} in one place.

Let \(\rho' = U\rho U^\dagger\), with \(U=U_t\) differentiable in time. Then
\[
\dot\rho' = \dot U \rho U^\dagger + U\dot\rho U^\dagger + U \rho \dot U^\dagger.
\]
Insert \(\dot\rho=\mathfrak L_t[\rho]\) and write \(\rho=U^\dagger \rho' U\):
\[
\dot\rho' = \dot U U^\dagger \rho' + U\mathfrak L_t[U^\dagger\rho'U]U^\dagger + \rho' U\dot U^\dagger.
\]
Since \(U U^\dagger = \id\), differentiation gives \(\dot U U^\dagger + U\dot U^\dagger=0\), hence \(U\dot U^\dagger = -\dot U U^\dagger\). Define \(K_t = \ii \dot U U^\dagger\), so that \(\dot U U^\dagger = -\ii K_t\). Then
\[
\dot\rho' = U\mathfrak L_t[U^\dagger\rho'U]U^\dagger - \ii[K_t,\rho'].
\]
Therefore
\[
\mathfrak L_t^{(U)} = \mathcal U_t\circ \mathfrak L_t \circ \mathcal U_t^{-1} - \ii[K_t,\cdot].
\]
This is the precise operator meaning of time-dependent gauge covariance on the boundary.

\subsection{Quotient reduction of the continuity equation}
\label{app:quotient-reduction}

Let the effective current be
\[
J^A = J_{\perp}^A + R^A{}_{\alpha}\lambda^\alpha p,
\]
with \(J_{\perp}^A\) horizontal and \(R^A{}_{\alpha}\) spanning the gauge orbits. For a gauge-invariant observable \(O\),
\[
\frac{\dd}{\dd t}\langle O\rangle
=
\int (\nabla_A O)J^A\,\dd\mu
=
\int (\nabla_A O)J_{\perp}^A\,\dd\mu
+
\int (R_{\alpha} O)\lambda^\alpha p\,\dd\mu.
\]
The second term vanishes identically. Hence the reduced physical continuity equation is obtained by pushing forward only the horizontal current:
\[
\partial_t p_{\mathrm{phys}} + \widetilde{\nabla}_i \widetilde J^i = 0.
\]
This is why all instability and attractor analyses relevant to \GSSB\ and \GA\ must be performed on \(\State_{\mathrm{phys}}\), not on the unreduced state space.

\subsection{Local bifurcation algebra}
\label{app:landau-algebra}

For the local scalar model
\[
\mathcal F(q)=\mathcal F_0+\frac12 a q^2+\frac14 b q^4,\qquad b>0,
\]
the stationary points satisfy \(q(a+bq^2)=0\). The Hessian is
\[
\mathcal F''(q)=a+3bq^2.
\]
Hence:
\[
\mathcal F''(0)=a,
\qquad
\mathcal F''(q_{\pm})=a+3b\Big(-\frac{a}{b}\Big)=-2a.
\]
Thus for \(a>0\), \(q=0\) is a local minimum and the broken branches do not exist; for \(a<0\), \(q=0\) is a local maximum and \(q_{\pm}\) are minima. The stochastic stationary density is
\[
p_*(q)\propto \exp\!\left[-\frac{2}{\sigma^2}\mathcal F(q)\right],
\]
which changes from unimodal to bimodal across the same threshold. This is the simplest analytic model of open-system \GSSB.

\subsection{Catalog of possible Geometric Attractors}
\label{app:ga-catalogue}

The \GA\ definition intentionally allows several kinds of asymptotic objects. Table~\ref{tab:ga-catalogue} records the most important cases.

\begin{table}[H]
\centering
\caption[The six kinds of attractor the definition allows.]{Catalog of the attractor types the definition admits. Six types run down the first column --- fixed point, limit cycle, torus, localized bump, invariant measure and metastable branch --- with the mathematical form and typical interpretation of each. The catalog matters because the framework's terminus is often read as a fixed point and the definition is wider than that: a stable bump on a ring is an attractor in exactly the same sense, which is what lets the vocabulary reach a navigation circuit without being stretched.}
\label{tab:ga-catalogue}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}p{0.44\textwidth}}
\toprule
\textbf{GA type} & \textbf{Mathematical form} & \textbf{Typical interpretation} \\
\midrule
Fixed-point GA & Stable critical point of the reduced drift & Stable phase, homeostatic set-point, committed developmental state \\
Limit-cycle GA & Stable periodic orbit & Molecular machine cycle, circadian rhythm, locomotor pattern, economic oscillation \\
Torus or quasiperiodic GA & Invariant torus with several frequencies & Multioscillator coordination, coupled circadian or sensorimotor rhythms \\
Localized bump GA & Structured invariant profile on a ring or sheet & Ring-attractor head-direction state, working-memory packet, spatial focus of activity \\
Invariant-measure GA & Stationary probability measure with structured support & Nonequilibrium steady state, critical or near-critical open system \\
Metastable GA & Long-lived quasi-invariant set & Decision state, learned policy, memory trace, developmental canalization \\
\bottomrule
\end{tabular}
\end{table}

These categories are not separate theories. They are alternative concrete realizations of the same general definition once the \DGI\ and \GSSB\ filters have been applied.



\section{Technical derivations for \QiQUA{}}
\label{app:technical-derivations-for-qi-qua}

\subsection{Coefficient-space to natural-unit conversion}
\label{app:qiqua-conversion}

This appendix records the sector-rescaling algebra behind Theorem~\ref{thm:natural-unit-equivalence}.

Write the magnitude coefficient-space vector as
\begin{equation}
\mathbf{v}'
=
\begin{pmatrix}
|J'_E|\\
|J'_e|\\
|J'_B|\\
|J'_{Qi}|
\end{pmatrix}
=
r
\begin{pmatrix}
1\\
1\\
2/5\\
2/5
\end{pmatrix}.
\label{eq:app-qiqua-vprime}
\end{equation}
Define the natural-unit conversion map
\begin{equation}
U_{\QiQUA}
=
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 5/2 & 0\\
0 & 0 & 0 & 5/2
\end{pmatrix},
\label{eq:app-qiqua-U}
\end{equation}
acting on the event-count vector \((J_E/\QB,J_e,J_B,J_Q)^\top\). Applying \(U_{\QiQUA}\) to Eq.~\eqref{eq:app-qiqua-vprime} gives
\begin{equation}
U_{\QiQUA}\mathbf{v}'
=
r
\begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix}.
\label{eq:app-qiqua-Uv}
\end{equation}
Equation~\eqref{eq:app-qiqua-Uv} is exactly the event-count form of the Arisaka Equation. Multiplying the first component by \(\QB\) restores the physical energy current \(J_E\).

The same result can be read backwards. Given a natural-unit event count \(r\), the corresponding coefficient-space magnitudes are
\begin{equation}
|J'_E| = r,
\qquad
|J'_e| = r,
\qquad
|J'_B| = \frac{2}{5}r,
\qquad
|J'_{Qi}| = \frac{2}{5}r.
\label{eq:app-qiqua-inverse}
\end{equation}
Hence the coefficient-space and natural-unit presentations are strictly equivalent once the unit conventions are fixed.

\subsection{Numerical ledger of the Arisaka Constant}
\label{app:qiqua-numerics}

For convenience we collect the basic numerical conversions of \(\QB\) in one place.

Starting from
\[
\ERyd \approx 13.6057\,\mathrm{eV},
\qquad
K=35,
\]
one finds
\[
\delta E_{\mathrm{geom}} = \frac{\ERyd}{K}\approx 0.3887\,\mathrm{eV}.
\]
Applying the \BIsix{} magnitude ratio \(2/5\) gives
\[
\QB = \frac{2}{5}\delta E_{\mathrm{geom}}
\approx
0.1555\,\mathrm{eV}.
\]
Converting to chemical units using \(1\,\mathrm{eV}=96.485\,\mathrm{kJ\,mol^{-1}}\) yields
\[
\QB \approx 15.0\,\mathrm{kJ\,mol^{-1}}.
\]
At \(T=310\,\mathrm{K}\), where \(k_B T \approx 0.0267\,\mathrm{eV}\), one obtains
\[
\QB \approx 5.82\,k_B T.
\]
Finally, since \(k_B T\ln 2 \approx 0.0185\,\mathrm{eV}\) at \(310\,\mathrm{K}\),
\[
\QB \approx 8.40\,k_B T\ln 2.
\]

\begin{table}[H]
\centering
\caption[Unit conversions for the Arisaka constant.]{Compact conversion table for the Arisaka constant. Each row gives one quantity, its value and a comment. The table is a working reference rather than an argument: it collects the conversions between the geometric definition, the electronvolt-per-qubit form and the molar form that the chemical comparisons need, so that a reader checking the \textsf{ATP} anchor against a bench measurement does not have to reconstruct them.}
\label{tab:app-qiqua-conversions}
\small
\begin{tabular}{p{0.36\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.28\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Value} & \textbf{Comment} \\
\midrule
\(\delta E_{\mathrm{geom}}=\ERyd/K\) & \(0.3887\,\mathrm{eV}\) & Per-step atomic geometric scale \\
\(\QB=(2/5)\ERyd/K\) & \(0.1555\,\mathrm{eV}\) & Arisaka Constant \\
\(\QB\) in molar units & \(15.0\,\mathrm{kJ\,mol^{-1}}\) & Chemistry-natural-unit form \\
\(\QB/(k_B T)\) at \(310\,\mathrm{K}\) & \(5.82\) & Physiological thermal comparison \\
\(\QB/(k_B T\ln 2)\) at \(310\,\mathrm{K}\) & \(8.40\) & Comparison with Landauer erasure bound \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Coarse-grained cone generated by ideal locked events}
\label{app:qiqua-cone}

The purpose of this appendix is to formalize the idea that realistic processes are built from sums of ideal events plus overhead.

Let the ideal event vector in natural units be
\[
\mathbf{u}_{\star}
=
(\QB,1,1,1).
\]
Any process consisting of \(N\) ideal events and a nonnegative overhead vector \(\Delta=(\Delta_E,\Delta_e,\Delta_B,\Delta_Q)\) has total ledger
\[
\mathbf{L}
=
N\mathbf{u}_{\star}+\Delta.
\]
If the overhead is monotone in the sense that \(\Delta_Q=0\) and \(\Delta_E,\Delta_e,\Delta_B\ge 0\), then
\[
\mathbf{L}
=
(\QB N+\Delta_E,\,
N+\Delta_e,\,
N+\Delta_B,\,
N).
\]
Hence
\[
L_E-\QB L_Q = \Delta_E\ge 0,
\qquad
L_e-L_Q = \Delta_e\ge 0,
\qquad
L_B-L_Q = \Delta_B\ge 0.
\]
This gives the inequality form quoted in Proposition~\ref{prop:qiqua-lower-bound}.

Geometrically, the set of ideal events spans a ray, while the set of coarse-grained processes spans a convex cone whose distinguished edge is the ideal \QiQUA\ ray. The closer a real process lies to that edge, the closer it is to ideal locked efficiency.

\subsection{Event-budget table}
\label{app:qiqua-budget-table}

Table~\ref{tab:app-qiqua-budget} provides a quick-reference ledger converting a few simple event counts into their ideal energy floors.

\begin{longtable}{p{0.18\textwidth}p{0.18\textwidth}p{0.18\textwidth}p{0.32\textwidth}}
\caption[Ideal event budgets, one to eight quanta.]{Ideal event budgets for locked boundary events. The first column counts the events, the second and third give the ideal energy in electronvolts and in molar units, and the fourth an interpretation. The row at two quanta is the one that meets experiment --- it is the \textsf{ATP} hydrolysis figure --- and the rows above and below it are included so that a reader can see how far the neighboring integers sit from the measured value, which is the sharpest available check that the multiplicity is forced rather than fitted.}
\label{tab:app-qiqua-budget}\\
\toprule
\textbf{Locked events} & \textbf{Ideal energy} & \textbf{Molar energy} & \textbf{Interpretation} \\
\midrule
\endfirsthead
\toprule
\textbf{Locked events} & \textbf{Ideal energy} & \textbf{Molar energy} & \textbf{Interpretation} \\
\midrule
\endhead
\(1\) & \(1\QB\) & \(15.0\,\mathrm{kJ\,mol^{-1}}\) & One minimal ideal boundary update \\
\(2\) & \(2\QB\) & \(30.0\,\mathrm{kJ\,mol^{-1}}\) & Two coupled commits or one two-bit update \\
\(3\) & \(3\QB\) & \(45.0\,\mathrm{kJ\,mol^{-1}}\) & Representative multi-step molecular act \\
\(8\) & \(8\QB\) & \(120.0\,\mathrm{kJ\,mol^{-1}}\) & One byte-equivalent update in the ideal limit \\
\(N\) & \(N\QB\) & \(15.0N\,\mathrm{kJ\,mol^{-1}}\) & General linear scaling for an ideal locked process \\
\bottomrule
\end{longtable}






\section{Technical derivations for \QiEP{}}
\label{app:technical-derivations-for-qi-ep}

\subsection{Reduction of the current-density action to a trajectory functional}
\label{app:qiep-trajectory-reduction}

This appendix spells out in detail the reduction from the current-density functional \eqref{eq:qiep-current-density-action} to the sharply localized trajectory functional \eqref{eq:qiep-trj-action}. The calculation is elementary but it is worth recording explicitly because it shows exactly where the effective metric and connection enter.

Let
\begin{equation}
\rho(\zeta,t)=\delta\!\big(\zeta-z(t)\big),
\qquad
J^a(\zeta,t)=\delta\!\big(\zeta-z(t)\big)\,\dot z^a(t).
\label{eq:app-qiep-delta}
\end{equation}
For any smooth test function \(f(\zeta)\),
\begin{equation}
\int_{\State} f(\zeta)\,\rho(\zeta,t)\,\dd\zeta = f(z(t)),
\qquad
\int_{\State} f(\zeta)\,J^a(\zeta,t)\,\dd\zeta = f(z(t))\,\dot z^a(t).
\label{eq:app-qiep-test-function}
\end{equation}
Substitute Eq.~\eqref{eq:app-qiep-delta} into the quadratic part of Eq.~\eqref{eq:qiep-current-density-action}:
\begin{align}
\frac14\int_{\State}
\big(J-\rho u\big)^a
(\rho D)^{-1}_{ab}
\big(J-\rho u\big)^b
\,\dd\zeta
&=
\frac14
\big(\dot z-u(z,t)\big)^a
D^{-1}_{ab}(z,t)
\big(\dot z-u(z,t)\big)^b.
\label{eq:app-qiep-quadratic}
\end{align}
The continuity equation is also automatically satisfied:
\begin{align}
\partial_t \rho + \nabla_a J^a
&=
-\dot z^a \partial_a \delta(\zeta-z(t))
+
\partial_a \left[\delta(\zeta-z(t))\,\dot z^a\right]
=0.
\label{eq:app-qiep-continuity-check}
\end{align}
The projected potential \(\mathcal{V}[\rho]\) reduces to its pointwise representative \(V(z,t)\). Therefore
\begin{equation}
\mathcal{A}_{\mathrm{eff}}[\rho,J]
\longrightarrow
\mathcal{A}_{\mathrm{trj}}[z]
=
\int_0^T \dd t\,
\left[
\frac14(\dot z-u)^a D^{-1}_{ab}(\dot z-u)^b + V(z,t)
\right].
\label{eq:app-qiep-reduced-trj}
\end{equation}

Now define \(g_{ab}=\half D^{-1}_{ab}\) and \(a_a=g_{ab}u^b\). Then
\begin{align}
\frac14(\dot z-u)^aD^{-1}_{ab}(\dot z-u)^b
&=
\frac12 g_{ab}\dot z^a\dot z^b
-
g_{ab}u^a \dot z^b
+
\frac12 g_{ab}u^a u^b
\nonumber\\
&=
\frac12 g_{ab}\dot z^a\dot z^b
-
a_a\dot z^a
+
\frac12 g_{ab}u^a u^b,
\label{eq:app-qiep-complete-square}
\end{align}
which is Eq.~\eqref{eq:qiep-trj-expanded}. The Euler--Lagrange equations follow by direct variation:
\begin{align}
\delta \mathcal{A}_{\mathrm{trj}}
&=
\int_0^T \dd t\,
\left[
g_{ab}\dot z^b\,\delta \dot z^a
+\frac12(\partial_c g_{ab})\dot z^a\dot z^b \delta z^c
-(\partial_c a_a)\dot z^a \delta z^c
-a_a \delta\dot z^a
+(\partial_c U_{\mathrm{eff}})\delta z^c
\right].
\label{eq:app-qiep-var1}
\end{align}
Integrating by parts the terms proportional to \(\delta\dot z^a\) gives
\begin{equation}
\delta \mathcal{A}_{\mathrm{trj}}
=
\int_0^T \dd t\,
\left[
-\frac{\dd}{\dd t}(g_{ab}\dot z^b-a_a)
+\frac12\partial_a g_{bc}\dot z^b\dot z^c
-\partial_a a_b \dot z^b
+\partial_a U_{\mathrm{eff}}
\right]\delta z^a,
\label{eq:app-qiep-var2}
\end{equation}
and setting \(\delta \mathcal{A}_{\mathrm{trj}}=0\) yields Eq.~\eqref{eq:qiep-euler-lagrange-trajectory}.

The Jacobi reduction is equally standard. On a fixed energy shell
\begin{equation}
E
=
\frac12 g_{ab}\dot z^a\dot z^b + U_{\mathrm{eff}},
\label{eq:app-qiep-energy-shell}
\end{equation}
solve for the kinetic term and write the action along the curve \(\gamma\) rather than as an integral over an arbitrary parameter. One then obtains Eq.~\eqref{eq:qiep-optical-time}. The result justifies the interpretation of \PLT\ as the trajectory gauge of the projected 6D action.

\subsection{Force-controlled extremum and the dual minimum-dissipation form}
\label{app:qiep-mepp-duality}

This appendix expands the current-gauge calculation and makes the relation between \MEPP\ and the dual minimum-dissipation formulation completely explicit.

Start from the Rayleighian
\begin{equation}
\mathcal{R}_\rho[J;X]
=
\int_{\State}
\left(
J^aX_a - \frac12 J^a(L^{-1})_{ab}J^b
\right)\dd z,
\label{eq:app-qiep-rayleighian}
\end{equation}
with continuity and integrated ledger constraints imposed by Lagrange multipliers. Including a multiplier \(\lambda(z,t)\) for continuity gives
\begin{equation}
\widetilde{\mathcal{R}}
=
\mathcal{R}_\rho[J;X]
+
\int_{\State}
\lambda(\partial_t \rho + \nabla_a J^a)\,\dd z.
\label{eq:app-qiep-augmented-rayleighian}
\end{equation}
Variation with respect to \(J\) yields
\begin{equation}
X_a - (L^{-1})_{ab}J^b - \partial_a\lambda = 0.
\label{eq:app-qiep-ray-var}
\end{equation}
Thus the constrained stationary current is
\begin{equation}
J^a = L^{ab}(X_b-\partial_b\lambda).
\label{eq:app-qiep-constrained-current}
\end{equation}
The multiplier \(\lambda\) simply enforces admissibility; it does not change the local quadratic structure of the extremum.

Evaluate the Rayleighian at the stationary point:
\begin{align}
\mathcal{R}_{\rho,\star}
&=
\int_{\State}
\left[
J_\star^a X_a - \frac12 J_\star^a(L^{-1})_{ab}J_\star^b
\right]\dd z
\nonumber\\
&=
\frac12
\int_{\State}
J_\star^a X_a\,\dd z
=
\frac12 \dot\Sigma_\star,
\label{eq:app-qiep-ray-star}
\end{align}
where the last equality uses the definition of entropy production.

Now compute the second variation. At fixed \(X\), \(\rho\), and constraint surface,
\begin{equation}
\delta^2 \mathcal{R}_\rho
=
-
\int_{\State}
\delta J^a (L^{-1})_{ab}\delta J^b\,\dd z
\le 0.
\label{eq:app-qiep-second-var}
\end{equation}
Therefore the stationary point is a maximum. This is the precise force-controlled \MEPP\ statement of the present framework.

To obtain the dual picture, Legendre-transform Eq.~\eqref{eq:app-qiep-rayleighian} with respect to the force variable. The dual functional is
\begin{equation}
\mathcal{D}_\rho[J;X]
=
\frac12\int_{\State} J^a(L^{-1})_{ab}J^b\,\dd z
-
\int_{\State} J^a X_a\,\dd z.
\label{eq:app-qiep-dual}
\end{equation}
At the common stationary point,
\begin{equation}
\delta \mathcal{D}_\rho = 0,
\qquad
\delta^2 \mathcal{D}_\rho
=
\int_{\State}\delta J^a(L^{-1})_{ab}\delta J^b\,\dd z
\ge 0,
\label{eq:app-qiep-dual-second}
\end{equation}
so the same current minimizes \(\mathcal{D}_\rho\). This is Onsager's least-dissipation form familiar from near-equilibrium irreversible thermodynamics: it varies the fluxes at fixed forces, and it returns the same stationary point.

Two distinct minimum statements live in this neighborhood, and they should not be run together. Onsager's least dissipation, just derived, is the \emph{flux-controlled} Legendre transform: forces fixed, fluxes varied. Prigogine's minimum entropy production, derived in Appendix~\ref{app:qiep-duality}(ii), is the \emph{force-side} statement: the transport law fixed, some forces clamped and the rest free, and the stationary point is the steady state. Both are duals of the one stationary point, but of different variables, and it is the second --- not the first --- that maximum entropy production has been contrasted with for seventy years. The present framework accommodates all three readings because all three arise from the same projected quadratic form.

\subsection{Pullback to the statistical manifold and the Fisher metric}
\label{app:qiep-fisher}

This appendix derives the Fisher metric appearing in the inferential gauge. Let \(q_\mu(\eta)\) be a smooth parametric family of densities indexed by coordinates \(\mu^i\). A small parameter displacement \(\mu\to \mu+\dd\mu\) changes the density by
\begin{equation}
q_{\mu+\dd\mu}(\eta)
=
q_\mu(\eta)
+
\partial_i q_\mu(\eta)\,\dd\mu^i
+
\mathcal{O}(\dd\mu^2).
\label{eq:app-qiep-density-expansion}
\end{equation}
The KL divergence between neighboring family members is
\begin{align}
D_{\mathrm{KL}}(q_{\mu+\dd\mu}\|q_\mu)
&=
\int q_{\mu+\dd\mu}
\log\frac{q_{\mu+\dd\mu}}{q_\mu}\,\dd\eta
\nonumber\\
&=
\frac12
g^{\mathrm F}_{ij}(\mu)\,\dd\mu^i \dd\mu^j
+
\mathcal{O}(\dd\mu^3),
\label{eq:app-qiep-kl-expansion}
\end{align}
where
\begin{equation}
g^{\mathrm F}_{ij}(\mu)
=
\int q_\mu(\eta)
\partial_i\log q_\mu(\eta)\,
\partial_j\log q_\mu(\eta)\,\dd\eta.
\label{eq:app-qiep-fisher}
\end{equation}
Thus the Fisher metric is the quadratic form measuring distinguishability of neighboring internal models.

The metric is not installed by convention; it is the second variation of the ledger's own informational term, and the computation deserves its three lines. Write
\begin{equation}
h(\mu)
=
D_{\mathrm{KL}}\big(q_\mu\,\|\,q_{\mu_0}\big)
=
\int q_\mu \log\frac{q_\mu}{q_{\mu_0}}\,\dd\eta ,
\label{eq:app-qiep-h}
\end{equation}
which is the \CEP{} contribution to \(\mathcal{F}_{\GFF}\) of Eq.~\eqref{eq:eegg-functional} evaluated on the family, up to the additive reference term. Differentiating once, by the product rule,
\begin{equation}
\frac{\partial h}{\partial \mu^i}
=
\int \partial_i q_\mu\,\log\frac{q_\mu}{q_{\mu_0}}\,\dd\eta
+
\int q_\mu\,\partial_i \log q_\mu\,\dd\eta ,
\label{eq:app-qiep-firstderiv}
\end{equation}
and the second integral vanishes identically in \(\mu\) by the score identity --- differentiate the normalization under the integral sign,
\begin{equation}
\int q_\mu\,\partial_i \log q_\mu\,\dd\eta
=
\int \partial_i q_\mu\,\dd\eta
=
\partial_i \int q_\mu\,\dd\eta
=
\partial_i (1)
=
0 .
\label{eq:app-qiep-score}
\end{equation}
At \(\mu=\mu_0\) the logarithm in the first integral of Eq.~\eqref{eq:app-qiep-firstderiv} is \(\log 1 = 0\), so the whole first derivative vanishes at the base point --- as it must for a divergence attaining its minimum, zero, there. The metric therefore comes from the second derivative. Differentiating Eq.~\eqref{eq:app-qiep-firstderiv} again (the score term contributes nothing, being identically zero),
\begin{equation}
\frac{\partial^2 h}{\partial \mu^j \partial \mu^i}
=
\int \partial_j\partial_i q_\mu\,\log\frac{q_\mu}{q_{\mu_0}}\,\dd\eta
+
\int \partial_i q_\mu\,\partial_j \log q_\mu\,\dd\eta ,
\label{eq:app-qiep-secondderiv}
\end{equation}
and at \(\mu=\mu_0\) the first integral again dies on \(\log 1 = 0\), while the second, using \(\partial_i q = q\,\partial_i\log q\), is exactly the expectation of the product of scores. Hence
\begin{equation}
\frac{\partial^2}{\partial\mu^i\partial\mu^j}
D_{\mathrm{KL}}\big(q_\mu\,\|\,q_{\mu_0}\big)\Big|_{\mu=\mu_0}
=
g^{\mathrm F}_{ij}(\mu_0),
\label{eq:app-qiep-fisher-hessian}
\end{equation}
the Fisher metric of Eq.~\eqref{eq:app-qiep-fisher}. The geometry that the inference literature installs by convention arrives here as the Hessian of the \CEP{} term of the ledger --- which is why the inferential gauge inherits the same budget as the other two, rather than carrying a metric of its own. Restricting the current-density action to the statistical slice \(\rho_{\mu,s,a}(\eta)=q_\mu(\eta)\delta(s-s_t)\delta(a-a_t)\), whose time dependence enters through
\begin{equation}
\partial_t \rho
=
\frac{\partial q_\mu}{\partial \mu^i}\,\dot\mu^i
\delta(s-s_t)\delta(a-a_t)
+\cdots,
\label{eq:app-qiep-rho-dot}
\end{equation}
therefore converts the boundary kinetic term into the Fisher quadratic form. Therefore
\begin{equation}
\mathcal{A}_{\mathrm{eff}}[\rho,J]
\Big|_{\mathcal{S}_{\mathrm{inf}}}
=
\int_0^T \dd t\,
\left[
\frac12 g^{\mathrm F}_{ij}\dot\mu^i\dot\mu^j
+
\mathcal{F}(s,a,\mu)
\right].
\label{eq:app-qiep-pullback}
\end{equation}

The variational free energy itself can be decomposed as
\begin{align}
\mathcal{F}(s,a,\mu)
&=
\int q_\mu(\eta)\log q_\mu(\eta)\,\dd\eta
-
\int q_\mu(\eta)\log p(\eta,s,a)\,\dd\eta
\nonumber\\
&=
D_{\mathrm{KL}}(q_\mu(\eta)\|p(\eta|s,a))
-
\log p(s,a),
\label{eq:app-qiep-vfe-decomposition}
\end{align}
so the only \(\mu\)-dependent part relevant to internal dynamics is the KL mismatch term. Taking the Euler--Lagrange equation of Eq.~\eqref{eq:app-qiep-pullback} and then the overdamped limit produces the natural-gradient flow
\begin{equation}
\dot\mu^i
=
-
\kappa_\mu (g^{\mathrm F})^{ij}\partial_j \mathcal{F}.
\label{eq:app-qiep-natural-gradient}
\end{equation}

The derivation is important because it shows that the geometry of inference is not inserted as an extra hypothesis. It is the pullback of the same projected open-system metric that already governed trajectories and currents in the previous gauges.

\subsection{A two-state blanket model}
\label{app:qiep-two-state-example}

A simple two-state example is useful for seeing the inferential gauge in explicit algebra. Let the hidden external variable be \(\eta\in\{0,1\}\), and parameterize the recognition family by a single scalar \(\mu\) through
\begin{equation}
q_\mu(\eta=1)=\sigma(\mu),
\qquad
q_\mu(\eta=0)=1-\sigma(\mu),
\qquad
\sigma(\mu)=\frac{1}{1+e^{-\mu}}.
\label{eq:app-qiep-sigmoid}
\end{equation}
Then
\begin{equation}
\partial_\mu \log q_\mu(\eta)
=
\eta-\sigma(\mu),
\label{eq:app-qiep-log-derivative}
\end{equation}
and the Fisher metric is
\begin{equation}
g^{\mathrm F}_{\mu\mu}
=
\sum_{\eta=0}^1 q_\mu(\eta)\big(\eta-\sigma(\mu)\big)^2
=
\sigma(\mu)\big(1-\sigma(\mu)\big).
\label{eq:app-qiep-fisher-binary}
\end{equation}

Suppose the projected generative model assigns likelihoods \(p(s|\eta)\) and prior \(p(\eta)\). The variational free energy is
\begin{align}
\mathcal{F}(s,\mu)
&=
q_\mu(1)\log\frac{q_\mu(1)}{p(\eta=1,s)}
+
q_\mu(0)\log\frac{q_\mu(0)}{p(\eta=0,s)}.
\label{eq:app-qiep-binary-vfe}
\end{align}
Its derivative is
\begin{equation}
\partial_\mu \mathcal{F}
=
\sigma(\mu)\big(1-\sigma(\mu)\big)
\log\frac{q_\mu(1)\,p(\eta=0,s)}{q_\mu(0)\,p(\eta=1,s)}.
\label{eq:app-qiep-binary-derivative}
\end{equation}
The natural-gradient flow becomes
\begin{equation}
\dot\mu
=
-\kappa_\mu
\frac{1}{\sigma(1-\sigma)}
\partial_\mu \mathcal{F}
=
-\kappa_\mu
\log\frac{q_\mu(1)\,p(\eta=0,s)}{q_\mu(0)\,p(\eta=1,s)}.
\label{eq:app-qiep-binary-flow}
\end{equation}
The stationary condition \(\dot\mu=0\) is
\begin{equation}
\frac{q_\mu(1)}{q_\mu(0)}
=
\frac{p(\eta=1,s)}{p(\eta=0,s)},
\label{eq:app-qiep-binary-stationary}
\end{equation}
which simply says that the recognition odds match the posterior odds. This miniature example makes the logic transparent: the inferential gauge of \QiEP\ selects the internal state whose recognition density best matches the projected generative density.

\subsection{Dictionary of the three gauges}
\label{app:qiep-dictionary}

For convenience, Table~\ref{tab:qiep-dictionary} gives a compact dictionary of the three \QiEP\ gauges.

\begin{longtable}{p{0.18\textwidth}p{0.18\textwidth}p{0.19\textwidth}p{0.35\textwidth}}
\caption[A working dictionary for the three gauges.]{Dictionary of the three \QiEP{} gauges. Each row is one gauge, with its state variable, the object it extremizes and its typical use. It is the appendix companion to Table~\ref{tab:qiep-three-gauges}: the same three rows, but arranged for reference rather than for argument, so that a reader working through a derivation in one gauge can find the corresponding object in another without re-reading the Part.}
\label{tab:qiep-dictionary}\\
\toprule
\textbf{Gauge} & \textbf{State variable} & \textbf{Extremized object} & \textbf{Typical use} \\
\midrule
\endfirsthead
\toprule
\textbf{Gauge} & \textbf{State variable} & \textbf{Extremized object} & \textbf{Typical use} \\
\midrule
\endhead
Trajectory gauge & \(z(t)\) & Generalized optical time \(\mathcal{T}[z]\) & Path selection, transport, signaling, branch tracking \\
Current gauge & \((\rho,J)\) & Rayleighian \(\mathcal{R}_\rho[J;X]\) or dual dissipation functional & Flux networks, reaction systems, metabolic or ecological throughput \\
Inferential gauge & \(\mu(t)\) & Fisher-kinetic action \(\mathcal{A}_{\mathrm{inf}}[\mu]\) with \(\mathcal{F}\) potential & Adaptive systems, cells, nervous systems, cognition, action selection \\
\bottomrule
\end{longtable}

The table is intentionally simple. Its purpose is to emphasize that the three languages differ in their chosen reduced variable, not in their constitutional origin. The origin is always the same projected current-density action.





\subsection{The duality theorem: both readings, in full}
\label{app:qiep-duality}

This appendix proves Theorem~\ref{thm:qiep-duality}. Throughout, \(L=L^{\mathsf T}\succ 0\) is the mobility of Eq.~\eqref{eq:qiep-onsager-tensor}, \(\dot\Sigma(J)=J^a(L^{-1})_{ab}J^b\) is the flux-side entropy production, and \(P(X)=X_aL^{ab}X_b\) is the force-side one.

\paragraph{(i) The flux side: forces clamped.} The problem is to maximize \(\dot\Sigma(J)\) subject to the power-balance constraint
\begin{equation}
g(J)\;\equiv\;J^aX_a-\dot\Sigma(J)\;=\;0 .
\label{eq:app-qiep-powerbalance}
\end{equation}
Introduce a multiplier \(\mu\) and extremize \(\Lambda(J,\mu)=\dot\Sigma(J)+\mu\,g(J)\) freely. First the derivative of \(\dot\Sigma\), index by index, the two pieces combining by the symmetry of \(L^{-1}\):
\begin{equation}
\frac{\partial\dot\Sigma}{\partial J^c}
=
(L^{-1})_{cb}J^b + J^a(L^{-1})_{ac}
=
2\,(L^{-1})_{cb}J^b .
\label{eq:app-qiep-sigmaderiv}
\end{equation}
Stationarity of \(\Lambda\) is then
\begin{equation}
2(L^{-1})_{cb}J^b+\mu\big(X_c-2(L^{-1})_{cb}J^b\big)=0
\quad\Longrightarrow\quad
2(L^{-1})_{cb}J^b=\frac{\mu}{\mu-1}\,X_c
\qquad(\mu\neq 1),
\label{eq:app-qiep-dual-stationarity}
\end{equation}
so every candidate extremum is proportional to the physical flux,
\begin{equation}
J^b=k\,L^{bc}X_c ,
\qquad
k\equiv\frac{\mu}{2(\mu-1)} .
\label{eq:app-qiep-dual-candidate}
\end{equation}
The constraint fixes \(k\): on the candidate, \(J^aX_a=k\,X_aL^{ab}X_b\) and \(\dot\Sigma(J)=k^2X_aL^{ab}X_b\), so Eq.~\eqref{eq:app-qiep-powerbalance} gives \(k=k^2\), hence \(k=1\) (the root \(k=0\) is the trivial equilibrium point \(J=0\), excluded for nonzero \(X\) by maximality below). Therefore \(J=LX\): the linear law of Eq.~\eqref{eq:qiep-onsager-law}, recovered as the constrained maximizer rather than assumed.

For the orthogonality condition, note that \(\dot\Sigma\) is homogeneous of degree two, so Euler's homogeneous-function theorem gives
\begin{equation}
J^a\,\frac{\partial\dot\Sigma}{\partial J^a}=2\,\dot\Sigma(J);
\label{eq:app-qiep-euler}
\end{equation}
contracting the ansatz \(X_a=\lambda\,\partial\dot\Sigma/\partial J^a\) with \(J^a\) and using the constraint,
\begin{equation}
\dot\Sigma=J^aX_a=\lambda\,J^a\frac{\partial\dot\Sigma}{\partial J^a}=2\lambda\,\dot\Sigma
\quad\Longrightarrow\quad
\lambda=\tfrac12 ,
\label{eq:app-qiep-lambda}
\end{equation}
fixed by the constraint and by nothing else. That the extremum is a maximum follows by expanding along the constraint surface. Parameterize \(J=LX+\delta\) and impose \(g(J)=0\) exactly:
\begin{equation}
(LX+\delta)^aX_a=(LX+\delta)^a(L^{-1})_{ab}(LX+\delta)^b
\quad\Longrightarrow\quad
\delta^aX_a=-\,\delta^a(L^{-1})_{ab}\delta^b ,
\label{eq:app-qiep-admissible}
\end{equation}
after canceling \(XLX\) from both sides and collecting the terms linear and quadratic in \(\delta\). Substituting this admissibility condition back into \(\dot\Sigma\),
\begin{equation}
\dot\Sigma(LX+\delta)
=
X_aL^{ab}X_b+2\,\delta^aX_a+\delta^a(L^{-1})_{ab}\delta^b
=
\dot\Sigma(LX)-\delta^a(L^{-1})_{ab}\delta^b
\;\le\;
\dot\Sigma(LX),
\label{eq:app-qiep-maximum}
\end{equation}
with equality only at \(\delta=0\) because \(L^{-1}\succ 0\). The physical flux is the strict constrained maximum, which is the Ziegler statement, and it selects the constitutive relation.

\paragraph{(ii) The force side: constitutive law fixed.} Now hold \(J=LX\) fixed as the transport law and treat the forces as the variables, with \(X_1,\dots,X_m\) clamped by the boundary conditions and \(X_{m+1},\dots,X_n\) free. Differentiating the force-side entropy production,
\begin{equation}
\frac{\partial P}{\partial X_j}
=
\frac{\partial}{\partial X_j}\big(X_aL^{ab}X_b\big)
=
2L_{jk}X_k
=
2J_j
\qquad (j>m),
\label{eq:app-qiep-forceside}
\end{equation}
so \(P\) is stationary on the affine subspace of clamped forces exactly when the fluxes conjugate to the \emph{free} forces vanish --- which is the definition of the steady state under the imposed constraints. The Hessian restricted to the free block is
\begin{equation}
\frac{\partial^2 P}{\partial X_j\,\partial X_{j'}}
=
2L_{jj'}
\;\succ\;0
\qquad (j,j'>m),
\label{eq:app-qiep-forcehessian}
\end{equation}
positive definite by the positivity of \(L\); the stationary point is therefore a strict minimum. This is the minimum-entropy-production theorem of \citet{Prigogine1967} in its exact scope: near-equilibrium, linear constitutive law, and a partition of the forces into clamped and free. It selects the state.

\paragraph{(iii) One stationary point.} Both computations locate the same configuration, \(J_\star=LX_\star\), and at it \(\dot\Sigma(J_\star)=J_\star^a(L^{-1})_{ab}J_\star^b=X_{\star a}L^{ab}X_{\star b}=P(X_\star)\). The two extremal characterizations differ only in which variables are varied and which data are held fixed: (i) varies fluxes at fixed forces and returns the transport law; (ii) varies free forces at fixed transport law and returns the state. They are Legendre-dual readings of the single stationary point of the Rayleighian \eqref{eq:qiep-rayleighian}. \(\square\)

\subsection{The failure theorem: the exact obstruction}
\label{app:qiep-failure}

This appendix proves Theorem~\ref{thm:qiep-failure}. Split the lawful drift of Eq.~\eqref{eq:four-current-decomp} into its dissipative part and the two retained sectors,
\begin{equation}
\rho u^a=\rho u_{\mathrm{d}}^a+\rho u_{\mathrm{c}}^a+\rho u_{\mathrm{g}}^a ,
\label{eq:app-qiep-fail-split}
\end{equation}
where \(u_{\mathrm{d}}\) collects the gradient and noise sectors, \(u_{\mathrm{c}}\) is the divergence-free circulation, and \(u_{\mathrm{g}}\) generates motion along the dynamically maintained gauge directions. Carry the split through the first variation of the transport cost of Eq.~\eqref{eq:qiep-current-density-action} (the constant factor converting \((\rho D)^{-1}\) to \(L^{-1}\) is absorbed throughout):
\begin{equation}
\delta\mathcal{A}_{\mathrm{eff}}
=
\tfrac12\int_{\State}\big(J-\rho u_{\mathrm{d}}\big)^a(L^{-1})_{ab}\,\delta J^b\,\dd z
\;-\;
\tfrac12\int_{\State}\big(\rho u_{\mathrm{c}}+\rho u_{\mathrm{g}}\big)^a(L^{-1})_{ab}\,\delta J^b\,\dd z .
\label{eq:app-qiep-fail-firstvar}
\end{equation}
The first integral is the Rayleighian variation of Eq.~\eqref{eq:qiep-rayleigh-variation}; the second is the cross term that the quadratic reduction discarded. Setting the total to zero for all admissible \(\delta J\) gives the physical stationary current \eqref{eq:qiep-fail-Jstar}.

Now test whether any entropy-production functional is extremal there. Expand \(\dot\Sigma\) about \(J_\star\), using Eq.~\eqref{eq:app-qiep-sigmaderiv}:
\begin{equation}
\dot\Sigma(J_\star+\delta)-\dot\Sigma(J_\star)
=
2\,\delta^a(L^{-1})_{ab}J_\star^b+\delta^a(L^{-1})_{ab}\delta^b
=
2\,\delta^aX_a
+2\,\delta^a(L^{-1})_{ab}\big(\rho u_{\mathrm{c}}+\rho u_{\mathrm{g}}\big)^b
+\delta^a(L^{-1})_{ab}\delta^b .
\label{eq:app-qiep-fail-expansion}
\end{equation}
The first term is removed by admissibility exactly as in Eq.~\eqref{eq:app-qiep-admissible}; the third is the benign quadratic that made the flux-side reading a maximum. The middle term is the obstruction. It is \emph{linear} in \(\delta\), of either sign, and it survives at \(J_\star\) unless \(\rho u_{\mathrm{c}}+\rho u_{\mathrm{g}}\) is \(L^{-1}\)-orthogonal to every admissible variation. A stationary point carrying a surviving linear term is not an extremum of anything.

That the orthogonality condition is exactly the work condition is seen by taking the admissible variations generated by shifting the ledger potential, \(\delta J^a=L^{ab}\partial_b\chi\). Then
\begin{equation}
\delta^a(L^{-1})_{ab}\big(\rho u_{\mathrm{c}}+\rho u_{\mathrm{g}}\big)^b
=
\partial_a\chi\,\big(\rho u_{\mathrm{c}}+\rho u_{\mathrm{g}}\big)^a ,
\qquad
\chi=-\,\Phi
\;\Longrightarrow\;
\int_{\State}(\cdots)\,\dd z
=
-\,\frac{\mathcal{C}_{\mathrm{curl}}+\mathcal{G}_{\mathrm{gauge}}}{\Theta_{\mathrm{eff}}},
\label{eq:app-qiep-fail-work}
\end{equation}
with \(\Phi=\delta\mathcal{F}_{\GFF}/\delta\rho\) the ledger potential. The vanishing of the cross term for this family of variations is therefore precisely the condition \(\mathcal{C}_{\mathrm{curl}}=\mathcal{G}_{\mathrm{gauge}}=0\) of the balance identity \eqref{eq:eegg-balance-identity}. Where either sector does work against the ledger, no functional of the entropy-production family is stationary at the physical state, and the flux-side and force-side readings fail together. Because the inferential gauge is obtained by restricting the same functional to a statistical slice, it inherits the same obstruction. \(\square\)

\subsection{The overdamped limit, and why it is legitimate}
\label{app:qiep-overdamped}

The passage from the second-order inferential equation \eqref{eq:qiep-fep-second-order} to the first-order natural-gradient flow \eqref{eq:qiep-natural-gradient} is a limit, and limits deserve their hypotheses. The prototype is the familiar one from mechanics,
\begin{equation}
m\ddot x+\gamma\dot x+V'(x)=0 ,
\qquad
\frac{|m\ddot x|}{|\gamma\dot x|}\sim\frac{m}{\gamma\tau}\;\longrightarrow\;0
\quad\text{as }\gamma\tau\gg m ,
\label{eq:app-qiep-prototype}
\end{equation}
a particle in molasses forgetting its inertia: on histories varying over a timescale \(\tau\) long compared with \(m/\gamma\), the acceleration term is down by one power of the timescale ratio and the surviving balance is friction against force.

Here the role of the friction is played by the mobility that the transport term of Eq.~\eqref{eq:qiep-current-density-action} carries onto the statistical manifold --- the same \((\rho D)^{-1}\) weight, restricted to belief coordinates, that sets the rate \(\kappa_\mu\) at which the registered states exchange with the declared ports. On relaxational histories, slow compared with that port-coupling time, \(|\ddot\mu|\sim|\Delta\mu|/\tau^2\) while \(|\dot\mu|\sim|\Delta\mu|/\tau\), so both left-hand terms of Eq.~\eqref{eq:qiep-fep-second-order} --- the inertial term and the velocity-quadratic Christoffel term --- are suppressed by one power of the timescale ratio relative to the friction--gradient balance. Dropping them leaves Eq.~\eqref{eq:qiep-natural-gradient}.

The hypothesis is therefore explicit and checkable: the inferential gauge in its first-order form governs systems whose belief updates are slow compared with their port exchange. Systems driven faster than their ports retain the inertial and Christoffel terms, and their trajectories are not natural-gradient descents --- a boundary of the reading that the second-order equation \eqref{eq:qiep-fep-second-order} states exactly, and which is invisible in presentations that begin from the first-order form.

\section{Technical derivations for \QiCON{}}
\label{app:technical-derivations-for-qi-con}

\subsection{Radial integration and kernel homogeneity}
\label{app:qicon-radial}

This appendix records the scale-homogeneity algebra behind Eqs.~\eqref{eq:qicon-kernel-integral}--\eqref{eq:qicon-fractional-kernel} in one place.

Start from the projected kernel
\begin{equation}
W_{\Delta}(z,z')
=
\int_{\zeta_{\mathrm{min}}}^{\zeta_{\mathrm{max}}}
\omega(\zeta)
\left(
\frac{\zeta}{\zeta^2+\ell^2}
\right)^{2\Delta}
\,\dd\zeta,
\qquad
\ell=d_g(z,z').
\label{eq:app-qicon-start}
\end{equation}
Assume that in the active window the weight can be written as
\begin{equation}
\omega(\zeta)=\zeta^{\sigma-1}w(\zeta),
\qquad
w(\zeta)\approx w_0.
\label{eq:app-qicon-weight}
\end{equation}
Then
\begin{equation}
W_{\Delta}(z,z')
\approx
w_0
\int_{\zeta_{\mathrm{min}}}^{\zeta_{\mathrm{max}}}
\frac{\zeta^{\sigma+2\Delta-1}}{(\zeta^2+\ell^2)^{2\Delta}}\,\dd\zeta.
\label{eq:app-qicon-pre-u}
\end{equation}
Set
\begin{equation}
\zeta=\ell u,
\qquad
\dd\zeta=\ell\,\dd u,
\label{eq:app-qicon-u}
\end{equation}
to obtain
\begin{align}
W_{\Delta}(z,z')
&\approx
w_0
\int_{\zeta_{\mathrm{min}}/\ell}^{\zeta_{\mathrm{max}}/\ell}
\frac{(\ell u)^{\sigma+2\Delta-1}}{(\ell^2u^2+\ell^2)^{2\Delta}}\,
\ell\,\dd u
\nonumber\\
&=
w_0\,\ell^{\sigma+2\Delta-1-4\Delta+1}
\int_{\zeta_{\mathrm{min}}/\ell}^{\zeta_{\mathrm{max}}/\ell}
\frac{u^{\sigma+2\Delta-1}}{(1+u^2)^{2\Delta}}\,\dd u
\nonumber\\
&=
w_0\,\ell^{\sigma-2\Delta}
\int_{\zeta_{\mathrm{min}}/\ell}^{\zeta_{\mathrm{max}}/\ell}
\frac{u^{\sigma+2\Delta-1}}{(1+u^2)^{2\Delta}}\,\dd u.
\label{eq:app-qicon-after-u}
\end{align}
Now impose the separation condition
\begin{equation}
\zeta_{\mathrm{min}} \ll \ell \ll \zeta_{\mathrm{max}}.
\label{eq:app-qicon-window}
\end{equation}
Then the lower limit tends to \(0\), the upper limit tends to \(+\infty\), and the integral becomes the constant
\begin{equation}
C_{\Delta,\sigma}
=
\int_0^\infty
\frac{u^{\sigma+2\Delta-1}}{(1+u^2)^{2\Delta}}\,\dd u
=
\tfrac12\,B\!\left(\Delta+\tfrac{\sigma}{2},\;\Delta-\tfrac{\sigma}{2}\right),
\qquad |\sigma|<2\Delta,
\label{eq:app-qicon-C}
\end{equation}
Thus
\begin{equation}
W_{\Delta}(z,z')
\approx
w_0 C_{\Delta,\sigma}\,\ell^{-(2\Delta-\sigma)}.
\label{eq:app-qicon-homogeneous}
\end{equation}
If
\begin{equation}
2\Delta-\sigma = d_\State+\alpha,
\label{eq:app-qicon-identification}
\end{equation}
one obtains
\begin{equation}
W_{\Delta}(z,z')
\propto
\frac{1}{d_g(z,z')^{d_\State+\alpha}},
\label{eq:app-qicon-final}
\end{equation}
which is the kernel defining the fractional jump operator in the main text.

\subsection{Characteristic function and stable-law tail}
\label{app:qicon-stable}

This appendix records the derivation of Eqs.~\eqref{eq:qicon-characteristic} and \eqref{eq:qicon-tail}. The one-dimensional leading \QiCON\ equation is
\begin{equation}
\partial_t p(x,t)
=
-\,D_\alpha(-\partial_x^2)^{\alpha/2}p(x,t).
\label{eq:app-qicon-fde}
\end{equation}
Using the Fourier convention
\begin{equation}
\widehat{p}(k,t)=\int_{-\infty}^{\infty}e^{-\ii kx}p(x,t)\,\dd x,
\label{eq:app-qicon-ft}
\end{equation}
and the multiplier property
\begin{equation}
\mathcal{F}\!\left[(-\partial_x^2)^{\alpha/2}p\right]
=
|k|^\alpha \widehat{p}(k,t),
\label{eq:app-qicon-multiplier}
\end{equation}
Eq.~\eqref{eq:app-qicon-fde} becomes
\begin{equation}
\partial_t \widehat{p}(k,t)
=
-\,D_\alpha |k|^\alpha \widehat{p}(k,t).
\label{eq:app-qicon-ode}
\end{equation}
This is a first-order ordinary differential equation with solution
\begin{equation}
\widehat{p}(k,t)
=
e^{-D_\alpha t|k|^\alpha}\widehat{p}(k,0).
\label{eq:app-qicon-solution}
\end{equation}
For the initial condition \(p(x,0)=\delta(x)\), one has \(\widehat{p}(k,0)=1\), hence
\begin{equation}
\phi_{X_t}(k)=\widehat{p}(k,t)=e^{-D_\alpha t|k|^\alpha},
\label{eq:app-qicon-char}
\end{equation}
which is the characteristic function of a symmetric \(\alpha\)-stable law.

To see the tail, write the inverse transform formally as
\begin{equation}
p(x,t)
=
\frac{1}{2\pi}
\int_{-\infty}^{\infty}
e^{\ii kx-D_\alpha t|k|^\alpha}\,\dd k.
\label{eq:app-qicon-inverse}
\end{equation}
For large \(|x|\), the dominant contribution comes from small \(|k|\). Expanding the exponential,
\begin{equation}
e^{-D_\alpha t|k|^\alpha}
=
1-D_\alpha t|k|^\alpha+\cdots,
\label{eq:app-qicon-expansion}
\end{equation}
and transforming term by term gives the leading nontrivial asymptotic
\begin{equation}
p(x,t)
\sim
\frac{D_\alpha t\,\Gamma(1+\alpha)\sin(\pi\alpha/2)}{\pi |x|^{1+\alpha}}.
\label{eq:app-qicon-tail}
\end{equation}
Thus the heavy-tail exponent is \(1+\alpha\).

\subsection{Box-counting derivation of the fractal dimension}
\label{app:qicon-fractal}

This appendix spells out the short derivation of Eq.~\eqref{eq:qicon-fractal-dimension}. By self-similarity,
\begin{equation}
X_t \overset{d}{=} t^{1/\alpha}X_1.
\label{eq:app-qicon-selfsimilar}
\end{equation}
Therefore the typical spatial increment over a time step \(\delta t\) is
\begin{equation}
\delta x \sim (\delta t)^{1/\alpha}.
\label{eq:app-qicon-dx}
\end{equation}
To resolve the path at spatial scale \(\varepsilon\), choose \(\delta x\sim \varepsilon\), which implies
\begin{equation}
\delta t \sim \varepsilon^\alpha.
\label{eq:app-qicon-dt}
\end{equation}
Over an observation interval of length \(T\), the number of distinguishable \(\varepsilon\)-scale segments is then
\begin{equation}
N_T(\varepsilon)\sim \frac{T}{\delta t}\sim T\,\varepsilon^{-\alpha}.
\label{eq:app-qicon-Nraw}
\end{equation}
However, no subset of a \(d_\State\)-dimensional ambient manifold can require more than \(\varepsilon^{-d_\State}\) boxes of side \(\varepsilon\). Hence the actual covering exponent is the smaller of \(\alpha\) and \(d_\State\):
\begin{equation}
N_T(\varepsilon)\sim \varepsilon^{-D_f},
\qquad
D_f=\min(d_\State,\alpha).
\label{eq:app-qicon-Df}
\end{equation}
This is the box-counting form used in the main text.

\subsection{Spectral density and the two-knee approximation}
\label{app:qicon-psd}

Starting from
\begin{equation}
S_X(\omega)
=
2A_0
\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}
\frac{\lambda^\gamma}{\lambda^2+\omega^2}\,\dd\lambda,
\qquad
0\le \gamma <1,
\label{eq:app-qicon-psd-start}
\end{equation}
set \(\lambda=\omega u\). Then
\begin{equation}
S_X(\omega)
=
2A_0\,\omega^{\gamma-1}
\int_{\lambda_{\mathrm{IR}}/\omega}^{\lambda_{\mathrm{UV}}/\omega}
\frac{u^\gamma}{1+u^2}\,\dd u.
\label{eq:app-qicon-psd-u}
\end{equation}

\subsection*{Middle band}
When \(\lambda_{\mathrm{IR}}\ll \omega \ll \lambda_{\mathrm{UV}}\), the integral may be extended to \((0,\infty)\):
\begin{equation}
\int_0^\infty \frac{u^\gamma}{1+u^2}\,\dd u
=
\frac{\pi}{2}\sec\!\left(\frac{\pi\gamma}{2}\right),
\qquad
0\le\gamma<1.
\label{eq:app-qicon-middle}
\end{equation}
Hence
\begin{equation}
S_X(\omega)
\approx
A_0\pi \sec\!\left(\frac{\pi\gamma}{2}\right)\omega^{\gamma-1}
=
\frac{\widetilde{A}}{\omega^\beta},
\qquad
\beta=1-\gamma.
\label{eq:app-qicon-middle-result}
\end{equation}

\subsection*{Infrared side}
When \(\omega\ll \lambda_{\mathrm{IR}}\), approximate \(\lambda^2+\omega^2\approx \lambda^2\):
\begin{equation}
S_X(\omega)
\approx
2A_0\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}\lambda^{\gamma-2}\,\dd\lambda
=
\text{constant}.
\label{eq:app-qicon-ir}
\end{equation}

\subsection*{Ultraviolet side}
When \(\omega\gg \lambda_{\mathrm{UV}}\), approximate \(\lambda^2+\omega^2\approx \omega^2\):
\begin{equation}
S_X(\omega)
\approx
\frac{2A_0}{\omega^2}
\int_{\lambda_{\mathrm{IR}}}^{\lambda_{\mathrm{UV}}}\lambda^\gamma\,\dd\lambda
\propto
\omega^{-2}.
\label{eq:app-qicon-uv}
\end{equation}

Combining the three asymptotics yields the two-knee form of Eq.~\eqref{eq:qicon-two-knee}.

\subsection{Compact diagnostic dictionary}
\label{app:qicon-dictionary}

For later use, Table~\ref{tab:qicon-dictionary} records the compact diagnostic dictionary of Part~\ref{part:qicon}.

\begin{longtable}{p{0.18\textwidth}p{0.18\textwidth}p{0.19\textwidth}p{0.33\textwidth}}
\caption[A working dictionary of \QiCON{} quantities.]{Compact dictionary of \QiCON{} quantities. Each row is one quantity, with its symbol, its scaling form and an interpretation. The exponents are the reason the table exists: the same $\alpha$ appears under several names in different literatures --- as a fractional exponent, a soft-mode exponent, a spectral slope and a fractal dimension --- and a reader coming from any one of those literatures needs the translation before the pillar's claim can be evaluated.}
\label{tab:qicon-dictionary}\\
\toprule
\textbf{Quantity} & \textbf{Symbol} & \textbf{Scaling form} & \textbf{Interpretation} \\
\midrule
\endfirsthead
\toprule
\textbf{Quantity} & \textbf{Symbol} & \textbf{Scaling form} & \textbf{Interpretation} \\
\midrule
\endhead
Fractional exponent & \(\alpha\) & \(\mathcal{L}_{\mathrm{eff}}\sim -D_\alpha(-\Delta_g)^{\alpha/2}\) & Sets dynamical scaling \(t\sim \ell^\alpha\). \\
Soft-mode exponent & \(\gamma\) & \(A_X(\lambda)\rho(\lambda)\sim \lambda^{\gamma-1}\) & Measures the low-lying modal crowding of the selected branch. \\
Spectral slope & \(\beta\) & \(\beta=1-\gamma\) & Gives the intermediate \(1/f^\beta\) exponent. \\
Fractal dimension & \(D_f\) & \(D_f=\min(d_\State,\alpha)\) & Effective dimension of the trajectory range inside the active window. \\
Infrared knee & \(f_{\mathrm{IR}}\) & \(f_{\mathrm{IR}}=\lambda_{\mathrm{IR}}/(2\pi)\) & Largest branch or attractor timescale. \\
Ultraviolet knee & \(f_{\mathrm{UV}}\) & \(f_{\mathrm{UV}}=\lambda_{\mathrm{UV}}/(2\pi)\) & Smallest retained microscopic timescale. \\
\bottomrule
\end{longtable}




\section{Technical derivations for \QiHAL{}}
\label{app:technical-derivations-for-qi-hal}

\subsection{Uniqueness of the \texorpdfstring{\MePMoS}{MePMoS} causal order}
\label{app:qihal-mepmos}

The dependency graph used in Theorem~\ref{thm:mepmos-uniqueness} can be written explicitly. Let the macro-variables be
\[
V=\{\mathcal{M}_n,\widehat{y}_n,a_n,s_{n+1},\mathcal{M}_{n+1}\}.
\]
The arrows are
\[
E=\{\mathcal{M}_n\to\widehat{y}_n,\;\widehat{y}_n\to a_n,\;a_n\to s_{n+1},\;s_{n+1}\to\mathcal{M}_{n+1},\;\mathcal{M}_n\to\mathcal{M}_{n+1},\;\widehat{y}_n\to\mathcal{M}_{n+1}\}.
\]
A topological sort must place each tail before its head. The chain
\[
\mathcal{M}_n \prec \widehat{y}_n \prec a_n \prec s_{n+1} \prec \mathcal{M}_{n+1}
\]
is therefore forced. Any violation creates acausality or collapses prediction into postdiction. This is the graph-theoretic reason why \MePMoS\ is the unique minimal ordering.

\subsection{Metric recovery in \texorpdfstring{\NHT}{NHT}}
\label{app:qihal-nht}

Equation~\eqref{eq:nht-map} implies
\[
\dd\phi_i = \omega_i\,\dd t + \frac{2\pi}{L_i}\dd y^i + \dd\lambda.
\]
Subtracting the trivial carrier term and gauge term gives
\[
\dd\widetilde{\phi}_i = \frac{2\pi}{L_i}\dd y^i.
\]
Hence
\[
\dd y^i = \frac{L_i}{2\pi}\dd\widetilde{\phi}_i,
\qquad
\dd s^2 = \sum_i (\dd y^i)^2 = \sum_i\left(\frac{L_i}{2\pi}\right)^2(\dd\widetilde{\phi}_i)^2.
\]
For a general latent metric \(g_{ij}(y)\), choose a local frame \(e^a{}_i(y)\) with \(g_{ij}=e^a{}_i e^a{}_j\) and encode the phase coordinates in that frame. The phase metric then reconstructs the curved geometry locally.

\subsection{Frequency-independent \texorpdfstring{\HAL}{HAL} readout}
\label{app:qihal-hal}

Starting from
\[
W_{ij}=e^{\ii(\phi_i-\phi_j)},
\qquad
x_i^{\mathrm{r}}(t)=e^{\ii(\omega_{\mathrm{r}}t+\psi_i)},
\]
one computes
\begin{align*}
\mathcal{O}[\psi|W]
&=\sum_{i,j}\overline{x_i^{\mathrm{r}}(t)}W_{ij}x_j^{\mathrm{r}}(t)\\
&=\sum_{i,j}e^{-\ii(\omega_{\mathrm{r}}t+\psi_i)}e^{\ii(\phi_i-\phi_j)}e^{\ii(\omega_{\mathrm{r}}t+\psi_j)}\\
&=\sum_{i,j}e^{\ii[(\phi_i-\phi_j)-(\psi_i-\psi_j)]}.
\end{align*}
The read carrier cancels exactly. Therefore ideal recall depends only on relational phase structure. Frequency dependence enters only through nonideal hardware, finite-bandwidth envelopes, or additional amplitude dynamics.

\subsection{The dyadic ladder and level count of \texorpdfstring{\BFFT}{BFFT}}
\label{app:qihal-bfft}

If the conformal window spans \([f_{\mathrm{IR}},f_{\mathrm{UV}}]\) and neighboring levels satisfy \(f_{n+1}=2f_n\), then
\[
f_n = 2^n f_0.
\]
Choosing \(f_0\approx f_{\mathrm{IR}}\), the largest admissible level index satisfies
\[
2^{N_{\mathrm{lev}}-1}f_{\mathrm{IR}}\lesssim f_{\mathrm{UV}},
\]
so
\[
N_{\mathrm{lev}}\lesssim 1+\log_2\left(\frac{f_{\mathrm{UV}}}{f_{\mathrm{IR}}}\right).
\]
Taking the integer part yields Eq.~\eqref{eq:bfft-level-count}. This makes the number of architectural levels a measurable consequence of the observed two-knee spectrum.

\subsection{Normal-form bifurcation for the \texorpdfstring{\DGI--\GSSB}{GRIP} switch}
\label{app:qihal-switch}

For the scalar version of Eq.~\eqref{eq:qihal-potential},
\[
V(m)=\frac12 a m^2 + \frac{b}{4}m^4 - hm,
\qquad b>0,
\]
stationary points satisfy
\[
a m + b m^3 - h = 0.
\]
At \(h=0\), one has \(m=0\) for all \(a\), but this point is stable only for \(a>0\). When \(a<0\), the nonzero branches
\[
m_{\pm}=\pm\sqrt{-\frac{a}{b}}
\]
appear and are stable. This is the usual pitchfork bifurcation. Small \(h\neq0\) unfolds the pitchfork and selects one branch. In the present architecture \(a\) is controlled by accumulated evidence, so the bifurcation becomes the mathematical normal form of exploration-to-commitment dynamics.



\section{Technical details for \SIP{} and \GFF{}}
\label{app:technical-details-for-sip-and-eegg}

\subsection{Equivalence of folded and unfolded packet forms}
\label{app:sip-fold-equivalence}

The folded five-field packet and the more explicit unfolded runtime data are equivalent. Starting from Eq.~\eqref{eq:sip-unfolded}, define the grouping map
\[
\mathfrak{G}:\widehat\SIP\mapsto
\Bigl[(\mathcal{Z},g),\;\Ports,\;(\Lind_{\mathrm{\GDOS{}}},\eta),\;(\mathcal{F}_{\GFF},\mathcal{J}),\;(\Obs,\Kill)\Bigr].
\]
The inverse map simply unpacks the generator bundle into mobility, curl, diffusion, jump, and gauge data, and unpacks the observable bundle into exported observables and falsifiers. No information is lost because each grouped entry is a direct product of the corresponding unfolded entries.

\begin{table}[H]
\centering
\caption[How unfolded data groups into the packet.]{The grouping map from unfolded runtime data to the canonical packet. The left column lists the data as it arises and the right the canonical field it belongs to. The map is many-to-one by design: the packet is a normal form, and the point of the table is to show that the grouping loses nothing, so that a reader who has assembled runtime data in some other order can check that it lands in the same place.}
\label{tab:appendix-sip-grouping}
\small
\begin{tabular}{p{0.42\textwidth}p{0.42\textwidth}}
\toprule
\textbf{Unfolded data} & \textbf{Grouped canonical field} \\
\midrule
\((\mathcal{Z},g_{AB})\) & State manifold \\
\(\Ports\) & Ports \\
\((M^{AB},\Omega^A,D^{AB},R^A{}_{\alpha}\lambda^{\alpha},\eta_{\mathrm{jump}})\) & Generator + closure \\
\((\mathcal{F}_{\GFF},\mathcal{J})\) & Ledger bundle \\
\((\Obs,\Kill)\) & Observable dictionary + kill criteria \\
\bottomrule
\end{tabular}
\end{table}

Therefore the five-field form is not a simplification by omission, but a simplification by bundling.

\subsection{Detailed derivation of the \GFF\ balance law}
\label{app:eegg-balance-derivation}

Starting from Eq.~\eqref{eq:eegg-functional}, differentiate with respect to time:
\begin{align}
\frac{\dd}{\dd t}\mathcal{F}_{\GFF}
&=
\int \frac{\delta\mathcal{F}_{\GFF}}{\delta p}\,\partial_t p\,\dd\mu_g
+
\int p\,\partial_t\Phi_{\GEP}\,\dd\mu_g
-
\Theta_{\mathrm{eff}}\int p\,\partial_t\ln p_{\mathrm{ref}}\,\dd\mu_g.
\label{eq:appendix-eegg-start}
\end{align}
Using \(\partial_t p=-\nabla_A J^A\) and integrating by parts gives
\begin{align}
\frac{\dd}{\dd t}\mathcal{F}_{\GFF}
&=
\int \nabla_A\!\left(\frac{\delta\mathcal{F}_{\GFF}}{\delta p}\right)J^A\,\dd\mu_g
-
\oint_{\partial\mathcal{Z}} \frac{\delta\mathcal{F}_{\GFF}}{\delta p}\,J^A n_A\,\dd\Sigma
\notag\\
&\quad
+
\int p\,\partial_t\Phi_{\GEP}\,\dd\mu_g
-
\Theta_{\mathrm{eff}}\int p\,\partial_t\ln p_{\mathrm{ref}}\,\dd\mu_g.
\label{eq:appendix-eegg-integrated}
\end{align}
Substituting the current split \eqref{eq:eegg-full-current} isolates the dissipative quadratic form, the curl contribution, the gauge contribution, and the explicit port/boundary terms. The last relative-entropy reference term is absorbed into the port power because changes in \(p_{\mathrm{ref}}\) arise from boundary condition or control updates. This yields Eqs.~\eqref{eq:eegg-balance-identity}--\eqref{eq:eegg-gauge-term}.

If the local Einstein relation \(D^{AB}=\Theta_{\mathrm{eff}}M^{AB}\) holds, then the noise contribution combines naturally with the entropic term so that \(\Theta_{\mathrm{eff}}\) acquires the interpretation of an effective information temperature. This is the most direct bridge between \GFF\ and standard stochastic thermodynamics \citep{Seifert2012}.

\subsection{Event-level reduction on the locked current ray}
\label{app:eegg-event-ray}

Let the current vector be written in the ordered basis \((J_e,J_B,J_{Qi},J_E)\). Under the Arisaka Equation the allowed event increments span the one-dimensional ray
\[
\delta\mathcal{J}=\delta j\,v,
\qquad v=(1,1,1,\QB)^{\mathsf T}.
\]
Any polynomial event cost therefore reduces to a polynomial in the single amplitude \(\delta j\). For example:
\begin{align*}
L_1(\delta\mathcal{J}) &= a_{\alpha}\,\delta J^{\alpha} = (a\cdot v)\,\delta j,\\
L_2(\delta\mathcal{J}) &= Q_{\alpha\beta}\,\delta J^{\alpha}\delta J^{\beta} = (v^{\mathsf T}Qv)\,(\delta j)^2,\\
L_3(\delta\mathcal{J}) &= T_{\alpha\beta\gamma}\,\delta J^{\alpha}\delta J^{\beta}\delta J^{\gamma} = T(v,v,v)\,(\delta j)^3.
\end{align*}
Thus all admissible local event penalties are automatically scalarized once anomaly-derived locking is imposed. This is why the scalar \GFF\ ledger and the vector current ledger are compatible rather than competing descriptions.

\subsection{Proof sketch for the viability-tube criterion}
\label{app:seos-viability-proof}

The persistence theorem rests on a familiar Lyapunov argument. Define \(W(t)=\Viab(\Xi_t)-\beta\int_0^t \Maint(s)\,\dd s\). Then Eq.~\eqref{eq:viability-drift} implies
\[
\dot W \le -\kappa\,\Viab + c.
\]
If \(\Maint>c/\beta\) on average, then \(W\) cannot grow indefinitely, and neither can \(\Viab\). Hence packet distance, boundary damage, and memory deficit all remain bounded. The tube is therefore forward invariant up to the chosen tolerance. If \(\Maint<0\) for sufficiently long times, the opposite occurs: the accumulated negative maintenance term drives \(W\) upward, and one or more packet coordinates leave tolerance. The proof is not sensitive to whether the target attractor is fixed, periodic, or quasiperiodic; only boundedness of the chosen packet norm matters.

\subsection{Compact packet cards across representative domains}
\label{app:sip-domain-cards}

\begin{longtable}{p{0.17\textwidth}p{0.17\textwidth}p{0.18\textwidth}p{0.18\textwidth}p{0.20\textwidth}}
\caption[Packet, ledger and persistence, domain by domain.]{Compact cards across representative domains. Each row is one domain, from a chemical dissipative structure to a neural circuit, with its packet core, the dominant terms in its free-energy ledger, its maintenance source and its persistence status. The cards are working templates rather than results: each is a filled-in instance of the recipe of Table~\ref{tab:application-recipe}, offered so that a reader in a new domain has a worked example of the same shape to copy.}
\label{tab:appendix-domain-cards}\\
\toprule
\textbf{Domain} & \textbf{Packet core} & \textbf{Dominant \GFF\ terms} & \textbf{Maintenance source} & \textbf{\SeOS\ status} \\
\midrule
\endfirsthead
\toprule
\textbf{Domain} & \textbf{Packet core} & \textbf{Dominant \GFF\ terms} & \textbf{Maintenance source} & \textbf{\SeOS\ status} \\
\midrule
\endhead
Chemical dissipative structure & Pattern amplitude, thermal gradient, vessel geometry & Gradient energy, entropic broadening, boundary forcing & External heating / cooling & Usually proto-\SeOS\ rather than full \SeOS \\
Protocell & Membrane state, internal metabolite packet, permeability & Assembly cost, mixing entropy, electrochemical coupling & Nutrient and amphiphile throughput & Full \SeOS\ if repair margin is positive \\
Bacterium & Membrane, metabolism, adaptive memory, action state & Nutrient uptake, ATP ledger, information update cost & Continuous nutrient uptake and repair metabolism & Full \SeOS \\
Neural circuit & Recurrent state, synaptic memory, phase code & Coherence cost, plasticity cost, metabolic supply & Organism-level metabolic and sensory ports & Full \SeOS; often \QiHAL-rich \\
Institution & Archive, policy state, communication network, resource stock & Coordination cost, archive maintenance, informational entropy & Resource inflow and repair of structure/records & Distributed higher-order \SeOS \\
\bottomrule
\end{longtable}




\section{Technical derivations for \OH{}, \IH{}, and \EIC{}}
\label{app:technical-derivations-for-oh-ih-and-eic}

\subsection{Derivation of the outer-hologram kernel from instruments}
\label{app:oh-kernel}

This appendix records the instrument-theoretic derivation behind Eqs.~\eqref{eq:oh-kernel}--\eqref{eq:oh-kernel-normalization}.

Let the effective reduced bulk-to-boundary map be represented by a completely positive trace-preserving map \(\Phi_{t,0}\), and let \(\{\Instr_y\}\) be an admissible instrument family satisfying
\[
\Instr_y \ge 0,
\qquad
\int_{\mathcal{Y}}\Instr_y\,\dd y = I_{\partial}.
\]
For a pure bulk representative \(\ket{X}\), define
\[
K_{\mathrm{\OH{}}}(y|X;t)
=
\Tr\!\left[
\Instr_y\,\Phi_{t,0}\!\bigl(\ket{X}\bra{X}\bigr)
\right].
\]
Positivity follows because the trace of the product of two positive operators is nonnegative. Normalization follows from
\[
\int_{\mathcal{Y}} K_{\mathrm{\OH{}}}(y|X;t)\,\dd y
=
\Tr\!\left[
\left(
\int_{\mathcal{Y}}\Instr_y\,\dd y
\right)
\Phi_{t,0}\!\bigl(\ket{X}\bra{X}\bigr)
\right]
=
\Tr\!\left[
\Phi_{t,0}\!\bigl(\ket{X}\bra{X}\bigr)
\right]
=
1,
\]
because \(\Phi_{t,0}\) is trace preserving. For a mixed bulk state \(\rho_6(X,t)\), linearity immediately yields Eq.~\eqref{eq:outer-hologram-density}.

The derivation shows that the \OH{} kernel is not a new postulate. It is the instrument-level unpacking of the \OGGEP\ boundary law.

\subsection{Observability and action-conditioned reconstruction}
\label{app:ih-observability}

This appendix gives the standard algebra behind Theorem~\ref{thm:ih-observability}. For the linearized dynamics
\[
\delta x_{t+1}=A_t\delta x_t+B_t\delta a_t,
\qquad
\delta s_t=C_t\delta x_t,
\]
one has
\begin{align*}
\delta s_0 &= C_0\delta x_0,\\
\delta s_1 &= C_1A_0\delta x_0 + C_1B_0\delta a_0,\\
\delta s_2 &= C_2A_1A_0\delta x_0 + C_2A_1B_0\delta a_0 + C_2B_1\delta a_1,
\end{align*}
and so on. Collecting terms gives
\[
\mathbf{s}_{0:T}
=
\mathcal{O}_T\delta x_0
+
\mathbf{u}_{0:T},
\]
where \(\mathbf{u}_{0:T}\) contains the known action-dependent contributions. If \(\mathcal{O}_T\) has full column rank, the least-squares estimate
\[
\widehat{\delta x}_0
=
(\mathcal{O}_T^{\top}\mathcal{O}_T)^{-1}\mathcal{O}_T^{\top}
(\mathbf{s}_{0:T}-\mathbf{u}_{0:T})
\]
is unique. If not, the nullspace is nontrivial and the hidden state cannot be uniquely reconstructed from the available history.

The mathematical moral is simple: action changes what can be seen, and memory stores the history needed to exploit that fact.

\subsection{Two-view triangulation from lower-dimensional sensory inputs}
\label{app:triangulation}

A familiar example of \IH{} formation is the reconstruction of a 3D point from 2D observations at multiple viewpoints. Let \(X\in \mathbb{P}^3\) be a homogeneous 3D point and \(\tilde u_i\in\mathbb{P}^2\) its homogeneous image in frame \(i\). Then
\[
\lambda_i \tilde u_i = P_i X,
\qquad
P_i = K_i[R_i|t_i].
\]
For two or more views, one forms the linear system
\[
A X = 0,
\]
where \(A\) is assembled from the cross-product equations \(\tilde u_i\times (P_iX)=0\). Nonzero camera baseline or self-motion is required for unique depth recovery up to the usual projective gauge. This is the most elementary concrete instance of the general claim in the main text: a lower-dimensional instantaneous observation does not determine the world; the world becomes reconstructable only after motion generates multiple views.

The same algebra appears, abstractly, in chemical gradient sensing, head-direction stabilization, and other active-sensing settings. The substrate changes; the observability logic does not.

\subsection{Internal measurement operator and selected content}
\label{app:internal-measurement}

Suppose the \IH{} state is expanded in an orthonormal content basis \(\{\ket{c}\}\):
\[
\rho_{\IH}
=
\sum_{c,c'}
\rho_{cc'}\,\ket{c}\bra{c'}.
\]
An internal coarse-graining may be represented by block projectors \(\Pi_{\alpha}=\sum_{c\in B_\alpha}\ket{c}\bra{c}\), where \(B_\alpha\) is a content block. Then
\[
p(\alpha)=\Tr(\Pi_\alpha\rho_{\IH})=\sum_{c\in B_\alpha}\rho_{cc}.
\]
If the \DGI\(\to\)\GSSB\(\to\)\GA\ sequence suppresses all but one dominant block \(B_{\alpha_\ast}\), the internal post-selected state becomes
\[
\rho_{\IH}^{(\alpha_\ast)}
=
\frac{\Pi_{\alpha_\ast}\rho_{\IH}\Pi_{\alpha_\ast}}
{\Tr(\Pi_{\alpha_\ast}\rho_{\IH})}.
\]
A subsequent write or action-bias map \(\mathcal{W}\) produces
\[
m_{t+1} = \mathcal{W}\!\left[\rho_{\IH}^{(\alpha_\ast)}\right],
\]
which makes the selected content causally effective in the next \MePMoS\ cycle. This is the algebraic heart of the internal-measurement idea.

\subsection{Lyapunov decrease of the \EIC{} functional with bounded fluctuations}
\label{app:eic-lyapunov}

For the noisy gradient flow
\[
\dot{\theta}
=
-\,G_{\theta}^{-1}\nabla_{\theta}\mathcal{F}_{\EIC}
+
\eta_t,
\]
one has
\[
\dot{\mathcal{F}}_{\EIC}
=
-\,\nabla_{\theta}\mathcal{F}_{\EIC}^{\top}
G_{\theta}^{-1}
\nabla_{\theta}\mathcal{F}_{\EIC}
+
\nabla_{\theta}\mathcal{F}_{\EIC}^{\top}\eta_t.
\]
If \(\eta_t\) is zero mean with covariance bounded by \(\Sigma_{\eta}\), the expected drift satisfies
\[
\mathbb{E}\,\dot{\mathcal{F}}_{\EIC}
\le
-\,\lambda_{\min}(G_{\theta}^{-1})
\mathbb{E}\,\|\nabla_{\theta}\mathcal{F}_{\EIC}\|^2
+
\mathbb{E}\,\|\nabla_{\theta}\mathcal{F}_{\EIC}\|\,\|\eta_t\|.
\]
Whenever the deterministic descent term dominates the fluctuation term, the expected \EIC{} functional decreases. In practice this is the appropriate statement for open systems subject to bounded noise: exact monotonicity may be broken locally, but average descent toward a basin is preserved.

\subsection{Representative \OH{}/\IH{}/\EIC{} cards}
\label{app:ohih-cards}

\begin{longtable}{p{0.17\textwidth}p{0.24\textwidth}p{0.24\textwidth}p{0.23\textwidth}}
\caption[Outer and inner holograms, system by system.]{Compact cards across representative systems. Each row is one system, from a molecular rotor to a rodent navigation system, with what plays the part of the outer hologram, what plays the inner one, and the resulting reading. The cards are ordered by increasing memory depth, and the value of the ordering is that it shows the inner column filling in gradually rather than switching on.}
\label{tab:ohih-cards}\\
\toprule
\textbf{System} & \textbf{\OH{}} & \textbf{\IH{}} & \textbf{\EIC{} reading} \\
\midrule
\endfirsthead
\toprule
\textbf{System} & \textbf{\OH{}} & \textbf{\IH{}} & \textbf{\EIC{} reading} \\
\midrule
\endhead
Molecular rotor & External torque, chemical potentials, occupancy transitions at the boundary & Conformational phase / occupancy state only & Minimal or degenerate; local cycle knowledge rather than a genuine world model \\
Chemotactic bacterium & Sampled ligand concentration along the trajectory & Adaptive methylation state encoding recent gradient history & Proto-\EIC{}: local chemical world reconstructed enough to bias future motion \\
Ring-attractor navigation system & Visual, vestibular, and proprioceptive heading evidence & Persistent heading bump and related attractor manifold & Allocentric heading \EIC{} with repeated correction by \OH{} events \\
Rodent navigation system & Multimodal environmental stream with self-motion updates & Place/head-direction/time-structured spatial model & Rich navigational \EIC{}; imagination and replay become possible \\
Human scientific cognition & Experimental data, language, symbolic and sensory streams & Multiscale internal model spanning space, time, causality, and abstraction & Consciousness-grade \EIC{}; the system measures its own model and can report it symbolically \\
\bottomrule
\end{longtable}




\section{Pagoda hierarchy and cross-domain technical details}
\label{app:pagoda-hierarchy-and-cross-domain-techni}

\subsection{Closure defects, effective floors, and error budgets}
\label{app:pagoda-closure}

This appendix expands Definition~\ref{def:admissible-floor}. Let \(z^{(\ell+1)}=C_{\ell+1}(z^{(\ell)})\) and define the exact induced derivative and closure defect by Eq.~\eqref{eq:closure-defect}. A convenient normalized defect index is
\begin{equation}
\varepsilon_{\mathrm{cl}}^{(\ell+1)}
:=
\frac{\left\langle \|\Delta_{\ell+1}\|^2\right\rangle^{1/2}}
{\left\langle \|F^{(\ell+1)}_{\mathrm{eff}}\|^2\right\rangle^{1/2}+\delta},
\qquad \delta>0.
\label{eq:appendix-closure-index}
\end{equation}
The index is small when the floor is dynamically autonomous. In data-driven applications one can estimate \(F^{(\ell+1)}_{\mathrm{eff}}\) from reduced trajectories and treat \(\varepsilon_{\mathrm{cl}}\) as a model-selection criterion for candidate higher-level packets.

A second useful quantity is the \emph{port novelty index}
\begin{equation}
\nu_{\mathrm{port}}^{(\ell+1)}
:=
\dim\mathrm{span}\,\Ports^{(\ell+1)}-
\dim\mathrm{span}\,\Portspull^{(\ell\to\ell+1)}.
\label{eq:appendix-port-novelty}
\end{equation}
If \(\nu_{\mathrm{port}}^{(\ell+1)}=0\), then the higher floor introduces no genuinely new mode of exchange and is usually better interpreted as a descriptive convenience rather than a new scientific level.

\subsection{Hitting-time proof of conditional inevitability}
\label{app:inevitability-proof}

Theorem~\ref{thm:conditional-inevitability} may be rewritten using visit counts. Let \(N_W(T)\) be the number of visits to a viable window \(W\) by time \(T\), and let \(S_n\) be the indicator that the \(n\)-th visit stabilizes. Under the assumptions of the theorem,
\[
\mathbb P(S_n=1\mid \text{visit}_n)\ge p>0.
\]
Hence
\[
\mathbb P(\text{no stabilization up to visit }N)
\le (1-p)^N.
\]
If \(N_W(T)\to\infty\) almost surely as \(T\to\infty\), then
\[
\mathbb P(\tau_W=\infty)
=\lim_{N\to\infty}(1-p)^N=0.
\]
The theorem therefore uses only three ingredients: positive-measure windows, recurrent exploration, and positive stabilization probability.

\subsection{Compact domain cards and observables}
\label{app:application-cards}

\begin{longtable}{p{0.14\textwidth}p{0.17\textwidth}p{0.18\textwidth}p{0.18\textwidth}p{0.21\textwidth}}
\caption[Application cards for four domains.]{Compact application cards for representative domains. Each row is one domain, with its key packet variables, candidate observables, the primary memory variable and the single most diagnostic test. The last column is the one to read first: it names, for each domain, the measurement whose outcome would most quickly distinguish a system running this runtime from one that merely looks as though it does.}
\label{tab:appendix-application-cards}\\
\toprule
\textbf{Domain} & \textbf{Key packet variables} & \textbf{Candidate observables} & \textbf{Primary memory variable} & \textbf{Most diagnostic test} \\
\midrule
\endfirsthead
\toprule
\textbf{Domain} & \textbf{Key packet variables} & \textbf{Candidate observables} & \textbf{Primary memory variable} & \textbf{Most diagnostic test} \\
\midrule
\endhead
Catalytic chemistry & concentrations, occupancy, rotor angle & product counts, spectra, stepping time, fluxes & conformational state & event-ledger closure plus burst statistics \\
Chemotactic cell & receptor state, internal methylation, trajectory & run/tumble statistics, local gradient, adaptation curve & adaptive receptor memory & motion-conditioned predictive quality \\
Organismal physiology & nutrient store, damage, repair flux, internal model & metabolic rate, motion, sensory traces, repair markers & regulatory and long-term physiological memory & maintenance-positive \SeOS\ inequality \\
Navigation circuit & neural manifold, latent heading/place state, action output & spikes/LFP, movement, sensory cues, reports & recurrent attractor state & DGI-to-GSSB-to-report timing \\
Scientific institution & agent states, archive, instruments, communication graph & data releases, publications, consensus updates & distributed archive and trained priors & \OH{}/\IH{} model-improvement under controlled measurement practice \\
\bottomrule
\end{longtable}

\subsection{Minimal dynamical forms across domains}
\label{app:application-minimodels}

For convenience, the minimal application equations from the main text are collected here in one place.
\begin{align*}
\text{Chemistry:}\quad
\dot n_i &= \sum_r \nu_{ir}R_r(n,m)-\nabla\cdot J_i+I_i-O_i+\xi_i,
&
\dot m &= f_m(n,m)+\eta_m,
\\[4pt]
\text{Life:}\quad
\dot N &= J_{\mathrm{in}}-J_{\mathrm{use}}-J_{\mathrm{leak}},
&
\dot B &= YJ_{\mathrm{use}}-\delta B-c_r(D),
\\
\dot D &= \sigma-r,
&
\dot m &= f_m(y,a,m),
\\[4pt]
\text{Brain:}\quad
\tau_x\dot x &= -x+W\phi(x)+Uy+Va+\zeta_x,
&
\tau_m\dot m &= f_m-K\nabla_m\mathcal E_{\EIC},
\\[4pt]
\text{Society:}\quad
\dot x_i &= f_i(x_i,A,d_i)+\sum_jW_{ij}\Psi(x_j-x_i)+\eta_i,
&
\dot A &= \Gamma(\{x_i\},d)-\Lambda A.
\end{align*}
These forms are not intended as finished empirical models. They are canonical placeholders showing how the same runtime grammar appears in very different domains.


\section{Falsifiability summary}
\label{app:falsifiability-page-budget-and-architect}

\subsection{Compact falsifiability matrix}
\label{app:falsifiability-matrix}

\begin{table}[H]
\centering
\caption[The falsifiability matrix, block by block.]{Compact falsifiability matrix for the integrated framework. Each row is one runtime block, with an observable proxy for it, a representative domain in which that proxy is measurable, and the failure mode that would refute the block. Read beside Table~\ref{tab:falsifiability} it is the same program arranged by structure rather than by signature, so that a reader who wants to know what a given pillar risks can find it in one row.}
\label{tab:appendix-falsifiability}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.19\textwidth}}
\toprule
\textbf{Runtime block} & \textbf{Observable proxy} & \textbf{Representative domain} & \textbf{Failure mode} \\
\midrule
\QiQUA & event-level energy/information accounting & chemistry, molecular motors & no coherent locked ledger \\
\QiEP & path-selection under perturbation & reaction routing, behavior, inference tasks & no common selector family \\
\QiCON & spectra/tails/fractal dimension & neural activity, ecological time series, market-like records & no stable multiscale signature \\
\QiHAL & memory depth and multiscale coordination & cells, brains, institutions & no measurable internal memory architecture \\
\OH{}/\IH{}/\EIC{} & model quality under action restriction & navigation, manipulation, scientific practice & internal model unaffected by motion constraints \\
\bottomrule
\end{tabular}
\end{table}

%% [v11.5] Internal page-budget and forward-architecture drafting tables
%% removed (a planning page-count and an F1-F4 to-do list, both contradicting
%% the finished ten-Part paper); Appendix L keeps only the falsifiability matrix.




\section{Chemistry, biochemistry, and protocellular open systems}
\label{app:chemistry-domain}

The main text argued that chemistry is the first decisive testbed for the universal-runtime claim, because chemistry is the domain in which one simultaneously sees energetic throughput, charge transfer, material reorganization, structural persistence, and information-bearing selectivity. A detector or a scattering experiment already displays lawful open-system projection, but chemistry adds a stronger demand: the system must often preserve a far-from-equilibrium structure while repeatedly performing useful cycles. That is exactly the setting in which \SIP{}, \GFF{}, \SeOS{}, GRIP--GA, and the \QiFour{} pillars should either become precise and useful or else fail. The point of the present appendix is therefore not decorative. It is to show, in slower detail than the main text allowed, how the \GDOS{} runtime is expected to be instantiated in reaction networks, enzyme cycles, membrane systems, autocatalytic sets, and protocell-scale organizations \citep{Landauer1961,Prigogine1967,NicolisPrigogine1977,Kauffman1993,Lane2015}.

\subsection{Why chemistry is the first nontrivial export layer}

In the present program, the chemical layer is not merely ``between'' particle physics and biology in an informal pedagogical sense. It is the first layer at which a finite open system can both preserve a persistent identity and implement selective throughput. A molecule, membrane, or protocell must trade among at least four demands: energetic affordability, material or structural maintenance, charge or redox consistency, and information-bearing discrimination. The first two demands are familiar in thermodynamics; the third is explicit in electrochemistry; the fourth appears in selectivity, proofreading, templating, signaling, and memory-like state retention. The claim of \QiQUA{} is that these demands are not independent ledgers once the underlying event is projected to the lawful boundary description of \GDOS{}. The current-locking law should therefore be tested here first, because chemistry gives one the cleanest mesoscopic event count.

\subsection{A chemistry-scale System Instance Packet}

Let a chemical open system possess \(N_s\) coarse-grained species variables \(n_i\), \(N_c\) conformational coordinates \(\xi_\alpha\), \(N_p\) port or membrane coordinates \(\phi_a\), and \(N_m\) slowly writable internal markers \(m_b\). The reduced state coordinates are taken to be
\begin{equation}
z^A
=
\bigl(
n_1,\dots,n_{N_s},
\xi_1,\dots,\xi_{N_c},
\phi_1,\dots,\phi_{N_p},
m_1,\dots,m_{N_m}
\bigr).
\label{eq:appJ-state}
\end{equation}
The corresponding runtime packet is
\begin{equation}
\SIP_{\mathrm{chem}}
=
\bigl(
\mathcal{M}_{\mathrm{chem}},
z^A,
G_{AB}^{(\mathrm{chem})},
\mathcal A_A^{(\mathrm{chem})},
\FGFF^{(\mathrm{chem})},
\Sigma^{AB}_{(\mathrm{chem})},
\mathcal O_{\mathrm{chem}}
\bigr).
\label{eq:appJ-sip}
\end{equation}
Each slot now has a direct interpretation.

First, \(\mathcal{M}_{\mathrm{chem}}\) is the accessible manifold of occupation levels, conformers, port states, and slow memory marks. It is not the full \(6N\)-dimensional microscopic phase space. Second, the induced metric \(G_{AB}^{(\mathrm{chem})}\) measures the local cost or distinguishability of moving on that reduced manifold; in practice this may be inferred from susceptibility matrices, covariance matrices, or information-geometric approximations. Third, \(\mathcal A_A^{(\mathrm{chem})}\) is the effective gauge or curl connection that survives coarse-graining. In chemical language it packages cyclic driving, nonconservative affinities, and phase-like loop effects. Fourth, \(\FGFF^{(\mathrm{chem})}\) is the common ledger that combines energy supply, entropy production, structural maintenance, and geometric constraints. Fifth, \(\Sigma^{AB}_{(\mathrm{chem})}\) records unresolved stochasticity from microscopic collisions, conformational substructure, and environmental variation. Finally, \(\mathcal O_{\mathrm{chem}}\) is the explicit dictionary of measurable outputs: concentrations, currents, rates, redox states, membrane potentials, binding fidelity, growth, division, or other experimentally accessible records.

The reduced continuity law takes the standard \GDOS{} form
\begin{equation}
\partial_t p(z,t)+\nabla_A J^A(z,t)=0,
\qquad
J^A=J^A_{\mathrm{grad}}+J^A_{\mathrm{curl}}+J^A_{\mathrm{noise}},
\label{eq:appJ-continuity}
\end{equation}
with
\begin{align}
J^A_{\mathrm{grad}} &= -p\,M^{AB}\nabla_B \Xi_{\mathrm{chem}},
\label{eq:appJ-grad}\\
J^A_{\mathrm{curl}} &= p\,\Omega^{AB}\nabla_B \Psi_{\mathrm{chem}},
\label{eq:appJ-curl}\\
J^A_{\mathrm{noise}} &= -\frac12 \nabla_B\!\left(\Sigma^{AB}_{(\mathrm{chem})} p\right).
\label{eq:appJ-noise}
\end{align}
The first term drives relaxation on the common scalar ledger, the second encodes looped or oscillatory turnover, and the third closes positivity under unresolved fluctuations. In a simple enzyme cycle the curl term is the reduced remnant of a driven sequence of binding and release events. In a membrane-transport problem it is the remnant of circulation through a driven electrochemical loop. In an autocatalytic or templating set it captures the fact that turnover may circulate around a network rather than merely descend a scalar potential.

\begin{table}[H]
\centering
\caption[The packet slots, read in chemistry.]{One concrete interpretation of the seven packet slots in chemistry and biochemistry. The left column names the slot and the right gives its typical chemistry-scale meaning. The word \emph{one} in the heading is deliberate: the interpretation is an instance, not a derivation, and a different chemical setting would fill the slots differently. It is included because the packet is abstract enough that a single worked filling makes the rest of the appendix legible.}
\label{tab:appJ-sip-slots}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.15\textwidth} X}
\toprule
\textbf{\SIP{} slot} & \textbf{Typical chemistry-scale meaning}\\
\midrule
\(\mathcal{M}_{\mathrm{chem}}\) & Coarse-grained manifold of concentrations, conformers, port states, and slow writable markers.\\
\(z^A\) & Occupancies \(n_i\), conformers \(\xi_\alpha\), membrane or redox variables \(\phi_a\), and memory-like variables \(m_b\).\\
\(G_{AB}^{(\mathrm{chem})}\) & Susceptibility or information metric describing the local cost of state displacement.\\
\(\mathcal A_A^{(\mathrm{chem})}\) & Nonconservative affinities, cycle phases, and port couplings surviving coarse-graining.\\
\(\FGFF^{(\mathrm{chem})}\) & Joint ledger of free-energy supply, entropy production, structural maintenance, and geometric constraint.\\
\(\Sigma^{AB}_{(\mathrm{chem})}\) & Stochastic forcing from collisions, conformational microstates, and environmental variation.\\
\(\mathcal O_{\mathrm{chem}}\) & Measurable outputs: rates, redox or ion currents, yields, fidelity, growth, division, and robustness.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Reaction-network representation and current ledgers}

Suppose the system contains reaction channels \(\rho=1,\dots,R\) with stoichiometric matrix \(S_{i\rho}\) and coarse-grained fluxes \(j_\rho\). Then
\begin{equation}
\partial_t n_i
=
\sum_{\rho=1}^{R} S_{i\rho} j_\rho
+
\eta_i,
\label{eq:appJ-stoich}
\end{equation}
where \(\eta_i\) summarizes unresolved noise and exchange with neglected degrees of freedom. For the present program the important point is not merely stoichiometry but \emph{ledger separation}. To each flux channel one assigns four projected event weights:
\begin{equation}
\Delta G_\rho,\qquad
q_\rho,\qquad
b_\rho,\qquad
\iota_\rho,
\label{eq:appJ-channel-weights}
\end{equation}
interpreted respectively as energetic throughput, charged or electrochemical transfer, structural-material restructuring, and information-bearing discrimination. The associated reduced currents are then
\begin{align}
\JE &= \sum_{\rho} \Delta G_\rho\,j_\rho,
\label{eq:appJ-JE}\\
\Je &= \sum_{\rho} q_\rho\,j_\rho,
\label{eq:appJ-Je}\\
\JB &= \sum_{\rho} b_\rho\,j_\rho,
\label{eq:appJ-JB}\\
\JQi &= \sum_{\rho} \iota_\rho\,j_\rho.
\label{eq:appJ-JQi}
\end{align}
At the microphysical level \(\JB\) descends from the baryonic channel of \GG{}. At chemistry scale it becomes the structural-material bookkeeping current: the channel that counts net rearrangement of matter-bearing structure across the chosen boundary description. The notation is inherited, but the interpretation is mesoscopic. Likewise \(\JQi\) is the effective information-throughput channel, not an arbitrary symbolic label.

A \emph{locked elementary cycle} is defined to be a coarse-grained event class for which the four currents are proportional to a common event count \(n_{\mathrm{cyc}}\). In chemistry natural units one then writes
\begin{equation}
\frac{\JE}{\QB}
=
\Je
=
\JB
=
\JQi
=
n_{\mathrm{cyc}}.
\label{eq:appJ-arisaka-chem}
\end{equation}
Equation~\eqref{eq:appJ-arisaka-chem} is not asserted to hold for every arbitrarily coarse aggregate without correction. Rather, it is the ideal bookkeeping relation for an admissible elementary boundary update in which energy transfer, charge/redox transfer, structural reconfiguration, and information-bearing discrimination are inseparable aspects of one projected event. Deviations from exact equality are therefore diagnostic. If a biochemical model claims to implement a one-step selective event while grossly violating \eqref{eq:appJ-arisaka-chem}, then either the proposed event class is not elementary or the chosen \SIP{} has omitted one of the relevant ports.

\subsection{\QiEP{} in biochemical cycles}

The chemical domain is also the natural place to see why \QiEP{} is not a mere restatement of one preexisting extremal principle. Consider a path \(z(t)\) on \(\mathcal{M}_{\mathrm{chem}}\). The chemistry-scale \GDOS{} path functional may be organized as
\begin{equation}
\mathcal S_{\mathrm{chem}}[z]
=
\int_{t_i}^{t_f}
\!\dd t\,
\left[
\frac12 g_{AB}(z)\dot z^A \dot z^B
+
\dot z^A \mathcal A_A^{(\mathrm{chem})}(z)
+
V_{\mathrm{diss}}(z)
+
\Lambda_{\mathrm{sel}}(z,\dot z)
\right].
\label{eq:appJ-path}
\end{equation}
The first term is the metric cost of motion on the reduced manifold. The second is the cycle or affinity term. The third records the dissipative penalty or throughput cost. The fourth stands for selectivity or fidelity constraints, such as proofreading, templating, or gated transport. Now consider three limiting regimes.

If the stochastic forcing is weak, the cycle structure modest, and the relevant task primarily geometric, then minimization of \(\mathcal S_{\mathrm{chem}}\) reduces to a shortest-time or shortest-cost path problem: this is the \PLT{} face. If the task is dominated by sustained driving with strong dissipation, the extremum reorganizes into a constrained entropy-production problem: this is the \MEPP{} face. If, however, the system contains internal variables \(m_b\) that encode predictions or internal templates, then part of \(\Lambda_{\mathrm{sel}}\) becomes an inference-like mismatch term between internal and external port states, and the same master functional acquires a free-energy-like form: this is the \FEP{} face. The physical claim of \QiEP{} is therefore that the apparent plurality of principles emerges because different boundary tasks emphasize different sectors of the same path functional. Chemistry provides the cleanest example because the same device may operate in all three modes depending on timescale and coarse-graining.

\subsection{\QiCON{} signatures in chemistry and protocells}

Chemistry is also where \QiCON{} becomes sharply testable. Enzymes, ion channels, autocatalytic sets, and membrane-active protocells often display multiple relaxation scales, conformational substates, and intermittent switching. The \GDOS{} expectation is that the reduced noise kernel is not generically single-scale. Instead one should often see
\begin{equation}
S_X(f)
\sim
\begin{cases}
S_0, & f\ll f_L,\\[4pt]
f^{-\alpha}, & f_L \ll f \ll f_H,\\[4pt]
f^{-\beta}, & f\gg f_H,
\end{cases}
\label{eq:appJ-spectrum}
\end{equation}
with \(f_L\) set by the longest architectural closure time of the system and \(f_H\) by the fastest internal port-gating or conformational switching scale. Likewise residence-time distributions may exhibit a scale window
\begin{equation}
P(\tau)\propto \tau^{-1-\mu},
\qquad
0<\mu<2,
\label{eq:appJ-levy}
\end{equation}
before finite-size or finite-memory cutoffs restore integrability. In the present framework such laws are not treated as mysterious empirical curiosities. They are the expected traces of a multiscale attractor lattice whose degrees of freedom have been only partially projected to the observer's boundary.

\subsection{From protocells to living systems}

A protocell or minimally living system is precisely the point at which chemical runtime acquires the full \SeOS{} interpretation. The boundary is no longer merely conceptual; it is a maintained membrane or other selective enclosure. The internal chemistry is no longer merely reactive; it becomes self-referential in the limited but important sense that part of the throughput is spent preserving the ports through which throughput occurs. This is the first unmistakable appearance of a selfish open system. The system persists not by violating thermodynamics but by routing gradients through a geometry that keeps its own boundary in being \citep{Schrodinger1944,Lane2015}. In this sense, life at its lowest layer is a chemical proof-of-principle of the \GDOS{} program.

\begin{table}[H]
\centering
\caption[Chemistry-scale diagnostics, claim by claim.]{Chemistry-domain diagnostics. Each row is one claim of the framework, with an operational observable that bears on it and what would count as support or as tension. The third column is written to be symmetric on purpose --- it says what would go against the claim as well as what would go for it --- because the chemistry scale is where the framework's one sharp number lives and it should not be tested only in the direction that confirms it.}
\label{tab:appJ-diagnostics}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.19\textwidth} >{\raggedright\arraybackslash}p{0.20\textwidth} X}
\toprule
\textbf{Claim} & \textbf{Operational observable} & \textbf{What would count as support or tension}\\
\midrule
\QiQUA{} current locking & Joint measurement of energy, redox/ion transfer, structural turnover, and selectivity events & Support: one elementary cycle class approximately collapses these four ledgers to a common event count. Tension: no sensible coarse-graining yields such locking.\\
\QiEP{} unification & Fit of short-path, dissipative, and inference-like objectives within one master cycle functional & Support: regime changes can be modeled by reweighting one functional. Tension: mutually incompatible objectives are required for the same cycle class.\\
\QiCON{} multiscale statistics & Power spectra, waiting-time tails, and state-residence histograms & Support: two knees and finite scale windows correlate with identifiable structural scales. Tension: fitted exponents float independently of architecture.\\
\SeOS{} persistence & Boundary-maintenance cost vs. throughput budget & Support: a calculable maintenance sector exists and predicts loss-of-identity thresholds. Tension: persistence has no ledger representation.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Concluding remark on the chemistry layer}

The strongest reason to emphasize chemistry in a landmark \GDOS{} paper is methodological. Chemistry is the first domain in which the theory must cease to be merely a philosophical unifier and become a quantitative runtime. Concentrations, currents, binding events, redox changes, membrane potentials, yields, and fidelity measures are all observable. If the framework cannot become precise here, then the later claims about life and mind would be premature. Conversely, if it succeeds here, then the ascent to biological and cognitive systems becomes natural rather than rhetorical.

\clearpage


\section{Neural, cognitive, and consciousness-oriented open systems}
\label{app:neural-domain}

The main text introduced \QiHAL{} and the identity \(\OH+\IH=\EIC\) as the architectural export of \GDOS{} to internal-model systems. The present appendix slows that argument down and states it in a more explicitly operational form. The goal is not to claim that one already possesses a final quantitative theory of consciousness. The goal is more modest and, in a way, more demanding: to show that once the \GG{}/\GDOS{} runtime is accepted, nervous systems are \emph{not} exceptional in requiring entirely new metaphysical principles. They are exceptional only in architectural depth. The same boundary logic that governs detectors and cells now runs through large recurrent systems that sense, remember, predict, and act over many timescales \citep{Shannon1948,AmariNagaoka2000,Friston2010,Tononi2004}.

\subsection{A neural System Instance Packet}

Let a neural open system be described at reduced scale by fast activation variables \(x_i\), phase variables \(\theta_\alpha\), synaptic or coupling variables \(w_{ij}\), slow memory traces \(m_b\), and action-readiness variables \(a_c\). One may then choose
\begin{equation}
z^A
=
\bigl(
x_1,\dots,x_{N_x},
\theta_1,\dots,\theta_{N_\theta},
w_{11},\dots,w_{N_xN_x},
m_1,\dots,m_{N_m},
a_1,\dots,a_{N_a}
\bigr),
\label{eq:appK-state}
\end{equation}
together with the packet
\begin{equation}
\SIP_{\mathrm{neural}}
=
\bigl(
\mathcal{M}_{\mathrm{neural}},
z^A,
G_{AB}^{(\mathrm{neural})},
\mathcal A_A^{(\mathrm{neural})},
\FGFF^{(\mathrm{neural})},
\Sigma^{AB}_{(\mathrm{neural})},
\mathcal O_{\mathrm{neural}}
\bigr).
\label{eq:appK-sip}
\end{equation}
Here \(G_{AB}^{(\mathrm{neural})}\) may be interpreted as the effective information or control metric on the reduced state space; \(\mathcal A_A^{(\mathrm{neural})}\) encodes phase transport, recurrent loops, and nonconservative input drive; \(\FGFF^{(\mathrm{neural})}\) combines metabolic cost, prediction or mismatch cost, structural regularization, and task constraints; \(\Sigma^{AB}_{(\mathrm{neural})}\) records unresolved variability and fluctuating input; and \(\mathcal O_{\mathrm{neural}}\) specifies the readout dictionary, such as spikes, local fields, imaging signals, behavioral reports, reaction times, or action trajectories.

The universal \GDOS{} current law again applies:
\begin{equation}
\partial_t p(z,t)+\nabla_AJ^A(z,t)=0,
\qquad
J^A=-p\,M^{AB}\nabla_B\Xi+p\,\Omega^{AB}\nabla_B\Psi-\frac12\nabla_B(\Sigma^{AB}p).
\label{eq:appK-current}
\end{equation}
The essential point is that a neural reduced model needs all three sectors. Pure gradient descent misses oscillation and phase transport. Pure Hamiltonian rotation misses irreversibility, gain control, and noise-driven switching. Pure diffusion misses learned structure and directed inference. A brain-scale runtime is therefore an excellent domain in which the gradient, curl, and noise sectors must coexist.

\begin{table}[H]
\centering
\caption[Neural variables, mapped to the runtime.]{One possible mapping from neural variables to the universal runtime grammar. The left column names the runtime object and the right a neural or cognitive interpretation of it. \emph{One possible} is the operative phrase: the mapping is offered as an orientation for readers from neuroscience, the paper does not derive a neural quantum of information, and no result here depends on this table being the right assignment.}
\label{tab:appK-runtime-map}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.21\textwidth} X}
\toprule
\textbf{Runtime object} & \textbf{Neural or cognitive interpretation}\\
\midrule
\(\mathcal{M}_{\mathrm{neural}}\) & Reduced manifold of activation, phase, memory, and action-readiness states.\\
\(G_{AB}^{(\mathrm{neural})}\) & Effective information or control geometry; local distinguishability and transition cost.\\
\(\mathcal A_A^{(\mathrm{neural})}\) & Recurrent phase transport, oscillatory coupling, and loop-induced nonconservative structure.\\
\(\FGFF^{(\mathrm{neural})}\) & Joint ledger of metabolic cost, mismatch cost, structural regularization, and reward or viability terms.\\
\(\Sigma^{AB}_{(\mathrm{neural})}\) & Trial-to-trial variability, background fluctuations, and unresolved microstate contributions.\\
\(\mathcal O_{\mathrm{neural}}\) & Readout dictionary from spikes or fields to behavior, report, or other recorded boundary observables.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{\DGI{}, \GSSB{}, and GA in neural dynamics}

The triad GRIP--GA acquires a particularly vivid form in neural systems. \DGI{} says that the reduced generator should preserve its lawful content under admissible reparameterizations of coding variables, population coordinates, or gauge-like phase conventions. One should be free to rewrite the same dynamics in firing-rate variables, latent coordinates, or phase-amplitude coordinates without changing the physical predictions for observables in \(\mathcal O_{\mathrm{neural}}\). \GSSB{} then states that concrete sensory context, task demands, developmental constraints, or injury can reduce the symmetry of the admissible generator, selecting some modes and suppressing others. GA is the observable outcome: a stable or metastable attractor family corresponding to a memory, percept, motor plan, or global action policy.

A practical way to write this is to separate the reduced drift into background-symmetric and context-selective parts:
\begin{equation}
\dot z^A
=
\mathcal V^A_{\mathrm{sym}}(z)
+
\mathcal V^A_{\mathrm{ctx}}(z;u)
+
\eta^A,
\label{eq:appK-symmetry}
\end{equation}
where \(u\) denotes input or task context. \DGI{} constrains how \(\mathcal V^A_{\mathrm{sym}}\) and \(\mathcal V^A_{\mathrm{ctx}}\) transform under admissible coordinate changes. \GSSB{} is the statement that nonzero \(u\), together with boundaries and long-run learning, reduces the effective symmetry of \(\mathcal V^A\). GA is the resulting landscape of selected basins. The framework therefore predicts that neural function should not be understood as motion on a static graph only; it is motion on a graph whose effective symmetry class is continually being selected by boundary conditions.

\subsection{\QiHAL{} and the minimal adaptive closure}

The main text introduced \MePMoS{} as the minimal adaptive loop of Memory, Prediction, Motion, and Sensing. Here the usefulness of that decomposition can be made explicit. Let
\begin{align}
m_{t+\Delta t} &= \mathcal M(m_t,x_t,o_t),
\label{eq:appK-memory}\\
\hat o_{t+\Delta t} &= \mathcal P(m_t,x_t,a_t),
\label{eq:appK-prediction}\\
a_{t+\Delta t} &= \mathcal A(m_t,x_t,\hat o_t,o_t),
\label{eq:appK-action}\\
o_{t+\Delta t} &= \mathcal S(a_t,\mathrm{world}_t)+\nu_t.
\label{eq:appK-sensing}
\end{align}
Equations~\eqref{eq:appK-memory}--\eqref{eq:appK-sensing} do not presuppose one specific neuroscience model. They state only that an adaptive open system capable of internal modeling must contain modules that play these four roles. \QiHAL{} then adds higher-order structure on top of that loop: multiscale memory, a graph of attractor basins, phase-coded communication among scales, and switching operations that let the system pass from one effective mode to another without catastrophic loss of coherence. In other words, the \MePMoS{} loop is the minimal closure, while the full holographic attractor lattice is the structured enlargement required for large-scale cognition.

\subsection{Outer hologram, inner hologram, and evolving inner cosmos}

The most delicate claim of the present program is not that the world contains measurement. That much was already established in \AFive{}. The more ambitious claim is that sufficiently complex open systems perform \emph{internal} measurement as well. To state this carefully, define the outer-hologram map
\begin{equation}
o_t
=
\Pi_{\mathrm{\OH{}}}\!\left[\mathrm{world}_t;a_t\right]
+
\nu_t,
\label{eq:appK-oh}
\end{equation}
where \(o_t\) is the boundary record delivered by the world conditional on the system's current action and sensory geometry. This is not passive reception; the action variable \(a_t\) explicitly enters the projection. Next define the inner-hologram dynamics
\begin{equation}
h_{t+\Delta t}
=
\mathcal K_{\mathrm{\IH{}}}(h_t,m_t,o_t,a_t),
\label{eq:appK-ih}
\end{equation}
which updates an internal reconstructive state \(h_t\) on the basis of past internal state, memory, present sensory evidence, and current action. Finally define the slowly accumulated inner world-model
\begin{equation}
\mathcal W_{t+\Delta t}
=
\mathcal U(\mathcal W_t,h_t,r_t),
\label{eq:appK-eic}
\end{equation}
with \(r_t\) the currently globally effective or reportable internal mode. The triple \((\Pi_{\mathrm{\OH{}}},\mathcal K_{\mathrm{\IH{}}},\mathcal U)\) is the operational meaning of \(\OH+\IH=\EIC\).

Within this framework, ``consciousness'' is not introduced as a separate ontological substance. It is introduced as a regime of internal measurement in which some internal attractor state \(r_t\) becomes sufficiently stable, globally effective, and action-relevant that it functions as a boundary-visible state \emph{for the system itself}. A simple operational criterion is
\begin{equation}
\tau_{\mathrm{dwell}}(r_t)>\tau_{\mathrm{cycle}},
\qquad
\Gamma_{\mathrm{global}}(r_t)>\Gamma_c,
\qquad
\mathcal R_{\mathrm{action}}(r_t)>\mathcal R_c,
\label{eq:appK-conscious}
\end{equation}
where \(\tau_{\mathrm{cycle}}\) is the characteristic internal \MePMoS{} cycle time, \(\Gamma_{\mathrm{global}}\) is a measure of cross-module causal reach, and \(\mathcal R_{\mathrm{action}}\) is the degree to which the state constrains future selection. Equation~\eqref{eq:appK-conscious} is not claimed to be a final empirical law. It is a placeholder for the kind of measurable criterion the present framework demands.

\subsection{\QiCON{} and the spectrum of neural timescales}

The neural domain is also one of the clearest places to test \QiCON{}. Empirically, brain activity often exhibits scale-rich spectra rather than one clean resonance. In the present framework this is not an accidental superposition of unrelated processes. It is the spectral signature of an attractor lattice whose modes are arranged across nested timescales. The simplest predicted form is again a two-knee law,
\begin{equation}
S_{\mathrm{neural}}(f)
\sim
\begin{cases}
S_0, & f\ll f_{\mathrm{global}},\\[4pt]
f^{-\alpha}, & f_{\mathrm{global}}\ll f\ll f_{\mathrm{local}},\\[4pt]
f^{-\beta}, & f\gg f_{\mathrm{local}},
\end{cases}
\label{eq:appK-two-knee}
\end{equation}
where \(f_{\mathrm{local}}\) is associated with local recurrent loops and \(f_{\mathrm{global}}\) with long-range integration cycles. The detailed exponents \(\alpha\) and \(\beta\) are not universal numbers; they are effective summaries of architecture. The critical claim is instead that knee locations should correlate with explicit anatomical or functional hierarchy rather than floating independently of it.

\begin{table}[H]
\centering
\caption[Candidate criteria for a reportable state.]{Candidate runtime criteria for reportable internal states. Five criteria run down the first column, with an operational reading of each and the reason it matters for the coupling of outer and inner holograms. They are labeled candidates because none is derived: the table's function is to make a hard question tractable by proposing criteria that can be measured and argued with, and its third column says what each would settle if it held.}
\label{tab:appK-reportable}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.17\textwidth} >{\raggedright\arraybackslash}p{0.25\textwidth} X}
\toprule
\textbf{Criterion} & \textbf{Operational reading} & \textbf{Reason it matters for \OH/\IH/\EIC{}}\\
\midrule
Metastable dwell time & State persists longer than the characteristic update cycle & Without dwell time the system cannot treat the state as an internal measurement record.\\
Broad causal reach & State influences multiple modules or scales & A purely local transient is not globally effective in the sense needed for report or coordinated action.\\
Action coupling & State constrains downstream policy or motor selection & Internal measurement matters only if it can reorganize future interaction with the world.\\
Reconstructive revision & State updates under new evidence without total collapse & The \IH{} must behave like a revisable model, not a frozen label.\\
Energy-budget support & State maintenance consumes and justifies a measurable metabolic budget & The state must live on the same \GFF{} ledger as the rest of the system.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Scope, caution, and falsifiability}

The present appendix does not claim that every neural correlate of consciousness is already derivable from first principles. It makes a narrower and testable claim. If reportable conscious-like states exist, then they should appear as a special class of globally effective attractor states in a multiscale internal-hologram architecture. If future experiments show that reportable states have no special dwell time, no broadened causal reach, no privileged relationship to action selection, and no identifiable role in the internal model, then the \OH/\IH/\EIC{} proposal would be in serious trouble. A landmark theory paper must say that clearly.

\clearpage


\section{Ecological, social, and technological collective systems}
\label{app:collective-domain}

The universal-runtime claim of \GDOS{} is strongest, and most vulnerable, when it is exported beyond laboratory physics or biochemistry into collective systems. The danger here is obvious: one can too easily use scientific vocabulary metaphorically. The purpose of the present appendix is therefore to draw a strict boundary between legitimate export and category error. The theory does \emph{not} claim that a society literally carries the same microphysical gauge charges as a single molecule. What it claims is that a collective system, once described as a finite open system with lawful ports, resources, memory, communication, and action channels, can be written on the same runtime grammar as any other reduced system. The reward for this discipline is that one may compare particles, cells, brains, and institutions without pretending they are identical. They are instead different realizations of one lawful open-system syntax \citep{Shannon1948,Kauffman1993,Dawkins1976,Vicsek1995,Friston2010}.

\subsection{A collective System Instance Packet}

Consider a collective system with population variables \(N_i\), resource variables \(R_a\), communication variables \(C_\mu\), memory or institutional variables \(M_\lambda\), and action or policy variables \(A_\kappa\). Define
\begin{equation}
z^A
=
\bigl(
N_1,\dots,N_{N_N},
R_1,\dots,R_{N_R},
C_1,\dots,C_{N_C},
M_1,\dots,M_{N_M},
A_1,\dots,A_{N_A}
\bigr).
\label{eq:appL-state}
\end{equation}
Then the runtime packet reads
\begin{equation}
\SIP_{\mathrm{coll}}
=
\bigl(
\mathcal{M}_{\mathrm{coll}},
z^A,
G_{AB}^{(\mathrm{coll})},
\mathcal A_A^{(\mathrm{coll})},
\FGFF^{(\mathrm{coll})},
\Sigma^{AB}_{(\mathrm{coll})},
\mathcal O_{\mathrm{coll}}
\bigr).
\label{eq:appL-sip}
\end{equation}
The interpretation of each slot is now mesoscopic. \(G_{AB}^{(\mathrm{coll})}\) measures the local cost or sensitivity of displacing the collective state; \(\mathcal A_A^{(\mathrm{coll})}\) packages nonconservative institutional loops, delayed-control cycles, or technological feedbacks; \(\FGFF^{(\mathrm{coll})}\) becomes the combined ledger of energetic or resource cost, entropy or uncertainty, structural maintenance, and coordination cost; \(\Sigma^{AB}_{(\mathrm{coll})}\) encodes unmodeled disturbances and heterogeneity; and \(\mathcal O_{\mathrm{coll}}\) is the explicit record dictionary: demographic rates, resource flows, communication delays, compliance, failure cascades, and resilience metrics.

\subsection{Dynamic equations and the meaning of the four channels}

At collective scale one may write the reduced equations schematically as
\begin{align}
\partial_t N_i &= B_i(z)-D_i(z)+\sum_j\!\left(F_{ji}(z)-F_{ij}(z)\right)+\eta_i,
\label{eq:appL-pop}\\
\partial_t R_a &= \Phi_a^{\mathrm{in}}(z)-\Phi_a^{\mathrm{out}}(z)-C_a(z)+\zeta_a,
\label{eq:appL-resource}\\
\partial_t M_\lambda &= \mathcal T_\lambda(M,C,A)-\mathcal L_\lambda(M)+\chi_\lambda,
\label{eq:appL-memory}
\end{align}
with analogous equations for communication and policy variables. The four-channel ledger then survives as a \emph{bundled} description. \(\JE\) becomes the net energetic or resource-throughput channel required to maintain the system. \(\Je\) becomes the signaling or control-carrier channel, descended from lower-level charged transport but now expressed as the effective communication or command throughput available at the chosen resolution. \(\JB\) becomes the structural-material maintenance channel: the one counting the throughput needed to preserve the physical or organizational body of the system. \(\JQi\) becomes the effective information-throughput channel: update, coordination, discrimination, and model revision.

The export rule is therefore strict: the same symbols may be used across scales only when the observer has supplied a lawful bundling map from lower-level physical events to higher-level ports. Without such a bundling map, the notation would indeed be metaphorical and should be rejected. This insistence on explicit bundling is exactly how \GDOS{} tries to remain scientific when exported to difficult domains.

\begin{table}[H]
\centering
\caption[The four channels, read collectively.]{Permissible interpretation of the four current channels in collective systems. Each row is one channel, with a collective-scale reading. \emph{Permissible} carries the fence: these readings are consistent with the framework but are not derived from it, and the paper does not claim that a social system locks its four currents through the same constant. The table is included because the question is asked often enough that leaving it unanswered invites a worse answer.}
\label{tab:appL-currents}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.14\textwidth} X}
\toprule
\textbf{Channel} & \textbf{Permissible collective-scale interpretation}\\
\midrule
\(\JE\) & Resource or energy throughput needed to keep the system operating and to power adaptation.\\
\(\Je\) & Effective signaling or control throughput available to coordinate distant parts of the system.\\
\(\JB\) & Material and structural maintenance throughput preserving the body, infrastructure, or persistent organization.\\
\(\JQi\) & Information-update throughput required for discrimination, learning, planning, and coordinated response.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{\QiEP{} for collective adaptation}

The value of the runtime language becomes clearer when one considers adaptation. Collective systems often appear to optimize one quantity in one regime and another quantity in another regime. A firm may seem to minimize time to response during crisis, maximize throughput efficiency during routine operation, and minimize predictive mismatch during planning. An ecosystem may appear to maximize persistence of material cycling while minimizing vulnerability to shocks. From the perspective of \QiEP{}, such plurality is expected. One may write a collective path functional
\begin{equation}
\mathcal S_{\mathrm{coll}}[z]
=
\int\!\dd t\,
\left[
\frac12 g_{AB}(z)\dot z^A\dot z^B
+
\dot z^A \mathcal A_A^{(\mathrm{coll})}
+
\Lambda_{\mathrm{maint}}(z)
+
\Lambda_{\mathrm{coord}}(z,\dot z)
+
\Lambda_{\mathrm{forecast}}(z)
\right],
\label{eq:appL-path}
\end{equation}
where the last three terms denote maintenance cost, coordination cost, and forecasting or model-mismatch cost. Short-response crises highlight the metric term and look PLT-like. Throughput-dominated steady operation highlights dissipation and looks MEPP-like. Long-horizon planning highlights mismatch reduction and looks FEP-like. The programmatic claim is again that one does not need separate constitutions for these cases; one needs only to understand which sector of the same path functional is dominant.

\subsection{\QiCON{}, heavy tails, and cascade structure}

Collective systems are famous for burstiness, heavy tails, and cascade phenomena. The \GDOS{} interpretation is not that such signatures are universally identical. It is that they are typical of multiscale attractor lattices with partial observation and delayed coupling. If communication, infrastructure, ecological turnover, and policy revision each operate on distinct timescales, then an observed aggregate signal should again display a knee-structured spectrum and possibly finite-window power laws. The generalized form is
\begin{equation}
S_{\mathrm{coll}}(f)
\sim
\begin{cases}
S_0, & f\ll f_{\mathrm{inst}},\\[4pt]
f^{-\alpha}, & f_{\mathrm{inst}}\ll f\ll f_{\mathrm{local}},\\[4pt]
f^{-\beta}, & f\gg f_{\mathrm{local}},
\end{cases}
\label{eq:appL-spectrum}
\end{equation}
where \(f_{\mathrm{inst}}\) is controlled by the slowest institutional or ecological revision cycle and \(f_{\mathrm{local}}\) by the fastest local exchange or signaling loop. The same logic applies to burst-size distributions and waiting times. The important claim is not that every collective system obeys the same exponent. The important claim is that the existence and location of the scale window should be constrained by explicit architecture.

\subsection{Where the framework can fail in collective domains}

Because collective systems invite overinterpretation, the falsifiers must be stated sharply. The framework fails in this domain if the chosen bundling map is not explicit, if the proposed \SIP{} packet does not improve prediction or diagnosis, if the current channels cannot be measured even approximately, or if the fitted scale-covariant laws bear no relationship to identifiable architecture. It also fails if the theory is used to smuggle teleology into a system that has no lawful memory, no meaningful action channel, and no persistent attractor structure. In such cases the honest conclusion would be that the export has exceeded its domain of validity.

\subsection{Why the collective appendix still matters}

Even with those cautions, the collective appendix is indispensable for a landmark paper. The reason is simple. If \GDOS{} is truly a universal language for open systems, it must tell the reader what \emph{sort} of export to attempt at the largest scales and what safeguards prevent abuse. A theory that stops at brains would still be a powerful theory. A theory that can say, in explicit runtime language, how ecological, social, and technological systems should be represented becomes something larger: a genuine bridge language across the five-story pagoda and beyond.

\clearpage


\section{Notation concordance and benchmark protocols}
\label{app:concordance-benchmarks}

This last appendix records the editorial and methodological consequences of the present polishing round. It serves three purposes. First, it fixes notation that had previously appeared in parallel drafts with minor variants. Second, it states the line-breaking and layout conventions adopted in the source so that later revisions can be compiled with fewer overfull boxes and fewer accidental notational collisions. Third, it translates the entire manuscript into a practical benchmark program across the five-story pagoda. In that sense the appendix is both a concordance and a forward-looking protocol.

\subsection{Notation concordance adopted in the polishing round}

The notational rules below should be treated as authoritative for the current revision.

\begin{table}[H]
\centering
\caption[Notation concordance: the preferred form of each object.]{Notation concordance. Each row names an object, gives the preferred notation for it and a comment. The table exists because the framework's objects appear in several companion papers under slightly different symbols, and a reader moving between them needs one place that says which form this paper uses and why. It is a maintenance object rather than a result, and it is placed last for that reason.}
\label{tab:appM-concordance}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.20\textwidth} >{\raggedright\arraybackslash}p{0.18\textwidth} X}
\toprule
\textbf{Object} & \textbf{Preferred notation} & \textbf{Comment}\\
\midrule
Physics natural-unit current ratio & \(\JpE:\Jpe:\JpB:\JpQi = 1:-1:-2/5:-2/5 = 5:-5:-2:-2\) & The order \((E,e,B,Qi)\) is aligned with \BIsix{}.\\
Chemistry natural-unit current lock & \(\JE/\QB=\Je=\JB=\JQi\) & This is the preferred form of the Arisaka Equation in chemistry-scale discussions.\\
Arisaka constant & \(\QB = (2/5)(\ERyd/K)\), \(K=35\) & The symbol \(\QB\) is preferred over earlier epsilon-based variants.\\
Information current & \(\JQi\) & Use \(J_{Qi}\), not shortened or ambiguous alternatives.\\
Triad order & \DGI{} \(\rightarrow\) \GSSB{} \(\rightarrow\) GA & This order is conceptually important and should be preserved in prose and figures.\\
\QiFour{} ordering & \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{} & Use this canonical ordering unless a local pedagogical reason requires otherwise.\\
Dual holography identity & \(\OH+\IH=\EIC\) & The equation is structural, not merely rhetorical.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Benchmark program}
\label{subsec:benchmark-pointer}

The five-story pagoda benchmark program (one candidate test per layer) is consolidated with the falsifiability program of Part~\ref{part:assessment}: see Table~\ref{tab:appendix-falsifiability} of Appendix~\ref{app:falsifiability-matrix} for the integrated falsifiability matrix that supersedes earlier per-layer benchmark tables.

%% [v11.1] Acknowledgment relocated to the end of the Conclusions Part via the standard acknowledgments macro (four-Part tail).

\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a hostile particle theorist, systems biologist, origin-of-life researcher, neuroscientist, and methodologist would each raise against the \QiFour{} runtime --- on the reality of the information sector, the Arisaka constant, the inevitability of life, the theory of consciousness, and the falsifiability of the whole --- answered, with pointers into the paper}
\label{part:objections}
\label{sec:common-objections}

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

A construction that claims to extend one boundary law into a universal runtime for chemistry, life, brains, and society --- and to price information as a gauge current with a definite energy quantum --- invites the most demanding scrutiny; it would not be honest scholarship to leave the obvious objections unstated. We collect here the strongest adversarial readings of the \QiFour{} thesis --- the ones a particle theorist, a systems biologist, an origin-of-life researcher, a neuroscientist, and a skeptical methodologist would each raise on a careful reading --- and answer each, with explicit pointers to the theorem, table, open problem, or companion paper that bears the weight. Where the honest answer is ``not yet,'' we say so and name the open item (in particular Open Problems O1--O9 and E1--E5 of \S\ref{sec:open-problems}). The objections are grouped by the discipline from which they are most naturally raised.

\bigskip

%% =====================================================================
%% Cluster 0 -- For the curious reader
%% =====================================================================
\bigskip
\section{For the curious reader: the big questions, plainly answered}
\label{sec:qa-curious}

The twenty-five questions that follow are not a specialist's. They are the
ones a thoughtful reader without a physics degree actually carries
into a paper like this one: whether information really costs anything and who pays, whether
life is inevitable or merely possible, what a memory is when you look at
it physically, whether consciousness has any business appearing in an
equation, and what is left of biology if a cell turns out to be running
the same machine as a star. Several of these have been settled in
popular science in directions this paper does not take, and a reader
is owed plain answers before the specialist objections begin. They
are met first, and plainly.

The hardest of the twenty-five is the fourth --- whether an account that reaches from particles to minds has
explained anything about minds, or has merely used the word. The response concedes the
difficulty rather than dissolving it, and says which part of it the
paper can carry and which part it cannot. Where a claim rests on a
step that is not closed, the response says so and names the step;
where the honest answer is that the framework does not reach the
question, it says that too.


\bigskip
\noindent\textbf{Question 1 (Q-Qi4-1) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Does information really have a price? That sounds like a metaphor.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is not a metaphor, and the demonstration is sixty years old and
has been measured. \\ Landauer showed in 1961 that erasing one bit in a
system at temperature $T$ must dissipate at least $kT\ln 2$ of energy.
Not as an engineering limit but as a consequence of the second law: if
erasure were cheaper one could build a machine extracting work from a
single heat bath, which is forbidden. The bound has since been confirmed
directly, in colloidal particles and single-electron devices. Bennett
used the same insight to resolve Maxwell's demon --- the demon does not
beat the second law because its memory must eventually be cleared, and
clearing costs exactly what the sorting saved. \\ So information is
already accountable in the same ledger as energy. What is new here is
the exchange rate. Landauer's price depends on the temperature of the
bath, so it is a different number in a warm cell than in a cold
detector. The claim in this paper is that there is a further,
temperature-independent cost per event, fixed by the geometry rather
than by the surroundings --- roughly a sixth of an electron-volt per
elementary informational event. \\ That is a much stronger claim than
Landauer's and correspondingly more exposed. It says every elementary
event of this channel is billed at one rate, whether it happens in an
enzyme, an electron transfer, or a memory write. One catalytic event
whose four-way accounting misses by more than a fifth would end it.
\end{objqabody}

\bigskip
\noindent\textbf{Question 2 (Q-Qi4-2) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You price a bit at about a sixth of an electron-volt, and the energy of the molecule life runs on is close to that. Isn't that suspicious?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should be, and the suspicion is the correct first response.
\\ Here is the situation without decoration. The constant is assembled
from the energy scale of the hydrogen atom and an integer fixed by a
closure condition several papers earlier, for reasons that had nothing
to do with biology and were fixed before biology was mentioned. It comes
out at about 0.155 electron-volts per elementary event. The free energy
released by the molecule that powers essentially every process in every
living thing on this planet is within one and a half per cent of that.
\\ Either that is a coincidence or it is the most interesting sentence
in the paper, and nothing about how the number was obtained settles
which. What settles it is whether the same constant works elsewhere.
The claim is not about one molecule; it is that every elementary event
of this kind is billed at the same rate --- an enzyme turnover, an
electron transfer down a respiratory chain, a molecular motor step, a
memory write in a neuron. Those are different systems with different
chemistry, and the framework says one number prices them all.
\\ So the honest position is: do not be persuaded by the agreement. Be
persuaded, if at all, by whether the second, third and fourth cases come
out at the same rate. Those measurements are ordinary biophysics rather
than frontier physics, and they are the reason the claim is worth
making at all.
\end{objqabody}

\bigskip
\noindent\textbf{Question 3 (Q-Qi4-3) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is life inevitable, or merely possible?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Neither in the form the question usually means, and the precise
version is worth more than either. \\ What is claimed is a statement
about a search rather than about a history. If a universe keeps
generating gradients, and if configurations capable of holding
themselves together occupy more than a vanishing fraction of what is
reachable, then such configurations are where the search ends up. That
is inevitability in the ensemble sense --- a statement about what happens
given enough exploration, not a prediction about any particular planet.
\\ What is emphatically not claimed: that any given world produces life.
Nothing here says that of the Earth, and nothing here would be
embarrassed by a sterile galaxy. Frequency is not the framework's
subject and it does not compute one. \\ The condition doing the work is
the one to watch, because it is where the claim could fail: that the
viable region has positive measure and is repeatedly sampled. That is
stated as a hypothesis rather than proved, it is flagged as such in the
assessment, and quantifying it is one of the named open problems. A
reader who wants the strongest version --- life is written into physics
--- will not find it here, and should be suspicious of anyone offering
it.
\end{objqabody}

\bigskip
\noindent\textbf{Question 4 (Q-Qi4-4) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is a memory, physically?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not a place where something is stored, which is the picture worth
giving up first. \\ Storage suggests a location where a thing is put and
from which it is later retrieved unchanged. Almost nothing about
biological memory works that way, and the account here explains why it
could not. What is held is not a set of items but a set of relations ---
differences between phases rather than the phases themselves. That is
why the same content survives being carried on a completely different
rhythm, so a chemical system, a nervous system and a machine can hold
the same thing on hardware with nothing in common. It is why damage
degrades a memory rather than deleting a list. And it is why ``where is
it stored'' has no good answer, not because the answer is hidden but
because the question presupposes the wrong kind of object. \\ What makes
something a memory, on this account, is structural: a quantity that
cannot be changed continuously. A count, once made, cannot be smoothly
undone --- there is no path from three windings to four --- and that
discreteness is what makes the record persist rather than drift away.
Memory lives on the part of the structure that closes into a loop, and
cost lives on the part that does not. \\ The consequence worth carrying
away is that remembering is not a special biological trick. It is what
any system does that has somewhere for a count to sit.
\end{objqabody}

\bigskip
\noindent\textbf{Question 5 (Q-Qi4-5) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is consciousness in your equations? And should I believe it if it is?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Something is, and it is much narrower than the word, and the
narrowness is the point. \\ This is the hardest question in the set, so
here is the boundary drawn as sharply as it can be. The framework
defines a measurable quantity: how much of a system's internal model of
its surroundings has been selected, stabilized, and made available to
the system's own next cycle of action. That is a graded quantity, it
applies to a bacterium and to a person by degree rather than by kind,
and it can in principle be estimated for either. The paper gives it a
name and studies it. \\ What it does not do --- and this is where every
account of this type either stays honest or does not --- is claim that
this quantity is what it is like to be the system. The question of why
there is experience at all, rather than the same processing occurring in
the dark, is untouched. The paper says so where the claim is made, and
the caution is not a formality: a reader who takes the quantity for an
explanation of experience has taken more than is offered, and the
paper's own assessment marks the gap. \\ So: believe the quantity to
whatever extent its predictions hold, which is a matter for
measurement. Do not believe that anything here has explained
consciousness. Nothing has, and the framework's willingness to say so is
the reason to trust the rest of it.
\end{objqabody}

\bigskip
\noindent\textbf{Question 6 (Q-Qi4-6) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``The free-energy principle has a difficult reputation. Why should I accept it here?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because here it is a consequence rather than a starting point, and
that changes what can be said against it. \\ The reputation is
deserved and worth stating. As usually presented, the principle says
that any self-organizing system minimizes a quantity called variational
free energy. Critics have argued for years that it is unfalsifiable ---
that the metric can be chosen after the fact, that the system boundary
can be drawn to suit, and that the existence of an internal model is
assumed rather than shown. Those are serious objections and they have
not been fully answered by the principle's proponents. \\ What is
different here is the direction of derivation. The principle is not
assumed; it is obtained as one reading of a functional that was
constructed from geometric data for other reasons. That has three
consequences, and each addresses one of the standing objections. The
metric stops being a modeling choice, because it comes from the
structure. The system boundary stops being a hopeful assumption,
because the projection defines it. And the existence of an internal
model becomes a testable condition rather than a premise. \\ The
caution belongs in the same breath. The inference reading fails exactly
where the entropy readings fail --- where circulation and gauge motion do
work against the ledger --- and the prospective-policy machinery that
much of the literature builds on top is not claimed here at all.
\end{objqabody}

\bigskip
\noindent\textbf{Question 7 (Q-Qi4-7) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If a cell runs the same machine as a galaxy, what is left of biology?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} All of it. What is claimed is a shared grammar, not a shared
subject, and confusing the two would make the claim both grander and
emptier than it is. \\ The framework says that certain questions have
the same form at every scale: what does this system spend, what does it
hold on to, what would kill it, where does it stop. Those questions
being shared does not make a cell a small galaxy any more than sharing
a grammar makes a legal contract a love letter. Everything that makes
biology biology --- which molecules, which pathways, which
contingencies of four billion years of history --- is untouched and
undetermined by anything here. \\ There is a real claim underneath and
it is worth stating carefully. It is that biology and physics would stop
being different \emph{kinds} of explanation while remaining entirely
different subjects. A biologist would still need to know which enzyme,
and no amount of geometry would supply it. What would change is that
the answer to ``why does this cost what it costs'' would be the same
kind of answer in both places. \\ The failure mode to watch for is the
one this kind of claim usually falls into: explaining biology in a way
that predicts nothing a biologist could check. The defense here is the
constant --- one number pricing an enzyme turnover, a motor step and a
memory write, measurable in each case separately.
\end{objqabody}

\bigskip
\noindent\textbf{Question 8 (Q-Qi4-8) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is the physical difference between a rock and a bacterium?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A margin, and the fact that the answer is a number rather than a
category is the substance of the claim. \\ The framework's version is
arithmetic. Something comes in; something is lost that cannot be
recovered; something must be spent repairing what has gone wrong. If the
first exceeds the other two on average, over a window long enough to
include the repairs, the thing persists as itself. If it does not, for
long enough, it stops. A rock has no repair term and does not need one;
it persists by being hard rather than by maintaining itself. A bacterium
maintains itself and would dissolve within minutes if it stopped.
\\ There is no vital principle in that sentence and no threshold of
complexity. There is a margin that can be positive or negative and that
can in principle be measured for a protocell, a coral reef, a company or
a state. \\ What quietly goes missing from the account is the moment at
which something becomes alive, and the absence is the result rather
than an omission. Anyone who has tried to say where the line falls ---
around a virus, a prion, a self-replicating crystal, an organelle that
used to be somebody else --- has found the line does not want to be
drawn. The framework's answer is that it is not there to be found. What
is there is a plane with things scattered across it, and our words are
names for regions of it rather than for kinds.
\end{objqabody}

\bigskip
\noindent\textbf{Question 9 (Q-Qi4-9) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is a thermostat conscious, then? Where is the line?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} There is no line, and a thermostat sits so far down the scale that
the question answers itself --- but the reason there is no line is more
interesting than either answer. \\ The framework's quantity is graded.
It measures how much of a system's internal model has been selected,
stabilized, and made available to the system's own next cycle. A
thermostat has a model of exactly one variable, holds it for as long as
the bimetallic strip takes to bend, and has no next cycle in any
meaningful sense. Its value is not zero as a matter of principle and it
is negligible as a matter of measurement. \\ Refusing to draw a line is
sometimes a way of avoiding a question, so it is worth saying why it is
not here. Every attempt to draw one has failed in the same way: whatever
criterion is proposed, cases are found just either side of it that
differ by nothing anyone can articulate. That is the signature of a
continuous quantity being forced into a binary. The framework takes the
continuity seriously instead. \\ What this does not do is tell you
whether a thermostat feels anything, and that limit should be repeated
rather than glossed. A graded physical quantity says how much of a
certain structure is present. Whether any amount of that structure is
accompanied by experience is a question the framework does not reach
and does not claim to.
\end{objqabody}

\bigskip
\noindent\textbf{Question 10 (Q-Qi4-10) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why do living things seem to fight entropy? Isn't that against the second law?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are not fighting it and never were, and the misunderstanding
is one of the most durable in popular science. \\ The second law says
the total entropy of an isolated system does not decrease. It says
nothing about any part. A refrigerator makes its contents colder and
more ordered every day by dumping more disorder into the kitchen than it
removes from the cabinet. Living things do exactly this: they take in
concentrated energy, use it, and radiate diffuse energy back out, and
every ordered thing they build is paid for out of the difference. The
Earth as a whole runs the same trade with the sun and deep space.
\\ So the appearance of struggle is an artifact of drawing the boundary
around the organism and forgetting the flow through it. There is no
countercurrent and nothing being resisted. \\ What this framework
contributes is a way of writing the trade with named terms rather than
describing it qualitatively --- intake, unrecoverable loss, repair cost
--- so that ``staying alive'' becomes an inequality that can be evaluated
for a given system rather than a metaphor. Whether the particular
inequality here is right is checkable. That the general picture is right
has not been in doubt among physicists since Schrodinger's lectures in
1943, and any account still describing life as defying the second law is
eighty years out of date.
\end{objqabody}

\bigskip
\noindent\textbf{Question 11 (Q-Qi4-11) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Does this explain the origin of life?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No, and the distinction between what it does and what that question
asks is worth being exact about. \\ What the framework supplies is a
set of conditions a system must satisfy to persist and to accumulate
structure, together with a price for the informational events involved.
That is a statement about what a candidate must look like, and it does
constrain: it says what the accounting has to balance, and it prices the
elementary events in units that a chemist could check. \\ What it does
not supply is a route. Which molecules, in which order, on which
mineral surface, under which conditions --- none of that follows from
anything here, and a framework that claimed it would be claiming to have
done chemistry it has not done. The origin of life is a historical
question about a particular planet, and constraints do not answer
historical questions. \\ There is one contribution worth naming because
it is unusual. If the price of an informational event is fixed by
geometry rather than by the local temperature, then the energy currency
any emerging system must use is fixed too --- not chosen by chemistry but
imposed on it. That converts ``why this molecule?'' from an accident of
early chemistry into a question with a possible answer. Whether the
answer is right is a laboratory matter, and the paper is careful to
present it as a claim rather than a result.
\end{objqabody}

\bigskip
\noindent\textbf{Question 12 (Q-Qi4-12) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should physics have anything to say about mind?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because minds are made of things that obey physics, which is a weak
premise almost nobody rejects, and because the specific structure at
issue is not mental. \\ The move here is narrower than it sounds. The
framework is about systems that maintain an internal model of what they
cannot directly reach, and correct it against what arrives at their
boundary. That description covers a bacterium holding a chemical trace
of whether conditions have been improving, a rodent maintaining a map
that survives its own turning, and a laboratory holding a theory of six
dimensions and checking it against instruments that report four. The
difference between these is depth, not kind. \\ That is a claim about
information processing, and information processing in physical systems
is a physical subject. It is not a claim about experience, and the paper
keeps the two apart deliberately and repeatedly. \\ Where the objection
does bite is on the risk of using the word to borrow credibility. A
framework that reaches from quarks to minds has an obvious temptation to
let the reach do the arguing. The protection is that the mind-adjacent
claims are the most conditional in the paper and are marked as such,
and that nothing else in the construction depends on them. A reader who
deletes the last Part entirely still has the four pillars and every
falsifier.
\end{objqabody}

\bigskip
\noindent\textbf{Question 13 (Q-Qi4-13) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If systems maximize entropy production, why does anything look designed?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because those are answers to different questions, and the apparent
tension dissolves once the question each is answering is named. \\ The
extremal principle picks out which of the available paths a system takes
given its constraints. It does not choose the constraints. Design-like
appearance comes from the constraints having been shaped, over long
periods, by the survival of whatever satisfied them --- and nothing about
that process is in tension with a path being extremal at every instant.
Water flowing downhill takes the steepest available route; that the
riverbed has an intricate shape is a fact about erosion over millennia,
not a violation of the flow. \\ There is a second point specific to this
framework. The extremal readings hold only where circulation and gauge
motion do no work against the ledger, and a system that has been shaped
by selection is precisely one that has arranged for that not to be true
--- it maintains circulation deliberately. So the framework predicts that
living systems will be exactly the systems where the simple extremal
statements fail, which is a checkable claim and a satisfying one.
\\ The appearance of design, on this account, is neither an illusion to
be explained away nor evidence of a designer. It is what accumulated
constraint looks like from outside, and the framework's contribution is
to say where the accumulation shows up in the equations.
\end{objqabody}

\bigskip
\noindent\textbf{Question 14 (Q-Qi4-14) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Can a machine be alive by your criteria?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By the persistence criterion, in principle yes, and the framework
is deliberate about not flinching from that. \\ The criterion is
arithmetic: intake exceeding unrecoverable loss plus repair cost, over a
window long enough to include the repairs. Nothing in it mentions
carbon, or reproduction, or having evolved. A machine that maintained
its own components, repaired its own damage and secured its own energy
would satisfy it, and no version of the criterion can be written to
exclude such a thing without smuggling in a preference. \\ Two
qualifications keep this from being a grand claim. First, no existing
machine comes close; every one depends on a maintenance and supply chain
that is not part of it, and drawing the boundary to include the chain
changes what is being assessed. Second, the persistence criterion and
the completeness index are different axes, and satisfying the first says
nothing about the second --- a thing can persist for a very long time
while running almost none of the machinery that makes a system
interesting. \\ There is a sharper point that the framework insists on
and that is easy to miss. The criterion says nothing about whether a
packet \emph{deserves} to persist. The same arithmetic describes a
healing wound and a tumor, a functioning institution and a captured one.
What separates them is which boundary you chose to draw, and the
framework is explicit that it does not choose for you.
\end{objqabody}

\bigskip
\noindent\textbf{Question 15 (Q-Qi4-15) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't our universe just one bubble among countless others?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It might be, and this paper is further from that question than
others in the cohort, but it has one thing to add. \\ The multiverse is
usually invoked to explain fine-tuning of the constants. There is a
version of the fine-tuning list that this paper bears on directly: the
entries about life. Those are often presented as the most striking ---
change this and there is no chemistry, change that and no stable
molecules --- and they are the entries where a selection effect is
sharpest, because nothing incapable of persisting could be present to
make a list. \\ What this framework adds is that the persistence
condition is arithmetic rather than delicate. A system persists if
intake exceeds loss plus repair. That is a much broader target than the
fine-tuning discussion usually assumes, and if the viable region has
positive measure rather than being a knife edge, the appearance of
extraordinary luck is reduced without any ensemble being needed.
\\ The condition is a hypothesis rather than a theorem, is flagged as
one, and quantifying it is a named open problem. So this is a
contribution to the argument rather than a resolution of it, and
presenting it as more would be exactly the overreach the question is
testing for.
\end{objqabody}

\bigskip
\noindent\textbf{Question 16 (Q-Qi4-16) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``The universe looks fine-tuned for life. Doesn't that point to a designer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It points to something requiring explanation, and this paper
contributes to the question from an unusual angle. \\ The cohort's main
answer is that the constants could not have been otherwise, which
removes the inference's premise. What this paper adds is about the other
half of the fine-tuning list --- the entries about life specifically.
\\ Much of what strikes people as tuned for life is a statement about
what it takes to persist, and persistence turns out to be an inequality
with named terms rather than a narrow coincidence. If the viable region
is not a knife edge, then a great deal of the apparent tuning is the
ordinary observation that things which cannot hold themselves together
are not around to be observed. That is a selection effect of a sharper
kind than the usual anthropic argument, because it follows from the
structure of persistence rather than from the requirements of
chemistry. \\ This is not an argument against a designer and should not
be recruited as one. Someone who believes on other grounds is untouched
by anything here. What changes is that part of the fine-tuning list
stops being evidence, because those entries could not have been observed
to be otherwise by anything capable of making a list.
\end{objqabody}

\bigskip
\noindent\textbf{Question 17 (Q-Qi4-17) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If everything is determined by geometry, is my future written? What happens to free will?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing in this paper touches your future, and this is the paper in
the cohort with the most reason to be careful about the question.
\\ The general answer holds: what is fixed is the rule book, not any
history within it, and a rule book with no alternatives says nothing
about which of an enormous number of permitted histories occurred.
\\ The specific answer is that this framework describes systems that
maintain an internal model and act on it, which is as close as physics
gets to the machinery people have in mind when they ask about agency ---
and what it says is that such a system's next action depends on a model
it built, from evidence it gathered, through a loop nothing outside it
computed. Whether that constitutes free will is a question about the
phrase rather than about the physics, and philosophers who have thought
about it longer than anyone here disagree. \\ What the framework does
supply, and it is worth having, is a reason the question is not
answerable by inspection. The state that determines what such a system
does next is distributed across a model that only that system holds,
and no shortcut computes it from outside. A universe can be lawful all
the way down and still contain systems whose next move is not available
in advance to anything but themselves.
\end{objqabody}

\bigskip
\noindent\textbf{Question 18 (Q-Qi4-18) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why is there something rather than nothing?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework does not touch it, and saying so plainly is more use
than a gesture. \\ Everything here is conditional --- given a structure
of a certain kind, systems that persist must satisfy this and pay that.
Why there is a structure at all sits outside the construction entirely.
That is not a gap further work closes; it is a different kind of
question. \\ What this paper can contribute is narrower and is not
nothing. If the price of an informational event is fixed, then the
question of why anything persists has an answer of the same kind as the
question of why anything falls: not a reason in the sense of a purpose,
but a constraint that makes the alternative unavailable. That converts
one member of the family of ``why'' questions into a member of the
family of ``what is forced'' questions. It leaves the deepest member
where it was. \\ The temptation, in a paper about emergence, is to say
that something is more stable than nothing and therefore wins. That move
is available, has defenders, and does not follow from anything here.
\end{objqabody}

\bigskip
\noindent\textbf{Question 19 (Q-Qi4-19) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``This explains an enormous amount from very little. Shouldn't that make you suspicious?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should and it does, and this paper carries the risk more heavily
than the others in the cohort. \\ The general warning applies: Eddington
derived the fine-structure constant from combinatorics and ended his
career defending it. But the specific risk here is different and worse.
A framework that reaches from gauge fields to metabolism to memory has
an enormous surface on which a coincidence can be mistaken for a
connection, and the reach itself is persuasive in a way that has nothing
to do with evidence. \\ The protection is the constant. It is one
number, fixed before biology entered the argument, and it prices every
elementary informational event at the same rate. That is an unusually
brittle commitment: an enzyme turnover, a motor step and a memory write
are different systems with different chemistry, and if their four-way
accounting does not come out at the same rate, the framework is finished
regardless of how much it appears to explain. \\ So keep the suspicion,
and aim it at the right place. Not at the reach, which is impressive and
proves nothing, but at whether the second and third systems come out at
the same price as the first. That is a bench measurement, and it is
where this paper lives or dies.
\end{objqabody}

\bigskip
\noindent\textbf{Question 20 (Q-Qi4-20) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should I take one person's framework seriously against thousands of physicists and biologists?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not because of who wrote it, and not despite it --- the same error
with opposite sign. \\ The practical question is fair. The useful filter
is exposure rather than headcount: a framework supported by thousands
that forbids nothing measurable is a worse use of your attention than
one supported by a single person that forbids specific things and names
the measurement that decides. Community size tracks funding history and
fashion and does not track correctness. \\ The cost of working outside a
community is real. Errors are found faster where more people look, and a
framework developed largely by one person carries a higher chance of a
mistake that a hundred readers would have caught quickly. This paper's
subject makes the risk worse, because it crosses fields, and the failure
mode of cross-field work is a claim that looks solid to each field
because each assumes the other checked it. \\ So do not take it on
trust, and do not take the breadth as evidence. Take the constant, which
needs no six-dimensional mathematics to evaluate: one number, claimed to
price elementary informational events in systems as different as an
enzyme and a neuron. Measure two of them. If the rates differ, the rest
does not matter.
\end{objqabody}

\bigskip
\noindent\textbf{Question 21 (Q-Qi4-21) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What would convince you that you are wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Specific things, and this paper's are unusually cheap to perform.
\\ One elementary catalytic event whose four-way accounting misses the
predicted rate by more than about a fifth would end the constant, and
with it the informational pillar. A clean violation of the thermodynamic
relations the framework says hold exactly in the lawful sector --- the
symmetry of a particular coupling matrix, a specific fluctuation
relation --- would break the variational pillar; these are bench
measurements in cold atoms or mesoscopic devices, not frontier physics.
A spectrum showing a single power law where the framework requires two
turns, with the turn positions fixed by the system's own slowest and
fastest retained rates, would break the third. And a memory-bearing
system that cannot be decomposed into the predicted normal form would
break the fourth. \\ What would not convince: a failure of the
consciousness-adjacent claims, which are already marked as the most
conditional in the paper and on which nothing else depends; or a
disagreement in a regime the framework says its readings do not cover
--- though that exception is itself constrained, since the framework
states exactly where the readings fail rather than retreating to it
after the fact.
\end{objqabody}

\bigskip
\noindent\textbf{Question 22 (Q-Qi4-22) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is the `self' a physical quantity here?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A boundary is, and the difference matters more than it might seem.
\\ What the framework can point at is a packet: a collection of
processes with an accounting that closes, maintaining itself against
loss, with a boundary defined by where the projection cuts rather than
by anything anatomical. That is a physical object in the only sense the
framework uses, and its persistence is computable. \\ What it cannot do
is tell you which packet is you. The boundary can be drawn in many
places --- a cell, an organ, an organism, an organism plus its gut
flora, a person plus their notebook --- and the arithmetic works for each.
The framework is explicit that it does not choose, and this is not
evasion: the choice is what the word ``self'' is doing, and physics has
no resource for making it. \\ There is a consequence worth sitting with.
The same arithmetic describes a healing wound and a tumor, a functioning
institution and a captured one. What distinguishes them is entirely
which boundary you drew around what. If that seems unsatisfying, the
dissatisfaction is pointing at something real --- the framework has
supplied the mathematics of persistence and nothing whatever about whose
persistence is worth wanting, and no amount of further physics will
supply the second.
\end{objqabody}

\bigskip
\noindent\textbf{Question 23 (Q-Qi4-23) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Does this say anything about whether there is life elsewhere?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Only in the weakest form, and it is worth being clear about how
weak. \\ The framework says nothing about frequency. It does not compute
how often life arises, because it does not compute histories --- it
computes constraints. A universe consistent with this framework could
be teeming or could be empty, and neither would embarrass it. Anyone
citing this program in support of an abundance estimate is
overreaching. \\ There is one contribution that is not nothing. If the
price of an informational event is fixed by geometry rather than by
local chemistry, then the energy currency any living system must use is
constrained wherever it arises. That is a claim about what alien
biochemistry would have to look like at the level of the accounting,
regardless of which molecules implement it --- and it is checkable here
before it is checkable anywhere else, because the same constraint
applies to terrestrial life. \\ The honest summary is that this is a
framework about what persistence requires, not about where it happens.
The question you asked is a question about frequency and initial
conditions, and the framework's silence on it is structural rather than
a gap awaiting further work.
\end{objqabody}

\bigskip
\noindent\textbf{Question 24 (Q-Qi4-24) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Emergence is a word people use when they cannot explain something. Is that what is happening here?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It often is, and the criticism is fair often enough that a paper
using the word owes an account of what it means by it. \\ The empty
usage runs: the whole is more than the sum of its parts, therefore
something new has appeared, therefore no further explanation is owed.
That is not an explanation and it has been used to avoid a great many.
\\ What is meant here is specific and checkable. A higher level is
legitimate only if three things hold: its description closes
approximately, so that it can be written without constant reference to
the level below; it persists long enough to be worth describing; and it
introduces genuinely new ports --- ways of exchanging with its
surroundings that the lower level does not have. Each of those is a
condition that can fail, and where they fail the framework says there is
no legitimate higher level. \\ The claim about levels is also weaker
than it is usually made. Nothing here overrules anything below. What a
higher level changes is never the law beneath it but the conditions
under which that law is running --- concentrations, boundaries,
attention, institutions. The old dilemma, that a higher level must
either reduce away entirely or acquire powers that break physics, turns
out to be a false one, and the resolution is unglamorous: it was never a
contest of causes but a difference of timescale.
\end{objqabody}

\bigskip
\noindent\textbf{Question 25 (Q-Qi4-25) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``I'm not a physicist and I've got this far. What is the one thing you want me to take away?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That information is not free, and that almost everything strange
about living things follows from that one fact once you take it
seriously. \\ For a century, energy has had a price in science and
information has not. Information was the thing you could invoke when the
physics ran out --- the system ``chose'', the molecule ``recognized'',
the network ``learned'' --- and no bill ever came. That was not
dishonesty. It was the absence of a rate. Without an exchange rate
between energy and information, no statement about what a memory costs
is checkable, and so no such statement is really a physical statement at
all. \\ What this paper supplies is the rate. And the thing worth
noticing is where it comes from, because it would have been much easier
and much weaker to measure it. A number fitted to biochemistry would
explain biochemistry and nothing else. This one is assembled from the
energy scale of the hydrogen atom and an integer fixed several papers
earlier for reasons that had nothing to do with life. \\ So take away
this. If information really has a price fixed by geometry, then the
familiar picture --- in which physics describes the world and information
theory describes our description of it --- has the relationship
backwards. There is one ledger, it has four columns, and the column we
have been treating as bookkeeping is as physical as the other three.
Nothing that remembers is getting anything for free.
\end{objqabody}

\section{The particle theorist and mathematical physicist}

\bigskip
\noindent\textbf{Objection 1 (O-Qi4-1) (the gauge skeptic)}\par\nobreak
\noindent\textbf{``Your whole edifice rests on $U(1)_{Qi}$ being a genuine gauge sector, yet you concede it is `constitutionally optional'. If the constitution does not force it, none of your four pillars is a theorem of the framework --- they are theorems of an unproven add-on.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the logical point and state the status plainly: the four pillars are theorems of the \emph{$U(1)_{Qi}$-active class} of admissible theories, not of the bare constitution. What the constitution forces is the four-component bulk $\SU(5)\times U(1)_B\times U(1)_{Qi}\times\text{gravity}$ as one consistent anomaly-free solution of \BIsix{}; whether the $U(1)_{Qi}$ factor is \emph{empirically realized} is a separate, falsifiable question, recorded as Open Problem~O9 and tested at high energy by the Kaxion search of \citet{ArisakaC1Kaxion} (a predicted $8.1~\mu$eV signature). So the pillars are conditional theorems with a sharp experimental gate, not unconditional ones --- and we would rather say so than overclaim.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-Qi4-2) (the differential geometer)}\par\nobreak
\noindent\textbf{```Information as a gauge current' is a metaphor dressed as physics. Show me the conserved Noether current, the coupling constant, the boson, and one experimental signature --- otherwise you have relabeled a bit-count as a charge.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The Noether current is the $U(1)_{Qi}$ current $J_{Qi}$ of the bulk action imported from \ASix{}/\ATwo{}; the boson is the Kaxion, with the cohort-canonical mass $8.1~\mu$eV ($\approx 1.95$~GHz), and the signature is its direct detection in \citet{ArisakaC1Kaxion}. What we honestly concede is the \emph{boundary} side: a turnkey protocol to count $J_{Qi}$ events in a cell or brain does not yet exist (Open Problem~O2). So the high-energy current is a standard gauge current with a named boson; the open work is its low-energy metrology, which we flag rather than hide.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-Qi4-3) (the bioenergetics-minded reviewer)}\par\nobreak
\noindent\textbf{``$\QB = (2/5)E_{\rm Ry}/K$ with $K=35$ is numerology: why the Rydberg, why divide by 35, why the 2/5? With that many discrete handles you can hit any biological energy.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The magnitude is no longer the soft spot. As now shown in Part~\ref{part:qiqua}, the elementary quantum $\QB$ obeys $2\,N_A\QB = 30.0$~kJ/mol against the measured standard ATP-hydrolysis free energy $30.5\pm0.5$~kJ/mol --- a $1.6\%$ prediction, the integer multiplicity (one phosphoanhydride bond $=$ two quanta) being forced, not fitted (\citet{ArisakaA11Origin}, \GGA{}$_7$). The $2/5$ is the \BIsix{} ratio, $E_{\rm Ry}$ and $K=35$ are imported from \ATwo{}/\AThree{}, and the chain has no continuous freedom. What \emph{is} genuinely open (Open Problem~O5) is the selection-rule \emph{proof} that the per-step quantum is $E_{\rm Ry}/K$ rather than $E_{\rm Ry}/J$ --- a discrete, not continuous, question --- and we mark exactly that, no more.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-Qi4-4) (the skeptical reviewer)}\par\nobreak
\noindent\textbf{``The Arisaka equation $J_e=J_B=J_{Qi}$ is true by your choice of units: you insert a $5/2$ in the definitions that exactly cancels the $2/5$ in the ray. It is a tautology wearing the costume of a law.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The numerical equality after unit-normalization is indeed a convention, and we now say so at the point of display (Part~\ref{part:qiqua}): the physical content of T-QUA.2 is not the equality but (i) the rank-one inflow structure --- the four currents descend from a \emph{single} one-dimensional source ray (Theorem~\ref{thm:qiqua-rank-one}), so they are not four independent quantities, and (ii) the existence of \emph{one} conversion factor $\QB$ rather than four. The non-trivial, falsifiable claim is that one elementary boundary event deposits the fixed ratio $(1,-1,-2/5,-2/5)$ across all four sectors; that ratio is fixed by \BIsix{}, not chosen.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-Qi4-5) (the QED-precision reviewer)}\par\nobreak
\noindent\textbf{``Your `two-knee $1/f$' has no numbers: the slope is free and the knee positions are whatever the slowest and fastest rates happen to be. A bounded power law is the generic Lorentzian-superposition result known since the 1950s. What exactly is falsifiable?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We have sharpened the claim to what is actually derived (Part~\ref{part:qicon}, Theorem~\ref{thm:qicon-two-knee}; F-Qi4-3): the structural prediction is the \emph{co-occurrence of exactly two} scale-breaking knees bracketing a finite power-law band --- not a single unbounded $1/f$ slope and not three or more breaks --- with the band slope tied to the conformal weight, and the two knees required to appear \emph{jointly} with the other \QiCON{} signatures (L\'evy tails, fractal support). The knee positions are conceded to be the system's slowest and fastest retained rates, not universal constants. That joint, bounded, two-break structure is the falsifier; a system showing a single unbroken slope, or breaks that do not co-occur with the other signatures, refutes it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-Qi4-6) (the mathematical physicist)}\par\nobreak
\noindent\textbf{``Theorem~T-Qi4-9.3 (rank-one inflow) assumes a single $\BIsix$ channel and concludes the source is rank one. That is `assume rank one, therefore rank one'. The only physics --- that a second channel is forbidden --- is offloaded to a citation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and we do not claim otherwise: the uniqueness of the four-form $X_4$ (hence the single inflow channel) is \emph{imported} from \AThree{}, where it is the load-bearing result; within \ASeven{}, T-Qi4-9.3 is the boundary consequence of that import, not an independent proof of channel-uniqueness. The honest reading is that the weight rests on \AThree{}'s \BIsix{} uniqueness theorem, which we cite at the point of use; T-Qi4-9.3's contribution is to carry that uniqueness to the four boundary currents.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-Qi4-7) (the string theorist)}\par\nobreak
\noindent\textbf{``You import the \GRIP{} cascade from \ASix{} as if it were proved, but its closure is conditional on \ASix{}'s open conjectures M10 and M11. Downstream you use it everywhere without re-flagging that.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fixed, and you are right that it mattered. The conditional status is now re-flagged at every load-bearing use --- the \QiHAL{} universality theorem (Theorem~\ref{thm:qihal-universality}), the pagoda factorization (Theorem~\ref{thm:pagoda-factorization}), and the synthesis theorem (Theorem~\ref{thm:gdos_synthesis}) each carry the inline caveat that the inherited \GRIP{}-cascade closure
is conditional on \ASix{}'s remaining residues --- since the capacity
theorem of \citealp{ArisakaP2N3} resolved M10's demand and its
rigidity theorems closed the warped and relative refinements at class
level, that means M11, now carried in spectral-flow form. Note the \emph{other} import, the ground-attractor existence and uniqueness, does not depend on M10 or M11 at all --- it is \ASix{}'s per-stratum corollary, carrying one stated assumption of its own (the moment-map identification) rather than either conjecture --- so not everything downstream is conjectural, and the two conditionings should not be conflated.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-Qi4-8) (the operator-algebra specialist)}\par\nobreak
\noindent\textbf{``Your SIP uniqueness proof is just Gelfand--Naimark plus Stinespring--Kraus: standard CPTP-channel identifiability. It says the packet data determine the dynamics --- near-tautological for a well-posed semigroup --- and cannot bear the weight of `one object, four faces'.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed on the mathematics: the SIP completeness theorem (T-Qi4-8.2) uses standard results and proves exactly what you say, that two packets agreeing on all observables are gauge-equivalent. We do not rest the cross-domain universality on that lemma; the universality is a \emph{constitutional} claim --- the same \GDOS{} boundary law and the same imported geometry apply at every scale --- for which the lemma supplies only the well-posedness floor. Where the prose over-read the lemma we have toned it (\S\ref{sec:sip}); the honest division is: standard machinery for well-posedness, constitution for universality.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-Qi4-9) (the falsifiability skeptic)}\par\nobreak
\noindent\textbf{``Your headline falsifiability move is a `joint signature' $\Xi_S=\sum w_a C_a$ with free weights and an undefined notion of `coherence'. A test with free knobs cannot be failed.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The weighted score $\Xi_S$ is a heuristic summary, not the falsifier, and we should not have let it carry that rhetorical load. The actual falsifiers are the \emph{named, weight-free} ones: the Arisaka bioenergetic quantum ($2\QB$ vs measured ATP, currently $+1.6\%$), the bounded two-knee co-occurrence (F-Qi4-3), the \MePMoS{}/\NHT{}/\HAL{}/\BFFT{} architecture in memory-bearing systems, and the high-energy $U(1)_{Qi}$/Kaxion signature. Each can fail on its own measurement. The joint-coherence statement is the \emph{additional} claim that these co-occur in a full-runtime system; it is suggestive, not the primary falsifier, and \S\ref{subsec:discussion-falsifiability} now reads that way.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-Qi4-10) (the numerology-hunter)}\par\nobreak
\noindent\textbf{``$K=35$ is asked to be a Diophantine identity, a logarithmic Planck-to-electroweak scale depth, and a linear count of atomic KK levels --- three incompatible roles for one integer.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The Diophantine identity $K=LJ=M^2+N^2+L^2=35$ is \AThree{}'s \OGNfive{} ledger; the scale depth and KK-level readings are the \emph{same} $K$ viewed through the renormalization-group dictionary of \ATwo{}, in which the fifth (scale) coordinate is the renormalization-group axis, so a logarithmic bulk depth and a level count are two descriptions of one ladder. We grant this is exactly the place a reader should demand the dictionary be explicit; the derivation lives in \ATwo{}/\AThree{}, and \ASeven{} imports the single integer rather than re-deriving it. If \ATwo{}'s dictionary fails to map the depth onto the level count, that is a falsifiable defect of \ATwo{}, not a hidden tuning in \ASeven{}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-Qi4-11) (the consistent-with-versus-derived-from reviewer)}\par\nobreak
\noindent\textbf{``Each pillar lists canonical results --- Maxwell, Noether, Onsager, Friston --- and asserts it `unifies' them. Listing predecessors and claiming unification is not unification.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We have replaced `unifies these' with `subsumes these as gauges or limiting cases of' throughout the pillar prefaces, which is the accurate verb. For \QiEP{} the reduction is genuine and explicit: \PLA{}, \MEPP{}, and \FEP{} are shown to be three gauges of one projected functional $F_{\GFF}=F_{\GEP}+F_{\CEP}$ (Theorem~T-EP, the common-origin theorem), each recovered as a limit. For the others the honest claim is a common variational template, not a reduction with recovered coefficients, and the text now says so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-Qi4-12) (the reductionist)}\par\nobreak
\noindent\textbf{``Why drag a six-dimensional bulk into an information theory at all? It looks like cosmic overkill to motivate some bookkeeping about cells and brains.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The bulk is not introduced by \ASeven{}; it is imported. The point of the program is that the \emph{same} constitution that fixes the Standard-Model parameters (\ANine{}) and the cosmic inventory (\ATen{}) also fixes the open-system runtime --- and that economy is the claim worth testing. If one wanted an information bookkeeping for cells alone, one would not need six dimensions; what six dimensions buy is that the bioenergetic quantum $\QB$, the family number $N=3$, and $\Omega_m=1/\pi$ all descend from one integer ledger. That is either a deep unification or a coincidence, and the cohort is built to let the data decide.
\end{objqabody}

\bigskip
\section{The systems biologist and biophysicist}

\bigskip
\noindent\textbf{Objection 13 (O-Qi4-13) (the bioenergeticist)}\par\nobreak
\noindent\textbf{``$\QB = 0.155$~eV is not the energy of one ATP. Standard ATP hydrolysis is $0.32$~eV and the in-vivo value is $0.5$--$0.66$~eV. You are off by a factor of two to four and you call it your headline result.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You caught a real error, now corrected. $\QB$ is the \emph{elementary} quantum; one ATP phosphoanhydride bond carries \emph{two} of them, so the prediction is $2\,N_A\QB = 30.0$~kJ/mol against the measured \emph{standard} free energy $30.5\pm0.5$~kJ/mol --- a $1.6\%$ match, not a factor-of-two miss (Part~\ref{part:qiqua}; \citet{ArisakaA11Origin}). The factor of two is the integer multiplicity, not a discrepancy. The in-vivo $\sim50$~kJ/mol you cite is not a constant --- it carries the mass-action term $RT\ln([\mathrm{ADP}][\mathrm{P_i}]/[\mathrm{ATP}])$ --- and even it is recovered as $(10/3)\QB$ once the \emph{E.~coli} $c_{10}$ proton stoichiometry is included.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-Qi4-14) (the biophysicist)}\par\nobreak
\noindent\textbf{``What would you actually put on the axes to measure `eV per qubit' in a cell? A qubit is not a biological observable.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The energetic side is measurable: $\QB$ is read off ATP-hydrolysis calorimetry as above. The \emph{informational} side --- counting $J_{Qi}$ events independently of the energy, charge, and structural ledgers --- is exactly what we flag as open (Open Problem~O2): there is at present no turnkey instrument for it. So `eV per qubit' is operationally half-grounded (the eV half) and half-promissory (the qubit half), and we label the promissory half rather than pretend it is settled.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-Qi4-15) (the stochastic-thermodynamicist)}\par\nobreak
\noindent\textbf{``You call $J_{Qi}$ conserved `like electric charge', but your own balance law has generically nonzero information sources, and nobody can measure the flux. A conservation law you cannot measure and that is not conserved is an accounting convention.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The conservation is a boundary-current statement modulo named sources, not a closed-system identity, and the prose has been disciplined to say `conserved along its boundary and on appropriate time-scales' rather than flatly `like charge'. The measurable proxy a systems biologist already uses --- stochastic-thermodynamic mutual-information flow (Sagawa--Ueda-type bounds) --- is the low-energy face of $J_{Qi}$, and connecting the two quantitatively is part of the O2 metrology program. Until that is done we do not claim $J_{Qi}$ has the same operational standing as an ammeter current.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-Qi4-16) (the nonequilibrium-thermodynamicist)}\par\nobreak
\noindent\textbf{``Your SIP and persistence theorems are autopoiesis and viability theory in new notation. What do they predict that Maturana--Varela, Aubin, or stochastic thermodynamics do not?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede the lineage openly --- the persistence criterion (Theorem~\ref{thm:seos-persistence}) is a Lyapunov/viability bound, and the SIP completeness theorem imports standard CPTP results. The added value is not a new persistence mechanism but a \emph{constitutional embedding}: the same ledger that fixes $\QB$ and $N=3$ also fixes the floor of the maintenance budget, so the persistence bound is tied to an integer-quantized energy scale rather than left as a free dissipation threshold. Whether that tie buys a novel, checkable number beyond the existing frameworks is the fair test, and we point to the bioenergetic quantum as the first such number.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-Qi4-17) (the molecular-machines reviewer)}\par\nobreak
\noindent\textbf{``Your `universal' decomposition into gradient, curl, noise, and gauge is just the Helmholtz--Hodge structure of \emph{any} Fokker--Planck operator. It is true of every stochastic system, alive or not, so it forbids nothing.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Right that the decomposition is generic; the content is not in the decomposition but in the \emph{ledger constraints} laid on it. \QiQUA{} asserts that the budgets carried by the four sectors are quantized in units of $\QB$ and locked in the fixed \BIsix{} ratio --- that is the part that can fail, and it is not a property of a generic Fokker--Planck operator. So the Hodge form is the universal canvas; the falsifiable claim is the $\QB$-quantized, ratio-locked ledger painted on it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-Qi4-18) (the ATP-synthase specialist)}\par\nobreak
\noindent\textbf{``Your `worked example' of ATP synthase is a citation plus the integer $c_{10}$. There is no closed bioenergetic audit.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is now a closed audit at the level the framework claims (Part~\ref{part:qiqua}): the standard bond energy is $2\QB$; the per-proton chemiosmotic step $e\,\Delta p$ at the measured \emph{E.~coli} proton-motive force $\approx 0.155$~V equals $\QB$; and the $c_{10}$ ring with $H^+/\mathrm{ATP}=10/3$ recovers the in-vivo cost $(10/3)\QB\approx50$~kJ/mol. The deeper claim --- that the rotor stoichiometry is an environment-tuned integer ($c_8$ mammals, $c_{10}$ \emph{E.~coli}, $c_{13}$--$c_{15}$ alkaliphiles/chloroplasts) selected by \GRIP{} to match the available $\Delta p$ --- is forward-referenced to a dedicated forthcoming study, and labeled as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-Qi4-19) (the DNA biophysicist)}\par\nobreak
\noindent\textbf{``You name-drop the $10.5$~bp/turn helical repeat, writhe, and supercoiling, but no DNA observable is ever computed. Linking number is a real conserved quantity; you only place the words near each other.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: in \ASeven{} the DNA case is illustrative, not a closed derivation. The mapping we find suggestive --- the linking-number identity $\mathrm{Lk}=\mathrm{Tw}+\mathrm{Wr}$ as a structural-current conservation in the baryonic sector --- is exactly that, suggestive, and the text now frames it as a motivating analogy rather than a worked result. We would rather under-claim DNA here than dress an analogy as a theorem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-Qi4-20) (the chemotaxis modeler)}\par\nobreak
\noindent\textbf{``Bacterial chemotaxis is quantitatively understood through Barkai--Leibler integral feedback. You recover none of that structure and then `predict' the Levy walk, which is a sixty-year-old observation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair on both counts. The framework supplies the runtime \emph{grammar} (adaptive methylation as externalized \HAL{} memory; run-and-tumble as a \MePMoS{} loop), not the integral-feedback adaptation law, which it does not derive. The heavy-tailed search statistic is a retrodiction, and the verb has been softened from `predict' to `is consistent with'. The honest status is organizational: we offer a common vocabulary across chemotaxis, neurons, and societies, not a replacement for the domain-specific kinetics.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-Qi4-21) (the falsification reviewer)}\par\nobreak
\noindent\textbf{``The $\pm20\%$ \QiQUA{} kill-test is hollow: it tests a unit identity, its tolerance band excludes $\QB$'s own value, and you exempt every coarse-grained --- i.e.\ every real --- system as `overhead'.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The test is meant for \emph{elementary} events --- a single catalytic turnover --- where the four ledgers can be measured independently and the equality is \emph{not} guaranteed by unit choice (the unit choice fixes the ratio, not the per-event counts). Coarse-grained systems are excluded precisely because their overhead is built from sums of elementary events and belongs to a later bookkeeping layer; that is a scope statement, not an escape hatch. We grant the $\pm20\%$ band needed recalibration to the elementary-event dispersion rather than the in-vivo $\Delta G$ spread, and Part~\ref{part:qiqua} now states the elementary-event scope explicitly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-Qi4-22) (the cell biologist)}\par\nobreak
\noindent\textbf{``Concretely: what does this runtime let me compute that nonequilibrium thermodynamics does not?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Three numbers, in increasing order of sharpness: the quantized event budget $E = N_{\rm lock}\QB$ for an elementary catalytic cycle; the bounded two-knee spectral shape of a full-runtime system; and the bioenergetic anchor $2\QB = 30.0$~kJ/mol, which nonequilibrium thermodynamics measures but does not predict. We are candid that much of the cross-domain content is organizational rather than predictive (Open Problems O1, O2), but the bioenergetic quantum is a genuine prediction a thermodynamicist would not write down.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-Qi4-23) (the origin-of-life skeptic)}\par\nobreak
\noindent\textbf{``You invoke Schr\"odinger and say life needs `no new physics', then introduce a new gauge sector and a new boson. Which is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both, stated precisely: the \emph{only} new physics is a single new $U(1)_{Qi}$ gauge sector and its Kaxion boson; everything else --- the open-system character of life, the information ledger, the persistence criterion --- is reinterpretation of known physics. The Schr\"odinger framing is that life needs no new \emph{biological} law beyond physics; it does need that one new gauge sector, which is a falsifiable high-energy commitment (\citet{ArisakaC1Kaxion}). The text now owns this in one sentence rather than implying a free lunch.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-Qi4-24) (the proton-accounting reviewer)}\par\nobreak
\noindent\textbf{``The thermodynamic $H^+/\mathrm{ATP}$ for \emph{E.~coli} $c_{10}$ is measured near $4.0$, not your structural $10/3$. Your `$e\,\Delta p=\QB$' coincidence is then shaky.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Which is why the rigorous anchor is the \emph{standard free energy}, $2\QB = 30.0$ vs $30.5$~kJ/mol, an assumption-free comparison that does not depend on the proton ratio at all. The per-proton picture ($e\,\Delta p\approx\QB$ with the structural $10/3$) is presented as suggestive mechanistic corroboration, and we explicitly note the thermodynamic-versus-structural $H^+/\mathrm{ATP}$ distinction; reconciling them quantitatively, with the $c$-ring optimization across organisms, is the job of the forthcoming dedicated study, not of \ASeven{}.
\end{objqabody}

\bigskip
\section{The origin-of-life researcher}

\bigskip
\noindent\textbf{Objection 25 (O-Qi4-25) (the probabilist)}\par\nobreak
\noindent\textbf{``Your conditional-inevitability theorem is a Borel--Cantelli triviality: a recurrent process that hits a positive-measure set with bounded-below probability hits it almost surely. That is true of coin flips. The biology is smuggled into the hypotheses.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree it is a conditional reduction, and the title carries the word `conditional' for that reason. The theorem's job is not to prove inevitability \emph{ex nihilo} but to isolate \emph{exactly} what inevitability requires: a finite invariant measure, positive-measure viable windows, and bounded-below per-visit stabilization (Theorem~\ref{thm:conditional-inevitability}). Naming those three conditions, and conceding (Open Problem~O8) that we do not yet bound them, is the honest contribution; the abstract and summary have been retuned from `inevitable' to `conditionally inevitable, given positive viable measure'.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-Qi4-26) (the prebiotic chemist)}\par\nobreak
\noindent\textbf{``The whole result hangs on $\mu(W_{\rm life})>0$, the rarity of a self-maintaining chemical recipe. That is the central unsolved problem, and you assert it as a sign condition.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes --- and we say in Open Problem~O8 that $\mu(W_{\rm life})$ is `not provided in the present paper'. The theorem converts `is life inevitable?' into `is the viable measure positive and repeatedly explored?', which is progress only in that it localizes the hard question precisely. It does not answer it. We explicitly decline to claim the measure is large, or even nonzero, on present evidence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-Qi4-27) (the major-transitions theorist)}\par\nobreak
\noindent\textbf{``Why five floors? Maynard Smith and Szathm\'ary count about eight transitions. Your `particle to mind' ladder is a curated reading of a generic adiabatic-elimination scheme.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Now stated outright in the pagoda section: the factorization theorem yields an \emph{arbitrary-depth} tower wherever the floor-creation conditions hold, and the number five is an empirical reading, not a theorem output (Open Problem~O7). A different coarse-graining --- including a finer one matching the major-transitions count --- satisfies the same existence theorem. We keep the five-story figure as an interpretive aid and label it as interpretation, not derivation.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-Qi4-28) (the replicator-first/metabolism-first reviewer)}\par\nobreak
\noindent\textbf{```Chemistry to life as \GRIPout{} to \GRIPin{}' relabels the origin of life without deriving autocatalysis, compartmentalization, or template replication. You define life as a positive maintenance margin and call the definition an origin.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: \ASeven{} states the \emph{condition} a living system satisfies (the \SeOS{} maintenance margin), not the \emph{mechanism} by which chemistry crosses into it. We do not derive the error threshold, the concentration problem, or chirality, and we do not adjudicate metabolism-first versus replicator-first. The framework's present contribution at this floor is descriptive vocabulary plus the bioenergetic anchor; the transition mechanism is named as future chemistry-floor work, not claimed.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-Qi4-29) (the stationarity reviewer)}\par\nobreak
\noindent\textbf{``Borel--Cantelli needs a fixed invariant measure, but your selling point is that \GRIPout{} events reorganize the landscape itself. You cannot have both a fixed measure and a measure that life keeps remaking.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A genuinely sharp point. The recurrence argument applies \emph{within} a stratum --- the fixed-landscape \GRIPin{} regime --- where an invariant measure is well-defined; the cross-stratum \GRIPout{} steps that remake the landscape are the separate, non-stationary question the theorem does not address. The honest consequence is that the infinite-trial idealization is the wrong limit for a finite, irreversibly evolving cosmos, and the physically relevant object is a finite-horizon hitting probability. We flag this as a limitation rather than papering over it; the finite-horizon estimate is future work.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-Qi4-30) (the compartmentalization reviewer)}\par\nobreak
\noindent\textbf{``Your chemistry-floor equation has no compartment variable, yet your life-floor criterion presupposes a maintained membrane. You assume the very thing the field cannot yet explain.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} True: compartmentalization enters \ASeven{} as an input boundary condition of the life floor, not as a consequence of the chemistry-floor dynamics. We do not derive self-organizing membrane closure from reaction--diffusion, and we should not be read as claiming to. It is named as an open boundary between floors --- one of the places where the framework provides the destination's description but not the journey's mechanism.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-Qi4-31) (the astrobiologist)}\par\nobreak
\noindent\textbf{``Your `conditional inevitability' is compatible with life arising exactly once in the whole universe. The $N\to\infty$ limit is meaningless for a finite cosmos, so the slogan is empty.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted, and this is precisely why the author's own framing is `inevitable \emph{under an Earth-like biosphere}', not `inevitable everywhere'. With no bound on $\mu(W_{\rm life})$ the theorem cannot distinguish ubiquitous from unique, and the abstract no longer implies it can. The meaningful quantity is the expected number of origins per Hubble volume per Hubble time, which requires the measure estimates of Open Problem~O8; we pose it and decline to answer it on present evidence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-Qi4-32) (the no-novel-prediction reviewer)}\par\nobreak
\noindent\textbf{``Every biological `test' in your falsifiability matrix is either known phenomenology or an untestably general failure mode. Where is the risky origin-of-life prediction?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The risky, quantitative biological prediction is the bioenergetic one: $2\QB = 30.0$~kJ/mol for the universal energy currency, already landing at $+1.6\%$, with the sharper forthcoming claim that the rotor-ring integer tracks the biosphere's proton-motive force. We concede the rest of the biological content is currently organizational or retrodictive (Open Problems O1, O2) and we do not dress retrodictions of $1/f$ noise or Levy walks as confirmations.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-Qi4-33) (the mind-window reviewer)}\par\nobreak
\noindent\textbf{``Your second conclusion conditions mind-inevitability on a recurrent evolutionary exploration of the mind-window with bounded-below probability. That bakes in a directional drive to mind that evolution explicitly denies.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree there is no orthogenetic drive, and the conditioning $p_{\rm mind}>0$ is an assumption for which \ASeven{} supplies \emph{no} selective mechanism. The theorem therefore says nothing about whether minds are favored or astronomically rare given life; it only says that \emph{if} the mind-window is recurrently explored with positive stabilization probability, minds occur almost surely in the ensemble. That `if' is doing visible work, and we leave it visible.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-Qi4-34) (the determinism reviewer)}\par\nobreak
\noindent\textbf{``You headline `deterministically inevitable' and then, mid-section, redefine determinism into harmlessness. That is a motte-and-bailey.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The body distinguishes the two meanings explicitly (\S\ref{subsec:discussion-limitations}): not that every micro-history is identical, but that higher-floor packets are attractor \emph{classes} rather than freak accidents, in the ensemble sense. The strong trajectory word has been removed from the abstract and summary in favor of `ensemble-inevitable, conditional on positive viable measure'. The motte is now the only structure we defend.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-Qi4-35) (the inevitability-of-ATP reviewer)}\par\nobreak
\noindent\textbf{``Is ATP-based life really inevitable, or is ATP just one chemical solution among many that you have retro-fitted to a constant?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The claim is conditional, not universal: \emph{given} an Earth-like chemiosmotic biosphere with a proton-motive corridor, ATP is the ground geometric attractor of the bioenergetic stratum (\GGA{}$_7$, $2\QB$), and the $c$-ring optimization shows the \emph{same} attractor re-tuned across niches by an integer rotor count. A biosphere with no proton-motive chemistry could select a different currency, which is why we condition on the biosphere. The non-trivial content is that, conditioned on that chemistry, the energy quantum is fixed by geometry rather than by contingency --- and that is what $2\QB = 30.0$~kJ/mol tests.
\end{objqabody}

\bigskip
\section{The neuroscientist}

\bigskip
\noindent\textbf{Objection 36 (O-Qi4-36) (the predictive-processing theorist)}\par\nobreak
\noindent\textbf{``Your \OH{}/\IH{}/\EIC{} loop is active inference plus the global workspace with new names. It states no experiment whose outcome differs from Friston, Tononi, or Llin\'as.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We concede \EIC{} reproduces the predictive-processing and global-availability structure, and we do not claim a knockout of those theories. The framework's distinctive content is the layer \emph{beneath} them --- a geometric constitution in which the free-energy update and the workspace access are boundary projections, with $U(1)_{Qi}$ phase-coded memory and the bounded two-knee spectrum as the candidate differential levers. Where it cannot yet discriminate experimentally from active inference or IIT, we now say so rather than imply a settled theory of consciousness.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-Qi4-37) (the control theorist)}\par\nobreak
\noindent\textbf{``Your `action-conditioned observability theorem' is the Kalman observability rank condition from 1960, generalized to time-varying systems. Dignifying it with a cohort ID invites a desk reject.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Now credited at the theorem (\S\ref{subsec:ih-motion-memory-necessary}): the rank condition is Kalman's, and the active-sensing reading is Bajcsy's and O'Regan--No\"e's. The contribution claimed is narrow and explicit --- importing the time-varying observability criterion into the inner-hologram reconstruction, where the system's own action sets the observation geometry. We do not present it as an original theorem, and the prose no longer does.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-Qi4-38) (the developmental neuroscientist)}\par\nobreak
\noindent\textbf{``Your falsifier F-Qi4-5 --- model degrades if action is restricted while passive input is preserved --- is the Held--Hein kitten-carousel result from 1963. It was confirmed sixty years ago, so it is retrodiction, not a live frontier.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded on the priority. F-Qi4-5 cites Held--Hein (and von Holst--Mittelstaedt's reafference principle) and concedes the qualitative active-versus-passive asymmetry is established. The genuinely open, sharper prediction is the \emph{quantitative} one --- model error as a graded function of the observability-Gramian deficit, not a binary active/passive contrast --- and that is what we now flag as the live test.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 39 (O-Qi4-39) (the consciousness reviewer)}\par\nobreak
\noindent\textbf{``Defining consciousness as `internal measurement of selected \IH{} content' buys an access criterion and then relabels the explanatory gap a `structural placeholder'. That renames the hard problem; it does not dissolve it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed, and now stated in those exact terms: the criterion the framework supplies is one of \emph{access} consciousness --- reportability and global availability of selected content, in the sense of \citet{Block1995} --- and the \emph{phenomenal} hard problem is left explicitly open (Open Problem~O3; the Conclusions now carry the access/phenomenal distinction). We claim a runtime condition for access-conscious content, not a solution to subjective experience, and we do not blur the two.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-Qi4-40) (the oscillations reviewer)}\par\nobreak
\noindent\textbf{``Your $U(1)_{Qi}$ `memory carrier' reduces to relative phase, and the holonomy is gauge-invariant precisely so that absolute phase is unobservable. A carrier built to be unobservable yields no neural test beyond ordinary phase coding.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The observable is indeed relative phase --- phase precession, theta--gamma coupling --- and we do not claim a measurement that separates a $U(1)_{Qi}$-gauged memory from ordinary relative-phase coding (Open Problem~O2). What the gauge structure adds is the constitutional \emph{account} of why relative phase, corrected by holonomy, is the lawful carrier of a quantity whose absolute value is meaningless. That is an explanatory claim, not an extra neural observable, and the frequency-independent readout theorem (T-Qi4-12.9) is honestly just the gauge-invariance of relative-phase correlators.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-Qi4-41) (the aperiodic-spectrum reviewer)}\par\nobreak
\noindent\textbf{``Your two-knee $1/f$ is indistinguishable from the standard FOOOF aperiodic-with-knee fit: free slope, fitted knees, generic $f^{-2}$ roll-off. No knee in Hz is ever produced.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The knee positions are conceded to be fitted, as in FOOOF; the distinguishing content is structural and joint (F-Qi4-3): the prediction of \emph{exactly two} co-occurring breaks bracketing a bounded band, the slope tied to the conformal weight, and the requirement that the spectral signature \emph{co-occurs} with the L\'evy and fractal diagnostics of the same selected branch. A single-knee aperiodic fit, or a two-knee fit whose breaks do not co-occur with the other \QiCON{} signatures, is the discriminating failure. We do not claim a universal knee frequency in Hz, and the falsifiability text no longer does.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-Qi4-42) (the hippocampus reviewer)}\par\nobreak
\noindent\textbf{``Your \BFFT{} derives a dyadic $1{:}2$ ladder, but the canonical hippocampal theta--gamma code nests at roughly $1{:}7$. The data falsify your $\lambda=2$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Now conceded in the \BFFT{} section: $\lambda=2$ is the minimal idealization, and measured cross-frequency coupling is non-dyadic --- theta--gamma at $1{:}5$--$1{:}8$ (\citet{LismanJensen2013}), an integer $m\approx7$. Proposition~\ref{prop:bfft-dyadic} already admits any integer $m>1$; the dyadic case is the theoretical floor of the nesting hierarchy, not a claim that biology is binary. The empirical ladder selects the architecture-specific $m$, and we no longer present $\lambda=2$ as a prediction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-Qi4-43) (the energetics reviewer)}\par\nobreak
\noindent\textbf{``$\QB$ `per qubit' applied to a neural memory write is unmotivated --- a single spike costs $10^4$--$10^5$ ATP for ion-gradient restoration, not one $\QB$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and the neural application is now scoped accordingly: $\QB$ is derived and validated at the \emph{chemistry} (ATP-hydrolysis) scale, and \ASeven{} does \emph{not} derive a neural per-bit quantum. A spike's energy budget is dominated by Na$^+$/K$^+$-ATPase, as you say; the statement that memory writes consume budget is qualitative (they do, via ATPase) and is not a claim of one $\QB$ per stored bit. We have removed any reading that prices neural information at $\QB$ directly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-Qi4-44) (the IIT reviewer)}\par\nobreak
\noindent\textbf{``Your graded continuum places a chemotactic bacterium at `proto-EIC'. That is the panpsychist move IIT is criticized for, smuggled past the reader in a table.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The continuum is a ladder of representational and architectural complexity, not a claim that every memory-bearing system is conscious --- the non-claims table states exactly that. Part~(d) of the universality theorem, which attaches `consciousness-grade' to the top rung, is a definitional specialization, not a theorem corollary, and it is now read as such. We do not assert micro-experience for bacteria; we assert that the same runtime grammar runs at every rung, with consciousness reserved for the high-$\QCI{}$, self-accessible end.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-Qi4-45) (the quantum-cognition skeptic)}\par\nobreak
\noindent\textbf{``You write conscious readout as a POVM, $p(c)=\mathrm{Tr}(\Pi_c\rho_{\IH})$, with Luders post-selection --- quantum-measurement formalism on a classical belief state. Either justify quantum neurons or drop the costume.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The instrument notation is the \emph{general} measurement language, which reduces to ordinary Bayesian conditioning (marginalization over a partition of the belief simplex) precisely when the internal observable algebra is commutative --- the relevant case for neural representations at brain temperature. We do not claim quantum neural states; the formalism is used because it specializes cleanly to the classical case while keeping the bulk-to-boundary projection language uniform. \S\ref{subsec:consciousness-internal-measurement} now states the commutative reduction so the notation is not mistaken for a quantum-brain claim.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-Qi4-46) (the thalamocortical reviewer)}\par\nobreak
\noindent\textbf{``You cite Llin\'as for `predictive coding' --- a mis-citation --- and never engage his specific $40$~Hz thalamocortical-resonance claim or IIT's $\Phi$-based dissociations.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The mis-attribution is corrected, and we concede the framework is presently \emph{silent} on the signature predictions of the thalamocortical-resonance theory and of IIT --- it neither reproduces nor contradicts the $40$~Hz gating or the $\Phi$ dissociations. Engaging them on their own empirical commitments is named as future work rather than implied by a citation. `Relocating consciousness after the runtime' is a statement about our exposition, not an empirical relation to those theories, and we no longer let it read otherwise.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-Qi4-47) (the localization reviewer)}\par\nobreak
\noindent\textbf{``Your conscious-readout boundary $\partial_{\rm int}$ has no anatomical referent and no stated threshold --- the appendix admits the criteria are placeholders. Any reportable state can be declared post hoc to have crossed it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept that $\partial_{\rm int}$ is defined dynamically (global availability, dwell-time exceeding the cycle time, action-coupling) rather than anatomically, and that the thresholds are presently placeholders, as the appendix says. At this stage the consciousness construction is therefore a \emph{research program} with an operational shape, not yet a fully falsifiable theory, and we mark it as such (Open Problem~O3). Supplying order-of-magnitude thresholds --- cycle times in milliseconds, a cross-module causal-reach metric --- is the work that would promote it to a test.
\end{objqabody}

\bigskip
\section{The methodologist and the working scientist}

\bigskip
\noindent\textbf{Objection 48 (O-Qi4-48) (the honest-disclosure auditor)}\par\nobreak
\noindent\textbf{``Your abstract and conclusions speak as though the runtime is established, while your own triage lists thirteen of fourteen core problems as genuinely open. The front matter contradicts the body.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The front matter has been brought into line with the triage: `inevitable' is now `conditionally inevitable', the consciousness claim is scoped to access (Open Problem~O3 governs), the two-knee falsifier states what is and is not predicted, and the imported \GRIP{} closure is flagged conditional on the remaining
conditioning (M11, spectral-flow form). The disciplined non-claims tables and the O1--O9/E1--E5 triage now govern the abstract, not the other way round. The candor was always in the body; it now reaches the summary.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-Qi4-49) (the single-falsifier reviewer)}\par\nobreak
\noindent\textbf{``Strip away the joint signatures. What is the \emph{one} experiment that, done tomorrow, could kill this?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The bioenergetic quantum. The framework forces $2\QB = 30.0$~kJ/mol for the standard ATP-hydrolysis free energy; it sits at $+1.6\%$ of the measured value. A measured standard free energy disagreeing with $2\QB$ beyond calorimetric precision, with no integer-multiplicity reinterpretation available, collapses the chemistry-floor anchor and with it the claim that the energy currency of life is geometrically fixed. That is a single number, already on the table, with a sharp tolerance.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 50 (O-Qi4-50) (the cohort reviewer)}\par\nobreak
\noindent\textbf{``Twenty-two interlocking papers look like a system engineered to absorb any refutation: knock one down and the others reabsorb it. Is the cohort falsifiable at all?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Each paper carries named, independent falsifiers --- \ANine{}'s Standard-Model parameters, \ATen{}'s $\Omega_m=1/\pi$, \citet{ArisakaC1Kaxion}'s $8.1~\mu$eV axion, \ASeven{}'s $2\QB$ and two-knee signatures --- and they share inputs, so a single measured failure propagates rather than being absorbed (a wrong $N=3$, for instance, would impugn every paper that uses the \OGNfive{} ledger). The interlocking concentrates falsification risk; it does not diffuse it. The cohort map (\S\ref{sec:cohort-a7}) shows where each load-bearing input sits.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-Qi4-51) (the Occam reviewer)}\par\nobreak
\noindent\textbf{``After importing \GDOS{}, \GRIP{}, the bulk, and the ledger, and conceding most floors to companion papers, what does \ASeven{} itself genuinely add?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Four pillar theorems (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}), the internal-measurement \EIC{} universality theorem, the Arisaka four-current lock, and the discovery that a memory-bearing system internalizes its own recoverable \GRIPout{} (the \QiHAL{} \GRIP{}-loop) --- the ten structural innovations of \S\ref{sec:ten-innovations}. We are explicit about what is imported (the constitution, the bulk, \GDOS{}, the \GRIP{} machinery) versus proved here, and the innovations are stated as the value added on top of those imports, not as a re-derivation of them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-Qi4-52) (the standing-of-the-math reviewer)}\par\nobreak
\noindent\textbf{``Much of your apparatus --- GKLS semigroups, Hodge decomposition, Lyapunov bounds, Lorentzian-superposition spectra --- is standard, competently relabeled. Where is the new mathematics?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We do not claim new mathematics; we claim a new \emph{constitutional reading} of standard mathematics. The contribution is which standard objects are forced to appear, in which order, with which integer-quantized constants --- not a new theorem in operator algebra or Morse theory. Where a result is textbook (CPTP identifiability, the two-knee Lorentzian superposition, the observability rank condition) we now cite the textbook source and claim only the application. The novelty is the architecture and the constants, not the lemmas.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-Qi4-53) (the working scientist)}\par\nobreak
\noindent\textbf{``Why should I, with limited time, invest in reading and testing this rather than the framework I already use?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because it makes a small number of sharp, risky, already-checkable commitments that your current framework does not: a geometrically fixed energy currency ($2\QB = 30.0$~kJ/mol, at $+1.6\%$), a bounded two-knee spectral co-occurrence, a named memory architecture, and a high-energy $U(1)_{Qi}$ axion at $8.1~\mu$eV --- and it ties them to the Standard-Model and cosmological predictions of the same cohort through one integer ledger. If those commitments fail, the failures are diagnostic and localized; if they hold, disparate domains share one accounting. The paper is built to be wrong in specific, visible ways, which is the only honest reason to spend time on it.
\end{objqabody}

\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{part:references}
\label{sec:references}

\noindent Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which theorem, table, or section each one underwrites.

\noindent The notes are written for two readers at once. For the newcomer, they trace the intellectual lineage of the \QiFour{} runtime: from the foundations of open-quantum-system dynamics, nonequilibrium thermodynamics, information geometry, and gauge theory, to the twenty-two-paper \GGTheory{} cohort in which the present paper supplies the four informational pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) that run on top of the \GDOS{} boundary law. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

\noindent The \GGTheory{} cohort papers (\AOne{}--\AEleven{}, together with the phenomenology P-series and the consequence C-series) are cited throughout under the canonical keys and titles fixed by the cross-paper coherence map of the program; the present paper's place in that cohort, and its dependence on the imported runtime constitution of \ASix{}, is set out in the entrance Part. The references are organized in two groups --- first the author's \GGTheory{} papers (Section~A below), then all other works (Section~B), each in alphabetical order by first author. Every cited entry closes with a note recording the section in which it is first cited.

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

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K.} (2026).
\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 preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The parent constitutional paper of the \GGTheory{} program. Establishes the four axioms \AX{1} \LC{}, \AX{2} \GGEP{}, \AX{3} \OGGEP{}, \AX{4} \MSMF{}, together with the seven postulates \POS{1}--\POS{7} (\GI{}, \UEP{}, \AF{}, \CP{}, \RG{}, \OGN{}, \MF{}) and the central Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ that ties the four boundary currents through one energy--information conversion
factor. Every theorem in the present paper that names its constitutional source inline cites this paper at the point of invocation; the \OGGEP{} axiom is the foundational dynamics from which the \QiFour{} pillars and the \EIC{} framework are derived as boundary consequences. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K.} (2026).
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})}.
Arisaka-GG-A2-GG6 preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The bulk-side foundational paper of the GG6 Trio. Establishes the 6D bulk metric $\dd s_6^2 = \e^{-2r}\eta_{\mu\nu}\dd x^\mu \dd x^\nu + \dd r^2 + \dd\theta^2$, the canonical four-component gauge group $\Gtot = \Spinone \times \SUfive \times \UoneB \times \UoneQi$, the magic-dimension ladder $\Spinone \cong \SLtwoH$, and the \OGNfive{} Diophantine cascade $(M,N,L;J,K) = (1,3,5;7,35)$ with the dual-lock identity $K = LJ = M^2 + N^2 + L^2 = 35$. These are the ultraviolet inputs that the boundary \GDOS{} dynamics of the present paper projects downward from; every \QiEP{} variational statement that invokes the bulk $X_4$ four-form ultimately rests on the trace-ratio lock $|a/c| = |a/d| = 5/2$ established here. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K.} (2026).
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})}.
Arisaka-GG-A3-BI6 preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The anomaly-inflow companion of the GG6 Trio. Establishes \BIsix{} as the four-component Green--Schwarz-modified Bianchi identity $\dd H_3 = X_4$ on the 6D bulk, with the four-component coefficient vector $(a,b,c,d) = (1,-1,-2/5,-2/5)$ and the trace-ratio lock $|a/c| = |a/d| = 5/2$ pinned by the \OGNfive{} integer ledger. Section~8.7 (``\BIsix{} as the \GGEP{} completion of GS: the geometrization reading'') is the canonical home of the supportive treatment of the 1984 Green--Schwarz construction that the present paper cites whenever the bulk anomaly-inflow structure is invoked. (This entry absorbs the legacy preprint code GD-2026-5, under which the \BIsix{} package circulated before the cohort renaming; older internal references to ``Paper V'' resolve here.) First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K.\ (2026).}
\textit{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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The historical companion to the
GG6 Trio. Pedagogical commentary on the long chain from Euclid
through Newton, Riemann, Maxwell, and Einstein to the \GG{} stage.
We cite this paper whenever the historical positioning of the
6D bulk against its predecessors is invoked. First cited in \S\ref{sec:cohort-a7}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{Arisaka, K.} (2026).
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT}.
Arisaka-GG-\AFive{} preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The boundary-law foundational paper of the \GDOS{} Trio, paired with the present paper (\ASeven{}) and \ASix{} as the three formal pillars of the open-system half of the \GGTheory{} program. Derives the CPTP bulk-to-boundary projector $\cD_{\OGGEP} = \cR \circ \mathrm{Tr}_{\rm env}$ from \AX{3} \OGGEP{}; constructs the System Instance Packet (\SIP{}) framework that the present paper uses to define the persistence class \SeOS{} and the \QiFour{} Completeness Index \QCI{}; and establishes the boundary Lindblad-form dynamics that the \QiFour{} pillars formalize on the information side. The Outer Hologram + Inner Hologram = Evolving Inner Cosmos (\OH+\IH=\EIC{}) framework on the boundary, developed in detail in the present paper's Part~\ref{part:internal}, descends directly from the boundary-projection structure proved in this companion paper. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K.} (2026).
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind}.
Arisaka-GG-A6-GRIP preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Immediate predecessor to the present paper and the geometric half of the \GDOS{} Trio. Formalizes the four Geometric Invariance Pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) as dual-layer principal theorems T1--T4 with named compatibility propositions $\mathfrak{C}_{\TGG}$, $\mathfrak{C}_{\CCI}$, $\mathfrak{C}_{\GFF}$, $\mathfrak{C}_{\GRIP}$. Derives the runtime constitution \GRIP{} ($\DGI{} \to \GSSB{} \to \GA{}$ cascade), the scalar ledger $F_{\GFF} = F_{\GEP} + F_{\CEP}$, the \SIP{} packet, and the Master Conservation Law $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$. These are treated in the present paper as standing inputs for the parallel \QiFour{} pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}), for the \SeOS{}/\QCI{}/\EIC{} framework of Part~\ref{part:internal}, and for the five-story pagoda hierarchy of Part~\ref{part:emergence}. It is also the source of the runtime definitions adopted here: the Ground Geometric Attractor, established there as the unique global minimum on an admissible stratum and characterized by the four pillars in Theorem~\ref{thm:qi4-gga-characterisation} of the present paper, and the two runtime regimes \GRIPin{} and \GRIPout{}. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{Arisaka, K.\ (2026).}
\textit{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}}.
Arisaka-GG-A7-Qi4 preprint (the present paper).
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The present paper. Formalizes the four
information-theoretic \QiFour{} pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{})
descending from \OGGEP{} on the memory-bearing $U(1)_{Qi}$ sector of the 6D bulk --- the quantum-information half of the \GDOS{} Trio. First cited in \S\ref{sec:cohort-a7}.

\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The first paper of the 4D-Physics
Trio. Operationalizes $\T{2}$ Operational Closure and $\T{7}$ 4D
Quantum Mechanics from $\GDOS{}$ into a concept-by-concept
reformulation of textbook quantum mechanics, with the central thesis that
Einstein was right in 6D and Bohr was right in 4D, both within
their respective domains. First cited in \S\ref{sec:cohort-a7}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K.} (2026).
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A9-SM preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The 28-parameter codebook of the Standard Model derived from the \OGNfive{} integer ledger. Cited here as the downstream paper that uses the \QiQUA{} rank-one inflow, the Arisaka constant $\QB$, and the \QiEP{} common-origin theorem to produce the closed-form Yukawa structure of the SM. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K.} (2026).
\emph{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A10-Cosmos preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Derives the cosmological parameters $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, $\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$ from the \GGConstitution{} and the 6D bulk. Cited here as the companion paper carrying part of the resolution of the individual-environment realization of the pagoda hierarchy (open problem O4). First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K.} (2026).
\emph{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}.
Arisaka-GG-A11-Origin. In preparation; to be posted at
\texttt{https://arisaka-gg.org/} on completion. No result of the present
paper depends on it.
\par\noindent\textbf{Relevance.} The capstone of the program and the proof venue for the vocabulary adopted here: \textsf{T-GGA-as-Ground-State}, the cascade-wide generalization of the per-stratum ground-state result of \ASix{} whose four-pillar characterization is Theorem~\ref{thm:qi4-gga-characterisation} of the present paper; the regime theorem \textsf{T-GRIP-In-Lyapunov-Descent}; and the \textsf{Cascade-Self-Sustenance} corollary whose microscopic mechanism is the \QiHAL{} \GRIP{}-loop (\S\ref{subsec:qihal-switch}). Its eighteen-layer cascade is the full-length version of the five-story pagoda of Part~\ref{part:emergence}. First cited in \S\ref{sec:cohort-a7}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K.} (2026).
\emph{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, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The KK resonance-cluster prediction at $M_{\rm lock} = (1/2) M_{\rm Pl} \e^{-35} \approx 3.85\,\TeV$. Cited here as the downstream paper that uses the \QiQUA{} Arisaka constant and the \QiCON{} two-knee spectral theorem to predict the LHC-falsifiable mass-and-width structure. First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second paper of the
Predictions Branch: the family number derived as a squeeze between
a CP-violation lower bound and a \GDOS{} Morse/budget upper bound,
with the conditioning declared by name. First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The third paper of the Predictions
Branch: identity, absolute stability, the two excited copies, the
positron as conjugate readout, pointlikeness, and field response
derived as six faces of ground-state-ness, with every fence declared; imports \PaperPTwo{}'s
$N=3$ verbatim. First cited in \S\ref{sec:cohort-a7}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K.\ (2026).}
\textit{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}}.
Arisaka-GG-P4-Real preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Foundations Branch: the
origin-of-$i$ member paper, deriving the single imaginary unit of
boundary physics from the two real structures of the \ATwo{} arena
(the compact phase circle and the causal clock). For the present
paper it is the origin of the boundary's complex structure whose
phase the $U(1)_{Qi}$ memory carrier stores; cited at roster level.
First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Foundations Branch: the CPT
member paper, reading C, P, and T as orientation reversals of the
three real bulk factors and carrying the matter--antimatter door;
its mirrors and its law-level arrow of time are kin of the one-way
information cascade that the \QiFour{} runtime drives in a single
direction; cited at roster level. First cited in
\S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Predictions Branch: the sister of
\PaperPThree{}, running the ground-attractor method on the baryonic
stratum. Its baryon number is a winding of the same compact circle
whose phase this paper reads as memory, which makes it the sharpest
available illustration of the difference between a quantity protected
by a symmetry and one protected by a topological class; cited at
roster level. First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} White-Paper Branch: the historical
and conceptual survey of how conservation moved from symmetry to
cohomology. It supplies the context in which \QiQUA{}'s four-current
locking is a statement about classes rather than about charges; cited
at roster level. First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} White-Paper Branch: the structural
treatment of protection by topology. The memory that \QiHAL{}
formalizes is holonomy-class data on a compact fiber, so the ladder of
protection mechanisms set out there is the classification this paper's
fourth pillar sits inside; cited at roster level. First cited in
\S\ref{sec:cohort-a7}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K.} (2026).
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}.
Arisaka-GG-C1-Kaxion preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The dark-matter predictions companion of the \GGTheory{} program. Predicts a $\sim\mu$eV Kaxion-like dark-matter candidate as the boundary realization of the $U(1)_{Qi}$ sector at cosmological scales. Cited here as the high-energy companion paper that tracks the direct-detection program for the informational $U(1)_{Qi}$ symmetry on which \QiQUA{}, \QiEP{}, \QiCON{}, and \QiHAL{} depend. First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second paper of the Cosmology
\& Astrophysics Branch: reads the persistent discrepancy between
early- and late-universe Hubble measurements as the two ends of one
geometric corridor of the 6D bulk, rather than as a systematic
error or new fluid. First cited in \S\ref{sec:cohort-a7}.

\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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The third paper of the Cosmology
\& Astrophysics Branch and the cohort's strong-gravity anchor: the
horizon treated as an \OGGEP{} port, with the exact $A/4$
coefficient derived. First cited in \S\ref{sec:cohort-a7}.

\bibitem[Arisaka(2026, S1-EP)]{ArisakaS1EP}
\textbf{Arisaka, K.} (2026).
\emph{One Functional, Three Principles: Maximum Entropy Production, Minimum Entropy Production, and the Free Energy Principle as Boundary Gauges of \GDOS{}}.
Arisaka-GG-S1-EP supplementary review preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The extended companion of Part~\ref{part:qiep}: the long-form histories of MinEP, \MEPP{}, and \FEP{} at full review depth, the complete proofs, the critical literatures assembled with their exact counterexamples, and the deployment of the three-gauge theorem with the grade of every step. Part~\ref{part:qiep} is self-contained; the supplement is where every compressed statement made there can be checked at length. First cited in \S\ref{sec:qiep-overview}.

\end{thebibliography}

\section*{B.~Other references}
\begin{thebibliography}{99}
\bibitem[Aguilera et~al.(2022)]{Aguilera2022}
\textbf{Aguilera, M., Millidge, B., Tschantz, A., \& Buckley, C.~L.} (2022).
How particular is the physics of the free energy principle?
\emph{Physics of Life Reviews}, \emph{40}, 24--50.
\par\noindent\textbf{Relevance.} Exact analysis showing that the conditions for a statistical boundary to exist in the stationary density of a generic coupled system hold only on a narrow parameter subspace. The reason the inferential gauge here takes its system--environment split from the declared ports of the packet rather than hoping to find one. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Amari(1998)]{Amari1998}
\textbf{Amari, S.-I.} (1998).
Natural gradient works efficiently in learning.
\emph{Neural Computation}, \emph{10}, 251--276.
\par\noindent\textbf{Relevance.} Amari's information-geometric formulation shows that optimization dynamics depend on the metric structure of probability/state manifolds. We cite it in the appendix to support POS-2's claim that ``extremal principles'' on the boundary can be written as geometry-driven flows (e.g.\ natural gradients) when \GDOS{} is expressed on statistical manifolds. First cited in \S\ref{subsec:qiep-fep}.

\bibitem[Amari and Nagaoka(2000)]{Amari2000}
\textbf{Amari, S.-I., \& Nagaoka, H.} (2000).
\emph{Methods of Information Geometry}.
American Mathematical Society.
\par\noindent\textbf{Relevance.} Information geometry provides the Fisher-metric viewpoint that underlies ``natural gradient'' flows. We cite it to motivate the \GDOS{} interpretation of inference dynamics as geometric gradient descent on a scalar functional, and to connect \GGTheory{}'s \GFF{}/\GDOS{} language to established geometric statistics. First cited in \S\ref{subsec:qiep-fep}.

\bibitem[Amari and Nagaoka(2000)]{AmariNagaoka2000}
\textbf{Amari, S.-I., \& Nagaoka, H.} (2000).
\emph{Methods of Information Geometry}.
American Mathematical Society.
\par\noindent\textbf{Relevance.} Information geometry provides a rigorous differential-geometric formulation of statistical models and inference via the Fisher metric and dual connections, serving as a mathematical precedent for writing \GDOS{} as natural-gradient flow on a curved state manifold. First cited in \S\ref{app:neural-domain}.

\bibitem[Anderson(1972)]{Anderson1972}
\textbf{Anderson, P.~W.} (1972).
More is different.
\emph{Science}, \emph{177}, 393--396.
\par\noindent\textbf{Relevance.} Anderson's argument that each level of complexity exhibits genuinely new organizing principles anticipates the pagoda hierarchy: the \GDOS\ framework preserves level-specific emergence while denying level-specific lawlessness. First cited in \S\ref{subsec:discussion-new}.

\bibitem[Ashby(1952)]{Ashby1952}
\textbf{Ashby, W.\ R.} (1952).
\emph{Design for a Brain}.
Chapman \& Hall, London.
\par\noindent\textbf{Relevance.} Frames adaptive behavior as homeostasis of essential variables under ultrastable dynamics; the cybernetic root of the FEP's identification of persistence with bounded surprisal. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Bajcsy(1988)]{Bajcsy1988}
\textbf{Bajcsy, R.} (1988). Active perception. \emph{Proceedings of the IEEE}, \emph{76}, 966--1005.
\par\noindent\textbf{Relevance.} Founding statement of active perception: self-generated action improves the conditioning of the perceptual inverse problem. The lineage in which the action-conditioned observability of the \IH{} sits. First cited in \S\ref{subsec:ih-motion-memory-necessary}.

\bibitem[Ballhausen et~al.(2009)]{Ballhausen2009}
\textbf{Ballhausen, B., Altendorf, K., \& Deckers-Hebestreit, G.} (2009).
Constant \(c_{10}\) ring stoichiometry in the \emph{Escherichia coli} ATP synthase analyzed by cross-linking.
\emph{Journal of Bacteriology}, \emph{191}, 2400--2404.
\par\noindent\textbf{Relevance.} Establishes the $c_{10}$ ring stoichiometry of \emph{E.~coli} ATP synthase, providing a precise molecular integer that enters the event-budget accounting of the Arisaka Equation at the chemistry scale. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Bauer(1931)]{Bauer1931}
\textbf{Bauer, H.} (1931).
Dissipative dynamical systems I.
\emph{Proceedings of the National Academy of Sciences of the USA}, \emph{17}, 311--314.
\par\noindent\textbf{Relevance.} Proves that the equations of motion of a linearly damped system admit no variational formulation in the original variables: friction cannot simply be appended to a Lagrangian. This is the theorem that stopped the classical action tradition at the boundary of every open system, and it is why the trajectory gauge of this Part has to be derived rather than inherited. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Belavin, Polyakov \& Zamolodchikov(1984)]{BelavinPolyakovZamolodchikov1984}
\textbf{Belavin, A.~A., Polyakov, A.~M., \& Zamolodchikov, A.~B.} (1984).
\emph{Infinite conformal symmetry in two-dimensional quantum field theory}.
Nuclear Physics B, 241, 333--380.
\par\noindent\textbf{Relevance.} Founding paper of 2D conformal field theory; with Cardy 1984 establishes that scale invariance at criticality is automatically promoted to full conformal invariance. The structural ancestor of the conformality face of \QiCON{}. First cited in Part~\ref{part:qicon}.

\bibitem[Bell(1964)]{Bell1964}
\textbf{Bell, J.~S.} (1964).
On the Einstein--Podolsky--Rosen paradox.
\emph{Physics}, \emph{1}, 195--200.
\par\noindent\textbf{Relevance.} Bell's theorem clarifies that the measurement problem is not merely philosophical: operational predictions about correlations constrain the kinds of ``hidden variable'' completions that remain possible. This motivates treating measurement as a constitutional issue rather than an interpretational afterthought. First cited in \S\ref{subsec:universal-observation}.

\bibitem[Benham(1979)]{Benham1979}
\textbf{Benham, C.~J.} (1979).
Torsional stress and local denaturation in supercoiled DNA.
\emph{Proceedings of the National Academy of Sciences of the USA}, \emph{76}, 3870--3874.
\par\noindent\textbf{Relevance.} Torsional stress and local denaturation in supercoiled DNA illustrate how geometric topology constrains molecular-scale dynamics, motivating the gauge-connection interpretation of structural currents in the chemistry-domain \SIP. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Bertini et~al.(2015)]{Bertini2015}
\textbf{Bertini, L., De~Sole, A., Gabrielli, D., Jona-Lasinio, G., \& Landim, C.} (2015).
Macroscopic fluctuation theory.
\emph{Reviews of Modern Physics}, \emph{87}, 593--636.
\par\noindent\textbf{Relevance.} The comprehensive account of the quasi-potential and its Hamilton--Jacobi equation as the exact variational characterization of nonequilibrium steady states, of which the entropy-production readings are the near-equilibrium quadratic shadow. The independent statement of the order confinement derived here. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Biehl et al.(2021)]{Biehl2021}
\textbf{Biehl, M., Pollock, F.\ A., and Kanai, R.} (2021).
A technical critique of some parts of the free energy principle.
\emph{Entropy}, \emph{23(3)}, 293.
\par\noindent\textbf{Relevance.} Proves the two blanket definitions inequivalent, the rewriting of expected flows invalid without unstated assumptions, and the Free Energy Lemma false at face value, with counterexamples. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Block(1995)]{Block1995}
\textbf{Block, N.} (1995). On a confusion about a function of consciousness. \emph{Behavioral and Brain Sciences}, \emph{18}, 227--247.
\par\noindent\textbf{Relevance.} The access/phenomenal distinction. The runtime criterion the present framework supplies is one of \emph{access} consciousness; the \emph{phenomenal} hard problem is left explicitly open. First cited in \S\ref{subsec:discussion-limitations}.

\bibitem[Bohr(1913)]{Bohr1913}
\textbf{Bohr, N.} (1913).
On the constitution of atoms and molecules.
\emph{The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science}, \emph{26}, 1--25.
\par\noindent\textbf{Relevance.} Bohr's quantized atomic model establishes the discrete energy levels whose spacing ultimately descends from the Rydberg energy $E_{\mathrm{Ryd}}$; this energy scale reappears in the Arisaka Constant $\QB = (2/5)\,E_{\mathrm{Ryd}}/K$ as the bridge between physics-natural and chemistry-natural units. First cited in Part~\ref{part:qiqua}.

\bibitem[Boltzmann(1872)]{Boltzmann1872}
\textbf{Boltzmann, L.} (1872).
\emph{Weitere Studien \"uber das W\"armegleichgewicht unter Gasmolek\"ulen}.
Sitzungsberichte der Akademie der Wissenschaften zu Wien, 66, 275--370.
\par\noindent\textbf{Relevance.} The $H$-theorem and the first statistical-mechanical derivation of the second law from microscopic dynamics. The variational tradition that \QiEP{} formalizes traces through this paper to Onsager's reciprocal relations and Prigogine's minimum-entropy-production principle. First cited in Part~\ref{part:qiep}.

\bibitem[Boyer(1979)]{Boyer1979}
\textbf{Boyer, P.~D.} (1979).
\emph{The binding-change mechanism for ATP synthesis}.
in Membrane Bioenergetics (Lee et al., eds.), Addison-Wesley.
\par\noindent\textbf{Relevance.} The binding-change rotary mechanism of ATP synthase, providing the molecular machinery that realizes chemiosmotic ATP synthesis. First cited in Part~\ref{part:qiqua}.

\bibitem[Breuer and Petruccione(2002)]{BreuerPetruccione2002}
\textbf{Breuer, H.-P., \& Petruccione, F.} (2002).
\emph{The Theory of Open Quantum Systems}.
Oxford University Press.
\par\noindent\textbf{Relevance.} This standard reference develops the mathematics of open quantum systems (reduced dynamics, master equations, Markovian limits). It anchors our claim that openness is not optional in realistic modeling, and it provides the mature language (CPTP maps and generators) that the \GGConstitution{} intends to elevate to first-class status. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Bruers et~al.(2007)]{BruersMaesNetocny2007}
\textbf{Bruers, S., Maes, C., \& Neto\v{c}n\'y, K.} (2007).
On the validity of entropy production principles for linear electrical circuits.
\emph{Journal of Statistical Physics}, \emph{129}, 725--740.
\par\noindent\textbf{Relevance.} The parity classification of the entropy-production principles by direct circuit analysis, and the identification of time-reversal-odd currents as the obstruction. Reached from below the resolution that the duality and failure theorems of Part~\ref{part:qiep} derive from above. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Bruers(2007)]{Bruers2007}
\textbf{Bruers, S.} (2007).
A discussion on maximum entropy production and information theory.
\emph{Journal of Physics A: Mathematical and Theoretical}, \emph{40}, 7441.
\par\noindent\textbf{Relevance.} Reconstructs both Dewar derivations, shows the 2003 machinery yields MinEP in an explicit model, and supplies the taxonomy of inequivalent MaxEP statements. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Bruineberg et al.(2022)]{Bruineberg2022}
\textbf{Bruineberg, J., Dolega, K., Dewhurst, J., and Baltieri, M.} (2022).
The Emperor's New Markov Blankets.
\emph{Behavioral and Brain Sciences}, \emph{45}, e183.
\par\noindent\textbf{Relevance.} Distinguishes Pearl blankets (epistemic instruments within a model) from Friston blankets (reified physical boundaries), showing the mathematics does not underwrite the metaphysical reading. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Choi(1975)]{Choi1975}
\textbf{Choi, M.-D.} (1975).
Completely positive linear maps on complex matrices.
\emph{Linear Algebra and Its Applications}, \emph{10}, 285--290.
\par\noindent\textbf{Relevance.} Choi's theorem characterizes complete positivity, the mathematical core of physical admissibility for open-system dynamics and measurement instruments. AX-3 and POS-\CP\ rely on complete positivity as the operational replacement for ``no negative probabilities'' across all open systems. First cited in \S\ref{sec:sip}.

\bibitem[Colombo and Wright(2021)]{ColomboWright2021}
\textbf{Colombo, M.\ and Wright, C.} (2021).
First principles in the life sciences: the free-energy principle,
organicism, and mechanism.
\emph{Synthese}, \emph{198(S14)}, 3463--3488.
\par\noindent\textbf{Relevance.} Reconstructs the FEP's argument chain, exposes its stipulative cross-disciplinary identifications, and makes precise the point that a principle unfalsifiable by design cannot serve as an empirical law. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Conant and Ashby(1970)]{ConantAshby1970}
\textbf{Conant, R.\ C.\ and Ashby, W.\ R.} (1970).
Every good regulator of a system must be a model of that system.
\emph{International Journal of Systems Science}, \emph{1(2)}, 89--97.
\par\noindent\textbf{Relevance.} The good-regulator theorem; the FEP's claim that organisms are models of their environments is this result restated probabilistically. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Crooks(1999)]{Crooks1999}
\textbf{Crooks, G.~E.} (1999).
Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences.
\emph{Physical Review E}, \emph{60}, 2721--2726.
\par\noindent\textbf{Relevance.} A cornerstone of stochastic thermodynamics connecting forward/backward path probabilities and nonequilibrium work. We cite it as a modern exemplar of how open-system probability flows carry rigorous, testable constraints---precisely the style enforced by POS-4 and POS-7. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Da Costa et al.(2021)]{DaCosta2021}
\textbf{Da Costa, L., Friston, K., Heins, C., and Pavliotis, G.\ A.} (2021).
Bayesian mechanics for stationary processes.
\emph{Proceedings of the Royal Society A}, \emph{477(2256)}, 20210518.
\par\noindent\textbf{Relevance.} The first mathematically careful Bayesian mechanics: for linear Gaussian processes with an imposed blanket, the synchronization map is exact and the expected internal flow is a genuine free-energy gradient flow. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Dawkins(1976)]{Dawkins1976}
\textbf{Dawkins, R.} (1976).
\emph{The Selfish Gene}.
Oxford University Press.
\par\noindent\textbf{Relevance.} The ``selfish gene'' perspective anticipates the information-centric view of biology; in the \GDOS{} framework, information is a conserved gauge current ($J_{Qi}$), not merely a metaphor. First cited in \S\ref{app:collective-domain}.

\bibitem[Dayan et al.(1995)]{DayanHinton1995}
\textbf{Dayan, P., Hinton, G.\ E., Neal, R.\ M., and Zemel, R.\ S.} (1995).
The Helmholtz machine.
\emph{Neural Computation}, \emph{7(5)}, 889--904.
\par\noindent\textbf{Relevance.} The entry of variational free energy into neural modeling: a recognition density trained to bound the log evidence of a generative model, mathematically identical to Friston's $F$. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[de Groot and Mazur(1962)]{deGrootMazur1962}
\textbf{de Groot, S.\ R.\ and Mazur, P.} (1962).
\emph{Non-Equilibrium Thermodynamics}.
North-Holland, Amsterdam.
\par\noindent\textbf{Relevance.} The standard treatise of linear irreversible thermodynamics; its Chapter~V states the MinEP theorem with the constancy and boundary-condition hypotheses explicit. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Dewar(2003)]{Dewar2003}
\textbf{Dewar, R.~C.} (2003).
Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states.
\emph{Journal of Physics A: Mathematical and General}, \emph{36}, 631--641.
\par\noindent\textbf{Relevance.} Dewar provides an information-theoretic route to maximum entropy production ideas in nonequilibrium systems. We cite it in POS-2 to represent the \MEPP{} lineage and to clarify which claims are principled (variational) versus heuristic. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Dewar(2005)]{Dewar2005}
\textbf{Dewar, R.~C.} (2005).
Maximum entropy production and the fluctuation theorem.
\emph{Journal of Physics A: Mathematical and General}, \emph{38}, L371--L381.
\par\noindent\textbf{Relevance.} Second derivation attempt, claiming Ziegler's orthogonality far from equilibrium via a Gaussian step later shown to confine the result to linear response. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Dewar(2009)]{Dewar2009}
\textbf{Dewar, R.~C.} (2009).
Maximum entropy production as an inference algorithm that translates
physical assumptions into macroscopic predictions: don't shoot the
messenger.
\emph{Entropy}, \emph{11}, 931--944.
\par\noindent\textbf{Relevance.} The author's concession that the Gaussian approximation is valid only close to equilibrium, and the retreat to MEP-as-inference. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Essex(1984)]{Essex1984}
\textbf{Essex, C.} (1984).
Radiation and the irreversible thermodynamics of climate.
\emph{Journal of the Atmospheric Sciences}, \emph{41}, 1985--1991.
\par\noindent\textbf{Relevance.} The structural objection that the dominant radiative entropy production is not extremized by the climate MEPP; the which-entropy-production problem. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Euler(1744)]{Euler1744}
\textbf{Euler, L.} (1744).
\emph{Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes}.
Bousquet, Lausanne and Geneva (Additamentum II).
\par\noindent\textbf{Relevance.} The first rigorous statement of the least-action principle: for a single particle at fixed energy, the trajectory extremizes the abbreviated action. Euler's appendix converted Maupertuis's metaphysical economy into mathematics and founded the calculus of variations as the language of mechanics. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Fermat(1662)]{Fermat1662}
\textbf{Fermat, P.\ de} (1662).
Synth\`ese pour les r\'efractions (letter to M.\ Cureau de la Chambre).
In \emph{{\OE}uvres de Fermat}, Vol.~II (P.~Tannery and C.~Henry, Eds.), Gauthier-Villars, Paris, 1894, 457--463.
\par\noindent\textbf{Relevance.} The founding document of the least-time tradition: the sine law of refraction derived from the single postulate that light takes the path of least time. Every extremal principle reviewed in this Part descends, through the action tradition, from this calculation, and the objection it drew --- why should nature optimize? --- is the question the projected functional finally answers. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Feynman(1948)]{Feynman1948}
\textbf{Feynman, R.~P.} (1948).
Space-time approach to non-relativistic quantum mechanics.
\emph{Reviews of Modern Physics}, \emph{20}, 367--387.
\par\noindent\textbf{Relevance.} The path integral makes the connection between action extremization and quantum amplitudes explicit (stationary phase in the classical limit). We cite it in POS-2 to link PLA to quantum mechanics within a single extremal framework. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Feynman and Vernon(1963)]{FeynmanVernon1963}
\textbf{Feynman, R.~P., \& Vernon, F.~L.} (1963).
The theory of a general quantum system interacting with a linear dissipative system.
\emph{Annals of Physics}, \emph{24}, 118--173.
\par\noindent\textbf{Relevance.} Introduces the influence functional for open quantum systems, a key tool for deriving reduced dynamics and decoherence. It is cited to connect \OGGEP{}/GDOS-style open evolution to established path-integral machinery. First cited in \S\ref{subsec:qiep-bulk-to-boundary}.

\bibitem[Friston et al.(2006)]{FristonKilnerHarrison2006}
\textbf{Friston, K., Kilner, J., and Harrison, L.} (2006).
A free energy principle for the brain.
\emph{Journal of Physiology---Paris}, \emph{100(1--3)}, 70--87.
\par\noindent\textbf{Relevance.} First use of the phrase as a principle; introduces action as the second route to minimizing free energy, the germ of active inference. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston et al.(2010)]{FristonAction2010}
\textbf{Friston, K., Daunizeau, J., Kilner, J., and Kiebel, S.\ J.} (2010).
Action and behavior: a free-energy formulation.
\emph{Biological Cybernetics}, \emph{102(3)}, 227--260.
\par\noindent\textbf{Relevance.} The founding continuous-state active-inference paper: action minimizes the same free energy that perception minimizes, by changing the data rather than the beliefs. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston et al.(2017)]{FristonProcess2017}
\textbf{Friston, K., FitzGerald, T., Rigoli, F., Schwartenbeck, P., and
Pezzulo, G.} (2017).
Active inference: a process theory.
\emph{Neural Computation}, \emph{29(1)}, 1--49.
\par\noindent\textbf{Relevance.} The discrete-state (POMDP) formulation with expected free energy over policies; the basis of essentially all empirical active-inference work. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston et al.(2021)]{FristonDaCostaParr2021}
\textbf{Friston, K., Da Costa, L., and Parr, T.} (2021).
Some interesting observations on the free energy principle.
\emph{Entropy}, \emph{23(8)}, 1076.
\par\noindent\textbf{Relevance.} The authorized reply: accepts that the critique identifies implicit assumptions and heuristic proofs, and supplies a sparse-coupling ansatz restricting the principle's antecedent conditions. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston et al.(2023a)]{FristonSimpler2023}
\textbf{Friston, K., Da Costa, L., Sajid, N., Heins, C., Ueltzh\"offer, K.,
Pavliotis, G.\ A., and Parr, T.} (2023).
The free energy principle made simpler but not too simple.
\emph{Physics Reports}, \emph{1024}, 1--29.
\par\noindent\textbf{Relevance.} The path-integral reformulation that retreats from the nonequilibrium steady-state density factorization to a paths-of-least-action statement of the principle. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston(2005)]{Friston2005}
\textbf{Friston, K.} (2005).
A theory of cortical responses.
\emph{Philosophical Transactions of the Royal Society B}, \emph{360(1456)}, 815--836.
\par\noindent\textbf{Relevance.} First full statement of the brain-level program: hierarchical cortical processing as empirical-Bayes inference minimizing free energy. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston(2010)]{Friston2010}
\textbf{Friston, K.} (2010).
The free-energy principle: A unified brain theory?
\emph{Nature Reviews Neuroscience}, \emph{11}, 127--138.
\par\noindent\textbf{Relevance.} Friston's free-energy principle is a prominent modern attempt to unify perception, learning, and action by variational optimization. In the \GDOS{} framework, FEP appears as one gauge of the universal \QiEP{} extremal engine rather than as a standalone foundation. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Friston(2013)]{Friston2013}
\textbf{Friston, K.} (2013).
Life as we know it.
\emph{Journal of the Royal Society Interface}, \emph{10(86)}, 20130475.
\par\noindent\textbf{Relevance.} The pivot to physics: random dynamical systems with Markov blankets at nonequilibrium steady state claimed to appear to perform Bayesian inference; origin of the contested derivations. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Friston(2019)]{Friston2019}
\textbf{Friston, K.} (2019).
A free energy principle for a particular physics.
arXiv:1906.10184 (monograph, 148 pp.).
\par\noindent\textbf{Relevance.} The full physics-phase construction --- particular partition, synchronization map, free-energy gradient flow --- and the specific target of the technical critiques. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Fuchs et~al.(2009)]{Fuchs2009}
\textbf{Fuchs, G.~W., Cuppen, H.~M., Ioppolo, S., Romanzin, C., Bisschop, S.~E., Andersson, S., van Dishoeck, E.~F., \& Linnartz, H.} (2009).
Hydrogenation reactions in interstellar CO ice analogues: A combined experimental/theoretical approach.
\emph{Astronomy \& Astrophysics}, \emph{505}, 629--639.
\par\noindent\textbf{Relevance.} Demonstrates that complex molecular synthesis proceeds under driven nonequilibrium conditions even in interstellar environments, supporting the universality of the \GDOS\ open-system framework beyond terrestrial chemistry. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Fuller(1971)]{Fuller1971}
\textbf{Fuller, F.~B.} (1971).
The writhing number of a space curve.
\emph{Proceedings of the National Academy of Sciences of the USA}, \emph{68}, 815--819.
\par\noindent\textbf{Relevance.} The writhing number is a topological invariant of closed space curves; it enters the geometric characterization of DNA supercoiling and provides a concrete example of a curl-type conserved quantity in the chemistry-domain current ledger. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Glansdorff and Prigogine(1954)]{GlansdorffPrigogine1954}
\textbf{Glansdorff, P.\ and Prigogine, I.} (1954).
Sur les propri\'et\'es diff\'erentielles de la production
d'entropie.
\emph{Physica}, \emph{20}, 773--780.
\par\noindent\textbf{Relevance.} Introduces the decomposition $dP/dt=d_XP/dt+d_JP/dt$, the first step of the attempted extension beyond the linear regime. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Glansdorff and Prigogine(1964)]{GlansdorffPrigogine1964}
\textbf{Glansdorff, P.\ and Prigogine, I.} (1964).
On a general evolution criterion in macroscopic physics.
\emph{Physica}, \emph{30}, 351--374.
\par\noindent\textbf{Relevance.} Proves $d_XP/dt\le 0$ without linearity or reciprocity and establishes that $d_XP$ is not an exact differential --- the precise sense in which MinEP admits no far-from-equilibrium generalization. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Glansdorff and Prigogine(1971)]{GlansdorffPrigogine1971}
\textbf{Glansdorff, P.\ and Prigogine, I.} (1971).
\emph{Thermodynamic Theory of Structure, Stability and Fluctuations}.
Wiley-Interscience, London and New York.
\par\noindent\textbf{Relevance.} The mature statement of the program: MinEP confined to the linear regime, the local-potential methods, and the stability theory from which dissipative structures emerged. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Goody(2007)]{Goody2007}
\textbf{Goody, R.} (2007).
Maximum entropy production in climate theory.
\emph{Journal of the Atmospheric Sciences}, \emph{64}, 2735.
\par\noindent\textbf{Relevance.} Climate-side critique: the missing dynamical constraints, the steady-versus-chaotic distinction, and the planetary data assessment. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Gorini et~al.(1976)]{Gorini1976}
\textbf{Gorini, V., Kossakowski, A., \& Sudarshan, E.~C.~G.} (1976).
Completely positive dynamical semigroups of \(N\)-level systems.
\emph{Journal of Mathematical Physics}, \emph{17}, 821--825.
\par\noindent\textbf{Relevance.} Along with Lindblad's work, this establishes the generator form required for physical open-system quantum evolution. It supports POS-\CP\ and the \OGGEP\ demand that open evolution be part of the foundational dictionary. First cited in \S\ref{subsec:qiep-bulk-to-boundary}.

\bibitem[Gray and Taylor(2007)]{GrayTaylor2007}
\textbf{Gray, C.~G., \& Taylor, E.~F.} (2007).
When action is not least.
\emph{American Journal of Physics}, \emph{75}, 434--458.
\par\noindent\textbf{Relevance.} The definitive modern account of the fact that the realized path renders the action stationary rather than minimal, with the kinetic-focus criterion for when the extremum is a saddle. Cited to fix the correct universal statement --- stationarity --- which the projected functional of this Part and its three gauges inherit. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Grinstein and Linsker(2007)]{GrinsteinLinsker2007}
\textbf{Grinstein, G., and Linsker, R.} (2007).
Comments on a derivation and application of the ``maximum entropy
production'' principle.
\emph{Journal of Physics A: Mathematical and Theoretical}, \emph{40}, 9717--9720.
\par\noindent\textbf{Relevance.} Identifies the error in the 2005 derivation and the unjustified self-organized-criticality claim of 2003; the pivotal critique. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Haken(1983)]{Haken1983}
\textbf{Haken, H.} (1983).
\emph{Synergetics: An Introduction} (3rd ed.).
Springer.
\par\noindent\textbf{Relevance.} Synergetics develops the slaving principle and order-parameter dynamics for self-organizing systems; these ideas are absorbed into the \GSSB\ and \GA\ stages of the master runtime pipeline. First cited in \S\ref{subsec:discussion-new}.

\bibitem[Hamilton(1834)]{Hamilton1834}
\textbf{Hamilton, W.~R.} (1834).
On a general method in dynamics.
\emph{Philosophical Transactions of the Royal Society of London}, \emph{124}, 247--308.
\par\noindent\textbf{Relevance.} Hamilton's formulation is the modern backbone of the principle of least action and canonical dynamics. We cite it in POS-2 to emphasize that extremal principles historically reorganized mechanics and are a natural candidate for a unification engine. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Hebb(1949)]{Hebb1949}
\textbf{Hebb, D.~O.} (1949).
\emph{The Organization of Behavior: A Neuropsychological Theory}.
Wiley, New York.
\par\noindent\textbf{Relevance.} Hebbian plasticity rule (``neurons that fire together wire together''), the founding model of synaptic learning dynamics. One of the four ancestral threads that \QiHAL{} unifies. First cited in Part~\ref{part:qihal}.

\bibitem[Held \& Hein(1963)]{HeldHein1963}
\textbf{Held, R., \& Hein, A.} (1963). Movement-produced stimulation in the development of visually guided behavior. \emph{Journal of Comparative and Physiological Psychology}, \emph{56}, 872--876.
\par\noindent\textbf{Relevance.} The kitten-carousel experiment: actively moving animals develop visually guided behavior while passively moved animals with identical optic flow do not --- the classic confirmation of the active-versus-passive asymmetry the action-conditioned observability theorem predicts. First cited in \S\ref{subsec:discussion-falsifiability}.

\bibitem[Helmholtz(1867)]{HelmholtzOptik}
\textbf{von Helmholtz, H.} (1856--1867).
\emph{Handbuch der physiologischen Optik}.
Voss, Leipzig; three volumes, completed 1867.
\par\noindent\textbf{Relevance.} Introduces unconscious inference, the doctrine that percepts are conclusions drawn from sensory premises and prior experience; the conceptual ancestor claimed by predictive coding and the free energy principle alike. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Helmholtz(1868)]{Helmholtz1868}
\textbf{Helmholtz, H.\ von} (1868).
Zur Theorie der station\"aren Str\"ome in reibenden
Fl\"ussigkeiten.
\emph{Verhandlungen des naturhistorisch-medicinischen Vereins zu Heidelberg}, \emph{5}, 1--7.
\par\noindent\textbf{Relevance.} The minimum dissipation theorem of Stokes flow, establishing the pattern that the minimum property belongs to the linear regime alone. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Higgs(1964)]{Higgs1964}
\textbf{Higgs, P.~W.} (1964).
Broken symmetries and the masses of gauge bosons.
\emph{Physical Review Letters}, \emph{13}, 508--509.
\par\noindent\textbf{Relevance.} Foundational paper on gauge-boson mass generation. \GG{} reproduces the mass relations from holonomy. First cited in \S\ref{sec:dgi-gssb-ga}.

\bibitem[Hopfield(1982)]{Hopfield1982}
\textbf{Hopfield, J.~J.} (1982).
Neural networks and physical systems with emergent collective computational abilities.
\emph{Proceedings of the National Academy of Sciences of the USA}, \emph{79}, 2554--2558.
\par\noindent\textbf{Relevance.} Hopfield networks provide the canonical model of attractor-based associative memory; in the \QiHAL\ framework, the Holographic Attractor Lattice generalizes the Hopfield picture to a metastable landscape of selected attractors under open-system dynamics. First cited in Part~\ref{part:qihal}.

\bibitem[Jarzynski(1997)]{Jarzynski1997}
\textbf{Jarzynski, C.} (1997).
Nonequilibrium equality for free-energy differences.
\emph{Physical Review Letters}, \emph{78}, 2690--2693.
\par\noindent\textbf{Relevance.} Jarzynski's equality shows that free-energy differences can be inferred from nonequilibrium work statistics. It supports the UEP/\GDOS{} viewpoint that variational/ledger principles remain meaningful and falsifiable far from equilibrium. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Jaynes(1957)]{Jaynes1957}
\textbf{Jaynes, E.~T.} (1957).
Information theory and statistical mechanics.
\emph{Physical Review}, \emph{106}, 620--630.
\par\noindent\textbf{Relevance.} Introduces the maximum-entropy (MaxEnt) principle as a constructive variational rule for statistical mechanics. We cite it in POS-2 to connect thermodynamic extremal principles to inference and to motivate treating free-energy/entropy functionals as legitimate variational objects when paired with operational constraints. First cited in Part~\ref{part:qiep}.

\bibitem[Jaynes(1980)]{Jaynes1980}
\textbf{Jaynes, E.~T.} (1980).
The minimum entropy production principle.
\emph{Annual Review of Physical Chemistry}, \emph{31}, 579--601.
\par\noindent\textbf{Relevance.} The scope critique that distinguishes Onsager's principle from Prigogine's and exhibits a two-temperature network in which the minimum principle fails. In the present framework that network is a case with surviving curl-sector work, and the failure is derived rather than cataloged. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Jiang et~al.(2001)]{Jiang2001}
\textbf{Jiang, W., Hermolin, J., \& Fillingame, R.~H.} (2001).
The preferred stoichiometry of c subunits in the rotary motor sector of \emph{Escherichia coli} ATP synthase is 10.
\emph{Proceedings of the National Academy of Sciences of the USA}, \emph{98}, 4966--4971.
\par\noindent\textbf{Relevance.} Confirms the preferred $c_{10}$ stoichiometry of the ATP synthase rotor in \emph{E.~coli}, a key experimental integer used in the event-budget verification of the Arisaka Equation. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Kalman(1960)]{Kalman1960}
\textbf{Kalman, R.~E.} (1960). On the general theory of control systems. \emph{Proc.\ First IFAC Congress}, Moscow, 481--492.
\par\noindent\textbf{Relevance.} The classical observability criterion: a linear (time-varying) system's initial state is recoverable iff the observability matrix has full column rank. The action-conditioned observability theorem behind the inner hologram is this criterion applied to internal-model reconstruction. First cited in \S\ref{subsec:ih-motion-memory-necessary}.

\bibitem[Kauffman(1993)]{Kauffman1993}
\textbf{Kauffman, S.} (1993).
\emph{The Origins of Order: Self-Organization and Selection in Evolution}.
Oxford University Press.
\par\noindent\textbf{Relevance.} A foundational systems-biology perspective emphasizing attractors, networks, and self-organization in evolution, connecting the attractor and phase-selection language to established biological dynamical-systems frameworks.

\
\
\
\
\
\
\
\
\
\ First cited in \S\ref{app:chemistry-domain}.

\bibitem[Kirchhoff(1848)]{Kirchhoff1848}
\textbf{Kirchhoff, G.} (1848).
Ueber die Anwendbarkeit der Formeln f\"ur die Intensit\"aten der
galvanischen Str\"ome in einem Systeme linearer Leiter auf Systeme, die
zum Theil aus nicht linearen Leitern bestehen.
\emph{Annalen der Physik und Chemie}, \emph{75}, 189--205.
\par\noindent\textbf{Relevance.} The earliest minimum-dissipation theorem for a driven steady state --- currents in a linear resistor network minimize the Joule heat production --- and the benchmark against which Jaynes's counterexample is measured. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Klein and Meijer(1954)]{KleinMeijer1954}
\textbf{Klein, M.\ J.\ and Meijer, P.\ H.\ E.} (1954).
Principle of Minimum Entropy Production.
\emph{Physical Review}, \emph{96}, 250--255.
\par\noindent\textbf{Relevance.} First statistical-mechanical verification of Prigogine's theorem, whose first-order-only agreement is the earliest quantitative marker of MinEP's status as a near-equilibrium approximation. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Kraus(1983)]{Kraus1983}
\textbf{Kraus, K.} (1983).
\emph{States, Effects, and Operations}.
Springer.
\par\noindent\textbf{Relevance.} Kraus' monograph systematizes the operational formulation of quantum measurement and dynamics via completely positive maps. AX-3 adopts this language constitutionally by requiring that measurement be defined in terms of instruments and that open evolution be CPTP. First cited in \S\ref{sec:sip}.

\bibitem[Lagrange(1788)]{Lagrange1788}
\textbf{Lagrange, J.-L.} (1788).
\emph{M\'echanique analitique}.
Veuve Desaint, Paris.
\par\noindent\textbf{Relevance.} The construction of the whole of mechanics on the variational calculus, with neither figures nor Maupertuis's theology. Lagrange's treatise is the point at which the action principle stopped being a curiosity about light and became the standard form of physical law. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Landauer(1961)]{Landauer1961}
\textbf{Landauer, R.} (1961).
Irreversibility and heat generation in the computing process.
\emph{IBM Journal of Research and Development}, \emph{5}, 183--191.
\par\noindent\textbf{Relevance.} Landauer's principle connects information processing with thermodynamic cost. We cite it in the undergrad-intuition appendix to emphasize that ``information'' is physical in open systems, reinforcing \GDOS{}'s treatment of boundary dynamics as energetics + entropy + information bookkeeping under constitutional constraints. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Landauer(1975a)]{Landauer1975a}
\textbf{Landauer, R.} (1975).
Inadequacy of entropy and entropy derivatives in characterizing
the steady state.
\emph{Physical Review A}, \emph{12}, 636--638.
\par\noindent\textbf{Relevance.} The blowtorch theorem: steady states of inhomogeneously heated bistable systems depend on kinetic details invisible to the entire entropy family of functionals. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Landauer(1975b)]{Landauer1975b}
\textbf{Landauer, R.} (1975).
Stability and entropy production in electrical circuits.
\emph{Journal of Statistical Physics}, \emph{13}, 1--16.
\par\noindent\textbf{Relevance.} Exhibits the failure of MinEP arbitrarily close to equilibrium in circuits with inductors --- later diagnosed as a violation of the tacit time-reversal-parity hypothesis. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Lane(2015)]{Lane2015}
\textbf{Lane, N.} (2015).
\emph{The Vital Question: Why Is Life the Way It Is?}
W.~W. Norton \& Company.
\par\noindent\textbf{Relevance.} Lane's emphasis on energy flow in the origin of life aligns closely with the \GDOS{} \GFF{} ledger approach. First cited in \S\ref{app:chemistry-domain}.

\bibitem[L\'evy(1937)]{Levy1937}
\textbf{L\'evy, P.} (1937).
\emph{Th\'eorie de l'addition des variables al\'eatoires}.
Gauthier-Villars, Paris.
\par\noindent\textbf{Relevance.} Founding text on stable distributions and the L\'evy stable laws. Structural ancestor of the heavy-tailed statistics of \QiCON{}. First cited in Part~\ref{part:qicon}.

\bibitem[Lindblad(1976)]{Lindblad1976}
\textbf{Lindblad, G.} (1976).
On the generators of quantum dynamical semigroups.
\emph{Communications in Mathematical Physics}, \emph{48}, 119--130.
\par\noindent\textbf{Relevance.} Lindblad's theorem provides the canonical generator form for Markovian \CPTP\ evolution. It is central to POS-\CP\ and to \OGGEP's claim that open-system dynamics belongs in the theory's syntax. First cited in \S\ref{subsec:qiep-bulk-to-boundary}.

\bibitem[Lisman \& Jensen(2013)]{LismanJensen2013}
\textbf{Lisman, J.~E., \& Jensen, O.} (2013). The theta--gamma neural code. \emph{Neuron}, \emph{77}, 1002--1016.
\par\noindent\textbf{Relevance.} The canonical hippocampal theta--gamma cross-frequency code, nesting at roughly $1{:}5$--$1{:}8$ --- a non-dyadic integer ratio, showing the dyadic ($\lambda=2$) \BFFT{} ladder is a minimal idealization, not a prediction of binary nesting. First cited in \S\ref{subsec:qihal-bfft}.

\bibitem[Lorenz et al.(2001)]{Lorenz2001}
\textbf{Lorenz, R.~D., Lunine, J.~I., Withers, P.~G., and McKay, C.~P.} (2001).
Titan, Mars and Earth: entropy production by latitudinal heat
transport.
\emph{Geophysical Research Letters}, \emph{28}, 415.
\par\noindent\textbf{Relevance.} The comparative-planetology success of the two-box MEP model worked out in the relevant section of that paper. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Maes and Neto\v{c}n\'y(2007)]{MaesNetocny2007}
\textbf{Maes, C.\ and Neto\v{c}n\'y, K.} (2007).
Minimum entropy production principle from a dynamical fluctuation
law.
\emph{Journal of Mathematical Physics}, \emph{48}, 053306.
\par\noindent\textbf{Relevance.} The modern derivation and delimitation: MinEP as the leading order of the occupation-time rate functional around detailed balance, with the frenetic contribution as the obstruction at higher order. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Maes and Neto\v{c}n\'y(2013)]{MaesNetocny2013}
\textbf{Maes, C.\ and Neto\v{c}n\'y, K.} (2013).
Minimum entropy production principle.
\emph{Scholarpedia}, \emph{8(7)}, 9664.
\par\noindent\textbf{Relevance.} Authoritative encyclopedic summary by the authors of the fluctuation-theoretic derivation; source for the consolidated validity conditions adopted in this Part. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{Maldacena, J.~M.} (1998).
The large \(N\) limit of superconformal field theories and supergravity.
\emph{Advances in Theoretical and Mathematical Physics}, \emph{2}, 231--252.
\par\noindent\textbf{Relevance.} Maldacena's AdS/CFT conjecture is the modern archetype for bulk--boundary duality and motivates the idea that geometry can encode quantum gauge dynamics. We cite it as a historical anchor for the \GGEP{}/\OGGEP{} posture: bulk geometry is not merely a classical background, but a structure from which boundary quantum physics and observables can be exported. First cited in \S\ref{sec:introduction}.

\bibitem[Mallat(2009)]{Mallat2009}
\textbf{Mallat, S.} (2009).
\emph{A Wavelet Tour of Signal Processing} (3rd ed.).
Academic Press.
\par\noindent\textbf{Relevance.} Wavelet and multiscale decomposition theory provides the mathematical background for the \BFFT{} dyadic protocol, which organizes multiscale memory and replay in the \QiHAL\ architecture. First cited in \S\ref{subsec:qihal-bfft}.

\bibitem[Mandelbrot(1982)]{Mandelbrot1982}
\textbf{Mandelbrot, B.~B.} (1982).
\emph{The Fractal Geometry of Nature}.
W.~H. Freeman.
\par\noindent\textbf{Relevance.} Mandelbrot established fractal geometry as a mathematical language for scale-invariant natural structures; the \QiCON\ pillar derives fractal attractor supports and anomalous dimensions as consequences of the open-system generator rather than as phenomenological observations. First cited in Part~\ref{part:qicon}.

\bibitem[Maturana and Varela(1980)]{MaturanaVarela1980}
\textbf{Maturana, H.~R., \& Varela, F.~J.} (1980).
\emph{Autopoiesis and Cognition: The Realization of the Living}.
Reidel.
\par\noindent\textbf{Relevance.} The autopoiesis concept---a self-producing network that maintains its own boundary---is the biological precursor of the \SeOS\ persistence class in the present framework. First cited in \S\ref{subsec:seos-qci-why-now}.

\bibitem[Maupertuis(1744)]{Maupertuis1744}
\textbf{Maupertuis, P.-L.~M.\ de} (1744).
Accord de diff\'erentes loix de la nature qui avoient jusqu'ici paru incompatibles.
\emph{M\'emoires de l'Acad\'emie Royale des Sciences de Paris}, 417--426.
\par\noindent\textbf{Relevance.} The first mechanical least-action statement --- mass times speed times distance minimized in all natural processes --- read by its author as evidence of design. Cited as the historical hinge between Fermat's optics and Euler's mathematics, and as the origin of the overreading habit that has shadowed every extremal principle since. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[Maxwell(1865)]{Maxwell1865}
\textbf{Maxwell, J.~C.} (1865).
\emph{A dynamical theory of the electromagnetic field}.
Philosophical Transactions of the Royal Society of London, 155, 459--512.
\par\noindent\textbf{Relevance.} The founding paper of classical electromagnetism and the first gauge theory in physics; the conserved current $\partial_\mu J_e^\mu = 0$ is the structural ancestor of the four-current lock of \QiQUA{}. First cited in Part~\ref{part:qiqua}.

\bibitem[Maynard Smith and Szathm\'ary(1995)]{MaynardSmithSzathmary1995}
\textbf{Maynard Smith, J., \& Szathm\'ary, E.} (1995).
\emph{The Major Transitions in Evolution}.
W.~H. Freeman.
\par\noindent\textbf{Relevance.} The major-transitions framework organizes evolutionary history by successive information-processing innovations; the pagoda hierarchy in the present paper can be read as a physics-grounded restatement of this program. First cited in \S\ref{subsec:pagoda-why}.

\bibitem[McCulloch \& Pitts(1943)]{McCullochPitts1943}
\textbf{McCulloch, W.~S., \& Pitts, W.} (1943).
\emph{A logical calculus of the ideas immanent in nervous activity}.
Bulletin of Mathematical Biophysics, 5, 115--133.
\par\noindent\textbf{Relevance.} Founding paper of theoretical neuroscience; first explicit threshold-logic model of distributed memory in a biological system. Structural ancestor of \QiHAL{}. First cited in Part~\ref{part:qihal}.

\bibitem[Metzler and Klafter(2000)]{Metzler2000}
\textbf{Metzler, R., \& Klafter, J.} (2000).
The random walk's guide to anomalous diffusion.
\emph{Physics Reports}, \emph{339}, 1--77.
\par\noindent\textbf{Relevance.} Foundational paper on anomalous diffusion and L\'evy walks; the \GDOS{} derivation of bacterial chemotaxis predicts the biased L\'evy walk as a geometric consequence. First cited in \S\ref{subsec:qicon-levy}.

\bibitem[Mitchell(1961)]{Mitchell1961}
\textbf{Mitchell, P.} (1961).
\emph{Coupling of phosphorylation to electron and hydrogen transfer by a chemi-osmotic type of mechanism}.
Nature, 191, 144--148.
\par\noindent\textbf{Relevance.} The chemiosmotic hypothesis: biological ATP synthesis is driven by a transmembrane proton current. The empirical bioenergetic side of the \QiQUA{} Arisaka equation at biological temperatures. First cited in Part~\ref{part:qiqua}.

\bibitem[Nakagaki et~al.(2000)]{Nakagaki2000}
\textbf{Nakagaki, T., Yamada, H., \& T\'oth, A.} (2000).
Maze-solving by an amoeboid organism.
\emph{Nature}, \emph{407}, 470.
\par\noindent\textbf{Relevance.} Demonstrates that even a unicellular organism can solve network-optimization problems, providing a low-floor example of adaptive open-system behavior that the \GDOS\ framework must accommodate below the neural level. First cited in \S\ref{subsec:qihal-any-open-system}.

\bibitem[Neal and Hinton(1998)]{NealHinton1998}
\textbf{Neal, R.\ M.\ and Hinton, G.\ E.} (1998).
A view of the EM algorithm that justifies incremental, sparse, and
other variants.
In M.~I.~Jordan (Ed.), \emph{Learning in Graphical Models}, 355--368. Kluwer, Dordrecht.
\par\noindent\textbf{Relevance.} Shows expectation--maximization is coordinate descent on one free energy functional; the template for the FEP's unification of inference and learning under a single objective. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Nicolis and Prigogine(1977)]{NicolisPrigogine1977}
\textbf{Nicolis, G., \& Prigogine, I.} (1977).
\emph{Self-Organization in Nonequilibrium Systems}.
Wiley.
\par\noindent\textbf{Relevance.} A classic monograph on dissipative structures and nonequilibrium self-organization. It provides historical and mathematical grounding for the idea that open systems generically develop attractors and phases under driving---a key theme of \GDOS{}. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Noether(1918)]{Noether1918}
\textbf{Noether, E.} (1918).
Invariante Variationsprobleme.
\emph{Nachrichten von der Gesellschaft der Wissenschaften zu G\"ottingen, Mathematisch-Physikalische Klasse}, 235--257.
\par\noindent\textbf{Relevance.} Noether's theorem is the clearest statement that symmetry is not optional: it has measurable conservation consequences. This motivates POS-\GI\ as the first admissibility rule and supports the constitutional emphasis on symmetry. First cited in Part~\ref{part:qiqua}.

\bibitem[O'Keefe and Dostrovsky(1971)]{OkeefeDostrovsky1971}
\textbf{O'Keefe, J., \& Dostrovsky, J.} (1971).
The hippocampus as a spatial map: Preliminary evidence from unit activity in the freely-moving rat.
\emph{Brain Research}, \emph{34}, 171--175.
\par\noindent\textbf{Relevance.} The discovery of hippocampal place cells is a foundational observation for allocentric spatial representation; in the \QiHAL\ framework, place-like states instantiate the Inner Hologram and the \NHT{} phase-coding architecture. First cited in \S\ref{subsec:ih-allocentric}.

\bibitem[O'Keefe \& Dostrovsky(1971)]{OKeefeDostrovsky1971}
\textbf{O'Keefe, J., \& Dostrovsky, J.} (1971).
\emph{The hippocampus as a spatial map: preliminary evidence from unit activity in the freely-moving rat}.
Brain Research, 34, 171--175.
\par\noindent\textbf{Relevance.} The discovery of place cells in the rat hippocampus; first explicit demonstration of distributed spatial-memory representation in a population of biological neurons. Empirical anchor of the \QiHAL{} program. First cited in Part~\ref{part:qihal}.

\bibitem[Onsager(1931)]{Onsager1931}
\textbf{Onsager, L.} (1931).
Reciprocal relations in irreversible processes. I.
\emph{Physical Review}, \emph{37}, 405--426.
\par\noindent\textbf{Relevance.} Onsager's reciprocity and variational viewpoint provide a foundational extremal structure for near-equilibrium nonequilibrium thermodynamics. We cite it as historical and mathematical background for POS-2 (UEP), supporting the statement that open-system dissipation can be generated by a variational principle (minimum dissipation / gradient flow). First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Onsager and Machlup(1953)]{OnsagerMachlup1953}
\textbf{Onsager, L., \& Machlup, S.} (1953).
Fluctuations and irreversible processes.
\emph{Physical Review}, \emph{91}, 1505--1512.
\par\noindent\textbf{Relevance.} Introduces an action-like functional for stochastic trajectories near equilibrium, providing a bridge between variational principles and stochastic thermodynamics. It supports POS-2 (UEP) in the open-system regime by showing how ``least action'' persists as a most-probable-path principle for noisy dynamics. First cited in \S\ref{sec:qiep-hist-pla}.

\bibitem[O'Regan \& No\"e(2001)]{ORegan2001}
\textbf{O'Regan, J.~K., \& No\"e, A.} (2001). A sensorimotor account of vision and visual consciousness. \emph{Behavioral and Brain Sciences}, \emph{24}, 939--1031.
\par\noindent\textbf{Relevance.} Sensorimotor-contingency theory: perceptual content is constituted by lawful action-dependent changes in sensory input --- the cognitive-science form of the action-conditioned observability used here. First cited in \S\ref{subsec:ih-motion-memory-necessary}.

\bibitem[Ozawa et al.(2003)]{Ozawa2003}
\textbf{Ozawa, H., Ohmura, A., Lorenz, R.~D., and Pujol, T.} (2003).
The second law of thermodynamics and the global climate system: a
review of the maximum entropy production principle.
\emph{Reviews of Geophysics}, \emph{41}, 1018.
\par\noindent\textbf{Relevance.} The canonical geophysical review of the climate MEPP, its planetary tests, and its turbulence applications. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Paltridge(1975)]{Paltridge1975}
\textbf{Paltridge, G.~W.} (1975).
\emph{Global dynamics and climate --- a system of minimum entropy exchange}.
Quarterly Journal of the Royal Meteorological Society, 101, 475--484.
\par\noindent\textbf{Relevance.} Founding paper of the maximum-entropy-production tradition (\MEPP{}); applied to the atmospheric system but generalized to all far-from-equilibrium open systems. One of the four gauges that \QiEP{} unifies. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Paltridge(1978)]{Paltridge1978}
\textbf{Paltridge, G.~W.} (1978).
The steady-state format of global climate.
\emph{Quarterly Journal of the Royal Meteorological Society}, \emph{104}, 927--945.
\par\noindent\textbf{Relevance.} Restates the 1975 extremum as maximum entropy production of the internal transport, the canonical form of the climate MEPP. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Parr et al.(2022)]{ParrPezzuloFriston2022}
\textbf{Parr, T., Pezzulo, G., and Friston, K.} (2022).
\emph{Active Inference: The Free Energy Principle in Mind, Brain, and
Behavior}.
MIT Press, Cambridge, MA.
\par\noindent\textbf{Relevance.} The consolidated textbook treatment of active inference as a process theory and modeling framework, the stratum of the program that stands independently of the physics-phase claims. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Polettini(2013)]{Polettini2013}
\textbf{Polettini, M.} (2013).
Fact-checking Ziegler's maximum entropy production principle beyond the linear regime and towards steady states.
\emph{Entropy}, \emph{15}, 2570--2584.
\par\noindent\textbf{Relevance.} The constitutive-versus-phenomenological dichotomy that separates what the flux-side and force-side readings each select, with counterexamples beyond linear response. The external statement of the boundary that Theorem~\ref{thm:qiep-failure} derives. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Prigogine and Lefever(1968)]{PrigogineLefever1968}
\textbf{Prigogine, I.\ and Lefever, R.} (1968).
Symmetry Breaking Instabilities in Dissipative Systems.\ II.
\emph{Journal of Chemical Physics}, \emph{48}, 1695--1700.
\par\noindent\textbf{Relevance.} The Brusselator: explicit demonstration that beyond a finite distance from equilibrium the stable regime is not the branch of least entropy production. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Prigogine(1945)]{Prigogine1945}
\textbf{Prigogine, I.} (1945).
Mod\'eration et transformations irr\'eversibles des syst\`emes
ouverts.
\emph{Bulletin de la Classe des Sciences, Acad\'emie Royale de Belgique}, \emph{31}, 600--606.
\par\noindent\textbf{Relevance.} The priority document: first statement and proof of the minimum entropy production theorem, including the monotonic decrease of $P$ during relaxation. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Prigogine(1947)]{Prigogine1947}
\textbf{Prigogine, I.} (1947).
\emph{\'Etude thermodynamique des ph\'enom\`enes irr\'eversibles}.
Desoer, Li\`ege.
\par\noindent\textbf{Relevance.} The doctoral thesis in book form and the theorem's canonical early exposition, containing the thermodiffusion example of the relevant section of that paper. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Prigogine(1967)]{Prigogine1967}
\textbf{Prigogine, I.} (1967).
\emph{Introduction to Thermodynamics of Irreversible Processes}.
Interscience.
\par\noindent\textbf{Relevance.} Prigogine's exposition (including the minimum entropy production theorem near equilibrium) is a standard historical source for entropy-production variational ideas. We cite it in POS-2 (UEP) to connect the PLA-to-dissipation bridge with classical nonequilibrium thermodynamics and to clarify when entropy-production extremals are mathematically justified. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Prigogine(1978)]{PrigogineNobel1977}
\textbf{Prigogine, I.} (1978).
Time, Structure and Fluctuations (Nobel lecture, December 8,
1977).
\emph{Science}, \emph{201}, 777--785.
\par\noindent\textbf{Relevance.} Prigogine's own retrospective, presenting the linear-regime theorem together with the breakdown of its variational structure as the gateway to dissipative structures. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Raja et al.(2021)]{Raja2021}
\textbf{Raja, V., Valluri, D., Baggs, E., Chemero, A., and Anderson, M.\
L.} (2021).
The Markov blanket trick: on the scope of the free energy principle
and active inference.
\emph{Physics of Life Reviews}, \emph{39}, 49--72.
\par\noindent\textbf{Relevance.} Argues the blanket is induced by the modeler's parcellation rather than discovered in the system, with the Watt governor as central counterexample to universal scope. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Ramstead et al.(2023)]{Ramstead2023}
\textbf{Ramstead, M.\ J.\ D., Sakthivadivel, D.\ A.\ R., Heins, C.,
Koudahl, M., Millidge, B., Da Costa, L., Klein, B., and Friston, K.} (2023).
On Bayesian mechanics: a physics of and by beliefs.
\emph{Interface Focus}, \emph{13(3)}, 20220029.
\par\noindent\textbf{Relevance.} The programmatic statement of Bayesian mechanics, comparing the FEP to a principle of least action on belief spaces and making the deflationary reading official. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Rao and Ballard(1999)]{RaoBallard1999}
\textbf{Rao, R.\ P.\ N.\ and Ballard, D.\ H.} (1999).
Predictive coding in the visual cortex: a functional interpretation
of some extra-classical receptive-field effects.
\emph{Nature Neuroscience}, \emph{2(1)}, 79--87.
\par\noindent\textbf{Relevance.} The canonical hierarchical predictive-coding model that the Gaussian gradient dynamics of the FEP reproduce; an algorithm that neither requires nor implies the principle. First cited in \S\ref{sec:qiep-hist-fep}.

\bibitem[Rayleigh(1873)]{Rayleigh1873}
\textbf{Rayleigh, Lord (J.\ W.\ Strutt)} (1873).
Some General Theorems relating to Vibrations.
\emph{Proceedings of the London Mathematical Society}, \emph{4}, 357--368.
\par\noindent\textbf{Relevance.} Introduces the dissipation function for linearly damped systems; Onsager's variational function is modeled on it. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Schnakenberg(1976)]{Schnakenberg1976}
\textbf{Schnakenberg, J.} (1976).
Network theory of microscopic and macroscopic behavior of master
equation systems.
\emph{Reviews of Modern Physics}, \emph{48}, 571--585.
\par\noindent\textbf{Relevance.} Defines the standard entropy production functional for Markov jump processes, used in the stochastic formulation of MinEP. First cited in \S\ref{sec:qiep-hist-minep}.

\bibitem[Schr\"odinger(1944)]{Schrodinger1944}
\textbf{Schr\"odinger, E.} (1944).
\emph{What Is Life? The Physical Aspect of the Living Cell}.
Cambridge University Press.
\par\noindent\textbf{Relevance.} Schr\"odinger's question ``What is life?'' frames the central challenge that \GDOS{} with \QiFour{} Pillars addresses: how a physical framework can account for the persistence, adaptation, and information-processing capacities of living systems without invoking new fundamental laws beyond those governing the open-system boundary. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Seifert(2012)]{Seifert2012}
\textbf{Seifert, U.} (2012).
Stochastic thermodynamics, fluctuation theorems and molecular machines.
\emph{Reports on Progress in Physics}, \emph{75}, 126001.
\par\noindent\textbf{Relevance.} Stochastic thermodynamics extends thermodynamic concepts to small fluctuating open systems and provides precise expressions for entropy production and nonequilibrium work. We cite it for the appendix toy models that connect GDOS-style open dynamics to measurable entropy production and for motivating POS-2's open-system extremal forms. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Shannon(1948)]{Shannon1948}
\textbf{Shannon, C.~E.} (1948).
A mathematical theory of communication.
\emph{Bell System Technical Journal}, \emph{27}, 379--423, 623--656.
\par\noindent\textbf{Relevance.} Shannon's information theory provides a quantitative language for ``entropy'' and information flow. Because \GUT\ aims to export a single operational grammar across sciences, information-theoretic quantities become natural candidates for inclusion in the open-system ledger (\GDOS) implied by \OGGEP. First cited in \S\ref{subsec:living-entry-point}.

\bibitem[Simon(1962)]{Simon1962}
\textbf{Simon, H.~A.} (1962).
The architecture of complexity.
\emph{Proceedings of the American Philosophical Society}, \emph{106}, 467--482.
\par\noindent\textbf{Relevance.} Simon's ``nearly decomposable'' hierarchical architecture is a conceptual ancestor of the pagoda hierarchy; the present paper gives it a geometric runtime basis through the story-lift theorem and the \DGI\ shinbashira. First cited in \S\ref{subsec:discussion-new}.

\bibitem[Singh and O'Neill(2022)]{SinghONeill2022}
\textbf{Singh, M.~S., and O'Neill, M.~E.} (2022).
The climate system and the second law of thermodynamics.
\emph{Reviews of Modern Physics}, \emph{94}, 015001.
\par\noindent\textbf{Relevance.} The modern consensus assessment: theoretical and modeling issues preclude broad acceptance of MEPP in climate science. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Stinespring(1955)]{Stinespring1955}
\textbf{Stinespring, W.~F.} (1955).
Positive functions on \(C^\ast\)-algebras.
\emph{Proceedings of the American Mathematical Society}, \emph{6}, 211--216.
\par\noindent\textbf{Relevance.} Stinespring's dilation theorem is the mathematical backbone of quantum channels: CP maps arise from unitary evolution on an enlarged space. We cite it to justify how \OGGEP{} can treat open-system boundary dynamics as fundamental while remaining consistent with microscopic unitarity in the bulk. First cited in \S\ref{sec:sip}.

\bibitem[Susskind(1995)]{Susskind1995}
\textbf{Susskind, L.} (1995).
\emph{The world as a hologram}.
Journal of Mathematical Physics, 36, 6377--6396.
\par\noindent\textbf{Relevance.} Sharpened the holographic principle of \citet{tHooft1993} into the covariant entropy bound. Structural input to the \OH+\IH=\EIC{} framework. First cited in Part~\ref{part:qihal}.

\bibitem[\'t Hooft(1993)]{tHooft1993}
\textbf{'t Hooft, G.} (1993).
\emph{Dimensional reduction in quantum gravity}.
In ``Salamfestschrift'' (eds.\ Ali, Ellis, Randjbar-Daemi), World Scientific, pp.~284--296. arXiv:gr-qc/9310026.
\par\noindent\textbf{Relevance.} Founding statement of the holographic principle: the information content of a region of space is bounded by the area of its boundary. Structural ancestor of \OH+\IH=\EIC{}. First cited in \S\ref{sec:introduction}.

\bibitem[Tononi(2004)]{Tononi2004}
\textbf{Tononi, G.} (2004).
An information integration theory of consciousness.
\emph{BMC Neuroscience}, \emph{5}, 42.
\par\noindent\textbf{Relevance.} IIT's integrated information measure $\Phi$ is compared with \GDOS{}'s \QiHAL{} universality theorem in the consciousness chapters. First cited in Part~\ref{part:qihal}.

\bibitem[Touchette(2009)]{Touchette2009}
\textbf{Touchette, H.} (2009).
The large deviation approach to statistical mechanics.
\emph{Physics Reports}, \emph{478}, 1--69.
\par\noindent\textbf{Relevance.} A modern review of large-deviation theory, clarifying how action-like functionals control typical and rare trajectories. It supports POS-\UEP by providing a general mathematical framework for extremal principles in open systems. First cited in \S\ref{sec:dgi-gssb-ga}.

\bibitem[Turner-Evans et~al.(2020)]{TurnerEvans2020}
\textbf{Turner-Evans, D., et al.} (2020).
The neuroanatomical ultrastructure and function of a biological ring attractor.
\emph{Neuron}, \emph{108}, 145--163.
\par\noindent\textbf{Relevance.} Reviews neural ring attractor circuits in \emph{Drosophila}, providing a direct biological realization of the persistent phase-coded memory and evidence-dependent switching described by \GA\ and the \DGI--\GSSB\ switch in the \QiHAL{} framework. First cited in Part~\ref{part:qihal}.

\bibitem[van Kampen(2007)]{vanKampen2007}
\textbf{van Kampen, N.~G.} (2007).
\emph{Stochastic Processes in Physics and Chemistry} (3rd ed.).
Elsevier.
\par\noindent\textbf{Relevance.} Develops stochastic-process tools used for mesoscopic physics and chemical kinetics. It provides background for interpreting \GDOS{} as the universal open-system bridge between microscopic laws and coarse-grained dynamics. First cited in \S\ref{subsec:from-paper3-to-runtime}.

\bibitem[Vicsek et~al.(1995)]{Vicsek1995}
\textbf{Vicsek, T., Czir\'ok, A., Ben-Jacob, E., Cohen, I., \& Shochet, O.} (1995).
Novel type of phase transition in a system of self-driven particles.
\emph{Physical Review Letters}, \emph{75}, 1226--1229.
\par\noindent\textbf{Relevance.} Introduces a minimal model of collective self-driven motion that has become the standard reference for emergent flocking; in the \GDOS{} framework it instantiates the collective-level \DGI--\GSSB--\GA\ pipeline applied to distributed social or biological systems. First cited in \S\ref{app:collective-domain}.

\bibitem[von Holst \& Mittelstaedt(1950)]{vonHolst1950}
\textbf{von Holst, E., \& Mittelstaedt, H.} (1950). Das Reafferenzprinzip. \emph{Naturwissenschaften}, \emph{37}, 464--476.
\par\noindent\textbf{Relevance.} The reafference principle: the nervous system separates self-produced (reafferent) from externally produced sensory change via an efference copy --- the mechanistic basis for why action is constitutive of internal-model formation. First cited in \S\ref{subsec:discussion-falsifiability}.

\bibitem[von Neumann(1932)]{vonNeumann1932}
\textbf{von Neumann, J.} (1932).
\emph{Mathematische Grundlagen der Quantenmechanik}.
Springer.
\par\noindent\textbf{Relevance.} von Neumann formalized quantum theory axiomatically but introduced measurement as an additional process distinct from unitary evolution. This motivates \GUT's insistence that measurement be defined as a constitutional axiom/postulate rather than left as a dualistic rule. First cited in \S\ref{subsec:universal-observation}.

\bibitem[Wang(1979)]{Wang1979}
\textbf{Wang, J.~C.} (1979).
Helical repeat of DNA in solution.
\emph{Proceedings of the National Academy of Sciences of the USA}, \emph{76}, 200--203.
\par\noindent\textbf{Relevance.} Establishes the helical repeat of DNA in solution ($\sim$10.5 bp/turn), a structural integer that illustrates how geometric periodicity constrains the molecular-scale \SIP\ and the topology of the structural-material current $\JB$. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Watson and Crick(1953)]{WatsonCrick1953}
\textbf{Watson, J.~D., \& Crick, F.~H.~C.} (1953).
Molecular structure of nucleic acids: A structure for deoxyribose nucleic acid.
\emph{Nature}, \emph{171}, 737--738.
\par\noindent\textbf{Relevance.} The double-helical structure of DNA is the foundational molecular template whose geometric and informational properties motivate the chemistry-to-life transition in the \GDOS\ pagoda hierarchy. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Wilson(1971)]{Wilson1971}
\textbf{Wilson, K.~G.} (1971).
Renormalization group and critical phenomena. I.
\emph{Physical Review B}, \emph{4}, 3174--3183.
\par\noindent\textbf{Relevance.} Wilson's RG is the prototype of scale-covariant thinking and motivates POS-\RG. Because \GUT\ aspires to unify across scales and domains, it demands that coarse-graining and operational measurement commute in a controlled way. First cited in Part~\ref{part:qicon}.

\bibitem[Wilson and Kogut(1974)]{WilsonKogut1974}
\textbf{Wilson, K.~G., \& Kogut, J.} (1974).
The renormalization group and the \(\epsilon\)-expansion.
\emph{Physics Reports}, \emph{12}, 75--199.
\par\noindent\textbf{Relevance.} A standard reference for the modern RG viewpoint: coarse-graining as a flow on theory space. POS-5 builds on this picture but adds the constitutional demand that RG must commute with the operational dictionary (\OGGEP{}), enabling a geometric RG coordinate in \GG{}. First cited in Part~\ref{part:qicon}.

\bibitem[Yoshida et~al.(2001)]{Yoshida2001}
\textbf{Yoshida, M., Muneyuki, E., \& Hisabori, T.} (2001).
ATP synthase---a marvellous rotary engine of the cell.
\emph{Nature Reviews Molecular Cell Biology}, \emph{2}, 669--677.
\par\noindent\textbf{Relevance.} Authoritative review of ATP synthase; the rotary mechanism and energy budget are the empirical data against which the Arisaka Equation is verified. First cited in \S\ref{subsec:qiqua-landauer}.

\bibitem[Ziegler(1961)]{Ziegler1961}
\textbf{Ziegler, H.} (1961).
Zwei Extremalprinzipien der irreversiblen Thermodynamik.
\emph{Ingenieur-Archiv}, \emph{30}, 410--416.
\par\noindent\textbf{Relevance.} Companion statement of the two extremum principles (force space and flux space) whose flux-space member is the maximum entropy production form used throughout this Part. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Ziegler(1983)]{Ziegler1983}
\textbf{Ziegler, H.} (1983).
\emph{An Introduction to Thermomechanics}, 2nd ed.
North-Holland, Amsterdam.
\par\noindent\textbf{Relevance.} Textbook consolidation of the orthogonality principle and its plasticity applications; the canonical reference for the orthogonality condition as this Part states it. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Ziman(1956)]{Ziman1956}
\textbf{Ziman, J.~M.} (1956).
The general variational principle of transport theory.
\emph{Canadian Journal of Physics}, \emph{34}, 1256--1273.
\par\noindent\textbf{Relevance.} States and proves the one uncontested MaxEP theorem: the stationary solution of the linearized Boltzmann equation maximizes entropy production under the self-consistency constraint. First cited in \S\ref{sec:qiep-hist-mepp}.

\bibitem[Zurek(2003)]{Zurek2003}
\textbf{Zurek, W.~H.} (2003).
Decoherence, einselection, and the quantum origins of the classical.
\emph{Reviews of Modern Physics}, \emph{75}, 715--775.
\par\noindent\textbf{Relevance.} Decoherence provides a mechanism for effective classicality in open systems and illustrates why openness and environment-induced selection are essential to operational measurement. This motivates \OGGEP\ and the elevation of open dynamics to the constitutional layer. First cited in \S\ref{subsec:universal-observation}.
\end{thebibliography}

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