%% v20.2 (August 22, 2026): THE TWENTY-TWO-PAPER DISCLOSURE.
%%   Every reference to a companion paper now carries that companion's own
%%   permanent page -- e.g. https://preprints.arisaka-gg.org/a3-bi6/ -- in place
%%   of the single program address every reference used to share.  Papers that
%%   are not cohort members keep their own locators: the 1985 Kamiokande paper,
%%   the three PsyArXiv works, and the CKM paper still in preparation are
%%   untouched.
%%   Sixty-four companion references carried no locator at all and now carry
%%   one, in this paper's own format.
%%   A11-Origin is posted with the rest, so the twenty-one papers that called it
%%   "in preparation" no longer do, and fifteen printed statements about its
%%   status -- footnotes, table rows, captions and the staged-submission
%%   paragraph -- were corrected to match.  The author's own reservation about
%%   its upper layers is kept verbatim; only its posting status changed.
%%   Two malformed sentences were repaired: A3's "UCLA preprint,." and P2's
%%   title running into the identifier.
%%   No claim, value, figure, table or grade differs from the previous version.
%% ===================================================================
%% ===================================================================
%% v15.7 (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.
%% ====================================================================
%% ===================================================================
%% v15.6 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY HAD NO AVAILABILITY LINE AT ALL.  A
%%     reader was told only "Arisaka-GG-A11-Origin preprint", with nothing said about where
%%     the paper can be found.  It now reads "In preparation.  To be
%%     posted ... on completion; no result of the present paper depends
%%     on it."
%% (2) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  9 printed sites; \label / \ref identifiers and changelog
%%     comments protected.
%%     THE FIRST RUN OF THIS SWEEP WAS WRONG AND THE GATE PASSED IT.  The
%%     suffix alternation read (e|ed|es|ing), and Python's | is
%%     first-match-wins, so "catalogued" matched the bare "e" and left the
%%     "d" stranded: the sweep wrote "catalogd".  No spelling audit has
%%     that string in its pair list, so nothing reported it; it was found
%%     by reading a rendered page.  Alternation reordered longest-first,
%%     the suffix map asserted exhaustive, and the loosened check re-run
%%     against text carrying the known defect to confirm it still fires.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.5 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the 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 v15.4 printed, except for the lines named above.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.3 -> v15.4, 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.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.2 -> v15.3, August 16, 2026.  THE FRONT-MATTER SHOWCASE.
%% Fourteenth of the twenty-four, and the first C-paper that had no
%% showcase at all -- C1 and C2 each had three pages and needed only
%% the exposure.
%%   p1  Figure 8, the horizon as a visible OGGEP port -- THE CLAIM.
%%       The same partial trace in three settings: undirected in a
%%       laboratory, forced by causal structure at a black hole,
%%       forced by expansion in the cosmos.  This is the paper's
%%       reason for existing -- the place where an abstract axiom
%%       becomes a geometry one can point at.
%%   p2  Figure 9, the conical-deficit origin of 1/4 -- THE RESULT.
%%       1/4 = 4pi/16pi, and neither number knows anything about
%%       black holes: the 4pi is the geometry of spheres, the 16pi
%%       is Newton's law in the weak-field limit, both fixed decades
%%       before anyone computed a horizon entropy.
%%   p3  Table 9, the ten kill criteria -- THE EXPOSURE.
%% STAGE 0, ALL FOUR CHECKS CLEAN: no \includegraphics so no asset
%% can be missing; no pre-existing savebox so Report 385's leading
%% defect is absent; one minipage{textwidth} and it already carries
%% \noindent so A11's margin defect is absent; and the three names
%% added here are letters only, after C2's \GGscTenC2 was read as
%% \GGscTenC plus a stray digit.
%% THE BOXES FILL AT THE BODY'S MEASURED SPACING: \baselinestretch
%% 1, \arraystretch 1.2, \tabcolsep 6 pt.  C3's body \arraystretch
%% is 1.2 rather than 1 -- the same value that bit A8-QM.
%% NEW PROSE IN THIS SHIP: three headlines and three showcase
%% captions, all on the new pages.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
%% ===================================================================
%%  C3-BH v14.24  (August 14, 2026)
%%  TWO ITEMS, BOTH AUTHOR-APPROVED.
%%  1. THE MUON g-2 OVERCLAIM, IN FIVE READER-FACING SITES.  This paper
%%     said the framework "was developed primarily to address ... the
%%     muon g-2", listed g-2 among "its precision tests at specific
%%     observables", named "the muon g-2 anomaly" as one of the
%%     "manifestations of the same 6D bulk geometry", and cited g-2 in
%%     the warrant for the bulk's existence.  A5-GDOS T-9.9 proves the
%%     framework CANNOT touch it: the amplitude splits as A_log(r) + F
%%     with dF/dr = 0, and "the bulk geometrizes A_log exactly, and F
%%     not at all".  Schwinger's alpha/2pi is pure F.  A5's honesty note
%%     and A9's Objection 47 both say in terms that no claim is made on
%%     a_mu; A8's table caption calls the row "the imported
%%     Standard-Model evaluation".  C3 was the sole dissenting paper.
%%     The fix is subtraction; where a replacement was needed rather
%%     than a deletion, the W-boson mass and the fine-structure constant
%%     take g-2's place -- both derived, both already quoted here.  The
%%     FIFTH site said "the muon magnetic moment", contained neither
%%     "g-2" nor "anomaly", and was missed by the grep that scoped this
%%     ship; it was caught by the guard printing every surviving "muon"
%%     site instead of counting them.
%%  2. THE PULL ROW WAS STALE AND ITS PULL WAS COMPUTED WRONGLY.  It
%%     printed a_mu^SM = 116592050e-11 with NO uncertainty against
%%     a_mu^exp = 116592070(15)e-11 and called it +1.4 sigma -- the pull
%%     taken against the experimental error alone, with the theory error
%%     (four times larger) omitted.  Current values: Theory Initiative
%%     WP25 gives 116592033(62)e-11; the E821+E989 average is
%%     116592071.5(14.5)e-11; the difference is 38.5 +- 63.7, i.e.
%%     +0.6 sigma.  The correction is in the framework's favour, since
%%     GG predicts the pure Standard-Model value.  The row label now
%%     states that the value is imported.
%%  Nothing else in the paper is touched.  Arithmetic evaluated in the
%%  script; every anchor asserted unique from \begin{document}.
%% ===================================================================
% ===================================================================
%  C3-BH v14.23 (August 12, 2026) -- cohort-wide sweep, two defects.
%
%  (1) GU-5 IS A THREE-PLUS-TWO PARTITION, exactly as OGN-5 is
%      (M,N,L;J,K).  The separator before k_B is a SEMICOLON:
%      {c, G_N, hbar; k_B, Q_B}.  2 site(s) corrected here.
%      NOT touched, by design: the four-constant Planck unit system
%      {c, G_N, hbar, k_B}, which is a different object and takes a
%      comma; a semicolon there would be wrong.
%
%  (2) Internal affairs -- companion versions, dates, page counts and
%      build status -- do not belong in a preprint.
%%      Nothing of this class was found in this paper.
%
%  No physics changes; no parameter, theorem or falsifier is touched.
% ===================================================================
%% ===================================================================
%%  C3-BH v14.22  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  10 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.
%% ===================================================================
%% ===================================================================
%%  C3-BH v14.21  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 103mm boxes and then
%%  set at 0.78textwidth, so its \footnotesize landed 26% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.619\textwidth, computed rather than chosen: the picture measures
%%  113.9mm 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.
%% ===================================================================
%% ===================================================================
%%  C3-BH v14.20  (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.
%%  Also corrected here: the caption claimed "the present paper
%%  highlighted", but the map has carried no self-reference marker
%%  since the v1.4 master made the figure identical across the
%%  cohort, so nothing was highlighted.  Now "among them".
%%  Recipe and variant analysis: [v13-support]/sweep22.py.
%% ===================================================================
%% ===================================================================
%%  C3-BH v14.18  (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.
%% ===================================================================
%% ===================================================================
%% v14.17 (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.
%% ===================================================================
%% ===== v14.16 (2026-07-23): PROBLEMS & WORKED SOLUTIONS, SHIP 2 (L-R) =====
%% Ship two, completing the C3 problems campaign: 21 problems added
%% to appendices L, M, O, P, Q, R (L:4 M:3 O:4 P:4 Q:3 R:3);
%% appendix N (translation table) carries none by design.  53
%% problems now span the course.  Format as ship one (P5 model);
%% numbering matches actual Step numbers; all numbers verified
%% programmatically: the 10^86 SV degeneracy; gravity-protects-the-
%% bound; the AdS3/CFT2 RT check (c/3 log, Brown-Henneaux); replica
%% dominance = min(S_rad, S_BH); the 3/7 constant-rate toy vs 0.646;
%% two doors vs one door; QES = GFF ledger row by row; Stinespring
%% CP/TP; the DPI easy legs; the three ledger limits; the 0.459-nat
%% Hawking-quantum price; the Kerr-Newman kappa limits; extremal 29.3%;
%% Smarr with charge; the 1.0-kHz superradiant window; Lambda = 3H^2;
%% the 2*pi*r_H thermal wavelength; the 10^122 with holographic
%% reading; the conformal-scalar zero; pi e^70; the flat-interval
%% tower identity.  The front cover and Executive Summary gain their
%% "problems" clauses, true of the whole course as of this ship.
%% ===== v14.15 (2026-07-23): PROBLEMS & WORKED SOLUTIONS, SHIP 1 (E-K) =====
%% The cohort-wide problems campaign (P5-CPT model, author-approved,
%% all recommendations adopted), ship one of two for C3:
%% 32 problems with complete worked solutions added to course
%% appendices E-K (E:6 F:6 G:4 H:5 I:4 J:3 K:4), each set placed
%% before its chapter's take-away box (read -> do -> summary),
%% numbered X.s.n by appendix letter and Step, in the P5 format
%% (bold statement; Solution in explicit Steps; "Physics intuition:"
%% close; cross-references building the course network).
%% Every numerical answer verified programmatically: the 4.6e4-Msun
%% tidal-parity mass; Sgr A* 53 vs 51.8+-2.3 muas; the sqrt(8/9)
%% ISCO energy; the 46.2-Msun Christodoulou floor (29% ceiling vs
%% 4.6% observed); exact Bekenstein-bound saturation; the 4/3 GSL
%% margin; Unruh 4.0e-20 K per g and 7.4e22 m/s2 per 300 K; the
%% 0.61-lunar-mass CMB crossover; the N=2 Page average 1/3 nat; the
%% 1-2^{-3/2} ratio re-derived; the ~230 bit/s return rate.
%% Ship two (appendices L-R) follows after sign-off; the front-cover
%% and Executive-Summary "problems" clauses land there, when true of
%% the whole course.  Also fixed (declared): the pre-v14.11
%% Executive-Summary line still describing "ten derivation
%% appendices ... island dictionary" -> the fourteen-chapter course.
%% ===== v14.14 (2026-07-23): EN-DASH PASS + BLANK-PAGE REPAIR (author) =====
%% (1) The Bekenstein--Hawking en-dash micro-pass: 10 single-hyphen
%%     body sites (incl. one section title and four table rows)
%%     normalised to the canonical en-dash; the Strominger--Vafa
%%     bibliography title keeps its original hyphen by design.
%% (2) Author-reported blank page 180 removed.  Root cause: the
%%     named-statement deck's heading + [H] table + caption formed
%%     one unbreakable block ~39pt taller than the page (also the
%%     source of the long-standing Overfull-vbox warning and a
%%     caption/folio collision on the deck page).  The deck table
%%     is now a [p!] float page and the V section carries a short
%%     lead-in paragraph.  Expected gate change: the Overfull-vbox
%%     line DISAPPEARS from the log (an improvement, declared).
%% ===== v14.13 (2026-07-23): PART XIV REFERENCES, A10 FORMAT (author-comm.)
%% The author's commission after the v14.12 sign-off: Part XIV
%% completed to the A10-Cosmos reference format.
%% (1) Banner subtitle to the A10 wording; three A10-style intro
%%     paragraphs (Relevance notes; the two readers, with the
%%     black-hole lineage; cohort-keys note).
%% (2) A/B split: A = the 18 cohort papers in cohort order
%%     (A1-A11, P1-P5, C1-C2); B = the 71 external references,
%%     ALPHABETICAL by first author.  bibsection suppressed per
%%     A10; the old bib-begin macro (duplicate toc line) dropped --
%%     the Part line stays via the banner.
%% (3) "First cited in ..." appended to every Relevance note (all
%%     89), computed programmatically: \S/Appendix link where the
%%     enclosing unit is labelled, else the Part.  Zero orphans
%%     verified (every key is cited in the body).
%% Relevance texts carried over verbatim; no entry text edited.
%% ===== v14.12 (2026-07-23): PART XIII GENERAL-READER Q&A (author-comm.) ==
%% The author's commission after the v14.11 sign-off: open Part XIII
%% with fundamental, philosophically profound Q&A for a broad
%% audience, drawn from Parts IX-XI.
%% (1) NEW section "Questions from the general reader": THIRTY
%%     questions Q1-Q30 with mini-personas and extensive plain-
%%     English answers, in four arcs (what/why; the paradox and the
%%     information; claims and honesty; what it means), each with a
%%     pointer to the full treatment.  New macro \PGenQ; answers
%%     reuse objqabody.  Register kept calibrated: Born-weight
%%     slogan flagged conditional (Q24), species sum conceded open
%%     (Q21), end-state open (Q15), no rival polemic (Q18),
%%     philosophy labelled as reading (Q25, Q29).
%% (2) Part banner: "Thirty Questions from the General Reader, in
%%     Plain English, followed by Fifty-three Adversarial
%%     Readings...". Part intro extended.  O-BH deck, numbering,
%%     and scorecard untouched; community sections renumber
%%     65-77 -> 66-78 (no \ref points at them).
%% (3) Cross-references updated at three sites: Executive Summary,
%%     the reading-guide weekend, Part XI's back-matter bridge.
%% No new lists, figures, tables, or citations; no canonical
%% numeric tokens introduced.
%% ===== v14.11 (2026-07-23): PART XII COURSE OVERHAUL (author-comm.) ======
%% The author's commission after the v14.10 sign-off: Part XII rebuilt
%% on the A10-Cosmos model as a graduate reading course.
%% (1) Banner -> \GGAppendixPart (A10's front cover); ALL Part XII
%%     sections now letter A..Y (dictionaries A-D, course E-R,
%%     registries S-Y); body sections end at 64; Part XIII resumes
%%     at 65 (no \ref pointed at the renumbered sections; verified).
%% (2) A10-style front-cover prose: three groups, the three-step
%%     course discipline (standard result -> what it assumes and
%%     cannot explain -> the GG reading pointing into the main text),
%%     stylistic commitments, prerequisites + recommended textbooks.
%% (3) FOUR NEW course appendices, textbook-grade, physics intuition
%%     on the spot: E Schwarzschild from Einstein's equations (with
%%     new Penrose-diagram and effective-potential figures; worked
%%     M87*/Sgr A* shadow and density numbers); F the four laws
%%     (Raychaudhuri derived; GW150914 area audit; the BCH analogy
%%     table); G Bekenstein's entropy and the GSL (teacup, bound
%%     derived, species problem); L the modern era (Strominger-Vafa
%%     worked, holographic principle, AdS/CFT, RT/QES, islands +
%%     replica wormholes, the five remaining problems stated
%%     precisely, and the map of where the main text picks up each).
%% (4) Course apparatus (A10's): \GGteachbox / \GGtakeaway /
%%     \GGreadalong added to every course appendix, new and existing.
%% (5) 11 new bibliography entries (MTW, Carroll, Poisson,
%%     Chandrasekhar, Frolov-Novikov, Birrell-Davies, Parker-Toms,
%%     Wald-QFT, Susskind-Lindesay, Harlow lectures, AHMST review),
%%     each with a Relevance note, appended to the single-list
%%     bibliography.
%% Declared deltas: +2 figures (tikz, no assets), +1 table,
%% +1 itemize (textbook list). All worked numbers verified by
%% script. No theorem envs touched; no existing labels changed.
%% ===== v14.10 (2026-07-23): PART XI (CONCLUSIONS) PASS (author-comm.) =====
%% The author's commission after the v14.9 sign-off: Part XI concluded
%% for a broader audience -- smooth natural flow, extensive physics
%% intuition (the BH as an open system, OGGEP, what is unique here),
%% profound philosophical implications, ~50% extension, and a closing
%% paragraph to end the paper on.
%% (1) Part introduction added (the last Part without one; the
%%     standing intro/close convention now covers Parts IV-XI).
%% (2) S59 + the geometry-imposed vs lab-chosen cut; "the universe's
%%     most perfect open quantum system"; why theorems are possible.
%% (3) S60 + the missing-environment paragraph (the working
%%     physicist's mixed state; the 6D register that supplies it).
%% (4) S61 + the horizon as a relation, not a place; the decoherence
%%     student's inheritance ("geometry's signature, not a new
%%     rulebook").
%% (5) S62 + what kind of probability survives (lawful ignorance
%%     through a port; the non-optional coarse-graining); Born-weight
%%     derivation pointed to the A8 companion with conditional status
%%     stated (one new citep of the A8 key); the view-from-nowhere
%%     paragraph deepened ("it gives the gap equations").
%% (6) S63 + the Rosetta metaphor taken literally (three scripts, one
%%     equation) and the fourth, particle-physics script.
%% (7) S64 + ledger-discipline sentence and the bridge to the back
%%     matter ("the paper asks to be checked, not believed"); the
%%     final paragraph replaced with the commissioned close.
%% No new lists; no label/ref changes; no canonical numeric tokens
%% touched; Author's Note and Acknowledgments untouched.
%% ===== v14.9 (2026-07-23): PARTS IX/X RESTRUCTURE (author-approved) ======
%% Option 1 of the IX/X redundancy memo, executed as approved: Part IX
%% is now the single home of ALL comparison; Part X is "Discussions:
%% Inventions, Open Questions, and Future Directions".
%% (1) X's five-framework comparison section + runtime-uniqueness
%%     subsection + table MOVED into IX's deeper-analysis section and
%%     fused with it (AdS/CFT limitations fused once, + cavity/brane
%%     clause + OGGEP-dictionary sentence; fuzzball GDOS-position
%%     paragraphs two -> one; islands "virtue" sentence deduped, IX's
%%     precise-statement display kept; ER=EPR kept; classical-GR and
%%     firewall subsections moved verbatim).
%% (2) IX synthesis de-duplicated: "What is new" itemize -> prose +
%%     pointer to Part X. REGISTER FIX caught in passing: the old
%%     bullet said the coefficient is "fixed by OGN-5" -- contradicting
%%     tab:bh_axioms and T-BH-1's OGN-5-independence; now stated
%%     correctly (regulator-independent, OGN-5-independent).
%%     "Role of the present paper" trimmed to a pointer (full text is
%%     X's role section).
%% (3) IX head-to-head itemize -> prose; X's two enumerates -> prose.
%% (4) X's "Open questions and future work" + "What the module
%%     explains, and what remains open" merged into one section; both
%%     labels (sec:open-questions-c3, sec:explains-open) kept.
%% (5) X falsifiability: opening division-of-labour sentence points
%%     the observational kill table to Part VIII; theory-side content
%%     and tab:falsification-dictionary kept; sec:falsifiers kept.
%% (6) Part X banner subtitle renamed; intros + "Part in one breath"
%%     closes added to both Parts (standing convention now covers
%%     Parts IV-X).
%% (7) Appendix section-counter hack 69 -> 67 (X: -2 numbered
%%     sections). Zero itemize/enumerate remain in Parts I-X.
%% (8) Hygiene (preview catch): three end-of-line hyphen breaks that
%%     rendered as "Bekenstein- Hawking" (x2) / "radial- interior"
%%     normalised to "Bekenstein--Hawking" / "radial-interior".
%% (9) Layout (preview catch): stale \clearpage before the
%%     falsifiability section removed -- it stranded a near-blank
%%     page in the restructured Part X.
%% No theorem envs in IX/X; all labels, citations, and table contents
%% preserved; net text reduction from de-duplication ~2 pages.
%% ===== v14.8 (2026-07-22): PARTS V-VIII PRESENTATION PASS (author Go) ====
%% The Part-IV treatment extended to Parts V-VIII on the author's
%% instruction: (1) Part introductions, (2) bullets -> book prose,
%% (3) the standing Part-close convention (physics intuition + bridge),
%% and for Part VIII, main-text narration of the three tables.
%% PART V: intro (the paper's summit; induced gravity the instrument;
%%   the four-step dial-free plan); 3-step strategy enumerate -> prose;
%%   close ("the 1/4 is not a count that could drift; it is a ratio
%%   that cannot" + bridge to VI).
%% PART VI: intro (the account settles in time; the one-sentence
%%   mechanism); five bullet blocks -> prose incl. the 11-item summary
%%   section retold as a four-paragraph story; close (information
%%   displaced, assumption expiring + bridge to VII).
%% PART VII: intro (the signature is a precise silence, budgeted);
%%   six bullet blocks -> prose (windows, contrasts, PBH fronts,
%%   analog systems, analog implications, de Sitter numbers); close
%%   after tab:bh_predictions ("silence is the prediction" + bridge).
%% PART VIII: narration added around all three tables -- cadence
%%   walk-through after tab:bh-exp-timeline; contract intro + three-
%%   reaches digest around tab:bh_kill (4 live / 1 model-level /
%%   4 beyond reach / 1 internal); two-halves intro + three-features
%%   digest around tab:bh_evidence (the -1.5 sigma H0 strain named,
%%   at C2's register; consistent = survival, not confirmation);
%%   close (one-sided record + bridge to IX).
%% Zero itemize/enumerate remain in Parts V-VIII. No theorem, table
%% content, grade, number, or citation touched.
%% ===== v14.7 (2026-07-22): PART IV PRESENTATION PASS (author Go) =========
%% The author's four Part-IV items:
%% (1) A proper Part introduction before Section 22 -- why OGGEP/GDOS
%%     is finally applied to the black hole as an open system (the
%%     framework's native object; lab trace chosen vs horizon trace
%%     forced -- the "contrast just stated" sentence now has its
%%     antecedent), plus a five-step map of the Part and a proper
%%     introduction of Figure 7.
%% (2) Figure 7 (fig:oggep-port) de-overlapped: the "6D bulk" title
%%     and "Hawking" label sat on top of each other at the top-left.
%%     Box widened (8 -> 10.2 units), title anchored west in
%%     sans-serif bold (Figure-8 style), "Hawking" -> "Hawking
%%     radiation" moved beside its arrows, interior note and bottom
%%     boundary box given more room; all figure text now \sffamily.
%% (3) Part IV's two bullet blocks -> book prose (the special-locus
%%     symptoms/diagnosis; the GDOS re-derivation's three moves).
%%     Zero itemize/enumerate remain in Part IV.
%% (4) The Part now closes with "The Part in one breath, and the door
%%     to the next": physics intuition ("a black hole is an open
%%     system, and geometry is its experimenter") + bridge to Part V.
%%     STANDING CONVENTION per the author: every Part ends with
%%     physics intuition + a bridge to the next Part.
%% No theorem, table, grade, number, or citation touched.
%% ===== v14.6 (2026-07-22): PART III BULLETS -> BOOK PROSE (author Go) ====
%% Same treatment Part I received at v14.4, applied to Part III "The
%% Open Problems" on the author's instruction. Six blocks converted to
%% flowing prose, every fact, number, name, and citation preserved:
%% (1) the four classes of resolution + verdict (the "Position n"
%%     references now read "the first/second/third/fourth position");
%% (2) microstates in ordinary thermodynamics vs no-hair;
%% (3) the five singularity replacements (organised by what each
%%     proposal removes: the endpoint, the interior, the crossing, the
%%     interior's independence, the observability);
%% (4) the four complementarity postulates (numbered in prose; the
%%     following paragraph's postulate references still resolve);
%% (5) the GG status of the four postulates;
%% (6) the five-point state-of-play consensus.
%% Zero itemize/enumerate remaining in Part III. No theorem, table,
%% grade, number, or citation touched.
%% ===== v14.5 (2026-07-22): PART II REBUILT ON THE A10 MODEL (author Go) ==
%% Author commission: make Part II comparable to A10-Cosmos Part II
%% (its Sections 10-13), adapting the inherited-framework treatment to
%% the black hole. The two-page "Mathematical Preliminaries" is replaced
%% by the five-section "The Inherited Framework: GG-Theory in One Part":
%% (1) One geometry, no dials [sec:bulk-arena + eq:BH_bulk kept; A10's
%%     throat/boundary figure adapted -> fig:bulk-boundary];
%% (2) The boundary projection [OGGEP dictionary + GDOS displays; the
%%     observer-indexed-vacuum subsection -> ssec:observer-vacuum,
%%     pointing at this paper's T-BH-3/T-BH-4 as the R14/R15 home];
%% (3) Zero free parameters [BI-6 display + OGN-5 cascade + dual lock;
%%     horizon payoffs: K=35 -> lock-scale suppression 1e-46, J=7 ->
%%     e^7 PBH ladder / F-BH-4];
%% (4) The runtime [sec:runtime-vocab kept, C3's No-Exit grade caveat
%%     preserved verbatim in content; GGA refinement, Qi-4 selector,
%%     A9 precedent, Wilson-loop-=-microstates identity, descent
%%     ladder imported as A10's T-Cosmos-1/2 (NO new theorem envs --
%%     T-BH-n = Theorem n alignment untouched); zero-parameter
%%     derivation tree adapted to the seven black-hole observables ->
%%     fig:derivation-tree-bh, borders = the paper's registers];
%% (5) What this buys the black hole [inheritance rule + problem-to-
%%     ledger planning table tab:problem-to-ledger-bh + register note].
%% Housekeeping: roadmap box/caption renamed; appendix section-counter
%% hack advanced 66 -> 69 (three net-new arabic sections). All macros
%% and citation keys verified against C3's preamble and bibliography;
%% A10 cosmology-specific macros/cites avoided.
%% ===== v14.4 (2026-07-22): FRONT-MATTER + PART-I PRESENTATION PASS =======
%% The author's eight review items of 2026-07-22, executed in one ship:
%% (1) Executive Summary extended ~half page (microstates + de Sitter
%%     paragraph; second-law price paragraph) -- page-4 orphan cured.
%% (2) \GGfrontmatteronepagetrue set: List of Figures joins the
%%     Overview-of-Contents page (cohort-standard switch, 13 companions).
%% (3) "Conventions, naming, and cross-references" moved from its
%%     dangling post-ToC page to the tail of the Preface.
%% (4) Figure 1 (timeline) now [H] directly under the Part-I banner;
%%     the Part-I opening rewritten as three paragraphs (white-paper
%%     register) introducing the figure smoothly.
%% (5) Figure 2 (funnel) shrunk ~25% and recoloured to the faint
%%     flowchart palette. NOTE: the A9/A10/A11 funnels still carry the
%%     stronger cohort palette -- recorded as a cohort-side style note.
%% (6) Part I's six bullet blocks rewritten as flowing prose (four
%%     laws; 1976-95 positions; Page-curve stages; AdS/CFT's three
%%     gifts; 2006-19 developments; islands limitations). Every fact,
%%     number, name, and citation preserved verbatim in content.
%% (7) Figure 3 (cohort map) enlarged to full text width; the long
%%     frozen cohort-shared caption trimmed to a short reader-facing
%%     one BY AUTHOR ORDER (divergence from the shared caption is
%%     deliberate; recorded for future cohort syncs).
%% (8) Figure 4 (roadmap) boxes shortened (8/8.5/10.5mm) and narrowed
%%     ~8% so the fixed-width render carries ~8-10% larger type with
%%     open gaps for the arrows.
%% No theorem, lemma, corollary, grade, canonical number, table, or
%% seed response touched (protect-asserted).
%% ===== v14.3 (2026-07-22): RB-13 CITATION MICRO-PASS (author ruling) =====
%% The real-bulk horizon-entry proposal (GG-RB-Proposal-C3-2026-07-21.md)
%% was ACCEPTED by the home-thread ruling of 2026-07-22 as A8 Statement
%% RB-13 (registrational; no series-entry number; proven series stands
%% at R15; custody count stands at five; A8 v13.8 is the registering
%% ship).  C3-side updates:
%% (1) Register-section lead and entry paragraph: propose-not-register
%%     language -> registered as Statement RB-13 (cite ArisakaA8QM).
%% (2) Named-statement deck: the (prop.) horizon-entropy row's key is
%%     now RB-13; grade cell "proposal to the series' home papers" ->
%%     "registered in A8 (registrational; no new theorem)".  The deck
%%     stays at 15 rows; one (prop.) row remains (the islands
%%     dictionary proposition, by design).
%% (3) Cohort-side record 3 folded in: the H0 evidence cell's "lift
%%     open (C2)" -> C2's preferred register ("lift: sign + rough third
%%     derived, rest a named ladder systematic"), matching A8 v13.8.
%% No theorem, grade, number, or seed response touched.
%% ===== v14.2 (2026-07-22): WARMTH-AND-ACCESSIBILITY POLISH (author Go) ====
%% Author instruction: port the white paper's (C3-WhitePaper-v1.0) best
%% plain-English devices into C3-BH; smoother, warmer, more accessible,
%% with zero loss of rigor.  Fifteen narrative insertions, no deletions
%% of substance; every theorem/lemma/corollary environment, every table,
%% every grade, every canonical number, and every seed objection response
%% is byte-identical to v14.1.
%% W1  Part I opener: the simplest objects / collapsed library.
%% W2  Bekenstein: the cup of tea; the scandalous formula (census,
%%     "states of what?", not one half, not 1/(4pi), but 1/4).
%% W3  Paradox stated: burn a photograph.
%% W4  Page curve: "derive the turnover".
%% W5  Counting problem: no features vs e^{10^77} of them.
%% W6  Strominger-Vafa: the protected corner.
%% W7  Conical replica: neither number knows anything about black holes.
%% W8  Novelty 2: a ratio cannot drift.
%% W9  AdS/CFT comparison: our universe declines to provide either;
%%     the observer anchor.
%% W10 External hostages: the Illinois line.
%% W11 Conclusions: mistaking a shadow for the object.
%% W12 What is a horizon: both observers right; objective does not mean
%%     observer-free (simultaneity, 1905).
%% W13 Determinism: Einstein's instinct / Bohr's / each correctly.
%% W14 Rosetta stones: one mechanism at two volumes.
%% W15 Port claim: bookkeeping convention -> causal wiring.
%% All insertions stay at or below the named-statement deck's register:
%% no claim upgraded, no grade touched, no new physics asserted.
%% ===== v14.1 (2026-07-22): REVIEWER-RESPONSE CLOSURE (author Go) =========
%% Author's blanket approval of 2026-07-22 ("I support any changes, as
%% long as my fundamental standpoints of GG-Theory are intact").
%% Closes every remaining in-paper item from the P3b ten-reviewer audit
%% and the Phase-1 deferred-registry list:
%% (1) Worked-table cells trued: M87* t_evap 10^96 -> 10^97 yr (5.8e96);
%%     PBH T_H 3e11 -> 2e11 K (2.45e11). All other cells re-verified.
%% (2) Duplicated closing sentence removed from the Foundations-Branch
%%     body paragraph (the frozen cohort-map caption keeps it; the
%%     caption is verbatim-shared across the nineteen papers).
%% (3) Part-XII banner course list now names the islands comparison.
%% (4) lem:tower (L-BH-Tower) internal pointer amended: the F-BH-9
%%     reference now routes to Table tab:bh_kill where the row lives
%%     (protected-statement amendment under the author's approval; one
%%     pointer, no physics content changed).
%% (5) Phase-1 "deferred by design" keys completed: L-BH-Shore (text
%%     key) registered for the Euclidean-arena license -- home
%%     paragraph sentence, deck row (Derived-conditional; quasi-free;
%%     saddle imported), source-ledger row, promotion-roadmap mention;
%%     C-BH-2 (text key) registered for the de Sitter same-theorem
%%     corollary -- home sentence at ssec:dS-induced, deck row
%%     (Derived-conditional, T-BH-1's premises + same-arena clause,
%%     exposure F-BH-7). Grades inherited verbatim from the existing
%%     text; nothing upgraded, nothing new claimed.
%% Cohort-side items remain recorded, not C3 edits: A8/P4 ruling on the
%% real-bulk proposal; charter cohort-count language; A8 g-2/W rows;
%% C2 H0-row cross-check.
%% ########################################################################
%% ##### v14.0 (2026-07-22): RELEASE -- THE C3-BH OVERHAUL COMPLETE ######
%% ########################################################################
%% The overhaul thread v13.0 -> v14.0 (charter GG-C3-Initial-Prompt),
%% author-signed at every ship, brought the cohort's least-updated paper
%% to the P5 structural standard. Consolidated record:
%% PHASE 1 -- THE SCIENCE (v13.1-13.7): hygiene + mid-2026 experiment
%%   record; T-BH-5 thermal clock face (KMS via T-P5-B1, Derived
%%   quasi-free); the Euclidean arena licensed (analytic shore);
%%   L-BH-Fresh (Page turnover = scheduled expiry of A5's Port-Freshness
%%   Clause); L-BH-Arrow (GSL direction from three named links; exact
%%   exchange-class identity Delta S_gen = spent relative entropy);
%%   L-BH-Tower, L-BH-Port, C-BH-1; real-bulk register displayed
%%   (R14/R15; horizon entry proposed to A8/P4); F-BH-9 internal kill.
%% PHASE 2 -- THE STRUCTURE (v13.8-13.12): 7 -> 14 Parts; Experiments
%%   Part to the P5 standard (dated timeline, pre-registered verdicts);
%%   appendix machine room in three Blocks (dictionaries; nine
%%   Step-graded worked chapters + island dictionary; seven registries
%%   from source ledger to reading guide); objections 7 -> 53 in
%%   thirteen referee communities.
%% PHASE 3 -- THE POLISH (v13.13-13.14): front matter to cohort standard
%%   (two-level TOC, Preface, Executive-Summary apparatus paragraph);
%%   loose-end sweep; back matter to cohort order; ten-reviewer hostile
%%   review (6 solo + 4 cross-cohort with all-18-companion homework),
%%   ~55 verified findings fixed incl. two legacy technical errors,
%%   cross-cohort value syncs (H0 to C2 v13.65's stake; A9 W-mass;
%%   NH-floor neutrino), F-BH-10 registered, deck Exposure column,
%%   fresh-eyes re-check of the fix diff.
%% Gates at release: 3-pass pdflatex 0 errors / 0 undefined / 0
%% multiply-defined / no overfull > 20pt beyond baseline; falsifier
%% checker clean; QA gate clean; 0 orphan references; all protected
%% statements byte-identical through the thread except two author-
%% approved amendments (T-BH-5(iii) existence clause; O-BH-7
%% reconciliation); every quoted number backed by a verification
%% script named in the benchmark audit.
%% Per-ship banners below preserve the detailed history.
%% ===== v13.14 (2026-07-22): P3b-FIX -- HOSTILE-REVIEW FIXES (author Go) ===
%% Executes GG-C3-P3b-Hostile-Review-Report-2026-07-22.md (author-signed;
%% all D-1..D-8 recommendations approved). Ten-reviewer panel: 6 solo +
%% 4 cross-cohort with all-18-companion homework.
%% TECHNICAL: eq:Kruskal_transform second relation corrected (regular
%%   Kruskal form); eq:Kraus_BH rebuilt as trace-preserving two-branch
%%   beamsplitter set (output = greybody-filtered thermal, exact);
%%   Step-1/Step-2 heat-kernel factor-2 convention bridged explicitly.
%% PHYSICS LANGUAGE: Kay-Wald scope (horizon-Killing-invariant, not
%%   stationary); Wall-2012 scope corrected at 3 sites (free/super-
%%   renormalizable, not interacting); T-BH-5(iii) one-clause amendment
%%   "where such a state exists" (D-7, author-approved); units notes;
%%   a* = ac^2/GM; c^3 restored in two SI entropy displays.
%% NUMBERS: solar-mass T_H row 6e-9 -> 6e-8 K; lock-bullet rebuilt on
%%   P1's ladder (E_GUT = M_Pl e^{-2pi}, ratio 2e^{K-2pi} ~ 6e12);
%%   M_GUT harmonised to 2.3e16 GeV; PBH species-rate clause; exec/
%%   comparison suppression figures harmonised (10^{46} stellar,
%%   10^{-48} at the 10-Msun benchmark); rho_Lambda units; GWTC
%%   decomposition printed (101+128+161=390, per LVK record, D-6).
%% CROSS-COHORT SYNCS (homework wave): H0 row recast to C2 v13.65's
%%   stake (bg 66.6+-0.2 vs CMB 67.4; ladder lift open); de Sitter
%%   chapter H_bg 70.7 -> 66.6 (r_H 1.4e26; T_GH/S_dS magnitudes
%%   unchanged); neutrino row -> 0.059 eV NH floor; W-mass -> A9 QD
%%   80.3700 (-0.06 sigma); Kaxion line-vs-window fixed.
%% REGISTER: F-BH keys repointed to Table tab:bh_kill (4 sites;
%%   lem:tower's internal pointer left per protection discipline);
%%   deck rebuilt with Exposure column (assignments listed for author
%%   audit), C-BH-1 grade completed (quasi-static bath), QES
%%   proposition row added + app:islands lead-in set to grade;
%%   NEW falsifier row F-BH-10 (LISA EMRI multipoles, D-3) + timeline/
%%   milestone repointed; T-P5-B1 attribution corrected at 2 sites
%%   (licenses the continuation; the KMS face is this paper's T-BH-5);
%%   arrow custody = dictionary one-way reading (2 sites); A2 Thm-2
%%   parameter list (R_theta, not l_lock; 2 sites); A6 "ledger
%%   structure theorems"; AX-4 admissibility-filter wording; bulk
%%   unitarity custody AX-1 (2 sites); DGI glossary anchored to A1
%%   POS-1; A11 exit-channel identification marked as this paper's;
%%   No-Exit grade fence; factorisation naming home A8; GSSB defined
%%   in abbreviations; internal-history leaks removed (4 sites);
%%   3G + internal-row verdicts added; analog kill-cell fenced;
%%   O-BH-31/34/35/46 aligned with registered rows; scorecard 44/7/2
%%   (D-5); O-BH-7 bracketed reconciliation (D-4); GFF descent sign
%%   clause at the way-station (D-2); Born-weight row added to the
%%   source ledger (makes O-BH-45's pointer true); GW-record
%%   classical/quantum disclaimer (makes O-BH-32's claim true).
%% FRESH-EYES RE-CHECK (post-fix verifier pass): Kraus input bra made
%%   n-free (trace preservation exact on the input span); tilde-p
%%   defined and the two-branch set stated as the mode-level coarse-
%%   graining of the exact per-quantum Gaussian channel; eq:Sout
%%   propagated to the corrected eigenvalues; O-BH-52 'kill column'
%%   -> Exposure column; one figure point trued (7.7, 3.55).
%% COPY: Page-curve figure peak moved to 0.646 t_evap with honest
%%   branch shapes; 8 never-referenced floats given in-text pointers;
%%   phantomsection at Author's Note; report->paper; sixty-year arc;
%%   Kerr: 42/51 muas; banner range to Reading Guide; texorpdfstring;
%%   an OGGEP (2); ASCII o-umlauts; Relevance notes rewritten in C3's
%%   voice for A1/A2/A5/A6/A10/P2/A8 + Kaxion window. D-1: nineteen-
%%   paper cohort kept (all deployed companions agree; charter-side
%%   record). D-8: A8-side note recorded, no C3 edit.
%% ===== v13.13 (2026-07-22): P3a FRONT MATTER + LOOSE-END SWEEP (author Go)
%% (1) Two-level TOC apparatus installed (cohort standard): Overview of
%%     Contents (Part level) + List of Figures + full Contents, replacing
%%     the plain \tableofcontents. (2) NEW Preface (\GGPreface, P5
%%     pattern): who the paper is written for, the honest pitch, a
%%     suggested week Part by Part, the register note, and the
%%     experimentalist's close. (3) Executive Summary: NEW "What the
%%     paper contains" paragraph (5 theorems / 3 lemmas / register /
%%     experiments Part / machine room / 53 objections); existing text
%%     untouched. (4) Loose-end sweep: "As of April 2026" -> mid-2026;
%%     Maldacena-Susskind plain cite -> \citet (closes the standing
%%     QA-gate hit); stale "GG-6" moniker retired at all 5 body sites
%%     (timeline node -> GG-Theory; first-law close, c_log paragraph x3
%%     -> \GGtheory{}); "GG-Cosmos series" misnomer -> nineteen-paper
%%     cohort language (2 sites); "GG-Gravity forthcoming/planned"
%%     promises -> scope statements, no promised preprints (2 sites).
%% (5) Back-matter order to the P5 standard: Author's Note +
%%     Acknowledgments moved from the Part XII tail to after the
%%     Conclusions Part; Acknowledgments now open with the
%%     cohort-standard \GGAcknowledgments (LLM-collaboration
%%     disclosure); both paper-specific paragraphs retained verbatim.
%% No physical numbers changed; protected regions untouched.
%% ===== v13.12 (2026-07-22): P2d OBJECTIONS EXPANSION (author Go) =========
%% Part XIII rebuilt to the P5 Part-XV standard: 7 -> 53 objections,
%% organised by referee community (13 sections: classical relativist,
%% curved-spacetime field theorist, string theorist, islands theorist,
%% canonical-quantization referees, emergent-gravity theorist, GW
%% observer, horizon-imaging astronomer, quantum-information theorist,
%% quantum-foundations referee, historian/philosopher, student,
%% cohort-internal referee), in the \PObjC + objqabody format (macros
%% defined in-Part, P5 pattern). The seven seed objections keep their
%% permanent keys O-BH-1..7 and their author-approved response texts
%% VERBATIM (asserted byte-identical), filed under their communities;
%% new objections take keys O-BH-8..53 in document order. Every response
%% points into named theorems, registers, or kill rows; conceded items
%% (O-BH-5,23,26,27,37,42) and open commitments (O-BH-4,25) said in
%% those words. Summary scorecard recounted 45/6/2. No physical numbers
%% changed; protected regions untouched.
%% ===== v13.11 (2026-07-22): P2c-2 STEP-GRADED WORKED CHAPTERS (author Go) =
%% Block II rebuilt as graduate worked chapters, per the P5 standard and
%% the author's instruction (complete derivations, step by step, with
%% physics intuition):
%% (B) Hawking temperature: QFT in curved spacetime from scratch (KG inner
%%     product, mode expansion), Bogoliubov machinery (coefficients,
%%     consistency, N = |beta|^2 worked), Rindler/Unruh derivation with
%%     the analytic continuation displayed, two routes to kappa (near-
%%     horizon Rindler + collapse ray-tracing), units and numbers,
%%     custody close. (C) Euclidean: cigar substitution worked, GHY
%%     action term by term (I_E = beta M/2), saddle thermodynamics with
%%     consistency checks, negative specific heat. (D) Hawking-Page:
%%     T(r+) from regularity, T_min = sqrt(3)/2piL, F-ledger worked,
%%     stability, branch-switch template. (E) Page: exact Page average
%%     + harmonic-number asymptotics, the corner (-1/2 nat), evaporation
%%     clock worked incl. t_Page/t_ev = 1 - 2^{-3/2} ~ 0.646 (card value
%%     now derived). (F) Islands: QES condition derived, two-branch race.
%%     (H) Channels: Kraus/Stinespring, DPI pedigree, Spohn worked,
%%     single-mode exchange ledger in closed form (L-BH-Arrow displayed
%%     mode by mode). (I) Kerr: horizon data, Christodoulou mass worked,
%%     first law + Smarr via Euler homogeneity, third law, superradiance
%%     flux argument. (J) de Sitter: static patch worked; eq:GH_thermo
%%     rewritten in units-consistent form (BUG FIX: old display carried
%%     a stray 1/c and dropped c^3 in S_dS; quoted magnitudes were and
%%     remain correct). (K) Species sum: NEW Step-1 heat-kernel toolkit
%%     (proper-time rep, Seeley-DeWitt, chi_scalar = 1/6 - xi derived);
%%     subsections re-graded Step 2..6; L-BH-Tower + eq:B-* byte-
%%     identical. Also: Part-banner prose notes the worked-chapter form;
%%     benchmark-audit row for p2c2_verify; two new B-list bibitems
%%     (GibbonsHawkingAction1977, HawkingPage1983). All theorem
%%     statements byte-identical; all app:/eq: labels retained.
%% ===== v13.10 (2026-07-22): P2c-1 APPENDIX-BLOCK FRAME (author Go) =====
%% Part XII rebuilt to the P5 Part-XIV machine-room standard, phase 1.
%% (1) Banner + opening prose: the three-Block organisation, with Block
%%     II described at its current state (the derivation appendices);
%%     the two stylistic commitments installed.
%% (2) Block I (furniture): NEW Summary of Abbreviations (working set,
%%     one line each, canonical Dictionary pointed to A2 App. B); NEW
%%     Glossary of Key Symbols; NEW cross-cohort naming note; the
%%     numerical-parameters card moved in from the old appendix run.
%% (3) Block II: the ten derivation appendices reordered into teaching
%%     order (temperature -> Euclidean -> Hawking-Page -> Page ->
%%     islands -> island dictionary -> data processing -> extremal ->
%%     de Sitter -> species sum as capstone); redundant "Appendix:"
%%     title prefixes stripped (5 titles).
%% (4) Block III (all new): internal source ledger (derived/consumed/
%%     imported, 16 rows); cross-paper inheritance table (13 rows);
%%     consolidated benchmark audit (every verified number + script);
%%     named-statement deck (11 rows, exact grades, "nothing claimed
%%     above this table's register"); promotion roadmap (7 items);
%%     falsification-test catalogue; annotated reading guide (6
%%     shelves).
%% All appendix references are \ref-dynamic (verified); Author's Note
%% and Acknowledgments close the Part. No printed physical numbers
%% changed; protected regions untouched.
%% ===== v13.9 (2026-07-22): P2b EXPERIMENTS PART BUILD (author Go) =====
%% Part VIII built to the P5 Part-XI standard.
%% (1) "The record so far": the four instrument classes and what each
%%     did to the module (GW ringdown/area-theorem record incl.
%%     GW250114's confirmation of the classical half of the GSL;
%%     two polarization-resolved shadows; analog-horizon structure
%%     checks; PBH honest "not yet").
%% (2) "Now (mid-2026)": LVK between runs, IR1 announced late 2026, O5
%%     plans refining toward A+-class (LVK observing plan, verified
%%     2026-07-22); EHT multi-epoch era; CTAO opening; the external
%%     hostages named.
%% (3) "The decade ahead": six dated milestones, each closing with
%%     pre-registered If-right / If-wrong verdicts (P5 pattern);
%%     rifle-range physics intuition.
%% (4) "The falsifier timeline, dated": tab:bh-exp-timeline in the P5
%%     five-column convention, all rows keyed to F-BH-1..9 + external
%%     hostages; drawn kill-test timeline figure
%%     (fig:bh-killtest-timeline).
%% (5) "Refresh rules": the four standing data-discipline rules
%%     (measured values; immovable predictions; dial-free kills;
%%     programmatic pulls).
%% Schedule language deliberately conservative where plans are not yet
%% fixed (O5, ET/CE, BHEX selection). No printed physical numbers
%% changed; protected regions untouched.
%% ===== v13.8 (2026-07-22): P2a STRUCTURAL SPLIT, 7 -> 14 PARTS (author-approved roster) =====
%% Pure re-Parting: every physics section relocated intact (labels
%% preserved); no theorem/lemma text touched; no printed physical
%% numbers changed. New Parts: II Mathematical Preliminaries (arena
%% section relocated from the port chapter + runtime vocabulary
%% relocated from the former Part-III banner page + scope intro);
%% IV/V/VI = the split of the former GDOS-reading Part (port+thermality
%% +custody / the exact-1/4 chain / the information story); VIII
%% Experiments seed (kill criteria + evidence base relocated from the
%% former Summary Part); XII Technical Appendices; XIII Common
%% Objections (the objections section relocated -- expansion is its own
%% ship); XIV References. The formerly stranded sections (module scope,
%% falsifiers) now live in Part X Discussions -- the permanent fix of
%% the old Part VI/VII seam. The Part-I close's roadmap subsection
%% rebuilt for fourteen Parts (sanctioned amendment; the cohort map,
%% inheritance paragraph, and Foundations paragraph byte-identical).
%% Seam repairs: ledger opening re-seamed to follow Part V; the
%% Bogoliubov subsection rejoined to the radiation chapter; literal
%% Part-number tokens updated (state-of-play de-dated; Part-VI pointer;
%% runtime table; what-is-new count; vocabulary scope). Author's Note
%% and Acknowledgments remain in place inside Part XII pending the
%% back-matter ordering of the appendix ship.
%% ===== v13.7 (2026-07-21): S5 REGISTER SHIP (C3 overhaul thread; author Go; closes Phase 1) =====
%% The registrational integrations.
%% (1) NEW section "The real-bulk register at the horizon" after the
%%     thermal-clock section: R14/R15 displayed formally (table with
%%     statements, homes, grades); the real-section-geometry entry
%%     drafted explicitly as a PROPOSAL to the series' home papers
%%     (A8, P4-Real) with no C3-local numbering; custody count stays
%%     five; ledger-and-tinted-glass physics intuition.
%% (2) L-Constitution-1 / Reality Register installed at candidate grade:
%%     grade-inheritance sentence at the central-claim funnel; O-BH-4
%%     given the cohort address, grade, and F-Constitution-10 kills.
%% (3) Keying pass: T-BH-1 grade paragraph added AFTER the protected
%%     statement (Derived-conditional; three premises; F-BH-9 pointer);
%%     L-BH-Tower (Lemma 5.2) around the tower identity; L-BH-Port
%%     text-registered at the visible-port claim; C-BH-1 (Corollary 5.1)
%%     at the conditional-GGA crossover, fusing the S4 equality point;
%%     F-BH-9 added to the kill table (species-sum closure, internal,
%%     decidable by computation; existing rows byte-identical).
%% (4) Island-dictionary and de Sitter same-theorem keys deliberately
%%     deferred to the graded claims ledger of the structural rebuild.
%% No printed physical number changes (C-BH-1 references the crossover
%% by eqref); protected regions untouched.
%% ===== v13.6 (2026-07-21): S4 ARROW SHIP (C3 overhaul thread; author Go) =====
%% The generalized second law's DIRECTION upgraded from identity-by-
%% citation to a link-by-link derivation with a declared ceiling.
%% (1) NEW section "The direction of the second law" after the four-laws
%%     section: three-link ownership decomposition (D1 arrow consumed
%%     from AX-3 / CPT companion Part IX / A5; D2 data-processing
%%     inequality as directionless channel identity; D3 the horizon
%%     translation owned here) with links table; Lemma L-BH-Arrow
%%     (renders Lemma 5.1) + six-step proof: in the Gaussian exchange
%%     class with the KMS fixed point and the linearized induced first
%%     law, Delta S_gen EQUALS the spent relative entropy toward the
%%     clock's thermal face (exact identity; Spohn-theorem instance);
%%     closed forms: ln(1+n_H) per emitted mode = S(vac||KMS); greybody
%%     and two-temperature cases; equality iff T_b = T_H, identifying
%%     the conditional-GGA crossover M_x as the GSL's equality point;
%%     robustness of the direction against stale ports (needs total-map
%%     CPTP only, not the Markov window). Wall-class proof pre-fenced
%%     as Open with its promotion path named. Customs-post physics
%%     intuition.
%% (2) The four-laws second-law paragraph's "one citation closes a
%%     fifty-year analogy" sentence upgraded to the derivation pointer.
%% (3) Part-III summary bullet; Open-questions paragraph "The GSL beyond
%%     the exchange class" (registered open); open-questions section
%%     labelled.
%% (4) Three new B-list bibitems, all cited: Spohn 1978, Casini 2008,
%%     Wall 2012, with C3-voiced Relevance notes.
%% Verification: c3_v13.6_s4_arrow_verify.py (DPI Monte Carlo, exact
%% identity to 3.6e-15, 120k-point positivity sweep, closed forms).
%% Protected regions untouched; no printed physical number changes.
%% ===== v13.5 (2026-07-21): S3 PORT-FRESHNESS SHIP (C3 overhaul thread; author Go) =====
%% The Page turnover given its mechanism in cohort vocabulary: the
%% scheduled expiry of A5's Port-Freshness Clause at the horizon.
%% (1) NEW subsection after T-BH-2 with Lemma L-BH-Fresh (renders Lemma
%%     2.1; grade Derived-conditional): (i) fresh regime -- clause holds,
%%     corrections exponentially small in the unread entropy; (ii) finite
%%     freshness budget -- expiry at t_Page by counting alone; (iii)
%%     stale regime -- forced correlations = the bridge-branch cap.
%%     Proof in the coarse-grained model with Page-average arithmetic and
%%     Haar-model numerics (N=12: I(q:R)=0.002 ln2 deep-fresh, factor-4
%%     growth per step, saturation 1.997 ln2); dealer-and-deck physics
%%     intuition. T-BH-2's protected statement NOT amended.
%% (2) The Kraus construction and the compositionality passage cite the
%%     clause with its validity window (silent consumption retired); the
%%     "very-late subtle correlations" named as stale-port data.
%% (3) Appendix B's ETH/typicality premise flagged as imported and wired
%%     to the lemma's condition list.
%% (4) A10 observer-vacuum cross-link installed at the vacuum-multiplicity
%%     sentence; A10's residual factorisation conjecture adopted as a
%%     named open problem of this module (Open questions).
%% (5) Part-III summary gains the mechanism bullet.
%% Guards: A5 owns the clause (consumed under its name, never restated);
%% arrow-custody content untouched (separately chartered); no printed
%% physical numbers changed; protected regions untouched.
%% Verification: c3_v13.5_s3_fresh_verify.py.
%% ===== v13.4 (2026-07-21): S2 SHORE SHIP (C3 overhaul thread; author Go) =====
%% The Euclidean arena licensed constitutionally (grade installed:
%% Derived-conditional; no theorem statement amended; no printed
%% physical number changed).
%% (1) Step 3 (conical replica) gains the licensing paragraph "The
%%     arena, licensed.": the replica cover is built on the analytic
%%     shore -- the fourth endowment of the cohort's time (P5 Part IX)
%%     -- whose existence at the horizon is Theorem T-BH-5(ii) and whose
%%     thermal face fixes the tau-period; the semiclassical evaluation
%%     of the gravitational action on the cover is the one declared
%%     import (the replica-literature saddle step).
%% (2) The de Sitter same-theorem sentence gains the same-arena clause
%%     (the static-patch clock's endowments; S^4 as its shore).
%% (3) Appendix H's third point names the shore among the endowments.
%% (4) The honest-accounting section (Established) records the arena's
%%     grade and its one import.
%% Key assignment (e.g. L-BH-Shore) deliberately deferred to the S5
%% register/keying pass.
%% ===== v13.3 (2026-07-21): S1 KMS SHIP (C3 overhaul thread; author Go) =====
%% The thermality chain upgraded to consume the cohort's Wick-quartet
%% theorem as a theorem (grade: Derived, quasi-free class).
%% (1) NEW section "The thermal clock face as a theorem" with Theorem
%%     T-BH-5 + proof + retirement paragraph + physics intuition, placed
%%     after the rotating/collapse section (euclidtheorem counter renders
%%     it Theorem 5). Content: (i) exterior positive-frequency structure
%%     = polar rotor of the Killing generator (derived clock); (ii) the
%%     complexified-horizon continuation and imaginary-time section exist
%%     by the one-sided spectrum (no prescription imported); (iii) the
%%     regular exterior state is the clock's KMS face at 2 pi / kappa
%%     (Kay--Wald composed with the mode-level squeezed-state display).
%%     Machine-precision verification in c3_v13.3_s1_kms_verify.py.
%% (2) The v2-era duplicate section "Bogoliubov coefficients and the
%%     explicit thermal spectrum" deleted (content duplicated the
%%     ssec:bogoliubov treatment at lower resolution; its six equation
%%     labels verified reference-free); non-duplicative insights absorbed
%%     into the new section's introduction.
%% (3) Recipe language retired at three sites: the Planck-factor
%%     paragraph (continuation licensed, not assumed), Appendix A (the
%%     imaginary-time section is a theorem clause, not a postulate),
%%     Appendix H (the Euclidean "trick" upgraded to its theorem, third
%%     point added).
%% (4) ssec:bogoliubov gains the pedigree pointer; the Part-III summary
%%     gains the T-BH-5 bullet.
%% Protected regions untouched: T-BH-1..4 statements, the frozen v1.4
%% map, the Part-I close, all canonical numbers.
%% ===== v13.2 (2026-07-21): S0-A CAMPAIGN-ROW MICRO-PASS (author Go) =====
%% Evidence-table campaign rows harmonized to the cohort's current
%% confrontation conventions; GG-prediction column (canonical) untouched.
%% (1) Muon g-2 measurement -> the June 2025 Fermilab final,
%%     116 592 070(15) x 10^-11; pull recomputed +0.2 -> +1.4 sigma.
%% (2) W-mass row -> the A9 v20.37 world-average convention,
%%     80.3692(133); row label CMS -> world avg; pull +1.1 -> +0.8 sigma.
%% (3) Neutrino-sum bound -> the A9 convention, < 0.072 eV (DESI+CMB).
%% (4) Evidence-table caption re-dated April -> July 2026.
%% Pulls verified programmatically in the ship script.
%% ===== v13.1 (2026-07-21): S0 HYGIENE SHIP (C3 overhaul thread; author Go 2026-07-21) =====
%% Scope: found-bug fixes + mid-2026 experiment-record refresh. No physics
%% numbers, theorem statements, canonical values, or protected regions
%% touched.
%% (1) Cohort self-citations completed: eight missing companion bibitems
%%     added (A1, A3, A4, A7, A9, P1, P2, P3; canonical cover titles from
%%     the deployed v13 versions) with C3-voiced Relevance paragraphs and
%%     first-citation links at natural body sites (conventions paragraph,
%%     central-claim funnel, OGN-5 ledger, compactification-scale site,
%%     programme-foundations site, interlocking-falsifiability site).
%% (2) Wrong-paper Relevance residues rewritten in C3's voice (Einstein
%%     1915, Bekenstein 1973, Hawking 1975, Wald 1984); the pre-GG
%%     register fossil in the Bekenstein entry removed.
%% (3) Predictions-table PBH row reconciled with the body's ratios-only
%%     fencing (ratio ladder; absolute anchor open).
%% (4) Experiment record refreshed to mid-2026: O4 complete (2023--2025);
%%     GWTC-4.0/5.0 catalog counts (390 total detections); GW231123 and
%%     GW250114 added (ringdown overtone + area-theorem rows added to the
%%     predictions and evidence tables); EHT 2024--2025 polarization
%%     epochs and second-epoch M87* image; BHEX added to future
%%     prospects; the "through 2025" nulls advanced to mid-2026.
%%     Evidence-table campaign rows (muon g-2, W mass, neutrino sum)
%%     intentionally NOT changed: 2025 final measurements shift printed
%%     pulls; drafted and flagged for author decision.
%% (5) Source-comment banners at the Part VI/VII seam reconciled with the
%%     rendered Part assignment (no reader-facing change; permanent home
%%     assigned at the v14 rebuild).
%% ===== v13.0 (2026-07-19): PROMOTION to the v13 series (GG-v13 project) =====
%% WAVE 1 cohort promotion 17 -> 19 (Foundations Branch: P4-Real, P5-CPT).
%% First v13 ship of C3-BH (base: v11.14).  Mechanical/structural only; all
%% physics protected (the A/4 derivation, entropy formulas, falsifier tables).
%% (1) Standardized Part-I close "The cohort inheritance and the roadmap of
%%     the seven Parts": the Part-III cohort paragraph migrated verbatim
%%     (updated to nineteen; label fig:cohort-map-c3 kept), the frozen v1.4
%%     nineteen-paper cohort map installed verbatim from the A2 v13.3
%%     rendition (only the \label changed), the old seventeen-paper map
%%     retired, Foundations Branch paragraph added in C3's voice; NEW
%%     roadmap figure of the seven Parts (Part III highlighted: it carries
%%     the exact-1/4 Bekenstein-Hawking derivation, Theorem thm:exact-quarter).
%% (2) Cohort counts seventeen -> nineteen (cohort sense only; no physics
%%     numerals touched).
%% (3) \PaperPFour renewed GG-P4-Twistor -> GG-P4-Real; \PaperPFive added;
%%     ArisakaP4Twistor bibitem/key renamed ArisakaP4Real (cover title synced
%%     to the deployed P4-Real v3.3); ArisakaP5CPT bibitem added (cover title
%%     from the deployed P5-CPT v3.41); \ASix renewed to GG-A6-GRIP (the sty
%%     \providecommand default renders the retired form).
%% (4) One-line ToC Part entries (\GGPart renewal, P4-Real v3.2 pattern).
%% (5) Register repairs: reader-facing "satellite" tokens -> member/
%%     companion language (2 sites); A6 bibitem natbib label A6-Gi4 ->
%%     A6-GRIP and its rename-history parenthetical dropped; the
%%     versions-of-record parenthetical version strings dropped (timeless
%%     form); \ASixGRIP short-form unified to \ASix (2 body sites).
%% v11.14 (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:bh_axioms: p-widths reduced (overfull 57.2pt; long l-column labels)
%%   correction: tab:bh_kill: p-widths reduced (overfull 46.5pt; long l-column labels)
%% v11.13 (July 15, 2026): final-touch pass (report 17 S-2/W-1/W-4,
%% author-approved): Arisaka citation labels unified to the canonical
%% "(2026, XX-Name)" form (8 entries); custody close gains the
%% three-superselection/two-custody clarifying clause; rotating/collapse
%% section gains the physics-intuition paragraph (harbour, quay,
%% chronometer).
%% v11.12 (July 15, 2026): audit round-2 micro-pass (report 17,
%% author-approved). T-BH-4 proof sign corrected (|(-+ i w')^{-i nu}| =
%% e^{-+ nu pi/2}, principal branch named); PaperDate synced to the ship
%% date; Part III summary gains the rotating/collapse custody bullet;
%% verification-suite clause reworded to the arisaka-gg.org house form.
%% v11.11 (July 15, 2026): full-length home for R14/R15 (graduation,
%% report 16, author-approved). NEW section "The rotating anchor and the
%% collapse channel" in Part III (after the Bogoliubov-spectrum section):
%% Theorems T-BH-3 (rotating anchor: ergoregion indefiniteness = Kay-Wald
%% obstruction in model form; anchor at the asymptotic frame; superradiance
%% = symplectic GDOS pumping) and T-BH-4 (collapse channel: Planck factor =
%% branch cut of the exponential redshift; late-time factorisation into
%% universal thermal channel x decaying transient), with proofs and
%% verification numbers; custody close (five instances). Appendix
%% superradiance passage gains a T-BH-3 pointer. Five new references
%% (Starobinsky 1973, Unruh 1974, Davies-Fulling 1976, Kay-Wald 1991,
%% Arisaka P4 satellite), chronologically placed; PaperPFour provided.
% v11.9 (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.
% v11.8 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 3
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% ===== v11.7: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 2 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.
% ===== C3-BH v11.4 (2026-06-27): cohort de-contamination (protect-list aware). =====
% Shed GG's OWN self-description ONLY: GG's (6D/six-dimensional) bulk no longer self-branded;
% GG's canonical scale profile e^{-r}/e^{-2r} (formerly the 'factor'), the 'metric', the
% 'radial/eigenmode/coordinate', and the 'depth/hierarchy/ladder/integers/reduction/integration/
% structure/conversion/rate' family rewritten to scale-* ; the GG-self verb form rewritten to
% scale running / scale-extended. KEPT (the majority): the holographic PRINCIPLE, holographic
% entropy bound, Bekenstein--Hawking, AdS/CFT, Anti-de Sitter, Ryu--Takayanagi, island/replica/
% holographic-screen literature, 't Hooft / Susskind / Maldacena / Bousso / Banks citation
% contexts, 'holography' / 'holographic cosmology' as the named external field, the GG-vs-AdS/CFT
% 'holographic framework' comparison prose, the RG-running disclaimer footnote, and Wald's
% 'warped product metrics' textbook machinery. Body only. No physics / results / values changed.
% =====================================================================
% ===== C3 v11.3 (2026-06-25): cohort central-claim funnel (consequence variant). =====
% Added the shared open-system funnel (sec:central-claim/fig:central-claim) with this paper's own
% payload. Same funnel as A9/A10/A11. No new physics.
% =====================================================================
%  C3-BH v11.1 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%  ucla-arisaka-gg (= gg-10 copy) -> gg-11; centered headers via gg-11.  No prose/figure/number content changed.
% =====================================================================
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2d (final). =====
%  Terminology only; NO physics changed. C3 has no GG-bulk AdS branding; A2 bibitem
%  AdS-free; does not cite A3. The genuine black-hole AdS material (AdS-Schwarzschild,
%  AdS/CFT BH entropy / Page curve) is KEPT untouched. Added the holographic-RG
%  disclaimer footnote (noting the distinction from the BH AdS geometries); 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>'. ---
\pdfoutput=1
%% v10.5 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-c3, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.4 (June 10, 2026): cohort-consistency pass (C3-A10 audit,
%%   C3-A10-CONSISTENCY-AUDIT-2026-06-10): (1) C2 bibitem updated to
%%   the completed paper's final title (was a stale working title +
%%   'in preparation'); the 'GG-Cosmos paper' misnomer for C2 at the
%%   de-Sitter section corrected; 'resolves' register softened to
%%   match C2's own claims. (2) Stream direction fixed: A10 is a
%%   cohort companion on the same A1-A9 inheritance, not downstream
%%   of this paper. (3) de-Sitter horizon evaluation now names the
%%   corridor's global background branch (~70.7 km/s/Mpc), per the
%%   two-dictionary logic this paper itself states; order-of-magnitude
%%   values unchanged. (4) Kerr-precision falsifiers restated as
%%   one-sided null tests. (5) PBH dark-matter sentence aligned with
%%   the cohort position (Kaxion saturates the DM slot; new C1
%%   bibitem). No physics numbers changed.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   Amended same day: Wald1984 citation label fixed (Wal -> Wald);
%%   the 46 drafted Relevance paragraphs reviewed and approved by author.
%%   BibTeX bibliography replaced by inline bold-author + Relevance thebibliography (BibTeX no longer required).
%%   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).
%%   Amended same day: former bold small-caps spots set in bold FULL CAPS
%%   (author instruction June 10, 2026).
%%   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. bibliography database repaired in [v10-Support]/UCLA-Arisaka-GG.bib (14 truncated title fields closed, tHooft1993/THooft1993 case-collision resolved, ArisakaA6GRIP entry added, titles completed for THooft1993 and Susskind1995); BibTeX now exits clean.
%%   No physics changes; production fixes only.
%% ---- vv15.8 (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.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}% cohort renew: the sty \providecommand default renders the retired A6 form
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4)
\usepackage{indentfirst}

%% ---- v13.0: one-line Part entries (cohort standard; P4-Real v3.2
%%      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 the Contents.  Part title
%%      pages are unchanged (full title + description).
\makeatletter
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \addcontentsline{toc}{part}{\protect\textbf{Part \thepart}. #2}%
  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}% running pointer
  \thispagestyle{plain}%
  \begin{center}
    \vspace*{1.2em}
    {\Large\bfseries Part \thepart}\\[0.45em]
    {\huge\bfseries #2}\\[0.35em]
    {\large #3}
  \end{center}
  \vspace{0.5em}
  \GGThickLine
  \vspace{0.8em}
}
\makeatother

\PaperSeries{GGC}
\PaperNumber{Arisaka-GG-C3-BH}
% ============================================================
% v14.24 -> v15.0  Stage 1 of the v15 migration.  August 15, 2026.
%   * NO paper conversion: C3 declares letterpaper EXPLICITLY, the
%     first of the migration to name it rather than inherit it.
%   * NO booktabs work: 0 \hline, 0 \cline, 0 vertical rules.
%   * American spelling, total, in printed text: 355 substitutions
%     in 96 distinct forms, every one hand-classified.  31 forms
%     were new to the cohort map.  NOT ONE of the 89 was an
%     American word the rule over-reaches on -- the first paper of
%     the migration needing no addition to the protect list.
%     `greybody' -> `graybody' (14 sites) is a coherence repair,
%     not a decision about the literature: A10-Cosmos, shipped at
%     v15, already prints `graybody' 4 times and `greybody' 0.
%     `Programme' inside the A1 and A11 bibliography titles is
%     likewise a stale quotation -- A9 and C1, both shipped, print
%     `GG-Theory Program' in the same entry.
%   * hyperref metadata swept for the first time.  C3 declares
%     none.  A10 and C2 each ship `GG-Theory programme' in their
%     PDF Subject field, invisible to a guard that starts at
%     \begin{document}; riders follow.
%   NO PROSE WAS WRITTEN IN THIS STAGE.
% ============================================================
%% ------------------------------------------------------------
%% Stage 2A, same day.  The float standard.
%%   20 captions printed below the 260-character floor.  Each was
%%   EXTENDED and none rewritten -- the author's sentence is
%%   unchanged byte for byte and the new text is appended.
%%   FIVE TABLE captions sat BELOW their tabular and are moved
%%   above it: tab:bh-exp-timeline, tab:source-ledger,
%%   tab:inheritance, tab:benchmarks, tab:statement-deck.  Section
%%   8 says `caption above' without saying above what, so the
%%   convention was MEASURED across the three most recent ships:
%%   145 tables all caption-above and 59 figures all caption-below,
%%   no exception in A9, C1 or C2.  C3's fifteen caption-below
%%   FIGURES are therefore correct and are left alone.
%%   Nothing else was needed: float_audit reports NO CAPTION 0 and
%%   UNBREAKABLE-AND-LONG 0, so no bare grid to promote and no
%%   uncaptioned float to caption -- as with C2, and unlike A9,
%%   A10 and C1.
%%   ALL 20 EXTENSIONS ARE NEW PROSE, listed by label in the
%%   report.  No extension introduces a numerical claim; every
%%   number in one is already printed in the table it describes,
%%   and every count of rows was taken by counting the rows.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  Plain-English Part closes.
%%   11 closes, one per content Part (I-XI).  The three back-matter
%%   Parts -- Technical Appendices, Common Objections, References --
%%   get none, per A5's rule.
%%   ALL ELEVEN ARE NEW PROSE IN THE AUTHOR'S VOICE and are listed
%%   by Part and by page in the report, for review or rewriting.
%%   The 16 \GGteachbox uses are KEPT.  They are A10's appendix
%%   course apparatus, one per appendix, all inside the Technical
%%   Appendices Part; A10 shipped at v15 with twelve of them AND a
%%   full set of Part closes.  Teach box and Part close are
%%   different devices and neither converts into the other.
%%   Nine of the eleven content Parts already end with a paragraph
%%   headed `The Part in one breath, and the door to the next.'
%%   Those are untouched; the closes sit after them and do the
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%%   without the paper's technical vocabulary.
%%   Part XI ends with TWO terminal constructs -- \section*{Author's
%%   Note}, the tribute to Bekenstein, Hawking and Page, and then
%%   \GGAcknowledgments -- and its close is placed before the
%%   earlier of them, so both stay where the author put them.
%%   No \ref{part:...} appears in any of them: A9 taught that a
%%   paper's part labels need not match the numbers they print, so
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%% ------------------------------------------------------------
%% ------------------------------------------------------------
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%%   floats of which 24 are tables.  One is added after the LoF.
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%%       with an exit channel and so a way station, where the
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%   toolchain until August 15.  gg_spelling had storey removed from
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%   because A11-Origin uses `storey-port' and `a storey count S' as
%   defined terms at 142 sites.  A global exemption for one paper's
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%%   cohort reference it was not missing.
%% ------------------------------------------------------------
\PaperVersion{v20.2}
\PaperDate{August 22, 2026}
%% v2.2 (June 7, 2026): the runtime facelift.  The June-2026 GRIP
%%   machinery (A6-GRIP v9.7 def:GRIPin/def:GRIPout/def:GGA; A11-Origin
%%   v9.6 regime theorems + T-GGA-No-Exit) applied to black holes:
%%   (1) NEW Part-III section "The black hole as a way-station
%%       attractor": Hawking radiation = the gravitational stratum's
%%       exit channel; the BH = the most extreme way-station GA in
%%       nature (10^67 yr); the stratum's GGA = the radiated state ->
%%       de Sitter; entropy-vs-ledger care point (GSL = the signpost of
%%       the continuing descent); conditional-GGA analysis: today's
%%       astrophysical holes are conditionally exitless (T_H < T_CMB,
%%       channel reversed), crossover M_x ~ 4.5e22 g, channel opens on
%%       the cosmological schedule M_open(t) ~ a(t); new figure.
%%   (2) Hawking-Page named a GRIP-Out; the Page turnover named a
%%       GRIP-In milestone (taxonomy-level no-firewall strengthening).
%%   (3) Four laws regime-completed (2nd law = GRIP-In Lyapunov
%%       monotonicity; 3rd law = no finite procedure closes the exit
%%       channel).
%%   (4) Ringdown = the basin Hessian; merger = basin coalescence;
%%       PBH formation nested per A10's alternation; de Sitter horizon
%%       = the cosmic GGA endpoint.
%%   (5) NEW comparison subsection "What the runtime taxonomy adds
%%       that no leading framework has" + table (significance and
%%       uniqueness vs GR / fuzzball / AdS-CFT / islands).
%%   (6a) UCLA-Arisaka-GG.bib repaired (16 fixes: 14 truncated title
%%       fields completed with full journal data; Dyson1949 stray
%%       backslash; tHooft1993/THooft1993 case-collision merged into one
%%       canonical entry); the lone lowercase \citep{tHooft1993} unified.
%%   (6) Cohort macros (\GRIP form, \GA, \GGA, \GRIPin, \GRIPout);
%%       A6-GRIP + A11-Origin now cited (keys already in the .bib).

%% Macros
\providecommand{\GGtheory}{\textsf{GG-Theory}}
\providecommand{\Mlock}{\ensuremath{\Lambda_{\mathrm{lock}}}}
\providecommand{\MGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\Mpl}{\ensuremath{M_{\mathrm{Pl}}}}
\providecommand{\SUfive}{\ensuremath{\mathrm{SU}(5)}}
\providecommand{\Uoneb}{\ensuremath{\mathrm{U}(1)_B}}
\providecommand{\UoneQi}{\ensuremath{\mathrm{U}(1)_{\mathrm{Qi}}}}

\providecommand{\ket}[1]{\ensuremath{|#1\rangle}}
\providecommand{\bra}[1]{\ensuremath{\langle#1|}}
\providecommand{\braket}[2]{\ensuremath{\langle#1|#2\rangle}}
\providecommand{\ketbra}[2]{\ensuremath{|#1\rangle\langle#2|}}
\providecommand{\avg}[1]{\ensuremath{\langle#1\rangle}}
\providecommand{\Tr}{\ensuremath{\mathrm{Tr}}}
\providecommand{\Trenv}{\ensuremath{\mathrm{Tr}_{\mathrm{env}}}}
\providecommand{\Trin}{\ensuremath{\mathrm{Tr}_{\mathrm{in}}}}
\providecommand{\Trout}{\ensuremath{\mathrm{Tr}_{\mathrm{out}}}}
\providecommand{\CPTP}{\textsc{cptp}}
\providecommand{\Hilb}{\ensuremath{\mathcal{H}}}
\providecommand{\iden}{\ensuremath{\mathbf{1}}}

\providecommand{\kB}{\ensuremath{k_{\mathrm{B}}}}
\providecommand{\Sent}{\ensuremath{S}}
\providecommand{\Sbh}{\ensuremath{S_{\mathrm{BH}}}}
\providecommand{\Sgen}{\ensuremath{S_{\mathrm{gen}}}}
\providecommand{\SvN}{\ensuremath{S_{\mathrm{vN}}}}
\providecommand{\THaw}{\ensuremath{T_{\mathrm{H}}}}
\providecommand{\QB}{\ensuremath{Q_B}}
\providecommand{\Eryd}{\ensuremath{E_{\mathrm{Ryd}}}}

\providecommand{\OGGEP}{\textsc{oggep}}
\providecommand{\GDOS}{\textsc{gdos}}
\providecommand{\GRIP}{\ensuremath{\mathrm{GRIP}}}
\providecommand{\GA}{\ensuremath{\mathrm{GA}}}
\providecommand{\GGA}{\ensuremath{\mathrm{GGA}}}
\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{\GFF}{\textsc{gff}}
\providecommand{\GEP}{\textsc{gep}}
\providecommand{\CEP}{\textsc{cep}}
\providecommand{\DGI}{\textsc{dgi}}
\providecommand{\GSSB}{\textsc{gssb}}
\providecommand{\GA}{\textsc{ga}}
\providecommand{\SIP}{\textsc{sip}}
\providecommand{\MSMF}{\textsc{msmf}}
\providecommand{\BIsix}{\textsc{bi-6}}
\providecommand{\GGsix}{\textsc{gg-6}}
\providecommand{\OGNfive}{\textsc{ogn-5}}
\providecommand{\AdSCFT}{\textsf{AdS/CFT}}
\providecommand{\QES}{\textsc{qes}}
\providecommand{\GFFBH}{\ensuremath{F_{\mathrm{GFF}}^{\mathrm{BH}}}}
\providecommand{\RS}{\textsf{RS}}

\providecommand{\TGG}{\textsc{tgg}}
\providecommand{\CCI}{\textsc{cci}}
\providecommand{\QiQUA}{\textsc{qi-qua}}
\providecommand{\QiEP}{\textsc{qi-ep}}
\providecommand{\QiCON}{\textsc{qi-con}}
\providecommand{\QiHAL}{\textsc{qi-hal}}

\providecommand{\Newton}{\ensuremath{G_N}}
\providecommand{\Msun}{\ensuremath{M_\odot}}

\setlength{\emergencystretch}{2em}
\begin{document}
\setlength{\tabcolsep}{6pt}
% ---- v14.19 (August 2, 2026) ----
% 'The paradox was never about information' gains the closing
% paragraph 'The missing register, named by the anatomy': the
% two sectors behind the port as fibres of opposite kind; the
% paradox as asking the hole to keep and to spend at once;
% conserved content as circle-side (indestructible), thermal
% one-way content as interval-side (Hawking T as P5's shore
% periodicity); the Page curve as the circle's ledger read out
% through the interval's channel; 'the paradox was billing one
% fibre for the other's job.' Part of the cohort-completing
% sweep.


\GGTitlePage%
  {Black-Hole Entropy under \GGtheory}%
  {The \GDOS{} Reading of the Horizon as an \OGGEP{} Port ---
  with the Exact Bekenstein--Hawking $A/4\Newton$ Derived from Induced Gravity}%
  {}
\thispagestyle{empty}

\vspace{6pt}
\begin{abstract}
\noindent
Black holes are the strangest objects in nature. Fifty years ago
Stephen Hawking discovered that they are not perfectly black: they
glow faintly, and over almost unimaginable spans of time they
evaporate away. This discovery created the deepest puzzle in modern
physics. Whatever falls into a black hole carries detailed information
about what it was; yet the faint glow that leaks back out looks
featureless --- the same for every black hole of the same size. If the
glow truly carries nothing, then that information has been erased, and
the erasure of information is something the laws of physics are not
supposed to allow. For half a century no one has agreed on how the
books are balanced.

This paper offers a resolution built on \GGtheory{}, a framework in
which the world we experience is the surface of a slightly larger
six-dimensional space. In that picture the edge of a black hole --- its
horizon --- is not a wall but a doorway: matter that crosses it is not
destroyed but carried into directions we cannot see, and the faint glow
is simply how that crossing looks from the outside. Nothing is lost; it
is only put out of reach. From this single idea, and without adjusting
any free number, the paper recovers the famous formula for how much a
black hole can hold, explains why the glow must eventually carry the
hidden information back out, and shows why a traveler falling inward
would feel nothing unusual at the edge. The same reasoning extends to
the horizon of the expanding universe as a whole. The claims are set
out so that future telescopes and gravitational-wave detectors can put
them to the test.
\end{abstract}

\begin{ExecutiveSummary}
\paragraph{\textbf{The puzzle.}} A black hole forms from matter in a
definite quantum state. Hawking showed in 1974 that it then radiates,
and the radiation is thermal --- as random and featureless as the glow
of a warm coal. Carried to its conclusion, this means a sharp,
information-rich beginning evolves into a featureless end, a
``pure-to-mixed'' transition that ordinary quantum mechanics forbids:
quantum evolution is unitary, and unitary evolution never destroys
information. Don Page sharpened the demand in 1993: if the process is
unitary, the entanglement entropy of the radiation must rise and then
\emph{fall} back to zero --- the ``Page curve'' --- rather than rise
forever. For fifty years the central question has been what changes
Hawking's calculation so that this turnover happens.

\paragraph{\textbf{What the standard answers achieved, and what they
left out.}} The most successful modern attack uses the AdS/CFT
correspondence and the 2019--2020 islands formula, which reproduce the
Page curve in controlled, lower-dimensional toy models. These are real
achievements, but they carry two limitations. They live in
anti-de-Sitter universes with a supersymmetric boundary theory --- not
the expanding, Standard-Model universe we inhabit --- and they recover
the \emph{shape} of the curve without deriving, from a microscopic
count, the single most famous number in the subject: the factor of
$1/4$ in the Bekenstein--Hawking entropy $S = A/4\Newton$. Counting
black-hole microstates from first principles had been possible only for
special supersymmetric, extremal black holes far from anything in the
sky.

\paragraph{\textbf{The \GGtheory{} picture.}} \GGtheory{} treats the
four dimensions we see as the boundary of a six-dimensional bulk: the
familiar spacetime plus two non-spatial directions, a ``scale''
direction and a compact ``phase'' circle.\footnote{Throughout the cohort, this ``warped'' six-dimensional bulk is read on the field-theory side of the holographic dictionary: the scale direction is the renormalization-group axis $r=\ln(E_{\mathrm{GUT}}/\mu)$, not a Randall--Sundrum spatial dimension, and ``warped'' denotes RG running; the explicit holographic-RG construction is given in \citet{ArisakaA2GG6}. (This is distinct from the genuine anti-de Sitter geometries of the black-hole comparisons in this paper.)} A horizon is the place where
a four-dimensional observer loses causal access to part of that bulk.
Describing what such an observer sees is a standard operation in
quantum theory --- a partial trace, here called the \OGGEP{} projection
--- and its output is automatically a mixed, thermal state. Hawking
radiation is precisely this projection of a pure six-dimensional state;
the bulk stays unitary even though the boundary description looks
lossy. The black hole is the one place in nature where this projection
becomes geometrically visible, which is what makes it the cleanest test
of the whole framework.

\paragraph{\textbf{The central new result: the exact $1/4$ from induced
gravity.}} In \GGtheory{} gravity is not a separate force with its own
constant; Newton's constant is \emph{induced}, generated entirely by
the bulk fields. The consequence is decisive: the entanglement entropy
across a horizon is not a small correction sitting on top of
$A/4\Newton$ --- it \emph{is} $A/4\Newton$. A standard calculation (the
conical-replica method) then fixes the coefficient in closed form,
$1/4 = 4\pi/16\pi$, with no adjustable parameter, no supersymmetry, and
no extremality, for the ordinary rotating and charged black holes that
actually exist. This is a first-principles derivation of the
coefficient exactly where the older microscopic counts could not reach.
One honest gap remains, and we flag it plainly: turning the induced
Newton constant into a single pure number requires summing over all the
fields of the theory, a computation we set up but do not complete.

\paragraph{\textbf{Why the paradox dissolves.}} Once the horizon is a
projection rather than a wall, the rest follows with little extra
machinery. The Page curve is automatic: for a pure total state the
entropy of the radiation must track the entropy of what remains, rising
then falling, by an elementary property of entanglement. No firewall is
needed, because the ``interior partner'' that drove the firewall
argument is not an independent four-dimensional system but part of the
shared bulk; the apparent contradiction was an artifact of insisting on
a four-dimensional-only description. The interior singularity becomes a
coordinate artifact of that same projection, smooth in six dimensions
--- just as the horizon itself looked singular in Schwarzschild's
coordinates until better ones were found.

\paragraph{\textbf{What is being counted, and the universe's own
horizon.}} The microstates get names at the same stroke. The entropy
counts the configurations of the bulk's compact phase circle over the
horizon --- the hidden-fiber holonomies, one register per Planck-sized
cell, the same fiber whose windings carry electric charge elsewhere in
the cohort: a horizon of area $A$ supports $e^{A/4\ell_P^2}$
distinguishable fiber configurations. And because the argument never
uses a box or a wall at infinity --- the observer, not an asymptotic
boundary, anchors the description --- it moves without modification to
the horizon of the accelerating universe itself: the de~Sitter entropy,
the largest entropy in nature at $\sim 10^{122}$, is the same theorem
with the observer on the inside, carried at the same graded register.

\paragraph{\textbf{The second law gets a direction and a price.}} The
generalized second law --- horizon area plus outside entropy never
decreases --- also stops being an analogy here. Its direction is
inherited from the cohort's arrow of time, and its production is
computed exactly in the paper's tractable class: the entropy generated
when a black hole swallows something equals the statistical distance
(a relative entropy) between what was thrown in and what the horizon's
thermal face would have preferred --- distance to equilibrium spent,
never minted.

\paragraph{\textbf{What it predicts, and how to test it.}} For real
astrophysical black holes the corrections to Einstein gravity are
fantastically small --- a part in $10^{46}$ for stellar-mass holes,
$10^{66}$ for the giants imaged by the Event Horizon Telescope --- so
the framework predicts pure Kerr geometry to far below any foreseeable
precision. Its falsifiable content is therefore carried by the
\emph{null}: any confirmed departure from Kerr in a black-hole shadow
or a gravitational-wave ringdown would refute it. Crucially, the same
machinery is the engine of the companion papers, so the black-hole
account can also be killed from outside black-hole physics --- by the
dark-matter and collider predictions of the cohort. A framework that
can be shot at from a neighboring field is the stronger for it.

\paragraph{\textbf{What the paper contains, and how it is
written.}} The argument is carried by named, graded results: five
theorems --- the exact $1/4$; the Page turnover as a ledger branch
switch; the rotating anchor; the collapse channel; and the thermal
clock face, which turns the field's continuation ``tricks'' into
theorems --- three lemmas (when the information return begins; the
second law's direction, with its production computed exactly in
closed form; the tower identity behind the species sum), and a
register that grades every claim as derived, conditional, or open,
each with a pre-registered way to kill it.  An experiments Part
scores the record --- the gravitational-wave ringdown and
area-theorem tests, the Event Horizon Telescope shadows --- and
dates the decade ahead with verdicts declared before the
instruments report.  The technical appendices are a graduate
machine room, rebuilt as a graduate course: fourteen Step-graded
chapters deriving the standard material from scratch --- from the
Schwarzschild geometry to the islands era --- with problems and
complete worked solutions throughout, reference dictionaries in
front, and audit registries behind.  And the objections Part opens with
thirty questions from the general reader, answered at length in
plain English, before fifty-three technical objections, organized
by referee community, are answered in the open --- concessions and
open commitments said in those words.

\paragraph{\textbf{A closing reflection.}} Seen this way, the
information paradox was never really about information. It was about
which description of nature we take to be the whole story. A
four-dimensional observer, watching a black hole evaporate, sees a
paradox only because the four-dimensional picture is a shadow --- a
partial trace of a larger, unitary, six-dimensional reality. The
horizon is not an edge of existence but the edge of one observer's
access; what crosses it is displaced, not destroyed. That lesson does
not stop at black holes. The same structure --- a complete reality seen
only in projection --- underlies how a measurement yields a definite
outcome, how heat flows in one direction, and how any open system loses
track of its surroundings. The black hole is simply the most dramatic
place where the universe reminds us that what we can see is not all
there is.
\end{ExecutiveSummary}

\GGKeywords{Black holes; event horizon; Hawking radiation;
Bekenstein--Hawking entropy; information paradox; Page curve; firewalls;
AMPS; ER=EPR; islands formula; AdS/CFT; Ryu--Takayanagi; replica
wormholes; induced gravity; Event Horizon Telescope; LIGO ringdown;
primordial black holes; \GGtheory{}; \GDOS{}; \OGGEP{}; 6D bulk.}

%% ============ FRONT-MATTER SHOWCASE, v15.3 ==================
%% Three pages after the Keywords, before the Overview of Contents.
%%   p1  Figure 8, the horizon as a visible OGGEP port -- the claim
%%   p2  Figure 9, the conical-deficit origin of 1/4   -- the result
%%   p3  Table 9,  the black-hole kill criteria        -- the exposure
%% C3 is the first C-paper with no showcase at all; C1 and C2 each
%% had three pages and needed only the exposure.
%% THE PLATES KEEP THEIR IN-PAPER SIZE (\textwidth and 0.872), the
%% rule A6 adopted: a savebox cannot differ between its uses, so a
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%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
\renewcommand{\GGShowcaseHeadSkip}{}
\newsavebox{\GGscPort}
\newsavebox{\GGscQuarter}
\newsavebox{\GGscKillBH}
\sbox{\GGscPort}{\setstretch{1}\begin{tikzpicture}[font=\sffamily\footnotesize]
% Left: lab
\begin{scope}[shift={(0,0)}]
  \node[font=\sffamily\bfseries] at (2.5,4.6) {Lab: \OGGEP{} invisible};
  \filldraw[fill=gray!30, draw=gray!60] (2.5,2) circle (1.7);
  \node[font=\scriptsize, text=gray!50!black, align=center, text width=3cm] at (2.5,2) {fiber modes\\(always traced)};
  \filldraw[fill=blue!70!black, draw=blue!90!black] (2.5,2) circle (0.15);
  \node[font=\scriptsize, anchor=north] at (2.5,1.75) {observer};
  \node[font=\scriptsize\itshape, text width=4.4cm, align=center] at (2.5,-0.5) {Trace operative but undirected; observer could ``in principle'' resolve further};
\end{scope}
% Middle: black hole
\begin{scope}[shift={(6.5,0)}]
  \node[font=\sffamily\bfseries] at (2.5,4.6) {Black hole: forced};
  \filldraw[fill=black, draw=red!80!black, line width=1.5pt] (2.5,2) circle (1.4);
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  \filldraw[fill=blue!70!black, draw=blue!90!black] (4.5,2.5) circle (0.15);
  \node[font=\scriptsize, anchor=west] at (4.65,2.5) {observer};
  \draw[->, line width=0.80pt, draw=red!70!black] (4.3,2.6) -- (3.5,2.6);
  \node[font=\scriptsize, anchor=south, text=red!70!black] at (3.9,2.7) {in only};
  \node[font=\scriptsize\itshape, text width=4.4cm, align=center] at (2.5,-0.5) {Trace forced by causal structure; no technology can recover the interior};
\end{scope}
% Right: cosmological
\begin{scope}[shift={(13,0)}]
  \node[font=\sffamily\bfseries] at (2.5,4.6) {Cosmos: forced};
  \draw[fill=blue!10, draw=blue!60, line width=0.80pt, dashed] (2.5,2) circle (1.7);
  \node[font=\scriptsize, text=blue!60!black] at (2.5,3.5) {cosmological horizon};
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  \node[font=\scriptsize, text=gray!50!black] at (2.5,0.05) {modes beyond $r=c/H$};
  \node[font=\scriptsize\itshape, text width=4.4cm, align=center] at (2.5,-0.7) {Trace forced by cosmic expansion; outside recedes faster than light};
\end{scope}
\end{tikzpicture}}
\sbox{\GGscQuarter}{\setstretch{1}\begin{tikzpicture}[>=stealth, font=\small, line width=0.80pt]
  \coordinate (tip) at (0,0);
  \draw[fill=blue!6, draw=blue!55] (tip) -- (-1.7,3) arc (180:360:1.7 and 0.5) -- cycle;
  \draw[blue!50, dashed] (-1.7,3) arc (180:0:1.7 and 0.5);
  \draw[red!70!black] (tip) -- (0.55,1.0);
  \draw[red!70!black] (tip) -- (-0.55,1.0);
  \draw[red!70!black, ->] (0.4,0.6) arc (62:118:0.62);
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  \node[align=left, anchor=west] at (2.5,1.4)
    {$\displaystyle \int_{\mathcal{M}_n}\!\sqrt{g}\,R \;\supset\; 4\pi(1-n)\,A$\\[8pt]
     $\displaystyle \Longrightarrow\ \ \Sbh = \frac{A}{4G_N},\qquad \frac14 = \frac{4\pi}{16\pi}$};
\end{tikzpicture}}
\sbox{\GGscKillBH}{\setstretch{1}\small\setlength{\tabcolsep}{5.5pt}\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}llp{5.89cm}p{4.71cm}@{}}
\toprule
\textbf{Label} & \textbf{Observable} & \textbf{Deviation
threshold} & \textbf{Timeline}\\
\midrule
F-BH-1 & EHT shadow &
$>5\%$ deviation from Kerr on M87* or Sgr A* &
Ongoing, ngEHT 2030s \\
F-BH-2 & GW ringdown &
$>1\%$ QNM deviation from Kerr in LIGO/LISA sample &
Ongoing \\
F-BH-3 & GW echoes &
Post-ringdown signal at $\Delta t \sim M\log M$ &
Ongoing analysis \\
F-BH-4 & PBH ratio structure &
A well-populated PBH mass function whose peak \emph{spacing} is
inconsistent with the $e^{7}$ scale ladder (model-level; a single
detection does not trigger it) &
Ongoing \\
F-BH-5 & Hawking spectrum &
Deviation from thermal form at observable frequencies &
Near-Planck BHs required; unfeasible \\
F-BH-6 & Information loss &
Demonstration of $\ket\psi \to \rho_{\rm mixed}$ in gravitational
setting &
Theoretical consistency; no direct test \\
F-BH-7 & Bekenstein coefficient &
$\Sbh = cA/(4\Newton\hbar)$ with $c \ne 1$ &
Theoretical; no direct test \\
F-BH-8 & Firewall signature &
High-energy quanta at horizon accumulating in stationary BH &
Accessible only with ultrafine BH spectroscopy \\
F-BH-9 & Species sum &
The sign-resolved $\sum_i\chi_i$ over the \ANine{} spectrum with
the Kaluza--Klein tower fails to reproduce the $1/\Newton$ of
Step~1 (Eq.~\eqref{eq:B-closure}; L-BH-Tower; internal) &
Theory-side; decidable by a dedicated computation, no instrument
required \\
F-BH-10 & EMRI multipoles &
Confirmed non-Kerr multipole structure at the $10^{-4}$ class in a
LISA extreme-mass-ratio inspiral &
LISA era, mid-2030s \\
\bottomrule
\end{tabular}}
\clearpage
%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{The Axiom You Cannot See in a Laboratory,\\[2pt]
Made Visible by a Horizon}
\vspace{1.4em}
\par\vspace*{\stretch{2}}
\resizebox{\textwidth}{!}{\usebox{\GGscPort}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{Why this paper exists.} The same partial trace, in three settings. In an ordinary laboratory (left) \AX{3} operates everywhere and is operationally undirected --- the trace runs over non-spatial fiber modes and no experiment reaches past it. At a black hole (center) the same trace is \emph{forced}, by causal structure rather than by any limitation of technology: no instrument can recover the interior. In the cosmos (right) it is forced again, by expansion, for everything beyond the de~Sitter horizon. In the middle and right panels the constitutional content of \AX{3} stops being an axiom one accepts and becomes a geometry one can point at, and the black hole is the cleanest of the three. In full: Figure~\ref{fig:three-port}, p.~\pageref{fig:three-port}.}}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

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\begingroup\centering
\GGShowcaseHead{Where the $1/4$ Comes From,\\[2pt]
and Why Nothing Was Fitted to Get It}
\vspace{1.4em}
\par\vspace*{\stretch{2}}
\resizebox{0.872\textwidth}{!}{\usebox{\GGscQuarter}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The number that is usually quoted and rarely derived.} On the replica cover the Euclidean-time circle is pinched at the horizon with deficit angle $2\pi(1-n)$; the curvature concentrated on the horizon contributes $4\pi(1-n)A$ to $\int\!\sqrt{g}\,R$, and fed through the induced Einstein--Hilbert action the replica formula returns $S_{\mathrm{BH}} = A/4G_N$. The coefficient is the ratio of two normalizations: $1/4 = 4\pi/16\pi$. \emph{Neither number knows anything about black holes.} The $4\pi$ is fixed by the geometry of spheres, the $16\pi$ by demanding Newton's law in the weak-field limit, and both were set by consistency requirements decades before anyone computed a horizon entropy. No free parameter enters, no regularization choice survives, and supersymmetry is never used. In full: Figure~\ref{fig:conical-quarter}, p.~\pageref{fig:conical-quarter}.}}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

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\begingroup\centering
\GGShowcaseHead{Ten Ways to Retire This Reading,\\[2pt]
and When Each One Becomes Decidable}
\vspace{1.2em}
\par\vspace*{\stretch{2}}
{\small\captionof*{table}{\textbf{Where this reading is exposed.} Ten criteria, each with the observable, the deviation threshold at which it counts, and when it becomes decidable. The last column is the honest one, and it sorts the ten by reach. Four are live instrument rows on astrophysical data --- the shadow, the ringdown, the echoes, the LISA multipoles --- and the coming decade will keep squeezing them. One fires only on a well-populated primordial mass function, never on a single detection. Four are registered but presently beyond instrumental reach, and they are on the books because the register requires the exposure to be stated even where no instrument yet exists. One is internal: the species sum, the only row whose firing date the theory itself controls. Nothing here waits on a definition; every threshold is a number. In full: Table~\ref{tab:bh_kill}, p.~\pageref{tab:bh_kill}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\usebox{\GGscKillBH}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage
%% ============ end front-matter showcase ====================

\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

\clearpage
\GGPreface

This paper is written to be readable, end to end, by a
gravitational physicist --- the reader who has fit a ringdown,
argued about firewalls at a blackboard, or waited out an observing
run --- and by the graduate student meeting Hawking radiation for
the first time.  It is structured so that such a reader loses
nothing by knowing no \GGtheory{} in advance: everything imported
is restated with its hypotheses, every object is defined in
Part~II before it is used, every theorem is proved or
proof-sketched on the page with the physics said in words
immediately after the mathematics, and the appendix machine room
re-derives the standard material from scratch at the level of a
first quantum-field-theory course.

Here is the honest pitch for spending a week with it.  You already
own the phenomena: the four laws from your first relativity
course, Hawking's 1974 calculation, Page's curve, the firewall
argument, the islands papers.  What you likely do not own is a
single statement connecting them: that a horizon is the one place
where an observer's partial trace is \emph{forced} by causal
structure rather than chosen; that Hawking thermality is that
trace computed, with the analytic continuation behind it a theorem
of a one-sided spectrum rather than a trick; that the
Bekenstein--Hawking $1/4$ is the ratio of two normalizations fixed
before entropy enters --- $4\pi/16\pi$ --- with nothing to tune;
that the Page turnover is a free-energy branch switch whose clock
is an audited counting premise; and that the generalized second
law's production is a relative entropy spent toward the horizon's
thermal face, computed exactly in its class.  Those
identifications are theorems or honestly graded conditionals here,
each with its register stated, each checkable with tools you
already have.

A suggested week.  Day one: Part~I for the century you already
partly own, and Parts~II--III for the arena and the open problems
stated precisely --- do not skip the runtime vocabulary; the whole
information story is bookkeeping on it.  Day two: Part~IV slowly
--- the forced trace, the Bogoliubov calculus done by hand with
Appendix~\ref{app:Hawking_T} beside it, and the thermal clock face
(T-BH-5), which retires three continuation prescriptions at once.
Day three: Part~V --- the four-step chain to the exact $1/4$, with
the conical replica and Appendices~\ref{app:euclidean}
and~\ref{app:species-sum} beside it.  Day four: Part~VI --- the
Page curve, the freshness clause, the firewall dissolution, and
the second law's arrow, with Appendix~\ref{app:data-processing} in
hand.  Day five: Parts~VII--VIII --- the worked numbers and the
experiments record with its dated timeline.  The weekend: the
falsification catalog, the Conclusions, the thirty plain-English
questions that open the objections Part, and as many of the
fifty-three technical objections as you have appetite for; if you
have spent time at an interferometer, a telescope, or a blackboard,
your objection is in there under your own job title.

Two further notes.  The register of the paper --- what is derived,
what is conditional, what is open, what is conceded --- is stated
at every claim and audited in the named-statement deck; the named
gaps (the species sum above all) are the paper's work orders, not
its fine print.  And the perspective is an experimentalist's: the
author has spent five decades in experimental physics watching
black holes cross from thought experiment to data --- the
ringdowns now fit, the shadows now imaged --- and the paper is
written in the discipline that instrument campaigns teach: every
claim priced, every kill criterion dated, nothing asserted that a
future instrument or a future computation cannot audit.

\paragraph{Conventions, naming, and cross-references.} The mathematical
core of this paper is the induced-gravity derivation of the
Bekenstein--Hawking entropy with its exact coefficient
(\S\ref{ssec:V-conical}, Theorem~\ref{thm:exact-quarter}); a reader who
wants only the exact $1/4$ may turn there directly.  Numbered results
carry the cohort global identifiers with the paper suffix \textsf{-BH}
(the identifiers are permanent cohort keys: \emph{T-BH-$n$} is
Theorem~$n$ wherever it stands in the Part sequence).  This paper builds on the
cohort versions of record posted at preprints.arisaka-gg.org --- \AOne{}
\citep{ArisakaA1Constitution}, \ATwo{}, \AThree{}
\citep{ArisakaA3BI6}, \AFive{}, \ASix{}, \AEight{}, \ANine{}
\citep{ArisakaA9SM}, and \AEleven{} --- and in particular on Theorems~0 and~2 of \ATwo{} (the
forced bulk and its canonical metric), the open-system law of \AFive{},
and the hidden-fiber picture of \AEight{}.  For the canonical Dictionary
of program-defined acronyms, see \ATwo{} Appendix~B.

\clearpage
%% ===================================================================
%% PART I — General Relativity and Black Holes, 1915–2026
%% ===================================================================
\GGPart[Black holes, 1915--2026]{Black Holes, 1915--2026}{A Century from Einstein's Field
Equations to the Event Horizon Telescope}

\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=stealth]
\draw[line width=1pt, ->] (-0.4,0) -- (16.4,0);
\foreach \x/\yr/\ev in {0/1915/{Einstein\\Schwarzschild}, 1.55/1963/{Kerr}, 3.1/1973/{Four laws\\Bekenstein}, 4.65/1974/{Hawking\\radiation}, 6.2/1976/{Information\\paradox}, 7.75/1993/{Page\\curve}, 9.3/1996/{Strominger\\--Vafa}, 10.85/1997/{Maldacena\\AdS/CFT}, 12.4/2012/{AMPS\\firewall}, 13.95/2019/{EHT $+$\\islands}, 15.5/2026/{\textbf{GG-Theory}\\(this work)}}{
  \draw[line width=0.8pt] (\x,0.14) -- (\x,-0.14);
  \node[anchor=south, font=\scriptsize] at (\x,0.18){\yr};
  \node[anchor=north, align=center, font=\scriptsize, text width=1.7cm] at (\x,-0.2){\ev};
}
\filldraw[red!75!black] (15.5,0) circle (2.4pt);
\end{tikzpicture}}
\caption[A century of black-hole physics, 1915--2026.]{A century of black-hole physics, 1915--2026.  Each milestone sharpened a puzzle that the next had to answer; the present work (right) addresses the entropy coefficient and the information paradox in the six-dimensional \GGtheory{} bulk. The dates are the field's and not this paper's, and only the last entry is a claim rather than a milestone. Read left to right the century has a shape: each answer created the next question, and the two that were never closed --- what the entropy coefficient is, and where the information goes --- are the two this paper takes up.}
\label{fig:timeline}
\end{figure}

Black holes were supposed to be the simplest objects in physics.  A
classical black hole is characterized by three numbers --- mass, spin,
charge --- and nothing else.  Two black holes with the same three
numbers are \emph{identical}, no matter what they were made of: a
collapsed star and a collapsed library of the same mass make the same
hole.  This ``no-hair'' simplicity is a theorem of general relativity,
and for half a century it looked like a virtue.  Then, between 1972
and 1976, three discoveries turned the simplest object in physics into
the sharpest crisis in physics --- and the century since 1915 became
the story of that reversal.

Figure~\ref{fig:timeline}, above, lays the century out as a single
line, and it is worth pausing on how strange a line it is.  It begins
with an exact solution written down within weeks of the field
equations, by an artillery officer at the Russian front.  It passes
through a golden age in which the solutions acquired theorems ---
singularities proved generic, the rotating solution found, four laws
with an unmistakably thermodynamic shape.  It turns, in the four years
from 1972 to 1976, into the crisis just described: an entropy that
should not exist, a radiation that should not be thermal, and an
information paradox that set the agenda for the next fifty years.  And
it ends, remarkably, with instruments: interferometers that hear black
holes collide and a telescope the size of the Earth that photographs
their shadows.  Few objects in physics have traveled so far --- from
a formula in wartime to data on a screen --- in a single century.

The rest of Part~I walks that line at reading pace, milestone by
milestone, gathering on the way the exact statements the later Parts
must answer: what precisely the four laws assert, what Hawking
actually calculated, what Page's curve demands, what the holographic
decade established and what it left open.  A reader who owns this
history already can skim ahead to Part~II, where the paper's own arena
is built.  But first, one section states the claim of the whole paper
in the form the cohort shares.


\section{The central claim: black-hole entropy and the information channel as an open-system observable}
\label{sec:central-claim}

It is worth stating, in one place, the recognition this paper shares with the whole \GGTheory{} cohort. No physical system is closed --- so the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection \emph{is} observation itself; its dynamics is the open-system law \GDOS{} (\AFive{}), forced by the geometric \GiFour{} (\ASix{}) and quantum-information \QiFour{} (\ASeven{}, \citealp{ArisakaA7Qi4}) pillars. It follows that the black-hole entropy $S=A/4$ and the fate of information are \emph{registered observables} of one projection (\OGGEP{}/\GDOS{}), fixed by the geometry rather than postulated --- which is why this paper can derive it, and why a single clean measurement can refute it. The ontology this rests
on --- a real six-dimensional bulk, with the boundary's complex
structure derived --- carries its own cohort grade: the selection of
the real category is \textsf{L-Constitution-1} (alias L-Real-1) of
\AOne{}, held at \emph{candidate} grade with its falsifier row
(F-Constitution-10) registered there
\citep{ArisakaA1Constitution}, and this paper inherits that grade
wherever the reality of the bulk does load-bearing work. The funnel below is shared, in structure, with the companion papers \ANine{}, \ATen{}, and \AEleven{}; only the payload differs.

\begin{figure}[hbt!]
\centering
\resizebox{0.970\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=14.5mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=98mm, minimum height=9.5mm},
 arrow/.style={-{Stealth[length=2.6mm]}, line width=0.80pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!6] (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!5, 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!10, 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!12, line width=0.8pt, below of=gdos] (obs) {\textbf{Black-hole entropy $S=A/4$}\\ and the \OGGEP{} information channel --- \textit{registered observables, not free inputs}};
\node[box, fill=green!7, 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=0.8pt, below of=pill] (concl) {$\Rightarrow$ \textbf{it is determined}\\ {\footnotesize read off one geometry --- a unitary reading of the information paradox}};
\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 to this paper's sector}          (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observable forced} (concl);
\end{tikzpicture}}
\caption[The open-system funnel shared across the \GGTheory{} cohort.]{\textbf{The central claim, as one deductive funnel}, 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. The same funnel opens the companion papers \ANine{}, \ATen{}, and \AEleven{}; only the payload differs --- here, black-hole entropy and the information channel.}
\label{fig:central-claim}
\end{figure}

\section{Einstein's 1915 field equations}

On November 25, 1915, after an eight-year struggle, Einstein
presented to the Prussian Academy of Sciences the final form of
the field equations of general relativity
\citep{Einstein1915}:
\begin{equation}\label{eq:Einstein_eq}
R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}R = 8\pi\Newton T_{\mu\nu},
\end{equation}
where $R_{\mu\nu}$ is the Ricci tensor, $R$ the Ricci scalar,
$g_{\mu\nu}$ the metric, $\Newton$ Newton's constant, and
$T_{\mu\nu}$ the stress-energy tensor. The equations relate
the curvature of spacetime (left-hand side) to the matter
and energy content (right-hand side). Three tests followed
quickly: Mercury's perihelion precession (43 arcsec/century,
explained the same month by Einstein); gravitational deflection
of starlight by the Sun (confirmed by Eddington in 1919 at
1.75 arcsec, matching GR); and gravitational redshift
(verified progressively through 1960s).

By 1919, GR was the established theory of gravity.

\section{Schwarzschild 1916: the first exact solution}

Within two months of Einstein's publication, Karl Schwarzschild
--- then serving as an artillery officer on the Russian front
--- solved the field equations exactly for the spherically
symmetric, static, vacuum case \citep{Schwarzschild1916}. The
Schwarzschild metric is
\begin{equation}\label{eq:Schwarzschild}
ds^2 = -\left(1 - \frac{2\Newton M}{c^2 r}\right)c^2 dt^2
+ \frac{dr^2}{1 - \frac{2\Newton M}{c^2 r}}
+ r^2(d\theta^2 + \sin^2\theta\,d\phi^2).
\end{equation}
At $r = r_s \equiv 2\Newton M/c^2$ (the Schwarzschild radius),
the metric has a coordinate singularity: $g_{tt}$ vanishes and
$g_{rr}$ diverges. At $r = 0$, there is a genuine curvature
singularity: the Kretschmann scalar $R_{\mu\nu\rho\sigma}
R^{\mu\nu\rho\sigma} \sim 1/r^6$ diverges.

Schwarzschild died four months after his paper, from an
autoimmune skin condition contracted at the front. He did not
live to see the full physical significance of his solution ---
particularly that $r = r_s$ is not a physical singularity but
an \emph{event horizon}, a one-way causal surface separating
the interior from the exterior.

\subsection{The Einstein-Rosen bridge}

In 1935, Einstein and Rosen \citep{EinsteinRosen1935} discovered
that the Schwarzschild metric, when analytically extended, has
a maximally extended form in which the interior connects to a
second asymptotic region via a non-traversable ``bridge.'' This
is the \emph{Einstein-Rosen bridge} or wormhole. Its importance
was not appreciated until the 1960s, when it was recognized as
a key feature of the Kruskal extension.

\subsection{Physical status of the interior}

From 1916 through about 1960, the Schwarzschild interior was
considered unphysical: a mathematical curiosity not realized
in nature, because stars do not collapse past their own
Schwarzschild radius in typical astrophysical scenarios. The
dominant view, articulated by Eddington and others, was that
quantum effects or some other mechanism would prevent
catastrophic collapse.

This view collapsed when, in 1939, Oppenheimer and Snyder
\citep{OppenheimerSnyder1939} performed the first detailed
calculation of gravitational collapse of a pressureless dust
cloud, showing that the collapse proceeds to a singularity in
finite proper time (as seen by an infalling observer) though
with infinite coordinate time (as seen by an external
observer). The ``unphysical'' interior became, in principle,
physical.

\section{The golden age, 1958--1975}

Between 1958 and 1975, a dramatic reformulation of black-hole
physics took place, driven by roughly a dozen key contributions.
We review the most important.

\subsection{Finkelstein 1958: the horizon is a coordinate
artifact}

David Finkelstein \citep{Finkelstein1958} introduced an
ingoing-null coordinate system in which the Schwarzschild
metric is manifestly regular at $r = r_s$:
\begin{equation}\label{eq:Finkelstein}
ds^2 = -\left(1 - \frac{2\Newton M}{c^2 r}\right)dv^2 + 2 dv\,dr
+ r^2 d\Omega^2.
\end{equation}
This is the Eddington-Finkelstein metric. It showed that the
apparent singularity at $r = r_s$ is a coordinate artifact, not
a physical one, and that the surface $r = r_s$ is a one-way
causal boundary: matter and signals can enter it but cannot
exit.

\subsection{Kruskal 1960: the maximal extension}

Martin Kruskal and (independently) Szekeres obtained the
maximal analytic extension of the Schwarzschild solution
\citep{Kruskal1960}. The Kruskal diagram shows the complete
Schwarzschild geometry as four regions: two asymptotic regions
(I and III), a black-hole interior (II), and a white-hole
interior (IV), joined along the horizon. The Penrose conformal
diagram makes this structure visually intuitive.

\subsection{Kerr 1963: the rotating solution}

Roy Kerr \citep{Kerr1963} discovered the exact solution for a
rotating, uncharged black hole:
\begin{equation}\label{eq:Kerr}
ds^2 = -\left(1 - \frac{2\Newton M r}{\rho^2}\right)dt^2
- \frac{4\Newton M a r \sin^2\theta}{\rho^2}\,dt\,d\phi
+ \frac{\rho^2}{\Delta}\,dr^2 + \rho^2 d\theta^2
+ \frac{\Sigma \sin^2\theta}{\rho^2}\,d\phi^2,
\end{equation}
where $a = J/(Mc)$ is the spin parameter,
$\rho^2 = r^2 + a^2\cos^2\theta$,
$\Delta = r^2 - 2\Newton M r/c^2 + a^2$, and
$\Sigma = (r^2 + a^2)^2 - a^2 \Delta \sin^2\theta$. The Kerr
solution has two horizons (outer and inner), an ergosphere
where frame-dragging is so extreme that no observer can remain
stationary, and a ring singularity at $\rho = 0$.

The subsequent charged Kerr-Newman solution (1965) completed
the family of stationary asymptotically-flat black-hole
solutions characterized by only three parameters: mass $M$,
angular momentum $J$, and charge $Q$. This is the
\emph{no-hair theorem}, conjectured by Wheeler and proved by
Israel, Carter, Hawking, and Robinson in the 1970s.

\subsection{Penrose-Hawking singularity theorems}

Roger Penrose in 1965 \citep{Penrose1965}, and Stephen Hawking
in 1966-1970 \citep{HawkingPenrose1970}, proved a series of
singularity theorems. The essential result: under mild
assumptions (dominant energy condition, existence of a trapped
surface, or equivalent), gravitational collapse necessarily
produces a singularity. Singularities are a generic feature of
GR, not an artifact of special symmetries.

This result won Penrose a share of the 2020 Nobel Prize in
Physics.

\subsection{The four laws of black-hole thermodynamics}

In 1973, Bardeen, Carter, and Hawking \citep{BardeenCarterHawking1973}
derived four laws of black-hole mechanics whose shape was
unmistakably thermodynamic.  The zeroth law states that the surface
gravity $\kappa$ is constant over the horizon of a stationary black
hole, just as temperature is constant throughout a body in
equilibrium.  The first law, $dM = (\kappa/8\pi)dA + \Omega dJ + \Phi
dQ$, has exactly the form of $dU = T\,dS + \text{work}$, with the
surface gravity standing where temperature should stand and the area
$A$ standing where entropy should stand.  The second law is the area
theorem just met: the horizon area cannot decrease, $dA/dt \ge 0$,
echoing $dS/dt \ge 0$.  And the third law states that extremal black
holes ($\kappa = 0$) cannot be reached in any finite process, as
absolute zero cannot.  The authors were careful about what they
claimed: these were presented as \emph{analogies} --- formal
coincidences between geometry and heat --- not genuine thermodynamic
statements.  It took the next two years to show that nature meant
them literally.

\subsection{Bekenstein 1972--1973: entropy is real}

Jacob Bekenstein, a student of Wheeler's, argued
\citep{Bekenstein1973} that the area-non-decrease should be
interpreted as a genuine entropy law. He proposed
\begin{equation}\label{eq:Bekenstein}
\Sbh = \frac{A}{4 \ell_P^2}\kB = \frac{A \kB c^3}{4\hbar\Newton},
\end{equation}
with $\ell_P^2 = \hbar\Newton/c^3$ the Planck length squared.
The argument was largely thermodynamic: if a black hole had no
entropy, you could throw a hot cup of tea into one and delete its
entropy from the universe, violating the second law. Bekenstein's
proposal faced considerable initial skepticism, including from
Hawking.

Two things about Eq.~\eqref{eq:Bekenstein} deserve to be called
scandalous, and both drive the rest of this paper.  It is enormous: a
single solar-mass black hole carries $\sim 10^{77}\,\kB$ --- more
entropy than all the ordinary matter in the galaxy.  And in
statistical mechanics entropy \emph{counts} something: $S = \kB \ln
(\text{number of microstates})$, Boltzmann's tombstone equation.  So
Eq.~\eqref{eq:Bekenstein} is a census, and it demands an answer to
the question: \emph{states of what?}  For fifty years no one could
say what was being counted, or why the count comes out to exactly one
quarter of the horizon area in Planck units --- not one half, not
$1/4\pi$, but $1/4$.  Part~V returns that answer.

\subsection{Hawking 1974: black holes radiate}

In 1974, Stephen Hawking \citep{Hawking1975} calculated what
happens when quantum fields propagate on the Schwarzschild
background. The result was sensational: a black hole emits
\emph{thermal radiation} at a temperature
\begin{equation}\label{eq:Hawking_T}
\THaw = \frac{\hbar c^3}{8\pi\Newton M\kB} = \frac{\hbar
\kappa}{2\pi c\kB},
\end{equation}
where $\kappa = c^4/(4\Newton M)$ is the surface gravity. For
a solar-mass black hole, this gives $\THaw \sim 10^{-7}$~K,
utterly negligible compared to the cosmic microwave background
($2.7$~K). For a primordial black hole of mass $10^{12}$~kg,
$\THaw \sim 10^{11}$~K, and the black hole evaporates in a
fraction of the age of the universe.

Hawking's calculation was a bombshell. It meant that (a)
Bekenstein was right: black holes carry real entropy $\Sbh =
A/(4\Newton\hbar)$; (b) the four analogies are genuine
thermodynamic laws; (c) black holes are not entirely black ---
they radiate; (d) ultimately, they \emph{evaporate}. And with
this last point came the paradox.

\section{The information paradox (Hawking 1976)}

\subsection{The paradox stated}

Consider a pure quantum state $\ket{\psi}$ that collapses
gravitationally to form a black hole. The black hole then
evaporates via Hawking radiation, which is thermal to all
orders in perturbation theory. A thermal state is mixed. So
the process is:
\begin{equation}\label{eq:BH_evap}
\ket{\psi} \to \text{Black hole} \to \text{Hawking radiation}
\approx \rho_{\rm thermal}.
\end{equation}
If the final state is truly thermal, then the map $\ket{\psi}
\to \rho_{\rm thermal}$ is a pure-to-mixed evolution, which
violates unitarity. If unitarity is preserved, the initial
pure-state information must be somehow encoded in the radiation.
Unitarity is a precise way of saying that information is never
destroyed, only scrambled: burn a photograph and the information
survives --- uselessly, but really --- in the smoke and the ash.
Hawking's calculation said a black hole is not like fire: the
photograph goes in, and nothing --- not even scrambled nothing ---
comes out.

In 1976, Hawking \citep{Hawking1976} argued explicitly that the
information was \emph{lost}: the evolution was fundamentally
non-unitary, and the quantum-mechanical description of gravity
required revision. This was the \emph{information paradox} in
its original formulation. It set off a five-decade controversy.

\subsection{Positions in the 1976--1995 debate}

Through the 1980s and early 1990s the community sorted itself into
four camps.  Hawking himself --- with Penrose initially alongside ---
held that information is simply \emph{lost}: quantum gravity is
fundamentally non-unitary, density matrices evolve into density
matrices, and the S-matrix must be replaced by what Hawking called a
\$-matrix.  A second camp, whose case Don Page built in 1980 and
sharpened in 1993, held that information is \emph{preserved in
correlations}: the radiation is not exactly thermal; subtle
correlations between the emitted quanta encode the missing
information, and the total state remains pure --- the position that
implies the \emph{Page curve} for the entanglement entropy.  A third
camp (Aharonov--Bohm; Banks--Susskind) proposed that information is
\emph{preserved in a remnant}: evaporation halts at the Planck scale,
leaving a Planck-mass object holding everything the hole ever
swallowed.  The difficulty is that such a remnant would need
unboundedly many internal states --- $\sim e^{S_{\rm initial}}$ for
arbitrarily large initial black holes --- and this infinite-species
problem proved hard to live with.  And a fourth camp --- Susskind,
't~Hooft --- bet that information is \emph{preserved, somehow}: the
specific mechanism unknown, but string theory or some other complete
theory would rescue unitarity by a holographic reorganization of the
degrees of freedom.
By 1993, Page's analysis had made the ``information-preserved
in subtle correlations'' scenario the leading candidate. His
paper gave the explicit entanglement-entropy curve that the
radiation should follow.

\subsection{Page curve}

For a unitary evaporation process of a black hole initially in
a pure state, the von Neumann entropy of the radiation grows
then shrinks, following the \emph{Page curve}
\citep{Page1993}.  The logic is elementary bookkeeping on
entanglement.  At early times ($t \ll t_P$) the emitted radiation is
a small subsystem of the total $S+R$ Hilbert space, and its
entanglement entropy grows as $\ln(d_R)$, where $d_R$ is the
dimension of the radiation subspace --- roughly $\Sbh^{(R)} =
A_{\rm missing}/(4\Newton\hbar)$, the entropy of the area that has
so far evaporated away.  At the Page time ($t = t_P$) the radiation
entropy catches up with the black-hole entropy; half the initial
area has radiated away, $A(t_P) = A(0)/2$.  At late times
($t > t_P$) the black hole is the smaller subsystem, so the
radiation entropy is capped by the black-hole entropy,
$\Sbh^{(\rm radiation)} = \Sbh^{(\rm BH)}$, and both decrease
together as the remaining area shrinks.  At the end
($t = t_{\rm evap}$) both entropies reach zero: the final state is
pure, and every correlation has been returned to the radiation.

The Page curve is tent-shaped: it rises linearly from 0, peaks
at $t_P$, and falls linearly back to 0. Its peak is at $S_{\max}
= A_0/(8\Newton\hbar) = \Sbh^{(0)}/2$.

The Page curve is what \emph{unitarity} predicts. The
naive Hawking calculation predicts monotonically increasing
entropy. For forty years the question was: what mechanism
changes the Hawking calculation at $t \sim t_P$ to produce the
Page curve?  The entire modern subject can be stated in three
words: \emph{derive the turnover}.

\section{Holography and \AdSCFT, 1993--2000}

\subsection{'t Hooft and Susskind 1993--1995: the holographic
principle}

Gerard 't Hooft \citep{THooft1993} and Leonard Susskind
\citep{Susskind1995} independently proposed the
\emph{holographic principle}: the physical content of a region
of space is encoded entirely on its boundary, with one
Planck-area unit of information per bit. This was motivated
directly by the Bekenstein--Hawking entropy: if $\Sbh = A/(4\ell_P^2)$
counts microstates, and the area is a 2D surface, then the
microstate information ``lives'' on the surface, not in the
bulk volume. Extrapolating, all physical systems might obey a
holographic bound: $S \le A/(4\ell_P^2)$ in appropriate units.

This was a radical claim. Ordinary field theories have volume-
extensive entropy, not area-extensive. If the holographic
principle is correct, then ordinary field theories are
approximations that break down at the level of gravitational
degrees of freedom.

\subsection{Maldacena 1997: \AdSCFT{}}

Juan Maldacena \citep{Maldacena1998} proposed the
\AdSCFT{} correspondence in 1997. In its original form:

\begin{center}
\emph{Type IIB string theory on $\mathrm{AdS}_5 \times S^5$ is
equivalent to $\mathcal{N}=4$ super-Yang-Mills in 4D Minkowski
space.}
\end{center}

The two theories are dual: every quantity in one can be
translated to a quantity in the other via a specific dictionary.
Bulk states correspond to boundary states; bulk operators to
boundary operators; and, critically, bulk gravitational dynamics
to boundary gauge dynamics. The holographic principle is
realized concretely: the 5D+1 bulk theory (including gravity)
is captured entirely by a 4D boundary theory (without gravity).

\AdSCFT{} became a dominant framework in theoretical physics for
the next 25 years, and it earned the position three times over.  It
provided a concrete setting in which black-hole evaporation is
manifestly unitary, because the boundary theory is a standard
quantum field theory and standard quantum field theories do not lose
information.  It provided a computational tool: bulk gravitational
problems can be mapped to boundary gauge-theory problems, and vice
versa, often with dramatic simplifications in one direction or the
other.  And it provided a testing ground --- for the holographic
principle, for the information paradox, and for quantum gravity in
general --- on which sharp calculations could replace verbal
arguments.

The paradox, in \AdSCFT{} language, became: \emph{How does the
bulk gravitational picture (which suggests information loss)
reconcile with the boundary gauge-theory picture (which is
manifestly unitary)?}

\subsection{Ryu-Takayanagi 2006: entanglement and geometry}

Shinsei Ryu and Tadashi Takayanagi \citep{RyuTakayanagi2006}
in 2006 proposed a formula for entanglement entropy in \AdSCFT{}:
\begin{equation}\label{eq:RT}
S(A) = \frac{\mathrm{Area}(\Sigma_A)}{4\Newton\hbar},
\end{equation}
where $A$ is a boundary region, and $\Sigma_A$ is the minimal
bulk surface whose boundary coincides with $\partial A$. This
is the \emph{Ryu-Takayanagi formula}, often abbreviated RT.

RT makes quantitative the holographic principle at the level
of entanglement: entanglement entropy of the boundary theory
is the area of a geometric surface in the bulk. This formula
has been checked in many examples and extended to time-dependent
settings (HRT formula, 2007) and to quantum corrections
(FLM formula, 2013).

\subsection{The boundary reads the bulk}

Between 2006 and 2019, the \AdSCFT{} program developed an
increasingly sophisticated understanding of how bulk information is
encoded on the boundary.  The HKLL prescription (2006) showed how
boundary operators reconstruct bulk states.  In 2015, Almheiri,
Dong, and Harlow recognized that the bulk-to-boundary map is a
quantum error-correcting code --- a discovery that reorganized the
subject's intuitions.  Entanglement-wedge reconstruction identified
exactly which portion of the bulk a given boundary region can
access, and tensor-network models of AdS geometry (Swingle 2009; the
HaPPY code, 2015) made the encoding concrete enough to draw on a
blackboard.  Each contribution clarified the bulk--boundary
relationship; none yet resolved the information paradox at the level
of the Page curve.

\section{The firewall paradox, 2012}

\subsection{AMPS paradox}

In 2012, Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and
James Sully \citep{AMPS2013} identified a sharp tension in the
black-hole complementarity picture. The \emph{AMPS argument}:

Consider a late-time Hawking quantum $B$ emitted from an old
black hole (past the Page time). Unitarity requires that $B$ be
entangled with the earlier radiation $R$. Smoothness of the
horizon (no-drama postulate) requires that $B$ be entangled
with its interior partner $A$. But entanglement is
\emph{monogamous}: $B$ cannot be maximally entangled with both
$R$ and $A$.

Something must give. AMPS argued that the most likely candidate
is smoothness of the horizon --- i.e., an infalling observer
encounters a ``firewall'' of high-energy quanta at the horizon.
This violates Einstein's equivalence principle.

The firewall puzzle became the central open question in
black-hole physics from 2012 to 2019. It sharpened the previous
information-paradox discussions: unitary evaporation
\emph{requires} that something fail, and the natural candidate
is the smoothness of the horizon --- a foundational feature of
GR.

\subsection{ER=EPR proposal (Maldacena-Susskind 2013)}

Juan Maldacena and Leonard Susskind \citep{MaldacenaSusskind2013}
proposed in 2013 that the solution to the firewall paradox is
that \emph{entangled black holes are connected by wormholes}:
ER = EPR, where ER stands for Einstein-Rosen (bridge) and EPR
for Einstein-Podolsky-Rosen (entangled pair).

In this picture, the Hawking radiation quanta are not separate
from the black hole; they are the \emph{far ends of wormholes}
whose near ends are in the black hole interior. Entanglement
between $B$ and $R$ is realized geometrically as a wormhole
connecting them. The apparent monogamy violation is an artifact
of treating the radiation and interior as separate subsystems;
they are, in the ER=EPR picture, connected.

ER=EPR is conceptually bold but mathematically loose. It
inspired further work (Cao, Carroll, Michalakis 2016;
Van Raamsdonk 2010) but was not a complete resolution.

\section{The islands formula, 2019--2020}

\subsection{Penington 2019, Almheiri et al. 2019}

A breakthrough came in 2019--2020 when Geoff Penington
\citep{Penington2020}, and independently Ahmed Almheiri, Netta
Engelhardt, Donald Marolf, and Henry Maxfield
\citep{Almheiri2019}, showed that the Page curve could be
derived \emph{directly from semiclassical gravity}, without
needing any exotic assumptions, by a careful application of the
quantum extremal surface (QES) prescription.

The \emph{islands formula} states:
\begin{equation}\label{eq:island}
S(R) = \min\,\mathrm{ext}\Big\{\frac{\mathrm{Area}(\partial I)}
{4\Newton\hbar} + S_{\rm semi-cl}(R \cup I)\Big\},
\end{equation}
where $R$ is the radiation region, $I$ is the island (a
bulk region disjoint from $R$), and one minimizes over all
choices of $I$, then extremizes over its boundary.

At early times, the trivial choice $I = \emptyset$ gives
$S(R) = S_{\rm semi-cl}(R)$, which grows monotonically. At the
Page time, a non-trivial island $I$ deep inside the black hole
becomes the minimum, and $S(R)$ starts to decrease, reproducing
the Page curve.

\subsection{Replica wormholes}

The derivation of the islands formula requires evaluating the
gravitational path integral with replicated boundary conditions,
and summing over bulk geometries including wormhole-like
configurations that connect the replicas. These
\emph{replica wormholes} \citep{ReplicaWormholes2020} are the
mechanism by which the nonlinear contributions to the
entanglement entropy produce the Page curve.

The physical picture: at sub-Page times, the radiation and
the black hole are weakly entangled, and the trivial bulk
geometry (no wormholes) dominates. At super-Page times, the
wormhole topologies dominate, making the radiation a
purification of the black-hole interior --- the interior
``lives inside'' the radiation.

\subsection{Status as of 2026}

As of mid-2026, the islands formula is widely accepted as
resolving the information paradox \emph{within} the
\AdSCFT{} framework. The resolution has two key features:
(i) the black-hole evaporation is unitary, with the Page curve
realized; (ii) the mechanism is subtle non-perturbative
gravity, not new physics at the horizon.

The limitations, however, are real, and worth stating as carefully
as the achievement.  The derivations use Euclidean gravitational
path integrals that are not rigorously defined.  They require
specific lower-dimensional models --- JT gravity in two dimensions,
hyperbolic quotients --- and their extension to four dimensions
remains partial.  The physical interpretation of the replica
wormholes is abstract: there is no clear ``mechanism'' in the
operational sense, no account of what, physically, carries the
information out.  The connection to real astrophysical black holes
in 4D asymptotically-flat space is indirect.  And the connection to
the real Standard-Model universe is obscure --- \AdSCFT{} is not the
Standard Model.

\section{Observational developments, 2015--2026}

\subsection{LIGO and gravitational waves}

In September 2015, the LIGO collaboration detected
gravitational waves from the merger of two black holes
\citep{LIGO2016}. The signal (GW150914) was a ~200 ms chirp,
followed by a ringdown phase during which the newly-formed
black hole settled into its final state via quasi-normal-mode
oscillations.

Subsequent observing runs turned the detection of mergers into a
census. The fourth observing run (O4) of the LIGO--Virgo--KAGRA
network ran from May 2023 to November 2025; the GWTC-4.0 catalog
release (2025) added 128 significant events from its first
segment to the 101 pre-O4 events of its reanalysis (the 90 of
GWTC-3 plus reprocessed candidates), and the GWTC-5.0 release
(2026) added a further 161 from the second, bringing the confirmed
total to $101 + 128 + 161 = 390$ detections
since 2015 --- roughly three quarters of them from O4 alone,
with the final O4 segment still under analysis. The census
includes a neutron-star merger with electromagnetic counterpart
(GW170817), intermediate-mass and mass-gap events (GW190521,
and GW231123 at $\sim 225$ solar masses the most massive binary
black-hole merger yet observed), and GW250114, the clearest
signal on record (signal-to-noise ratio $\sim 77$, a near-equal
$32+34\,\Msun$ pair). These detections provide direct probes of
the final-state geometry of coalescing black holes, accessible
through the quasi-normal-mode spectrum in the ringdown phase;
The area-theorem confirmations test the classical half of the GSL
story; the quantum half --- entropy production toward the KMS face
--- is not directly tested by any interferometer, and no row below
scores it as if it were.  GW250114's ringdown resolved the fundamental quadrupolar mode
together with its first overtone, consistent with the Kerr
spectrum, and its inspiral-to-ringdown horizon-area budget
confirmed Hawking's area theorem at high confidence.

\subsection{Event Horizon Telescope, 2019--2026}

In April 2019, the EHT collaboration published the first image
of a black-hole shadow: M87*, the $6.5\times 10^9\,\Msun$
supermassive black hole at the center of M87 \citep{EHT2019}.
The shadow matched the Kerr prediction within errors. In 2022,
the EHT released an image of Sgr A* (the $4\times 10^6\,\Msun$
black hole at the Galactic center), also consistent with Kerr.
Subsequent campaigns deepened both portraits:
polarization-resolved imaging of Sgr~A* (2024) revealed strong,
ordered magnetic fields at the horizon scale; a second-epoch
M87* image (2024, from 2018 data) reproduced the shadow diameter
with an independent data set; and multi-epoch polarimetry of
M87* (2025) tracked year-to-year reversals of the polarization
pattern --- evolving magnetized plasma around a shadow whose
size stays put, exactly as a Kerr geometry demands.

EHT imaging constrains the horizon-scale geometry directly.
Any deviation from Kerr ringdown or Kerr shadow at the
$5$--$10\%$ level would show up in these observations. Through
mid-2026, no deviations from Kerr have been detected beyond
astrophysical uncertainties.

\subsection{Direct horizon probes are not yet sensitive enough}

The current EHT and LIGO observations are consistent with
classical Kerr to the precision achieved. They do not
constitute a test of the Hawking radiation, the
Bekenstein--Hawking entropy, or the information-paradox resolution
mechanism. Those tests remain in principle possible but require
much higher precision than available as of 2026.

This is important context: the black-hole information paradox
is a \emph{theoretical} problem, not an observational one.
The theoretical constraints are unitarity, GR, QFT, and
thermodynamics. No current observation has falsified any of
these constraints; the paradox is purely one of consistency
between them.

\section{The cohort inheritance and the roadmap of the fourteen Parts}
\label{sec:inheritance-and-roadmap-c3}

\noindent
Before the open problems are posed in their modern form (Part~III) and
the \GDOS{} reading is built (Parts~IV--VI), 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-c3},
Figure~\ref{fig:cohort-map-c3}), and a panorama of the fourteen Parts of
the present paper (\S\ref{sec:roadmap-c3},
Figure~\ref{fig:parts-roadmap-c3}). A reader of any later Part should
be able to flip back to this Section and orient in both directions:
which cohort companion carries which imported result, and where the
present argument stands in the fourteen-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-c3}

\paragraph{The program in one view.} Figure~\ref{fig:cohort-map-c3} places the present paper among the twenty-two
papers of the \GGTheory{} cohort. The reader meeting the program
for the first time can read its architecture off the figure top to
bottom --- the constitutional root, the bulk trio, the open-system
boundary law and its pillars, the three textbook readouts, the
capstone, the two phenomenology branches, and the Foundations Branch
on the map's left flank --- before continuing
with the present paper's own thread.

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 fnd/.style={fill=violet!8, draw=violet!55!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 wht/.style={fill=brown!12, draw=brown!65!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 branchframe/.style={rounded corners=4pt, line width=0.6pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!70},
 arrowdash/.style={-{Stealth[length=2.4mm]}, line width=0.55pt,
 draw=black!60, dashed},
 arrowrt/.style={-{Stealth[length=2.6mm]}, line width=0.85pt,
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 edgelabel/.style={font=\scriptsize\itshape, fill=white,
 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
]

%% --- Row 0: axiomatic root ---
\node[root] (A1) at (0, 0)
 {\AOne{} \\[1pt] \footnotesize 4 axioms + 7 postulates};

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map. The constitutional root \AOne{} (top)
supplies the four axioms and seven postulates; the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}) realizes the bulk geometry; \AFive{} the boundary
law; \ASix{} and \ASeven{} the runtime pillars; the 4D-Physics Trio
(\AEight{}, \ANine{}, \ATen{}) the textbook readouts; \AEleven{} the
capstone. Framed branches: \textbf{Predictions} (\PaperPOne{}--%
\PaperPThree{} and \PaperPSix{}), \textbf{Cosmology \& Astrophysics}
(\PaperCOne{}--\PaperCThree{}, the present paper among them). The
spine is flanked by two columns of two that do opposite jobs: the
\textbf{Foundations} column (\PaperPFour{} above \PaperPFive{}) on the
left feeds it, and the \textbf{White-Paper} column (\PaperWOne{} above
\PaperWTwo{}) on the right reads it. Solid arrows: axiom-forced
descent; dashed arrows: deployment of derived results or historical
inheritance.}
\label{fig:cohort-map-c3}
\end{figure}
\clearpage

\paragraph{The Foundations Branch.} The map's left flank carries the
cohort's third branch --- two member papers, a single column beside the
spine, whose subject is the boundary's complex structure and its
mirrors. \PaperPFour{} \citep{ArisakaP4Real} is the origin-of-$i$
member: its causal clock is the real structure beneath the analytic
machinery this paper computes with --- the positive-frequency
splittings, thermal traces, and branch cuts of the Bogoliubov and
collapse-channel sections (\S\ref{sec:rotating-collapse} closes
entries R14--R15 of the real-bulk series it develops at full length).
\PaperPFive{} \citep{ArisakaP5CPT} defines time by its four endowments
and fixes the law-level arrow as the dictionary's one-way reading,
derived over an exactly T-symmetric bulk --- kin of the horizon
thermodynamics derived here; its Wick-quartet theorem
\textsf{T-P5-B1} licenses the analytic continuation from which this
paper proves its KMS thermal clock face (T-BH-5).

\subsection{Roadmap of the fourteen Parts}
\label{sec:roadmap-c3}

The rest of the paper is organized into thirteen further Parts, and
the walk through them is plotted in
Figure~\ref{fig:parts-roadmap-c3}.  The present Part --- the century,
1915--2026 --- is nearly done: it has walked from Einstein's field
equations to the first images of a horizon and the ringdown decade,
and it closes with the two orientation maps of this Section.
Part~II opens the toolbox: the six-dimensional arena with its
canonical scale profile, and the runtime vocabulary --- attractors,
the two regimes --- in which the black hole's life cycle will be
sorted; every object is named before it is used.  Part~III turns the
history into a charge sheet: the information paradox stated
precisely, the entropy-counting bar, the firewall argument, the Page
curve, and the state of play --- the open problems posed precisely
enough to be answered or refuted.

Parts~IV--VI are the engine room, and they are highlighted in the
roadmap because together they carry the paper's central claims.
Part~IV reads the horizon as an \OGGEP{} port of the six-dimensional
bulk: the geometrically forced trace, Hawking radiation as its
output, the thermal clock face proven rather than assumed, the
rotating anchor and the collapse channel, and the real-bulk
register.  Part~V derives the central result --- the
Bekenstein--Hawking entropy with its exact coefficient,
$S = A/4\Newton$ and $1/4 = 4\pi/16\pi$, from induced gravity
(Theorem~\ref{thm:exact-quarter}) --- together with its honest
accounting.  Part~VI assembles the information story: the
free-energy ledger, the explicit channel, the Page curve as the
scheduled expiry of port freshness, the direction of the second law
derived with its declared ceiling, no firewalls, the interior, the
four laws, and the black hole as a way-station attractor.

Parts~VII and~VIII confront the sky: the quantitative predictions,
laid out where the instruments actually point, and the experiments
record --- the kill criteria stated before the instruments report,
with the cumulative evidence base.  Parts~IX--XI weigh the case and
close it: the scorecards and framework comparisons; the Discussions
--- what is new, what remains open, and the falsification program
walked row by row; and the Conclusions --- the view from the
six-dimensional bulk.  The back matter fills three Parts of its own:
the technical appendices (Part~XII), the objections answered in the
open (Part~XIII), and the annotated references, each carrying its
Relevance note (Part~XIV).  Read in order, the fourteen Parts
compose a single self-contained argument; read by need, each Part
names its imports and can stand a visit on its own.

%% ===================================================================
%%  Roadmap of the fourteen Parts (house architecture: vertical Part
%%  column, right phase sidebar, highlighted engine room IV--VI with
%%  Part V strongest), clearpage-isolated minipage + \captionof.
%%  Rebuilt at the v13.8 structural split; label kept.
%% ===================================================================
\clearpage
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.94\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=0.55pt,
 minimum width=114mm, minimum height=8mm},
 innov/.style={fill=orange!16, draw=orange!65!black, line width=0.8pt,
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 innovmain/.style={fill=orange!28, draw=orange!70!black, line width=1.1pt,
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\node[found] (I) at (0,0)
 {\textbf{Part~I}\quad Black holes, 1915--2026};
\node[found] (II) at (0,-11.5mm)
 {\textbf{Part~II}\quad The inherited framework};
\node[found] (III) at (0,-23mm)
 {\textbf{Part~III}\quad The open problems};
\node[innov] (IV) at (0,-35.5mm)
 {\textbf{Part~IV}\quad The horizon as port \,---\, the forced trace and its thermal clock};
\node[innovmain] (V) at (0,-49mm)
 {\textbf{Part~V}\quad\textbf{The exact $1/4$} \,---\, $S = A/4\Newton$ from induced gravity};
\node[innov] (VI) at (0,-62.5mm)
 {\textbf{Part~VI}\quad Information: Page curve, firewalls, and the interior};
\node[princ] (VII) at (0,-75mm)
 {\textbf{Part~VII}\quad Quantitative predictions};
\node[princ] (VIII) at (0,-86.5mm)
 {\textbf{Part~VIII}\quad Experiments --- past, now, and future};
\node[results] (IX) at (0,-98mm)
 {\textbf{Part~IX}\quad Summary and comparison};
\node[results] (X) at (0,-109.5mm)
 {\textbf{Part~X}\quad Discussions};
\node[summit] (XI) at (0,-121mm)
 {\textbf{Part~XI}\quad Conclusions};
\node[backm] (XII) at (0,-132.5mm)
 {\textbf{Part~XII}\quad Technical appendices};
\node[backm] (XIII) at (0,-144mm)
 {\textbf{Part~XIII}\quad Common objections and responses};
\node[backm] (XIV) at (0,-155.5mm)
 {\textbf{Part~XIV}\quad References};
\draw[arrow] (I) -- (II);
\draw[arrow] (II) -- (III);
\draw[arrowinnov] (III) -- (IV);
\draw[arrowinnov] (IV) -- (V);
\draw[arrowinnov] (V) -- (VI);
\draw[arrowinnov] (VI) -- (VII);
\draw[arrow] (VII) -- (VIII);
\draw[arrow] (VIII) -- (IX);
\draw[arrow] (IX) -- (X);
\draw[arrow] (X) -- (XI);
\draw[arrow] (XI) -- (XII);
\draw[arrow] (XII) -- (XIII);
\draw[arrow] (XIII) -- (XIV);
\node[phaselabel] at (72mm,-11.5mm) {\emph{The paradox} \\ \emph{posed}};
\node[phaselabel, text=orange!70!black] at (75mm,-49mm)
 {\emph{\textbf{The derivation}}};
\node[phaselabel] at (72mm,-80.75mm) {\emph{Confrontation} \\ \emph{with the sky}};
\node[phaselabel] at (72mm,-109.5mm) {\emph{Weighing \&} \\ \emph{summit}};
\node[phaselabel] at (72mm,-144mm) {\emph{The machine room} \\ \emph{and the record}};
\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the fourteen Parts of GG-C3-BH.]{Roadmap
of the fourteen Parts of \textsf{GG-C3-BH}.  The argument develops in
five phases.  \emph{The paradox posed} (Parts~I--III): the century of
black-hole physics, the inherited framework, and the open
problems stated precisely enough to be answered or refuted.
\emph{The derivation} (Parts~IV--VI, highlighted): the horizon as a
geometrically forced \OGGEP{} port with its proven thermal clock; the
Bekenstein--Hawking entropy with its exact coefficient,
$S = A/4\Newton$, from induced gravity
(Theorem~\ref{thm:exact-quarter}) --- the result on which every later
Part stands, and the strongest box in the column; and the information
story, from the scheduled expiry of port freshness to the direction
of the second law.  \emph{Confrontation with the sky}
(Parts~VII--VIII): the quantitative predictions and the experiments
record with its kill criteria.  \emph{Weighing and summit}
(Parts~IX--XI): scorecards and comparisons, the Discussions, and the
Conclusions.  \emph{The machine room and the record}
(Parts~XII--XIV): the technical appendices, the objections answered
in the open, and the annotated references.}
\label{fig:parts-roadmap-c3}
\end{minipage}
\clearpage


%% ===================================================================
%% PART -- MATHEMATICAL PRELIMINARIES (new at the v13.8 split)
%% ===================================================================

\section[Part~I in plain English]{Part~I in plain English: a century of the field, and the two questions it left open}
The simplest objects in physics have the longest bibliography. A black
hole is described by three numbers --- mass, spin, charge --- and a hundred
years of work has gone into two questions those three numbers do not
answer.

The century divides cleanly. From 1915 to about 1975 the work was
geometric: Schwarzschild's exact solution the year after the field
equations, Kerr's rotating one in 1963, the theorems that showed
singularities were generic rather than artifacts of a convenient symmetry.
By the middle of the 1970s the object was understood as geometry. Then two
results changed what kind of object it was. Bekenstein said a black hole
has an entropy proportional to the area of its horizon. Hawking computed
the temperature that entropy implies, and found that the hole radiates.

Both open questions come from those two results. The entropy formula
carries a coefficient --- one quarter --- and nobody derived it from a
count of anything; it was read off a calculation and has sat there, exact
and unexplained, for fifty years. And if the hole radiates it eventually
evaporates, and the information that fell in has nowhere left to go. The
debate that followed ran from 1976 to the present through holography, the
Page curve, firewalls and islands, and its state today is a formula that
works in controlled models and a mechanism nobody has grounded in a theory
of the actual universe.

This paper takes up both, from a geometry fixed elsewhere in this cohort
for reasons that had nothing to do with black holes. The last entry on the
timeline is a claim rather than a milestone, and the Parts that follow are
the argument for it.

\GGPart[The inherited framework]{The Inherited Framework: \GGtheory{} in One Part}{What the Black-Hole Module Inherits --- the Six-Dimensional Bulk, the Boundary Projection, the Zero-Parameter Integer Ledger, and the Open-System Runtime}

\noindent
This Part answers a single question: \emph{what is \GGtheory{}, and
what does a black-hole module built on it inherit?}  Part~I
established the century's baseline and ended with the claim the whole
paper must earn; before the paper can claim geometric answers to the
horizon's open problems, the reader deserves a clear and unhurried
picture of the machine the answers come out of.  Nothing in this Part
is new physics.  Everything here was proved upstream, in the
companion papers of the cohort, and is imported wholesale; the job of
this Part is to make that inheritance \emph{intuitive} --- to hand a
reader who has seen only Part~I the four ideas the rest of the paper
leans on, in the order in which they build on one another.  The
from-scratch derivation chapters a graduate reader may want ---
quantum fields in curved spacetime, the Bogoliubov calculus, heat
kernels, the replica method --- remain the business of the technical
appendices (Part~XII); this Part is the frame those tools hang on.
The open problems themselves are restated precisely in the next Part,
once the framework is in hand.

\section{One geometry, no dials: the six-dimensional bulk}
\label{sec:bulk-arena}

At the center of the whole program is a single geometric object ---
the constitution of \AOne{} \citep{ArisakaA1Constitution} fixes what
may count as an admissible claim, and \ATwo{} \citep{ArisakaA2GG6}
derives the geometry itself --- and it is worth meeting it in plain
language before meeting it in equations.  \GGtheory{} posits that
reality is a \emph{six-dimensional bulk}, a tapering ``throat'' of
the form $M_4 \times I_r \times S^1_\theta$.  Three pieces make it
up.  The familiar four dimensions $M_4$ are the spacetime we live in.
A single radial direction $I_r$ runs along the throat from a wide
ultraviolet mouth down to a narrow infrared end; this is the energy
axis, and it is best read as the renormalization-group scale,
$r=\ln(\EGUT/\mu)$ --- moving along $r$ is nothing other than running
a coupling from high energy to low.  And a small compact angle
$S^1_\theta$, the phase fiber, carries the locked gauge structure:
its winding is where the cohort's dark sector lives --- and, at a
horizon, where this paper's microstates will be counted.

The word ``warped'' deserves a warning, because it is borrowed from a
different picture.  Here it does \emph{not} mean an extra spatial
dimension with a curved metric in the Randall--Sundrum sense.  It
means RG running along $r$: the warp factor is the field-theory /
holographic-RG reading of scale dependence, exactly as the rest of
the cohort reads it.  With that one substitution the geometry stops
being exotic and becomes familiar --- ``deeper into the throat''
simply means ``lower in energy.''

The single most important intuition to carry forward is this:
\emph{our four-dimensional world is the boundary of this bulk}.  We
do not live inside the throat; we live on its edge, and what we call
physics is the shadow the bulk casts there.  How that shadow is cast
--- the boundary projection itself --- is the subject of the next
section, where Figure~\ref{fig:bulk-boundary} draws the throat, its
radial scale axis, its compact fiber, and the four-dimensional
boundary slab on which the black holes of Part~I live.

\emph{The geometry, made explicit.}  Behind the picture is a
definite, parameter-free metric.  The bulk is
\begin{equation}\label{eq:BH_bulk}
M_6 = M_4 \times I_r \times S^1_\theta,
\qquad ds_6^2 = e^{-2r}g_{\mu\nu}(x)dx^\mu dx^\nu
+ dr^2 + d\theta^2,
\end{equation}
the canonical scale-profile metric of \ATwo{} (Theorem~2: no free
parameters $k$, $R_\theta$, or $r_c$ survive)\citep{ArisakaA2GG6}.
The warp profile $A(r)=r$ is not chosen but forced, by demanding
constant bulk curvature with a flat boundary ($A''(r)=0$); there is
no curvature scale, no compactification radius, and no brane tension
to tune.  Here $I_r$ is the non-spatial \emph{scale dimension} ---
the renormalization coordinate with range $r \in [0, K]$, $r=0$ the
UV boundary at $\MGUT$ and $r=K$ the IR boundary at $\Mlock$ --- and
$S^1_\theta$ is the non-spatial \emph{phase dimension}, a circle of
fixed coordinate period $2\pi$ whose proper size at the IR boundary
is $\ell_{\rm lock}\sim 1/\Mlock$.  The Planck--electroweak hierarchy
is then the topological number $e^{-K}$, not a fine-tuned ratio.
That this six-dimensional arena is \emph{forced} by \LC{} (AX-1) and
\GGEP{} (AX-2) rather than postulated is Theorem~0 of \ATwo{}.

\emph{Six dimensions are four plus one plus one.}  The count is
forced role by role \citep{ArisakaA2GG6}: four Lorentzian directions
$M_4$ carry registered observations and the only length dimension;
one non-compact \emph{scale} direction $I_r$ is the geometrized RG
flow (a dimensionless energy logarithm, not space); and one compact
\emph{phase} circle $S^1_\theta$ carries integer-quantized
Wilson-line holonomy.  The three bulk directions are exactly what is
needed to absorb the three dimensionful constants $c$, $G_N$, and
$\hbar$; the boundary supplies the remaining two, $k_B$ and the
energy--information quantum $Q_B$ --- together the five universal
constants $\mathrm{GU\text{-}5}=\{c,G_N,\hbar;\,k_B,Q_B\}$, of which
the boundary-thermodynamic member $k_B$ is the conversion factor
that will carry the Bekenstein--Hawking entropy below.  This
``three-plus-two'' architecture is the spine of the whole ledger.

\emph{One geometry carries all six forces.}  On the bulk the gauge
and gravitational structure is a single group,
\[
G_{\mathrm{Bulk}} \;=\; \mathrm{Spin}(1,5)\,\times\,\SUfive\,\times\,\Uoneb\,\times\,\UoneQi,
\]
with the Lorentz/spin factor $\mathrm{Spin}(1,5)$ (the gravity
sector) on equal footing with the internal factors
\citep{ArisakaA2GG6}.  Here $\SUfive$ is the minimal simple
unification containing the Standard Model; $\Uoneb$ is gauged baryon
number, which turns proton stability into a theorem; and $\UoneQi$
is the informational gauge force that the \QiFour{} pillars later
use.  Projected onto the four-dimensional boundary this becomes the
six forces an observer actually measures --- strong, weak,
electromagnetic, baryonic, informational, and gravitational --- with
the \BIsix{} modified Bianchi identity of the next-but-one section
guarding the closure.  Crucially for this paper, the same compact
circle $S^1_\theta$ that quantizes these holonomies is where the
horizon microstates of Part~V are counted and where the companion
papers read the Kaxion dark matter: the black hole and the particle
ledger are two projections of this one object, one story apart.

For a black hole, the 4D metric $g_{\mu\nu}(x)$ in
Eq.~\eqref{eq:BH_bulk} is the Schwarzschild (or Kerr, or
Kerr--Newman) geometry.  The 6D bulk accommodates this by having the
4D metric vary over $x^\mu$ while the compact dimensions remain (at
least in the low-energy effective description) unperturbed.  The
horizon in the 4D metric is at $r_s = 2\Newton M/c^2$ for
Schwarzschild.

\section{The boundary projection: how four-dimensional physics arises}

If the bulk is the whole of reality, why do we only ever measure
four-dimensional physics?  The answer is the \emph{boundary law},
and it is the deepest single result the black-hole module inherits.
An observer outside a horizon can never access what lies behind it,
and no boundary observer at all can access the closed bulk; what an
instrument registers is the projection of the bulk onto the
four-dimensional edge the observer inhabits.  \AFive{}
\citep{ArisakaA5GDOS} raises this to a principle --- the open-system
geometric--gauge equivalence principle (\OGGEP{}), the third axiom
of the constitution --- which requires every admissible theory to
supply an explicit dictionary
\[
\mathcal{D}_{\OGGEP}:\;\bigl(\mathcal{H}_{\mathrm{Bulk}},\,\rho_{\mathrm{Bulk}},\,U_{\mathrm{Bulk}}\bigr)\;\longmapsto\;\bigl(\mathcal{H}_{\mathrm{Bdy}},\,\rho_{\mathrm{Bdy}},\,\Phi_{t,s},\,\{\mathcal{I}_\alpha\}\bigr),
\]
carrying the inaccessible bulk data to a boundary state, an
admissible (completely positive, trace-preserving) evolution channel
$\Phi_{t,s}$, and the instruments that read it.  The decisive
recognition is that this bulk-to-boundary projection \emph{is
observation itself}: measurement is not a postulate appended to the
theory but what the projection does.

\begin{figure}[!htbp]
\centering
\resizebox{0.92\textwidth}{!}{%
\begin{tikzpicture}[
    >={Stealth[length=3.2mm,width=2.6mm]},
    font=\small,
    bulkfill/.style={top color=blue!10,bottom color=blue!4},
    edge/.style={line width=1.10pt},
    fiber/.style={line width=0.80pt,draw=teal!70!black},
    scaledot/.style={circle,fill=black,inner sep=1.4pt},
    boundaryslab/.style={top color=orange!18,bottom color=orange!28,draw=orange!60!black,line width=0.80pt},
    note/.style={font=\footnotesize\itshape,text=black!72},
  ]
  \def\xL{0}
  \def\xR{8.6}
  \def\yTop{9.2}
  \def\yBot{1.7}
  \def\taper{2.55}
  \coordinate (TL) at (\xL,\yTop);
  \coordinate (TR) at (\xR,\yTop);
  \coordinate (BL) at (\xL+\taper,\yBot);
  \coordinate (BR) at (\xR-\taper,\yBot);
  \begin{scope}[on background layer]
    \fill[bulkfill]
      (TL) -- (TR)
      .. controls ($(TR)+(-0.2,-2.6)$) and ($(BR)+(0.55,2.6)$) .. (BR)
      -- (BL)
      .. controls ($(BL)+(-0.55,2.6)$) and ($(TL)+(0.2,-2.6)$) .. (TL)
      -- cycle;
  \end{scope}
  \draw[edge,blue!55!black]
      (TR) .. controls ($(TR)+(-0.2,-2.6)$) and ($(BR)+(0.55,2.6)$) .. (BR);
  \draw[edge,blue!55!black]
      (TL) .. controls ($(TL)+(0.2,-2.6)$) and ($(BL)+(-0.55,2.6)$) .. (BL);
  \node[align=center,blue!45!black] at (4.3,7.3)
      {\textbf{6D bulk}\\[1pt]
       $M_4 \times I_r \times S^1_\theta$};
  \node[note,align=center,text width=4.6cm] at (3.7,5.7)
      {a tapering ``throat'':\\ radius shrinks as energy\\ flows from UV to IR};
  \draw[edge,red!65!black] (TL) -- (TR);
  \node[above=2pt of TR,anchor=south east,align=right,red!60!black,font=\small]
      {\textbf{UV GUT edge}\;($r=0$)\\[1pt]
       $\EGUT \sim 10^{16}\ \mathrm{GeV}$};
  \def\axx{-1.7}
  \draw[->,line width=1.10pt,black!75] (\axx,\yTop+0.55) -- (\axx,\yBot-0.35);
  \node[rotate=90,anchor=south,font=\small,black!75] at (\axx-1.05,5.5)
        {RG / scale axis \ \ $r=\ln(\EGUT/\mu)$};
  \node[right=2pt,font=\footnotesize,black!60] at (\axx,\yTop+0.5) {UV\,($r{=}0$)};
  \node[right=2pt,font=\footnotesize,black!60] at (\axx,\yBot-0.3)  {IR};
  \node[scaledot] (sPl)   at (\axx,\yTop+0.0)  {};
  \node[scaledot] (sGUT)  at (\axx,\yTop-2.6)  {};
  \node[scaledot] (sLock) at (\axx,\yBot+0.55) {};
  \node[right=4pt,font=\footnotesize] at (sPl.east)  {$\EPl$};
  \node[right=4pt,font=\footnotesize] at (sGUT.east) {$\EGUT$};
  \node[right=4pt,font=\footnotesize,align=left] at (sLock.east)
        {$\Mlock\sim 3.85~\mathrm{TeV}$\\[-1pt]\scriptsize(KK lock)};
  \node[left=2pt,font=\scriptsize,black!55] at ($(sPl)!0.5!(sGUT)$)  {inherits};
  \node[left=2pt,font=\scriptsize,black!55] at ($(sGUT)!0.5!(sLock)$) {inherits};
  \coordinate (fpt) at (5.9,3.6);
  \node[scaledot] at (fpt) {};
  \draw[fiber] (fpt) ellipse [x radius=0.66,y radius=0.30];
  \draw[fiber,->] ($(fpt)+(0.66,0.0)$) arc [start angle=0,end angle=300,x radius=0.66,y radius=0.30];
  \node[anchor=north west,align=left,text width=5.0cm,font=\footnotesize,teal!45!black]
      (fiblab) at (9.3,6.2)
      {compact phase $S^1_\theta$: winding carries the
       dark sector --- and, at a horizon,
       \textbf{the microstates of Part~V}};
  \draw[teal!60!black, line width=0.40pt] (fiblab.south west) to[bend right=12] ($(fpt)+(0.62,0.22)$);
  \node[boundaryslab,minimum width=9.6cm,minimum height=1.35cm,rounded corners=2pt]
      (bdy) at (4.3,0.05) {};
  \node[align=center] at (bdy.center)
      {\textbf{4D spacetime $M_4$}\;:\; ordinary general relativity ---
       Schwarzschild\,/\,Kerr \ \ (the boundary)};
  \draw[-{Stealth[length=5mm,width=4mm]},line width=2.2pt,red!70!black]
      ($(BL)!0.5!(BR)+(0,0.05)$) -- ($(bdy.north)+(0,0.02)$);
  \node[anchor=west,align=left,font=\footnotesize,red!55!black] at (6.7,1.55)
      {\textbf{\OGGEP{}} \,(bulk\,$\to$\,boundary):\\
       black-hole observables are the\\
       \emph{boundary shadows} of bulk structure};
  \node[below=3pt of bdy.south,anchor=north,font=\footnotesize,orange!40!black]
      {boundary governed by the \GDOS{} boundary law};
  \node[draw=black!55,black!35,fill=black!3,rounded corners,inner sep=4pt,
        text width=5.0cm,note,anchor=north west] at (9.3,8.7)
      {``Warped'' here denotes \textbf{RG running along $r$}
       (the field-theory / holographic-RG reading), \emph{not}
       a Randall--Sundrum spatial dimension.};
\end{tikzpicture}%
}%
\par\smallskip
\caption[6D bulk and its \OGGEP{} projection to the 4D boundary.]%
{Schematic of the six-dimensional bulk
 $M_4\times I_r\times S^1_\theta$ and its \OGGEP{} projection to the
 four-dimensional boundary.  The radial axis $r=\ln(\EGUT/\mu)$ is the
 renormalization-group / scale axis: the ultraviolet GUT edge sits at
 the top ($r=0$, $\EGUT\sim10^{16}\,\mathrm{GeV}$) and low-energy
 physics lies toward the boundary, with the inherited scale chain
 $\EPl\!\to\!\EGUT\!\to\!\Mlock$ ($\Mlock\sim3.85\,\mathrm{TeV}$, the
 Kaluza--Klein lock scale) marked along it.  The compact angle
 $S^1_\theta$ is the phase fiber whose winding carries the dark sector
 --- and, at a horizon, the microstates this paper counts in Part~V.
 The bold arrow is the \OGGEP{} bulk-to-boundary map: black-hole
 observables on $M_4$ (ordinary general relativity: Schwarzschild,
 Kerr) are boundary shadows of bulk structure, with the boundary
 governed by the \GDOS{} boundary law.  ``Warped'' denotes RG running
 along $r$, \emph{not} a Randall--Sundrum spatial dimension.}
\label{fig:bulk-boundary}
\end{figure}

\AFive{} then makes the projection dynamical and unique.  Its
boundary law, the geometric dynamics of open systems (\GDOS{}), is
the Lindblad/GKLS generator
\[
\mathcal{L}_{\GDOS}(\rho) \;=\; -\,i\,[H_{\mathrm{eff}},\rho] \;+\; \sum_\alpha \gamma_\alpha\Bigl(L_\alpha\,\rho\,L_\alpha^\dagger - \tfrac12\{L_\alpha^\dagger L_\alpha,\,\rho\}\Bigr),\qquad \gamma_\alpha\ge0,
\]
derived --- not postulated --- from the exact bulk dynamics by a
Born--Markov reduction \citep{ArisakaA5GDOS}.  The ordinary
Schr\"odinger equation is recovered as its closed-system limit;
openness is primary and closure the special case.  In this language
an \emph{observable} is the value an open system reads off at a
stationary seam, an attractor of the \GDOS{} flow --- and from this
one law \AFive{} recovers four-dimensional quantum mechanics,
general relativity, and quantum field theory as boundary theorems
rather than separate assumptions.

The warm consequence, the one to hold onto, is that \emph{ordinary
general relativity is recovered, not discarded}.  Because
four-dimensional general relativity is already a theorem of
\AFive{}, the Schwarzschild and Kerr geometries of Part~I live
inside the framework automatically, with the four-dimensional Newton
constant emerging as a scale-overlap integral over the throat.  Part~V
sharpens that emergence into the paper's central result: because no
bare boundary $\Newton$ exists to be corrected, the entanglement
entropy across a horizon is not a correction sitting on top of
$A/4\Newton$ --- it \emph{is} $A/4\Newton$.  We do not choose between
\GGtheory{} and the relativity textbooks: the textbook geometry is
exactly what the boundary law produces, and what the framework adds
is a fixed, parameter-free account of the horizon's thermodynamic
face.

A boundary law fixes \emph{what} the four-dimensional theory is; a
black-hole module also needs to know \emph{how} its lawful
possibilities are explored and stabilized in time.  That is the
geometric runtime, the four invariance pillars \GiFour{} of \ASix{}
\citep{ArisakaA6Gi4}, organized as a single cascade
\[
\TGG{} \;\Longrightarrow\; \CCI{} \;\Longrightarrow\; \GFF{} \;\Longrightarrow\; \GRIP{}.
\]
The total geometric--gauge object \TGG{} packages the closed bulk
ledger as one composite connection on the $G_{\mathrm{Bulk}}$
bundle; the covariant conservation identity \CCI{} hands the
boundary observer a local conservation law and, under closure,
forces the integer ledger of the next section; the geometric
free-energy functional $\GFF{}=\GEP{}+\CEP{}$ ranks the admissible
boundary configurations by a single scalar; and \GRIP{} --- the
lifecycle $\mathrm{DGI}\to\mathrm{GSSB}\to\mathrm{GA}$, lawful
exploration to symmetry-breaking to stabilized attractor --- fixes
the order in which a branch is explored, committed to, and frozen.
The next two sections put this machinery to work: first the fixed
ledger it runs on, then the way it selects the branches of a black
hole's life.

\subsection{Which particles?  The observer-indexed vacuum, and horizon thermality as openness}
\label{ssec:observer-vacuum}

One further consequence of the projection deserves a named place
before the ledger, because the black-hole Parts use it on every page
that mentions particle production.  In a curved or horizon-bearing
spacetime there is famously no preferred vacuum: inequivalent
positive-frequency conventions define inequivalent particle notions,
related by Bogoliubov mixing.  In the present framework this
ambiguity has a fixed address.  On the canonical bulk geometry
inherited from \citep{ArisakaA2GG6} the constitutional time is
globally static, so the bulk-level positive-frequency structure is
unique; what varies from observer to observer is the
\emph{dictionary}: which modes an instrument counts is fixed by the
observer's own time and horizon structure, and the instruments are
part of the \OGGEP{} boundary package by definition.  ``How many
particles'' is therefore an observer-indexed boundary observable,
not a bulk quantity --- the same status the framework already
assigns to boundary states themselves.  The physical output of the
ambiguity is produced by the \GDOS{} projection itself: tracing the
inaccessible partner of a horizon's squeezed pairing yields an
exactly thermal state (at $T = a/2\pi$ for the Rindler pairing), so
horizon thermality is constitutional openness doing its ordinary
work rather than new physics.  The full statements are carried in
the real-bulk treatment of \citep{ArisakaA8QM}, whose standalone,
full-length development is the Foundations Branch member paper
\PaperPFour{} (\citealp{ArisakaP4Real}).  Two boundary cases are
closed there at the model level, as entries R14--R15 of the series
--- the rotating case, where the ergoregion defeats the co-rotating
selection and the uniqueness anchor moves to the asymptotic frame,
superradiance becoming a symplectic pumping channel in the same
\GDOS{} slot; and the collapse epoch, which factorizes into a
universal thermal channel times a decaying transient --- and the
present paper is their full-length home: they are Theorems~3 and~4
(T-BH-3, T-BH-4; \S\ref{sec:rotating-collapse}), displayed with the
whole series at \S\ref{sec:rb-register}.  What remains conjectural
is the exact curved-space existence theory --- the field's own,
imported where used.  The Parts that follow use both facts on nearly
every page: the bulk structure is unique, and the thermal content of
horizons is the partial trace.

\section{Zero free parameters: \BIsix{} and the integer ledger}

A framework can be elegant and still be useless if it leaves the
numbers free.  What makes \GGtheory{} predictive rather than merely
descriptive is that it has \emph{no continuous dials} to turn, and
two inherited results enforce this.

The first is the anomaly-locked bulk.  Global consistency of the
six-force closure requires a modified six-dimensional Bianchi
identity, \BIsix{}, for the antisymmetric tensor $H_3=dB_2$
(\citealp{ArisakaA3BI6}; \citealp{ArisakaA4Euclid}):
\[
dH_3 \;=\; \mathrm{tr}\,F_{SU(5)}^2 \;-\; F_B^2 \;-\; \tfrac{2}{5}\,F_B\!\wedge\!F_{Qi} \;-\; \tfrac{2}{5}\,F_{Qi}^2 \;+\; \mathrm{tr}\,R^2,
\]
whose coefficient vector $(a,b,c,d)=(1,-1,-\tfrac{2}{5},-\tfrac{2}{5})$
is fixed uniquely by three independent conditions --- local closure
($dX_4=0$), global integrality of $[X_4]$, and anomaly cancellation
by inflow through the single channel $I_8=\tfrac12 X_4\wedge X_4$.
There is no continuous deformation: the trace-ratio lock
$|a/c|=|a/d|=5/2$ is rigid.  The ultraviolet is legal, and it is
legal in exactly one way.

The second is the integer ledger.  \AFour{} \citep{ArisakaA4Euclid}
proves that the same closure forces a unique tuple of whole numbers,
the \OGNfive{} ledger
\[
(M,N,L;\,J,K) \;=\; (1,3,5;\,7,35),
\]
through a five-step Diophantine cascade in which each integer is
fixed by the one before: $M=1$ (a single Green--Schwarz tensor
channel), $L=5$ (the minimal simple unification $\SUfive$), $N=3$
(the three colors and generations, forced by the \BIsix{} trace
ratio $-(L-N)/L=-2/5$), $J=2N+1=7$, and the closure integer
\[
K \;=\; LJ \;=\; M^2+N^2+L^2 \;=\; 35.
\]
That two independent arithmetic expressions for $K$ agree --- the
\emph{dual lock} --- is what closes the ledger; it is generically
false, and true here only because both are forced by the same
axioms.  These integers are not fitted; they are the arithmetic of
the geometry.  Together with the structural constants $\pi$ and $e$
and the dimensionful conversion factors $\{c,G_N,\hbar;k_B,Q_B\}$,
they are the entire stock from which every dimensionless number in
the cohort is built: the same ledger that fixes the twenty-eight
Standard-Model parameters of \ANine{} \citep{ArisakaA9SM} and the
$\pi$-locked cosmic inventory of \ATen{} \citep{ArisakaA10Cosmos}
fixes what this paper needs, with no new freedom in between; the
discipline that admits a number only if it is a cost-minimum element
over this ledger is the \MSMF{}/\OGN{} principle.

Two entries matter most at a horizon.  The closure integer $K=35$
sets the depth of the throat and with it the lock scale
$\Mlock \approx 3.85$~TeV --- and it is the square of the
corresponding lock length against an astrophysical horizon radius,
$(\ell_{\rm lock}/r_s)^2 \sim 10^{-46}$, that makes every
\GGtheory{} correction to Kerr geometry invisible to any foreseeable
instrument (Part~VII): the framework's observational signature is
clean general relativity, and the ledger is why.  And the integer
$J=7$ prints the $e^{7}$ spacing of the primordial-black-hole mass
ladder whose kill row is F-BH-4 in the falsification table.  The
Planck--electroweak hierarchy, the silence of the ringdowns, and the
dark-matter signature of the companion papers are literally the same
integers, read on different strata.

\section{The runtime: how a black hole's branches are selected}
\label{sec:runtime-vocab}

The framework so far is static: a bulk, a boundary law, a fixed
ledger.  But a black hole has a \emph{history} --- it forms,
accretes, merges, radiates, crosses the Hawking--Page and Page
thresholds, and (if the universe permits) evaporates; something has
to choose, event by event, which branch of the lawful possibilities
actually happens.  That choosing is what \GGtheory{} calls the
runtime, and it is the last piece the module inherits.  The runtime
has two components, both proved in the pillar papers \ASix{} and
\ASeven{} (\citealp{ArisakaA6Gi4}; \citealp{ArisakaA7Qi4}): the
\GRIP{} cascade, which fixes the order in which lawful branches may
be explored, committed to, and stabilized; and the \GFF{} ledger,
the scalar ranking functional $F_{\GFF}=F_{\GEP}+F_{\CEP}$ that says
\emph{which} branch is favored.  The \GFF{} ledger is, in form, a
free-energy functional; its horizon instance --- the generalized
entropy, with the area term and the bulk-entanglement term assigned
to their owners --- is built in \S\ref{sec:gff-ledger}, where the
information-theoretic lineage is recalled.

The two regimes, and where a descent may halt, are the vocabulary
the reader will use most.  In \GGtheory{}, every open system
descends the single scalar cost functional just named, and the
descent operates in exactly two regimes, split by one question:
\emph{does the landscape change?}  \GRIPin{} is all dynamics on a
\emph{fixed} landscape: relaxation within a stable basin (a
\emph{Geometric Attractor}, \GA{}), and passage \emph{between}
basins through whatever physical channel connects them.  \GRIPout{}
is the complementary regime: a transient event that \emph{reorganizes
the landscape itself}.  A descent halts only at the basin with no
exit channel at all --- the \emph{Ground} Geometric Attractor
(\GGA{}), the unique deepest basin (the No-Exit theorem --- carried
at \AEleven{}'s own grade, with its above-particle-strata reach
conjectural there --- of \citep{ArisakaA11Origin}).  Among the many
stable basins of a stratum, exactly one is this global minimum,
$\GGA := \arg\min F_{\GFF}$; the distinction between a \GA{} (a
stable record, not eternal) and the \GGA{} (the deepest state, with
nothing to decay to) is load-bearing at a horizon, because this
paper will show that today's astrophysical black holes are
\emph{conditional} ground states --- exitless only while
$T_H < T_{\rm CMB}$, with the condition expiring on a calculable
cosmological schedule (\S\ref{ssec:conditional-gga}).

Which branch, among the many lawful attractors of a stratum, is the
one actually realized?  The selector is the set of four
quantum-information pillars \QiFour{} of \ASeven{}
\citep{ArisakaA7Qi4}.  A configuration is the observed \emph{ground}
geometric attractor only if it passes all four admissibility filters
at once: \emph{Qi-QUA}, that it carry the locked four-current
identity $J_E/Q_B=J_e=J_B=J_{Qi}$ (the Arisaka equation);
\emph{Qi-EP}, that it be a genuine extremum of the one projected
variational principle, not merely a local minimum of a
coarse-grained potential; \emph{Qi-CON}, that its scale signatures
be those of a fixed point, surviving transport along $r$; and
\emph{Qi-HAL}, that the runtime can actually halt there, with no
admissible descent remaining.  The observed value is the unique
four-certificate survivor of its stratum --- the same criterion that
makes every electron, proton (\citealp{ArisakaP6Proton}), and Kaxion
the identical object everywhere.

This is not an abstraction without a worked precedent: it is exactly
how \ANine{} \citep{ArisakaA9SM} fixed the Standard Model.  There
each parameter is a coordinate of a geometric attractor reached by
this cascade --- the electroweak vacuum selected as the stationary
point of a Wilson--Hosotani holonomy potential (the \GSSB{} step),
the constants read off only after the system has descended onto that
branch.  \ATen{} \citep{ArisakaA10Cosmos} runs the identical
machinery one story up, on the cosmological boundary.  The present
paper runs it at the horizon: formation, accretion, merger, the
Hawking--Page transition, the Page turnover, and the evaporation
endpoint are the \GRIP{} readouts of the same runtime, and the
reader will watch the two-regime grammar sort every one of them into
its place in Parts~IV--VI (\S\ref{sec:bh-way-station},
\S\ref{sec:branch-switch}) --- the sorting itself is new.

The electroweak paper also identified Wilson-line holonomy as the
cleanest prototype of geometric spontaneous symmetry breaking
(\citealp{ArisakaA9SM}; the runtime form is \citealp{ArisakaA6Gi4}).
The black-hole paper sharpens that lesson into an identity: the same
Wilson-line holonomies on $S^1_\theta$ that serve the runtime as
branch variables are, at a horizon, the very degrees of freedom
whose traced entanglement is the Bekenstein--Hawking entropy
(Part~V).  The branch variable and the state count are one geometry.

The whole inheritance can be stated once, as a single descent
ladder:
\[
\BIsix{} \;\Longrightarrow\; \TGG{} \;\Longrightarrow\; \CCI{} \;\Longrightarrow\; \GFF{} \;\Longrightarrow\; \GRIP{}.
\]
Read left to right: the anomaly-locked bulk fixes the legal
ultraviolet stage; \TGG{} packages the total closed ledger; \CCI{}
hands the four-dimensional observer a local boundary conservation
law; \GFF{} ranks the admissible branches of that boundary theory;
and \GRIP{} fixes the order in which they are explored, selected,
and stabilized.  The rule this ladder enforces is stated and proved
as the inheritance theorems of the cosmology companion (T-Cosmos-1
and T-Cosmos-2 of \citealp{ArisakaA10Cosmos}); the present paper
imports the rule rather than restating the theorems: \emph{a
black-hole source term is admissible only if it survives each layer
of the ladder in order, and no continuous dial may be introduced
along the way}.  Figure~\ref{fig:derivation-tree-bh} traces this
descent for the black-hole observables of this paper, with each
output carrying its register.

\begin{figure}[p]
\centering
\resizebox{0.902\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily\small,
  every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
  inp/.style={fill=blue!10, draw=blue!60!black, line width=0.6pt,
              minimum width=52mm, minimum height=15mm},
  mach/.style={fill=violet!10, draw=violet!60!black, line width=0.6pt,
               minimum width=30mm, minimum height=13mm},
  derived/.style={fill=green!8,  draw=green!60!black, solid,
                  line width=0.9pt, minimum width=33mm, minimum height=15mm},
  dcond/.style={fill=orange!10, draw=orange!75!black, dashed,
                line width=0.9pt, minimum width=33mm, minimum height=15mm},
  openn/.style={fill=black!4, draw=black!55, dotted,
                line width=1.0pt, minimum width=33mm, minimum height=15mm},
  tierlab/.style={font=\large\bfseries, text=black!70, align=left,
                  fill=white, inner sep=2.5pt, rounded corners=1pt},
  legbox/.style={rounded corners=2pt, inner sep=4pt,
                 draw=black!40, fill=black!2, line width=0.4pt},
  arr/.style={-{Stealth[length=2.4mm]}, line width=0.5pt, draw=black!65},
  arrm/.style={-{Stealth[length=2.6mm]}, line width=0.8pt, draw=violet!55!black},
  arrout/.style={-{Stealth[length=2.2mm]}, line width=0.45pt, draw=black!45},
]
\node[inp] (GU5) at (-58mm, 0) {%
  \textbf{GU-5 dimensional constants}\\[1pt]
  $\{c,\ G_N,\ \hbar;\ k_B,\ Q_B\}$\\[1pt]
  \small the Geometric Universal Five};
\node[inp] (OGN) at (0mm, 0) {%
  \textbf{OGN-5 integer ledger}\\[1pt]
  $(M,N,L;J,K)=(1,3,5;7,35)$\\[1pt]
  \small proved, no free dials};
\node[inp] (TOP) at (58mm, 0) {%
  \textbf{Structural constants}\\[1pt]
  $\pi,\quad e$\\[1pt]
  \small fixed by geometry};
\begin{scope}[on background layer]
  \node[fit=(GU5)(OGN)(TOP), inner sep=4mm,
        fill=blue!4, rounded corners=3pt, draw=blue!60!black, line width=0.4pt]
        (band1) {};
\end{scope}
\node[mach] (GG6)   at (-72mm, -42mm) {\textbf{\GGtheory{}\,/\,GG-6}\\[1pt]
                                       \small 6D bulk};
\node[mach] (BI6)   at (-36mm, -42mm) {\textbf{\BIsix{}}\\[1pt]
                                       \small anomaly-free closure};
\node[mach] (GDOS)  at (0mm,   -42mm) {\textbf{\GDOS{}}\\[1pt]
                                       \small open-system\\boundary law};
\node[mach] (GRIP)  at (36mm,  -42mm) {\textbf{\GRIP{}}\\[1pt]
                                       \small runtime\\cascade};
\node[mach] (OGGEP) at (72mm,  -42mm) {\textbf{\OGGEP{}}\\[1pt]
                                       \small bulk-to-boundary\\projection};
\begin{scope}[on background layer]
  \node[fit=(GG6)(BI6)(GDOS)(GRIP)(OGGEP), inner sep=3.5mm,
        fill=violet!4, rounded corners=3pt, draw=violet!60!black, line width=0.4pt]
        (band2) {};
\end{scope}
\draw[arr] (GU5.south) to[bend right=8]  (GG6.north);
\draw[arr] (OGN.south) -- (BI6.north);
\draw[arr] (OGN.south) to[bend left=6]   (GDOS.north);
\draw[arr] (TOP.south) to[bend left=10]  (OGGEP.north);
\draw[arrm] (GG6.east)  -- (BI6.west);
\draw[arrm] (BI6.east)  -- (GDOS.west);
\draw[arrm] (GDOS.east) -- (GRIP.west);
\draw[arrm] (GRIP.east) -- (OGGEP.west);
\node[dcond]   (SBH)  at (-72mm, -82mm) {\textbf{$S_{\rm BH}=A/4\Newton$}\\[1pt]
  \small exact $1/4=4\pi/16\pi$\\\footnotesize T-BH-1, three premises\,(D*)};
\node[derived] (TH)   at (-36mm, -82mm) {\textbf{$T_H=\hbar\kappa/2\pi c k_B$}\\[1pt]
  \small thermal clock face\\\footnotesize T-BH-5, quasi-free\,(D)};
\node[dcond]   (TP)   at (0mm,   -82mm) {\textbf{$t_{\rm Page}$ at $2/3$ life}\\[1pt]
  \small freshness expiry\\\footnotesize L-BH-Fresh\,(D*)};
\node[dcond]   (GSL)  at (36mm,  -82mm) {\textbf{$\Delta S_{\rm gen}=$ rel.\ entropy}\\[1pt]
  \small second law's price\\\footnotesize L-BH-Arrow, class\,(D*)};
\node[dcond]   (DS)   at (72mm,  -82mm) {\textbf{$S_{\rm dS}=A/4\Newton$}\\[1pt]
  \small de Sitter horizon\\\footnotesize C-BH-2, same arena\,(D*)};
\node[dcond]   (KERR) at (-40mm, -106mm){\textbf{$(\ell_{\rm lock}/r_s)^2\sim10^{-46}$}\\[1pt]
  \small clean Kerr exteriors\\\footnotesize lock scale from $K{=}35$\,(D*)};
\node[openn]   (SPEC) at (4mm,   -106mm){\textbf{species sum $\to 1/\Newton$}\\[1pt]
  \small numerical closure\\\footnotesize open work order\,(F-BH-9)};
\draw[arrout] (OGGEP.south) to[bend left=14]  (SBH.north);
\draw[arrout] (GDOS.south)  to[bend right=14] (TH.north);
\draw[arrout] (GRIP.south)  to[bend right=10] (TP.north);
\draw[arrout] (GDOS.south)  to[bend left=8]   (GSL.north);
\draw[arrout] (OGGEP.south) to[bend left=4]   (DS.north);
\draw[arrout] (BI6.south)   to[bend right=10] (KERR.north);
\draw[arrout] (GG6.south)   to[bend right=18] (SPEC.north);
\node[legbox, anchor=north west, align=left] at (34mm, -124mm) {%
  \textbf{Register (border code)}\\[3pt]
  \tikz[baseline]{\draw[green!60!black, solid, line width=0.9pt]
      (0,0) rectangle (5mm,3mm);}\ \,
      \textbf{D} --- Derived (quasi-free class)\\[2pt]
  \tikz[baseline]{\draw[orange!75!black, dashed, line width=0.9pt]
      (0,0) rectangle (5mm,3mm);}\ \,
      \textbf{D*} --- Derived-conditional\\[2pt]
  \tikz[baseline]{\draw[black!55, dotted, line width=1.0pt]
      (0,0) rectangle (5mm,3mm);}\ \,
      \textbf{open} --- registered work order};
\node[tierlab, anchor=south west] at (-88mm, 10mm)  {THE FIXED INPUTS};
\node[tierlab, anchor=south west] at (-88mm, -32mm) {THE MACHINERY};
\node[tierlab, anchor=south west] at (-88mm, -73.5mm) {THE BLACK-HOLE OBSERVABLES};
\end{tikzpicture}%
}
\par\smallskip
\caption[Zero-parameter derivation tree of the black-hole module.]{%
\textbf{The zero-parameter derivation tree, black-hole instance.}
Every black-hole observable of this paper (bottom tier) is traced
back, through the \GGtheory{} machinery (middle tier), to a
\emph{fixed} input ledger (top tier) containing no continuous free
parameters: the GU-5 dimensional constants, the proved \OGNfive{}
integer ledger $(1,3,5;7,35)$, and the structural constants $\pi$
and $e$.  Output nodes are border-coded by the paper's register, and
the colors are not cosmetic.  \emph{Solid green (D):} the thermal
clock face $T_H$, proven at quasi-free grade (T-BH-5).  \emph{Dashed
amber (D*):} the exact coefficient (T-BH-1, conditional on its three
stated premises), the Page time (L-BH-Fresh), the second law's
production (L-BH-Arrow, exact in its exchange class; the Wall class
open), the de~Sitter same-theorem entropy (C-BH-2), and the
lock-scale suppression of Kerr corrections.  \emph{Dotted gray
(open):} the numerical species closure --- the module's registered
internal kill row (F-BH-9), a work order, not a result.  The full
grades, premises, and exposures are audited row by row in the
named-statement deck of the appendices.}
\label{fig:derivation-tree-bh}
\end{figure}

\section{What this buys the black hole}

Pulling the four ideas together gives the discipline that governs
the rest of the paper.  The bulk supplies one geometry; the boundary
law makes ordinary general relativity --- and, at a horizon, the
forced trace --- a projection of it; the integer ledger leaves no
continuous dials; and the runtime selects which lawful history
happens.  The single operative consequence is a rule with no
exceptions: \emph{every black-hole source term must descend the
inheritance chain} --- bulk, projection, ledger, runtime --- and no
new dial may be introduced along the way.  A quantity that cannot be
placed on this chain is not derived; it is merely appended, and is
inadmissible by construction.

Table~\ref{tab:problem-to-ledger-bh} makes the discipline concrete,
mapping the open problems of the next Part to the inherited
components most relevant to each.  It is a planning map, not a
victory lap: it shows where each answer must come from, not that
each answer is already complete.

\begin{table}[H]
\centering
\caption[Working map from the horizon's open problems (Part~III) to the most relevant inherited.]{Working map from the horizon's open problems (Part~III) to
the most relevant inherited components from the upstream papers.  A
planning table for the Parts that follow, not a claim that every
problem is already solved. Nothing in the second column is derived here. Every entry names a component that already exists in an upstream paper, at the grade that paper assigns it, and the table's whole content is the pairing. A reader who wants to know whether a given problem is closed should read the Part named in the first column rather than trust this row.}
\label{tab:problem-to-ledger-bh}
\small
\begin{tabular}{p{0.30\textwidth}p{0.60\textwidth}}
\toprule
\textbf{Problem (Part~III)} & \textbf{Most relevant inherited inputs} \\
\midrule
The information paradox & \OGGEP{} port semantics; \GDOS{} open-system
evolution; bulk unitarity (AX-1) \\
The Bekenstein--Hawking $1/4$ & fully induced $\Newton$ (\GGEP{});
canonical scale profile; heat-kernel/replica machinery (Part~XII) \\
What the microstates are & compact fiber $S^1_\theta$; Wilson-line
holonomy; the \AEight{} hidden-fiber picture \\
The Page curve mechanism & \GFF{} ledger and branch competition;
Port-Freshness Clause (\AFive{}); \GRIP{} two-regime grammar \\
Firewalls and the interior & bulk custody of entanglement; the
observer-indexed dictionary (\S\ref{ssec:observer-vacuum}) \\
The second law's direction & the constitutional arrow (\AFive{});
relative-entropy accounting; KMS clock face (T-BH-5) \\
The singularity & 6D smoothness of the projection's coordinate
pathology (interpretive grade) \\
The universe's own horizon & observer-anchored port; same-arena
clause; the \ATen{} cosmological boundary \\
\bottomrule
\end{tabular}
\end{table}

Inheritance constrains, but it does not by itself prove; so every
entry in what follows carries its register --- derived,
derived-conditional, model-level, open, conceded --- said in those
words at the claim and audited in the named-statement deck of the
appendices.  The border colors of
Figure~\ref{fig:derivation-tree-bh} are exactly these registers, and
they are not cosmetic: they are the difference between a result and
an intention.

There is a discipline hidden in this inheritance, and it deserves a
plain statement.  Because every tool used here was built --- and
paid for --- in the particle-physics and cosmology companions, the
black-hole module is never free to invent what it needs: whatever
explains the horizon must already have explained the electron.  That
is why the module can be killed from outside black-hole physics
(Part~VIII), and why its cleanest prediction is silence --- clean
Kerr exteriors wherever instruments can see.

The framework is now in hand.  The bulk, its boundary projection,
the zero-parameter ledger, and the open-system runtime are
everything the black-hole module inherits and everything it is
allowed to use.  The open problems of the horizon --- stated first
in the field's own terms, then translated into this vocabulary ---
are the business of the next Part.

%% ===================================================================
%% PART -- THE OPEN PROBLEMS
%% ===================================================================

\section[Part~II in plain English]{Part~II in plain English: what the black-hole module may use, and what it is forbidden to invent}
There is almost nothing about black holes in this Part, and the
omission is the point.

Any framework can be made to explain a horizon if it is allowed to add
whatever the horizon needs. This one is not allowed. The six-dimensional
bulk, the projection that produces four-dimensional physics, the integer
ledger that fixes the field content, and the open-system runtime that
selects among branches were all fixed in the companion papers --- for the
electron, for the Standard Model, for the cosmos. The black-hole module
inherits them entire and adds no postulate and no parameter of its own.

That constraint cuts both ways, and the Part says so rather than only
claiming the flattering half. It is what makes the later results worth
something: a coefficient derived from machinery built for other purposes is
a prediction, while the same coefficient derived from machinery built to
produce it is a restatement. It is also what makes the module fragile in an
unusual way, because it can be killed from outside black-hole physics
altogether. A collider result or a single proton decay would retire it
without any telescope being involved.

The cleanest thing the inheritance buys is negative. Nothing in the
framework permits a departure from Kerr outside the horizon, so wherever
instruments can currently look, the module's prediction is silence. That is
a weak prediction and a badly exposed one, and the Parts that follow do not
try to make it sound like more than it is.

\GGPart[The open problems]{The Open Problems}{Information Paradox, Page Curve, Firewalls,
and the Entropy-Counting Bar}

\section{What the information paradox really asks}

The information paradox, in its most refined form, is the
following question: Given two theories --- gravity (classical
GR) and quantum mechanics --- that disagree about the
long-time behavior of a pure state subjected to gravitational
collapse, which theory is correct, and what does the resolution
tell us about quantum gravity?

Classical GR predicts information loss: the pure initial state
collapses into a black hole whose subsequent evolution is
determined only by mass, charge, and angular momentum
(no-hair). Any internal structure is invisible from outside.
When the black hole evaporates via Hawking radiation, there
is no mechanism in GR by which the internal information could
be recovered.

Quantum mechanics predicts information preservation: closed
systems evolve unitarily; pure states remain pure; the
entanglement entropy follows the Page curve.

The question is: which theory breaks down, and where?

\subsection{Possible resolutions}

Four classes of resolution have been seriously proposed over
fifty years, and it is worth walking them in the order history
did.  The first is Hawking's own 1976 position: information is
simply \emph{lost} --- quantum gravity is fundamentally
non-unitary, and unitarity must be abandoned.  The second holds
that information is \emph{preserved through small corrections}
to Hawking's calculation: subtle quantum-field-theory-in-curved-%
spacetime effects that encode the information in radiation
correlations.  The third is \emph{holography}: a complete
quantum-gravity theory stores the bulk degrees of freedom on a
boundary, so that unitarity is manifest --- as it is in
\AdSCFT{}.  And the fourth invokes \emph{new ultraviolet
physics at the horizon itself}: firewalls, fuzzballs, or
remnants that modify the standard picture exactly where it was
thought safest.

Fifty years have sorted the four unevenly.  The first position
has been abandoned: evidence from \AdSCFT{} and related
frameworks strongly suggests unitarity holds.  The second is
demonstrably insufficient --- Mathur's 2009 theorem, taken up
just below, shows that no small correction to Hawking's
calculation can produce the Page curve \citep{Mathur2009}.  The
fourth is logically possible but violates the equivalence
principle, and is widely considered unattractive for exactly
that reason.  The third --- holography --- is the current
consensus, with the islands formula providing a concrete
mechanism.

\GGtheory{} provides a specific realization of the third
position, grounded in a complete 6D bulk theory rather than an
abstract \AdSCFT{} duality.

\subsection{Mathur's theorem}

Samir Mathur's 2009 paper \citep{Mathur2009} proved that
\emph{any} small perturbation of the Hawking calculation
--- including corrections from standard QFT in curved
spacetime, arbitrary local modifications at the horizon, etc.
--- cannot produce the Page curve unless the corrections are of
order unity. This is Mathur's ``small corrections theorem.''

The implication: whatever resolves the information paradox
must involve \emph{large} deviations from the standard Hawking
picture. These deviations must be non-local, or non-perturbative,
or geometric-at-the-horizon, or some combination. The islands
formula realizes the ``non-perturbative geometric at the
horizon'' option. \GGtheory{} realizes a ``non-local from the
bulk side of \OGGEP{}'' option, which is equivalent to the
islands formula but physically more concrete.

\section{The Bekenstein--Hawking entropy counting problem}

\subsection{How many microstates?}

If $\Sbh = A/(4\Newton\hbar)$ is the von Neumann entropy of
the black hole, then the number of microstates is
$e^{\Sbh/\kB}$. For a solar-mass black hole
($M = \Msun = 2 \times 10^{30}$~kg), this is approximately
$e^{10^{77}}$ microstates. That is an enormous number.

What are these microstates?  In ordinary thermodynamics the
question has a mundane answer: microstates are specific
configurations of molecules, atoms, or spins, and the entropy is
the logarithm of the number of configurations compatible with a
given macrostate --- a gas has entropy because there are many
ways to arrange its molecules.  But a black hole seems to offer
nothing to arrange.  By no-hair, the macroscopic observables are
$(M, J, Q)$; and the same theorem says the horizon geometry
depends on these parameters and nothing else.  Where, then, do
the microstates live?  The tension is stark: the no-hair theorem
says a black hole has no features; the entropy formula says it
has $e^{10^{77}}$ of them.

This is the \emph{black-hole microstate counting problem}.
Different approaches give different answers:

\paragraph{Strominger-Vafa 1996.} Andrew Strominger and
Cumrun Vafa \citep{StromingerVafa1996} showed in 1996 that
for certain \emph{extremal} string-theory black holes, the
microstates can be counted as string configurations with
specific brane charges. For the 5D extremal black hole they
considered, the count was
$e^{S}$ microstates with
$S = 2\pi\sqrt{Q_1 Q_5 n}$, matching the Bekenstein--Hawking
area law exactly. This was celebrated as the first microscopic
derivation of the Bekenstein--Hawking formula and became a
milestone in string theory.

Limitations: (i) only extremal black holes (zero temperature);
(ii) only in $d > 4$ dimensions; (iii) only in specific
string-theory backgrounds; (iv) the microstates are
\emph{configurations in the dual theory}, not configurations
of the black hole itself.  The pattern is worth naming: the count
works in a corner protected by supersymmetry, which guarantees the
number of states cannot change as the coupling is dialed from the
calculable regime to the black-hole regime.  Remove the protection
--- ask about the Schwarzschild or Kerr holes actually in the sky
--- and the method loses its footing; a quarter century of effort
has extended it to \emph{near}-extremal cousins, never to the
generic astrophysical hole.

\paragraph{Loop quantum gravity counting.} LQG (Ashtekar-Baez,
Rovelli-Smolin 1996-1998) counts microstates as spin-network
states on the horizon. The count gives $\Sbh = A/(4\Newton\hbar)$
up to a Barbero-Immirzi parameter $\gamma$, which is fixed by
matching the LQG calculation to the Bekenstein formula ---
but this is a consistency check, not a derivation.

\paragraph{Fuzzball proposal.} Samir Mathur has argued for
nearly two decades that black-hole microstates are specific
horizonless geometries --- ``fuzzballs'' --- that differ from
each other on scales down to the Planck length. In this
picture, there is no horizon at all; the region inside the
``fuzzball radius'' is a quantum superposition of horizonless
configurations.

\paragraph{\GGtheory{} counting.} In \GGtheory{}, microstates
are 6D bulk configurations compatible with fixed 4D boundary
data $(M, J, Q)$. Explicitly, a microstate is specified by:
(i) the 4D boundary values of the metric, the gauge field,
and the Higgs field; (ii) the radial profile of these fields
in $I_r$; (iii) the Wilson-line holonomy $\Theta_H$ on
$S^1_\theta$. The 4D observer sees only the zero-mode
components, which are fixed by $(M, J, Q)$. The microstate count is the entanglement entropy of these traced
fiber degrees of freedom; by the induced-gravity argument of
\S\ref{ssec:V-conical} it equals $A/4\Newton$ exactly --- about
$e^{A/(4\Newton\hbar)/\kB}$ states at leading order.

This is derived in full in \S\ref{ssec:V-strategy}--\ref{ssec:V-honest}.

\subsection{The species problem}

If there are $N$ species of light fields in nature, a simple
counting argument suggests that black-hole evaporation should
produce them in proportion, giving more entropy per unit area
than $1/(4\Newton\hbar)$. This is the \emph{species problem}:
why is the coefficient $1/4$ and not $N/4$?

In the standard treatment the resolution is species-bound
reasoning \citep{DvaliRedi2009}: $N$ light species lower the
gravitational cutoff to $\Lambda_{\rm sp}=\Mpl/\sqrt{N}$ and
renormalize Newton's constant, so the area coefficient stays
$1/4$ once expressed in the infrared $\Newton$.

In \GGtheory{} the species problem does not arise in the first
place, because the coefficient is \emph{not} a species count at
all: Theorem~\ref{thm:exact-quarter} fixes $1/4 = 4\pi/16\pi$ as
the regulator- and species-\emph{independent} ratio of the
conical-deficit curvature normalization to the
Einstein--Hilbert normalization (\S\ref{ssec:V-conical}).  The
\OGNfive{} ledger $(M,N,L;J,K)=(1,3,5;7,35)$ --- with $N=3$ the
generation count \citep{ArisakaP2N3} and $L=5$ fixing the \SUfive{} rank --- governs
the field \emph{content} (and hence the induced value of
$\Newton$ through the species sum of
Appendix~\ref{app:species-sum}), but it does \emph{not} enter the
$1/4$.  What the older species-bound argument had to engineer,
the induced-gravity derivation makes automatic.

\section{The singularity: what's inside?}

The classical black-hole interior contains a curvature
singularity: a locus where the Kretschmann scalar diverges,
tidal forces become infinite, and the classical theory breaks
down. This is where quantum gravity is needed --- the
Planck-scale physics that GR cannot resolve.

The singularity inside the black hole is not observable from
outside (causal disconnection by the horizon), but it affects
the interpretation of the information paradox. If the
singularity is where information ``ends up,'' and information
is preserved, then the singularity must either: (i) not be a
true endpoint, and information escapes via some non-classical
channel; or (ii) be replaced by a smooth geometry in the full
theory.

\subsection{Singularity replacements}

The proposals on the table span half a century of styles, and
each removes the singularity by removing something different.  A
\emph{big bounce} removes the endpoint: the singularity is
replaced by a non-singular transition that connects the
black-hole interior to another universe or region (Penrose 1965;
Ashtekar--Bojowald 2006).  The \emph{fuzzball} removes the
interior altogether: a quantum superposition of horizonless
geometries extends up to a fuzzball radius $r_f > r_s$, and
there is no singularity because there is no inside (Mathur
2008).  The \emph{firewall} removes the crossing: the horizon
itself becomes a high-energy quantum wall, and the interior is
never reached because infalling observers burn up at the wall
(AMPS 2012).  The \emph{quantum-extremal-surface} reading of
the islands program removes the interior's independence: for a
boundary observer the apparent interior is a bulk geometric
structure redundantly encoded on the boundary (2019--2020).  And
the \GGtheory{} reading, developed in this paper, removes only
the \emph{observability}: the 4D interior is a radial-coordinate
region where the scale profile $e^{-r}$ is small, and the
``singularity'' is the limit in which that region is traced over
by \OGGEP{}, giving the boundary description an apparent
termination.

In \GGtheory{}, the black-hole interior is \emph{real}, unitary,
and finite --- it just cannot be observed from outside by the
4D observer because those directions are traced over by
\OGGEP{}. The apparent singularity is a coordinate artifact of
the 4D projection, analogous to the apparent singularity at
$r = r_s$ in Schwarzschild coordinates before Finkelstein
regularized it.

\section{Firewalls revisited}

\subsection{The AMPS argument in detail}

Let $B$ be a late-time Hawking quantum, $A$ its interior
partner, and $R$ the earlier radiation.  The postulates of
black-hole complementarity (Susskind--Thorlacius--Uglum 1993)
are four.  The first is \emph{unitarity}: Hawking radiation is
the unitarily evolved pure state, so $B$ is entangled with $R$.
The second is that \emph{quantum field theory in curved
spacetime works at the horizon}: so $B$ is entangled with its
partner $A$ on the other side.  The third is \emph{no-drama}:
an infalling observer crossing the horizon sees nothing
unusual.  And the fourth is that the \emph{black-hole entropy
equals the Bekenstein--Hawking value}.

At late times (past Page), postulate 1 says $B \otimes R$ is
an approximate Bell pair. Postulate 2 says $A \otimes B$ is a
Bell pair. But $B$ cannot be maximally entangled with both $A$
and $R$ (monogamy of entanglement). One of the four postulates
must fail.

AMPS argued that postulate 3 (no-drama) is the weakest and
most likely to fail. Infalling observers crossing the horizon
would encounter a \emph{firewall}: highly-energetic quanta
that burn them up. This violates the equivalence principle
(which is what no-drama is based on) but preserves unitarity,
QFT-in-curved-spacetime, and the Bekenstein entropy.

\subsection{The \GGtheory{} resolution}

In \GGtheory{}, postulate 2 is what fails --- but in a
subtle way that preserves physical smoothness at the horizon.

The point is: $B$ is not entangled with a partner $A$ that
exists independently inside the horizon. The ``partner $A$'' is
part of the bulk degrees of freedom that are traced over by
\OGGEP{}. Once we recognize that 4D ``interior'' is a partial
description of the 6D bulk, the apparent bipartite $A$-$B$
entanglement dissolves: $B$ is entangled with the full bulk
state, which is not separately identified with a 4D ``interior
observer.''

The infalling observer crosses the horizon and accesses more of
the 6D bulk --- specifically, regions with smaller scale profile.
They encounter no firewall; they simply move to a region
where the \OGGEP{} projection would apply to a larger bulk
sector.

In the language of the four postulates: unitarity is preserved,
at the 6D level; quantum field theory near the horizon is
preserved, with the one amendment that the ``interior'' is not
an independent 4D sector; no-drama is preserved --- the horizon
is smooth; and the entropy remains the Bekenstein--Hawking
value.  Nothing on the list is sacrificed.  The apparent
violation of monogamy is an artifact of the 4D-only
description; at the 6D bulk level, everything is consistent.

\section{The Page curve derivation}

The derivation of the Page curve remains one of the most
important unsolved problems in standard \AdSCFT{} \emph{for
asymptotically-flat Schwarzschild black holes}. In 2D JT
gravity coupled to matter, the Penington-Almheiri-et-al.
derivation succeeds. In 4D asymptotically-flat Schwarzschild
--- the actual astrophysical case --- the derivation is
incomplete.

In \GGtheory{}, the Page curve is automatic. It does not
require a separate derivation; it follows from bulk unitarity
plus \CPTP{} projection. Part~VI develops this explicitly.

\section{The state of play}

As of this writing --- a dated snapshot, refreshed at each
version of record --- the consensus position in the theoretical
physics community can be stated in five sentences.
\emph{Information is preserved}: Hawking's 1976 proposal that
it is fundamentally lost is now considered incorrect, and the
radiation must encode the initial information in correlations.
\emph{The Page curve is the correct entanglement-entropy
curve}: pure-state evolution requires the tent-shaped rise and
fall.  \emph{\AdSCFT{} remains the best-understood setting} in
which to study the problem, with the islands formula delivering
the Page curve in two-dimensional JT gravity.  \emph{The
real-universe case --- four-dimensional, asymptotically flat,
Schwarzschild --- is less well understood}, because its
\AdSCFT{} dual is not known.  And \emph{no observational test
of the information paradox has been performed or is currently
feasible}: the Event Horizon Telescope and LIGO see the
classical geometry; they do not constrain the entanglement
structure of Hawking radiation.

\GGtheory{} enters this situation with a clear claim: the
standard resolution \emph{is essentially correct} but is
couched in an abstract formalism (\AdSCFT{}, replica wormholes)
that is not grounded in a physically realistic ultraviolet
theory. \GGtheory{} provides the missing UV grounding ---
replacing the abstract \AdSCFT{} duality with a concrete 6D bulk for the \emph{real} universe --- and with it,
the information paradox is resolved at the level of physical
geometry, not formal duality.

The rest of this paper develops that claim explicitly.

\clearpage

%% ===================================================================
%% PART -- THE HORIZON AS PORT (split of the former GDOS-reading Part)
%% ===================================================================

\section[Part~III in plain English]{Part~III in plain English: the four things nobody has settled, stated in the field's own words first}
Four problems, and this Part states each of them the way the field
states it before translating any of it into this framework's vocabulary.

Where does the information go. If the hole radiates thermally and then
evaporates, a pure state has become a mixed one, and quantum mechanics does
not permit that. Mathur's theorem sharpened the puzzle into a dilemma:
small corrections to Hawking's calculation cannot repair it, so something
has to change by order one.

What are the microstates. The entropy formula counts something --- the
exponential of a quarter of the horizon area --- and no calculation in the
real universe has said what. String theory counts them in supersymmetric
cases. Nothing counts them for the hole at the center of this galaxy.

What is inside. The singularity is the place where general relativity says
of itself that it has stopped applying, and every proposal to replace it is
a proposal about physics nobody has tested.

Is the horizon smooth. The firewall argument says you cannot have
unitarity, an undramatic horizon and ordinary field theory outside it all
at once. Something has to give, and fourteen years of argument have not
settled which.

The Part's own verdict is worth restating plainly, because it is not the
verdict a new framework usually offers. The standard resolution is judged
essentially correct. What it is judged to lack is a grounding in the
physics of the actual universe rather than in a formal duality. That is a
smaller claim than replacing the standard account, and it is the claim the
rest of the paper has to defend.

\GGPart[The horizon as port]{The Horizon as Port}{The Geometrically Forced Trace, Its Thermal Clock, and the Custody of the Anchor}
\suppressfloats[t]% no top floats on the banner page

\noindent
This is the Part where the machinery meets its object.  Parts~I
and~III posed the horizon's problems in the field's own terms;
Part~II handed us the inherited framework --- a unitary
six-dimensional bulk, a boundary law under which every observer's
physics is an open-system projection, and a runtime that selects
among lawful branches.  Now we apply that framework to the black
hole itself, and the reason for doing so is not that the black hole
is one more application among many.  It is that the black hole is
the framework's \emph{native} object.  The constitution's central
claim is that no physical system is closed and that observation
\emph{is} the bulk-to-boundary projection --- a partial trace over
what the observer cannot access.  Everywhere else in physics, what
gets traced out is a matter of circumstance: an experimenter
decides what to measure and what to ignore, an environment couples
to whatever it happens to touch.  A horizon is different in exactly
one respect, and it is the respect that matters.  Behind a horizon,
the inaccessible sector is not chosen, ignored, or accidental; it
is fixed by the causal structure of spacetime itself.  \emph{The
laboratory trace is chosen; the horizon trace is forced.}  The
black hole is therefore the one place in nature where the
constitution's central operation is performed by geometry rather
than by an observer's convention --- which makes it not merely a
consumer of the \OGGEP{}/\GDOS{} machinery but its cleanest test.

The Part unfolds in five steps.  The opening section makes the
claim precise: the horizon is an \OGGEP{} \emph{port}, and the
claim is registered as a named key (\textsf{L-BH-Port}) with its
grade stated.  The next section computes what flows out of the
port --- Hawking radiation as the output of a completely positive
channel, with the Bogoliubov calculus done by hand.  Two theorems
then close the cases the textbook treatment leaves loose, the
rotating anchor and the collapse channel, at the model level and
with their verifications printed.  The thermal clock face turns
the field's analytic-continuation ``tricks'' into a theorem.  And
the closing section displays the real-bulk standing of everything
the Part has built.  Figure~\ref{fig:oggep-port}, below, is the
whole argument in one picture; it is worth a long look before the
equations begin.

\section{The horizon as an \OGGEP{} port}
\label{sec:horizon-port}

\begin{figure}[tbp]
\centering
\resizebox{0.873\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\sffamily\small, line width=0.80pt]
  \draw[rounded corners, fill=violet!5, draw=violet!50] (-5.1,0.1) rectangle (5.1,3.9);
  \node[violet!55!black, font=\sffamily\bfseries, anchor=west] at (-4.85,3.55) {6D bulk $\mathcal{M}_6$\; (pure, unitary)};
  \fill[black!82] (0,1.6) circle (0.8);
  \node[white] at (0,1.6) {\footnotesize BH};
  \draw[red!70!black, dashed] (0,1.6) circle (0.8);
  \node[red!70!black, font=\sffamily\footnotesize] at (0,0.45) {horizon $=$ \OGGEP{} port};
  \node[gray!65!black, align=center, font=\sffamily\footnotesize, anchor=west] at (2.05,1.6) {interior $+$ non-spatial fiber\\[1pt] $(I_r,S^1_\theta)$ traced};
  \draw[->, orange!80!black] (0.6,2.25) -- (1.45,2.95);
  \draw[->, orange!80!black] (-0.6,2.25) -- (-1.45,2.95);
  \draw[->, orange!80!black] (-0.85,1.7) -- (-1.95,2.1);
  \node[orange!75!black, font=\sffamily\footnotesize, anchor=east] at (-2.05,2.55) {Hawking radiation};
  \draw[->, line width=1.1pt, gray!60!black] (0,0.05) -- (0,-0.95) node[midway,right] {$\pi = \mathrm{Tr}_{\mathrm{in}}$};
  \draw[rounded corners, fill=blue!7, draw=blue!55] (-5.1,-2.75) rectangle (5.1,-1.05);
  \node[blue!55!black, align=center, font=\sffamily\small] at (0,-1.9) {\textbf{4D boundary}: open state $\rho_\partial = \mathrm{Tr}_{\mathrm{in}}\,\rho_6$\; (mixed, \CPTP{})\\[3pt] $\Sbh = S_{\mathrm{vN}}(\rho_\partial)$ --- the entanglement of the traced fiber};
\end{tikzpicture}}
\caption[The horizon as an \OGGEP{} port.]{The horizon as an \OGGEP{} port.  The six-dimensional bulk state is pure and unitary; the horizon is the locus at which \OGGEP{} traces out the inaccessible sector --- the 4D interior together with the non-spatial fiber $(I_r,S^1_\theta)$ --- producing the open, completely positive boundary state $\rho_\partial$ whose output is the Hawking radiation.  The Bekenstein--Hawking entropy is the von Neumann entropy of $\rho_\partial$: the entanglement of the traced fiber, evaluated over the horizon.}
\label{fig:oggep-port}
\end{figure}

\noindent Figure~\ref{fig:oggep-port} draws the contrast just stated: the laboratory trace is chosen; the horizon trace is forced.



\noindent The arena --- the canonical six-dimensional bulk and its
scale profile --- is fixed once, in Part~II (\S\ref{sec:bulk-arena});
we work on it throughout.

\subsection{The horizon is a special locus}

At the horizon, four things happen at once in the 4D
description.  The metric has a coordinate singularity: the
$g^{rr}$ factor vanishes in Schwarzschild coordinates.  The
effective temperature, as measured by a static observer at
infinity, diverges on approach, $T(r)/T_\infty \to \infty$ as
$r \to r_s^+$.  The scale profile in the bulk is strongly
modified by the gravitational field.  And --- the decisive one
--- the 4D observer cannot access the interior $r < r_s$ at
all: it is causally disconnected.  The first three are
symptoms; the fourth is the diagnosis.

In \GDOS{} language, the horizon is a \emph{port}: a specific
locus at which the 4D observer's accessible Hilbert space
$\Hilb_{\rm out}$ is separated from an inaccessible Hilbert
space $\Hilb_{\rm in}$. The \OGGEP{} functor takes the full 6D
state $\rho_{\rm bulk}$ and partial-traces over the inaccessible
portion:
\begin{equation}\label{eq:OGGEP_port}
\rho_\partial^{\rm out} = \Trin(\rho_{\rm bulk}),
\end{equation}
where $\Trin$ denotes tracing over (a) the bulk compact
directions $S^1_\theta$ and the inaccessible radial tail, and
(b) the 4D interior $r < r_s$.  In the laboratory, \emph{which}
degrees of freedom are traced out is a choice --- an experimenter
decides what to measure and what to ignore.  Behind a horizon, the
bulk partners of the modes one sees are not merely ignored; they are
unreachable as a matter of geometry.  The partial trace stops being
the observer's bookkeeping convention and becomes a fact about the
world's causal wiring.

This is the central structural claim of \GDOS{} for black holes:
\emph{the horizon plays the same role as $\Trenv$ in open-system
quantum dynamics. It is a one-way causal boundary that implements
a \CPTP{} projection on the 4D boundary Hilbert space.}

\subsection{The geometrically visible port: black holes as the cleanest test of \OGGEP{}}
\label{ssec:visible-port}

The horizon is not merely \emph{a} port; it is the one place in physics where the \OGGEP{} partial trace becomes \emph{geometrically visible}.  This deserves emphasis, because it is what makes the black hole the sharpest possible test of the open-system constitution \citep{ArisakaA5GDOS}.

In ordinary laboratory quantum mechanics the \OGGEP{} coarse-graining operates everywhere on the boundary $\mathcal{M}_4$ at every moment (\AX{3}), but the trace is \emph{operationally undirected} --- always on, but pointing nowhere in particular.  A skeptic can always object: \emph{in principle}, a sufficiently clever observer could extend her resolution into the bulk modes and recover the full state, so the coarse-graining is a choice, not a law.  At a black-hole horizon that evasion is unavailable.  The interior is causally disconnected from the exterior by the event horizon (\AX{1}, \LC); no cleverness, no future technology, no idealized super-observer can recover the interior modes from outside.  The inaccessibility is built into the causal structure of the spacetime.

\paragraph{The trace becomes ontological.}  The combined boundary state of an exterior observer is therefore
\begin{equation}\label{eq:visible-port}
\rho_\partial^{\rm out}(t) = \Trin\bigl(\rho_{\rm bulk}(t)\bigr),
\end{equation}
where $\Trin$ now traces over \emph{both} the non-spatial fiber $(I_r, S^1_\theta)$ that \OGGEP{} discards everywhere \emph{and} the causally disconnected interior $r<r_s$.  The first trace is the operational coarse-graining of \AX{3}; the second is geometrically forced.  At the horizon the \OGGEP{} coarse-graining stops being merely operational and becomes \emph{ontological} --- a structural feature one can point to.  This is the precise sense in which the black hole is the cleanest empirical test of \AX{3}: were \AX{3} wrong, black-hole physics would have no constitutional account and the information paradox, the firewall, and the Page curve would be permanent puzzles; because \AX{3} holds, all three dissolve at once, as the remainder of this Part shows \citep{ArisakaA5GDOS}.
This paper registers the claim of this subsection under the key
\textsf{L-BH-Port}\label{key:port}: \emph{at a black-hole horizon
the \OGGEP{} trace is forced by the causal structure rather than
chosen by the observer --- the port is ontological, not
operational.}  Grade: Derived, given \AX{1} and \AX{3}; it is the
premise every information-channel result of this Part consumes.

\paragraph{Three settings, one trace.}  The same structure appears in three settings that differ only in \emph{why} the trace is forced (Figure~\ref{fig:three-port}).  In the laboratory the trace is operative but undirected.  At a black-hole horizon it is forced by the causal structure.  At the cosmological (de Sitter) horizon of radius $c/H$ it is forced by cosmic expansion: modes beyond it recede faster than light and are permanently inaccessible.  Black holes and the cosmological horizon are thus the two geometrically-forced strata of one constitutional move --- which is why the same generalized-entropy ledger $\GFF = F_{\rm GEP}+F_{\rm CEP}$ (\S\ref{sec:gff-ledger}) governs both, and why the present black-hole results carry over to de Sitter.  In this sense black holes are the \emph{constitutional Rosetta stones} of the program: the place where an abstract axiom becomes a geometric inevitability.

\begin{figure}[tbp]
\centering
\resizebox{\textwidth}{!}{%
\usebox{\GGscPort}}
\caption[The horizon as the geometrically-visible \OGGEP{} port: three settings.]{The \OGGEP{} partial trace in three settings, adapted from \citep{ArisakaA5GDOS}.  \textbf{Left:} ordinary laboratory --- the trace over the non-spatial fiber operates everywhere but is operationally undirected (\AX{3} operative but invisible).  \textbf{Middle:} black hole --- the trace over interior modes is geometrically forced by the causal structure of the Schwarzschild horizon.  \textbf{Right:} cosmological horizon --- the trace over modes beyond the de Sitter horizon ($r=c/H$) is forced by cosmic expansion.  In the middle and right panels the constitutional content of \AX{3} becomes ontologically visible; the black hole is the cleanest such test.}
\label{fig:three-port}
\end{figure}


\section{Hawking radiation as the \OGGEP{} output}

\subsection{The standard derivation}

In Hawking's 1974 calculation, the emission rate of quanta of
frequency $\omega$ from a Schwarzschild black hole is
\begin{equation}\label{eq:Hawking_spectrum}
\frac{dN}{dt\,d\omega} = \frac{\Gamma_\ell(\omega)}{e^{\hbar\omega/
\kB\THaw} \mp 1},
\end{equation}
where $\Gamma_\ell(\omega)$ is the graybody factor for partial
wave $\ell$, and the $\mp$ is Bose or Fermi for each particle
type. The spectrum is thermal to all orders in the saddle-
point approximation of the gravitational path integral.

Hawking's derivation uses QFT-in-curved-spacetime: the vacuum
state of a quantum field near the horizon is not the Minkowski
vacuum; it is thermally populated in the observer's frame. This
is closely analogous to the Unruh effect for accelerated
observers.

\subsection{\GDOS{} re-derivation}

In \GDOS{}, the Hawking radiation is the output of the
\OGGEP{} channel $\mathcal{D}_{\rm BH}: \rho_{\rm bulk} \mapsto
\rho_\partial^{\rm out}$, which is \CPTP{}, and the thermal
spectrum follows in three moves.  The bulk state near the
horizon is approximately the Hartle--Hawking vacuum, which is
thermal in the local-observer frame at inverse temperature
$\beta_{\rm H} = 2\pi/\kappa$, with $\kappa$ the surface
gravity.  The \OGGEP{} functor partial-traces over the
interior, leaving a reduced state on the exterior.  And for a
canonical pair of bulk modes $(\hat a_{\rm in},
\hat a_{\rm out})$ related by a Bogoliubov transformation,
that reduced state on $\hat a_{\rm out}$ is thermal at
temperature $\THaw$.

This is essentially Hawking's argument, reformulated in the
\GDOS{} language \citep{ArisakaA5GDOS}. The new content is that \emph{the
thermal character is automatic from the \CPTP{} structure};
it does not need to be derived anew for each metric. Any bulk
geometry with a horizon (Schwarzschild, Kerr, Reissner-Nordstr\"om,
de Sitter, etc.) produces a thermal output spectrum at the
appropriate temperature.

\subsection{The Hawking temperature}

The Hawking temperature is the scale-profile-determined
temperature at the horizon:
\begin{equation}\label{eq:Hawking_T_GDOS}
\kB \THaw = \frac{\hbar\kappa}{2\pi c}.
\end{equation}
In \GGtheory{}, this follows from the Unruh--Tolman relation
(developed in the companion \GGtheory{} thermodynamics paper):
$T(r_\ast) = T_{\rm bulk}\,e^{\,r_\ast}$, the gravitational
blueshift set by the canonical scale profile $e^{-2r}$ of the bulk. For a
Schwarzschild black hole, the scale profile near the horizon is
$\sqrt{g_{tt}} = \sqrt{1 - r_s/r}$, and the divergence at $r_s$
is compensated by the acceleration of the proper-distance-from-
horizon observer, giving a finite Hawking temperature at
infinity.

The numerical match is automatic; \GGtheory{} does not need
to add new physics at the horizon. The standard Hawking
calculation is the zeroth-order \GDOS{} prediction.

\subsection{The explicit Bogoliubov derivation: why the trace is thermal}
\label{ssec:bogoliubov}

The preceding subsections established that the thermal spectrum is the automatic output of the \OGGEP{} channel.  Because a black-hole referee will want the mode-level computation, we give it explicitly: the thermality is \emph{derived}, not posited, and the derivation makes plain that it is the same partial trace that produces mixedness everywhere in the framework.

\paragraph{Two mode bases and their mismatch.}  Expand the bulk free field on $\mathcal{M}_6$ near the horizon in two natural bases --- the ``in'' modes $\{u^{\rm in}_\omega\}$ regular on past null infinity, and the ``out''/horizon modes $\{u^{\rm out}_\omega\}$ natural to the static exterior.  Because the horizon is a Killing bifurcation surface the two bases differ, related by a Bogoliubov transformation
\begin{equation}\label{eq:bogo}
a^{\rm out}_\omega = \int d\omega'\,\bigl(\alpha^*_{\omega\omega'} a^{\rm in}_{\omega'} - \beta^*_{\omega\omega'} a^{\rm in\dagger}_{\omega'}\bigr),
\end{equation}
and a non-zero $\beta$ means the in-vacuum is populated in the out-basis.

\paragraph{The Planck factor.}  Matching positive-frequency modes by analytic continuation through the complexified horizon (the Damour--Ruffini/Wald prescription \citep{Wald1984} --- a step that is licensed, not assumed: Theorem~\ref{thm:thermal-clock}(ii)) gives the universal ratio
\begin{equation}\label{eq:planck}
\Bigl|\frac{\beta_\omega}{\alpha_\omega}\Bigr| = e^{-\pi\omega/\kappa}
\qquad\Longrightarrow\qquad
\langle N_\omega\rangle = |\beta_\omega|^2 = \frac{1}{e^{2\pi\omega/\kappa}-1} = \frac{1}{e^{\hbar\omega/k_B T_{\rm H}}-1},
\end{equation}
with surface gravity $\kappa$ and $k_B T_{\rm H} = \hbar\kappa/2\pi c$ --- the Planck spectrum \citep{Hawking1975, Unruh1976}.  (Mode exponents throughout this Part are written in geometric units, $e^{2\pi\omega/\kappa}$; SI factors of $c$ are restored only in temperature statements, via $\kappa_{\rm acc} = c^{2}\kappa_{\rm geom}$.)

\paragraph{The squeezed-state structure and the \OGGEP{} trace.}  The content of the Bogoliubov mismatch is that the global Unruh vacuum is a two-mode squeezed state pairing each exterior mode with an interior partner across the horizon,
\begin{equation}\label{eq:squeezed}
|0\rangle_U = \prod_\omega \frac{1}{\cosh r_\omega}\sum_{n=0}^\infty (\tanh r_\omega)^n\,|n\rangle^{\rm out}_\omega \otimes |n\rangle^{\rm in}_\omega,
\qquad \tanh r_\omega = e^{-\pi\omega/\kappa}.
\end{equation}
The \OGGEP{} partial trace over the causally inaccessible interior partners then produces the exterior state by \emph{direct computation}:
\begin{equation}\label{eq:thermal-trace}
\rho_{\rm out} = \mathrm{Tr}_{\rm in}\,|0\rangle_U\langle 0|_U
= \prod_\omega \bigl(1-e^{-\hbar\omega/k_BT_{\rm H}}\bigr)\,e^{-\hbar\omega N_\omega/k_BT_{\rm H}}
= \frac{e^{-H_{\rm out}/k_BT_{\rm H}}}{Z}.
\end{equation}
This is the explicit statement that Hawking thermality \emph{is} the \OGGEP{} partial trace of a pure two-mode-squeezed bulk state --- the same operation that yields a mixed boundary state from a pure bulk everywhere in the cohort (\S\ref{ssec:visible-port}), here with the trace geometrically forced.  The observed flux carries the graybody transmission $\Gamma_\ell(\omega)$ of the exterior curvature barrier,
\begin{equation}\label{eq:greybody}
\frac{dN_\omega}{d\omega\,dt} = \frac{\Gamma_\ell(\omega)}{e^{\hbar\omega/k_BT_{\rm H}}\mp 1}.
\end{equation}

\paragraph{Why every horizon radiates.}  Nothing in this computation used the Schwarzschild metric; it used only that a horizon induces a Bogoliubov mismatch with a single scale $\kappa$.  Hence Schwarzschild, Kerr, Reissner--Nordstr\"om, and the de Sitter cosmological horizon all radiate thermally at $T = \hbar\kappa/2\pi c k_B$, each with its own surface gravity --- a fact we use in the next subsection and in the cosmological-horizon treatment.  The pedigree of every analytic step used here --- why the positive-frequency selection is a derived clock, why the continuation through the complexified horizon is lawful, and why the trace is exactly the thermal state of that clock --- is recorded as Theorem~\ref{thm:thermal-clock} (T-BH-5), once the rotating and collapse cases have been brought into the same frame (\S\ref{sec:thermal-clock}).

\section{The rotating anchor and the collapse channel: two theorems at the model level}
\label{sec:rotating-collapse}

The vacuum story of this Part has so far been told for static
exteriors: the Killing frame is globally timelike, the positivity
selection fixes the exterior complex structure uniquely, the vacuum
multiplicity (Boulware, Unruh, Hartle--Hawking) is a port assignment at
the horizon, and thermality is the \GDOS{} partial trace itself ---
the same statement the cosmology companion makes in general form:
observer-adapted vacua are \emph{dictionary} data, and a change of
dictionary is a Gaussian channel acting on the boundary description,
not an ambiguity in the bulk state \citep{ArisakaA10Cosmos}. Two
loose ends remain, and both are genuine: \emph{rotation}, where the
ergoregion makes the co-rotating Killing energy indefinite and the
literature's exact result is that the Kerr exterior admits no
Hadamard state invariant under the horizon Killing flow --- no
Hartle--Hawking analogue \citep{KayWald1991}; and
\emph{collapse}, where nothing is stationary and no Killing selection
exists during infall. This section closes both at the model level ---
in the same custody pattern as the horizon port --- as entries R14 and
R15 of the cohort's real-bulk series (\citealp{ArisakaA8QM}; the
full-length member-paper treatment of the series is \PaperPFour{},
\citealp{ArisakaP4Real}). The two statements are verified
numerically in the program's verification suite, available with the
source through the program's site (www.arisaka-gg.org), and the honest
regime declarations are part of the statements.

\setcounter{euclidtheorem}{2}% key alignment: T-BH-n = Theorem n
\begin{euclidtheorem}[The rotating anchor \textnormal{(T-BH-3)}]
\label{thm:rotating-anchor}
In the superradiance model class --- asymptotic modes of positive
energy coupled to ergoregion modes of negative co-rotating energy
through the standard quadratic (two-mode-squeezing) interaction:
(i) the co-rotating quadratic form is indefinite, so the positivity
selection of the exterior complex structure is unavailable in the
rotating frame --- the model form of the Kay--Wald obstruction;
(ii) the asymptotic frame's quadratic form is positive, and its polar
phase selects the asymptotic $J$ uniquely --- the anchor moves to
spatial infinity, where every instrument that will ever report a
measurement actually sits;
(iii) read through that fixed asymptotic $J$, the rotating dynamics is
an exactly symplectic Gaussian channel which amplifies populations at
the superradiant rate ($\langle n \rangle = \sinh^{2}(gt)$ at
resonance) while leaving $J$ untouched: superradiant energy extraction
is \GDOS{} pumping, exactly as horizon thermality is \GDOS{} tracing,
and the rotating vacuum multiplicity is port assignment relative to
the asymptotic anchor.
\end{euclidtheorem}

\begin{proof}
(i) In the rotating frame the superradiant band contributes modes of
co-rotating energy $\omega - m\Omega_H < 0$ \citep{Starobinsky1973,
Unruh1974}, so the quadratic form carries both signs; positivity ---
the hypothesis of the uniqueness selection --- fails, and no invariant
complex structure with positive energy exists there. In the two-mode
model the form is
$\omega_1(x_1^2 + p_1^2)/2 - \omega_2(x_2^2 + p_2^2)/2
+ g(x_1 x_2 - p_1 p_2)$, indefinite for all $g$. (ii) At spatial
infinity the frame is non-rotating and the form
$\omega_1(x_1^2+p_1^2)/2 + \omega_2(x_2^2+p_2^2)/2$ is positive; the
polar phase of its generator is the standard $J$, unique by the
positivity selection. (iii) The Heisenberg flow of the full generator
is symplectic by construction; the interaction term
$g(a^{\dagger}b^{\dagger} + ab)$ generates two-mode squeezing, so the
evolution read on the asymptotic $J$-space is a Gaussian
completely-positive channel that grows $\langle a^{\dagger}a\rangle =
\sinh^{2}(gt)$ at resonance while the covariance-independent reading
structure --- the asymptotic $J$ --- is fixed by definition of the
frame. Numerically: the form's indefiniteness, the polar-phase
uniqueness at infinity, the exact symplecticity ($10^{-9}$), the state
validity (uncertainty relation), and the $\sinh^{2}(gt)$ law
($4\times10^{-14}$) are all verified. What is \emph{not} claimed: the
Hadamard analysis of the exact Kerr exterior --- which asymptotic
constructions are physically admissible there --- is the field's own
active area \citep{KayWald1991}, imported where used. The custody
clauses are insensitive to that choice: the choice \emph{is} the port.
\end{proof}

\begin{euclidtheorem}[The collapse channel \textnormal{(T-BH-4)}]
\label{thm:collapse-channel}
For quadratic field dynamics static in the far past and carrying the
exponential horizon redshift $x \sim e^{-\kappa u}$ at late times ---
the moving-mirror class \citep{DaviesFulling1976}, the model form of
gravitational collapse:
(i) the in-frame complex structure is selected uniquely at past
infinity, where the dynamics is static;
(ii) the out/in Bogoliubov overlaps reduce to
$I_{\pm} = \int_0^{\infty} x^{i\omega/\kappa - 1} e^{\pm i \omega' x}
\, dx$, and $|I_{+}/I_{-}|^{2} = e^{-2\pi\omega/\kappa}$ exactly: the
Planck factor at $T_{\mathrm{H}} = \kappa/2\pi$ is the branch-cut
asymmetry of the exponential redshift --- a real, classical, kinematic
datum;
(iii) for a trajectory static early and exponential late, the same
ratio converges to the Planck value as $\omega'$ grows: the late-time
out-field sees only the exponential region, so the out-state
factorizes into the universal thermal channel (at $T_{\mathrm{H}}$,
independent of the early trajectory) times a transient carrying the
collapse details, whose contribution decays. The collapse epoch
populates states through a Gaussian channel and never touches the
origin of the complex structure.
\end{euclidtheorem}

\begin{proof}
(i) is the static selection applied at past infinity. (ii) is the
classical evaluation underlying \citet{DaviesFulling1976} and
\citet{Hawking1975}: with $\nu = \omega/\kappa$,
$I_{\pm} = \Gamma(i\nu)\,(\mp i\omega')^{-i\nu}$ up to a common
factor (principal branch), and $|(\mp i\omega')^{-i\nu}| = e^{\mp \nu\pi/2}$, so the
ratio of moduli squared is $e^{-2\pi\nu}$ --- verified by direct
quadrature (with the oscillatory endpoint handled analytically) to
better than $10^{-3}$ relative at $\nu = \tfrac12, 1, 2$. (iii) The
factor $e^{\pm i\omega' x}$ confines the integral to
$x \lesssim 1/\omega'$; for $\omega'$ large this lies entirely in the
exponential region, and the early-trajectory contribution decays as an
endpoint term --- verified on a static-then-exponential trajectory,
where the relative deviation from the Planck ratio falls from $0.13$
at $\omega' = 3\kappa$ to below $10^{-2}$ at $\omega' = 30\kappa$. The
full gravitating statement --- that a collapsing star's late-time
radiation carries exactly this ratio --- is Hawking's derivation
\citep{Hawking1975}, imported with attribution; what the model class
proves is the factorization \emph{structure}: universal late-time
channel, decaying transient, custody preserved.
\end{proof}

\paragraph{The custody close.} With these two statements the custody
pattern of the cohort --- \emph{the dictionary owns the sectors; the
geometry owns the $i$} --- holds in five instances: the charge angle,
the horizon port, the infrared cloud, the rotating anchor, and the
collapse channel --- the first three superselection structures
proper (an unreadable relative phase between sectors), the last two
anchor-and-channel instances of the same division of custody. In every one, what varies between admissible
descriptions is a c-number port datum owned by the operational
dictionary --- a winding class, a horizon boundary condition, a soft
cloud, a stationary congruence, an in-vacuum --- and in every one the
derived complex structure passes through untouched. Black holes, on
this reading, are not where the quantum formalism breaks; they are
where its custody division shows most plainly.

\paragraph{Physics intuition.} A rotating black hole is a harbor
with a spinning quay: no clock mounted on the quay keeps honest time
everywhere (that is the ergoregion), so the ship's chronometer is set
out at sea, where the water is calm --- the asymptotic frame --- and
the spinning quay can then do only one thing to the cargo: pump it,
which is superradiance. Collapse is the harbor being built while the
ship watches: however violent the construction, what reaches the ship
at late times is a universal thermal wake at the horizon temperature,
plus a transient that carries the construction noise and dies away.
In both cases the pumping and the wake are freight --- states,
populations, energy --- and never touch the chronometer. The
dictionary owns the freight manifests; the geometry owns the clock.


\section{The thermal clock face as a theorem}
\label{sec:thermal-clock}

Every thermality statement in this Part has run on one analytic
device, bought in three pieces: a positive-frequency selection (which
modes are ``particles''), an analytic continuation through the
complexified horizon (the Planck factor of \S\ref{ssec:bogoliubov}),
and a periodicity in imaginary time (the Euclidean arguments of
Appendices~\ref{app:Hawking_T} and~\ref{app:euclidean}).  In the
older literature each piece is a prescription --- serviceable,
correct, and unexplained: the thermal character appears to be a
statement about a cleverly chosen vacuum.  It is not.  It is a
statement about the partial trace, made with a clock the bulk already
owns --- and the cohort now holds the theorem that says so.  The
Wick-quartet theorem \textsf{T-P5-B1} of the CPT companion
(\citealp{ArisakaP5CPT}) proves that for quasi-free fields on a
static arena four data are one datum, each determining the others
with orientations included: the positive-energy complex structure
(the clock's rotor), the harmonic-conjugate splitting
(positive-frequency selection), the imaginary-time contraction
semigroup (Wick rotation), and the holomorphic time tube (the lawful
meaning of $it$) --- all four notations of a single analytic fact,
the one-sidedness of the clock's spectrum.  This section consumes
that theorem at the horizon.

\begin{euclidtheorem}[The thermal clock face \textnormal{(T-BH-5)}]
\label{thm:thermal-clock}
For quasi-free bulk fields on the static exterior of a bifurcate
Killing horizon of surface gravity $\kappa$ --- the arena of
\S\ref{ssec:bogoliubov}, with rotation handled by the asymptotic
anchor of Theorem~\ref{thm:rotating-anchor}:
(i) the exterior positive-frequency structure is the polar rotor of
the Killing generator, the derived complex structure of the causal
clock (\citealp{ArisakaP4Real}), selected uniquely by the positivity
clause --- an output of the constitution, not an input to the
calculation;
(ii) by \textsf{T-P5-B1} the exterior evolution extends to a
contraction semigroup in imaginary Killing time, so the analytic
continuation through the complexified horizon used in
Eq.~\eqref{eq:planck} and the imaginary-time section used in
Appendices~\ref{app:Hawking_T} and~\ref{app:euclidean} exist as
theorems of the one-sided spectrum; no continuation prescription is
imported;
(iii) the regular quasi-free state of the exterior algebra ---
where such a state exists --- is the
thermal (KMS) state of that clock at the period fixed by the
horizon's smooth Euclidean cap, $\beta = 2\pi/\kappa$
\citep{KayWald1991}; the Bogoliubov ratio
$|\beta_\omega/\alpha_\omega| = e^{-\pi\omega/\kappa}$ and the
thermal trace of Eq.~\eqref{eq:thermal-trace} are that KMS condition
displayed mode by mode.
The thermal clock face on which this paper's Hawking temperature
runs is therefore proven, not assumed: the quartet supplies the
face; the geometry contributes only the reading, $2\pi/\kappa$.
\end{euclidtheorem}

\begin{proof}
(i) is the positivity selection already used throughout this Part
(the static exterior Killing frame is globally timelike outside the
ergoregion; for rotation the selection moves to the asymptotic
frame, Theorem~\ref{thm:rotating-anchor}(ii)), read through the
polar decomposition of \textsf{T-P5-B1}: a skew generator splits
uniquely into rotor times speed, and the rotor is the clock.  (ii)
is \textsf{T-P5-B1} applied verbatim: on the physical exterior modes
the generator's spectrum is one-sided, so the evolution continues to
the bounded half-plane and the contraction semigroup exists; a
two-sided spectrum would admit no such shore (the Paley--Wiener
obstruction).  (iii) composes two quasi-free-grade results: the
uniqueness-and-thermality theorems on bifurcate Killing horizons
\citep{KayWald1991}, which fix the regular exterior state as KMS at
$2\pi/\kappa$, and the mode-level display of
\S\ref{ssec:bogoliubov}, whose reduced two-mode-squeezed state
satisfies the KMS ratio
$\langle a a^{\dagger}\rangle/\langle a^{\dagger} a\rangle
= e^{\beta\omega}$ exactly.  Numerically: the polar split, the KMS
ratio, and the Planck occupation are verified to machine precision
($10^{-15}$ relative) at $\omega/\kappa = \tfrac12, 1, 2$, and the
one-sidedness obstruction is displayed explicitly, in the
program's verification suite (www.arisaka-gg.org).  What is \emph{not}
claimed: the gravitating, back-reacting extension beyond the
quasi-free class --- imported where used, exactly as in
Theorems~\ref{thm:rotating-anchor} and~\ref{thm:collapse-channel}.
\end{proof}

\paragraph{What the theorem retires.}  Three habits of the horizon
literature become clauses rather than assumptions.  The
``prescription'' of matching positive-frequency modes by
continuation is clause (ii).  The ``trick'' of demanding regularity
in imaginary time is the existence half of clause (iii) --- the
section exists because the spectrum is one-sided; the geometry's
regularity datum only sets the period.  And the choice-of-vacuum
language (``the thermal character of the Hartle--Hawking state'')
inverts: thermality is not a property a vacuum happens to have but
the KMS face every regular quasi-free exterior state must show,
because the clock it faces is derived.  The custody division of
\S\ref{sec:rotating-collapse} is unchanged --- T-BH-5 adds no sixth
instance; it names the clock that all five instances leave
untouched.  And the fiber supplies no rival: the phase circle's
winding spectrum is two-sided, so it damps no direction and admits
no half-plane --- among the bulk's directions, time alone has a
shore, which is why the horizon's thermality is the \emph{clock's}
face and never the fiber's.

\paragraph{Physics intuition.}  A harbor thermometer does not
manufacture temperature; it reads the water it is dipped in.  The
older derivations bought the thermometer in parts --- a dial here, a
damping law there, a calibration at the end --- and the parts always
fitted, without anyone saying why.  T-P5-B1 is the statement that
the parts are one instrument --- rotor, shore, and holomorphic tube,
three notations of the single clock the causal structure hands every
static observer --- and the thermal face is what this paper proves
on that instrument at the horizon (T-BH-5, composing the quartet
with \citealp{KayWald1991}).  The black hole contributes exactly one number to the
reading --- its surface gravity.  Everything else about Hawking
thermality was already in the clock.



\section{The real-bulk register at the horizon}
\label{sec:rb-register}

The cohort maintains a running register of results in which a
quantity the boundary literature treats as intrinsically complex is
exhibited as real bulk data read through the derived dictionary ---
the \emph{real-bulk series}, whose home is the quantum-mechanics
companion \citep{ArisakaA8QM} and whose full-length member treatment
is \PaperPFour{} \citep{ArisakaP4Real}.  This paper holds two entries
of that series, claimed at \S\ref{sec:rotating-collapse} and until
now never displayed.  We display them, together with the further
entry that this paper's entropy chain has earned --- proposed to the
series' home papers and, on their ruling of July 2026, registered as
Statement \textsf{RB-13} of the series home (\citealp{ArisakaA8QM};
registrational: no new theorem, no new custody instance, and no
series-entry number --- the proven series stands at R15).

\begin{table}[H]
\centering\small
\caption[The real-bulk series entries held by this module.]{The real-bulk series entries held by this module.  Each
exhibits a celebrated ``quantum'' datum as real, classical, or
kinematic bulk structure, with the derived complex window
untouched. Two entries, and both are model-level rather than general. The real-bulk series claims that a datum usually called quantum is in fact real, classical or kinematic bulk structure; what is verified here is that claim inside a solvable model, at the numerical tolerance the last column prints. That is a demonstration and not a theorem about generic black holes, and the grade column says so rather than leaving it to be inferred.}
\label{tab:rb-register}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.07\textwidth}p{0.47\textwidth}p{0.16\textwidth}p{0.18\textwidth}@{}}
\toprule
\textbf{Entry} & \textbf{Statement} & \textbf{Where proven} &
\textbf{Grade} \\
\midrule
R14 & The rotating anchor: the ergoregion defeats the rotating-frame
positivity selection for a real reason (an indefinite quadratic
form); the asymptotic frame's polar phase selects $J$ uniquely;
superradiance is exactly symplectic --- real --- Gaussian pumping &
Theorem~\ref{thm:rotating-anchor} (T-BH-3) & Model level, verified
($10^{-9}$, $4\times10^{-14}$) \\
R15 & The collapse channel: the Planck factor is the branch-cut
asymmetry of the exponential redshift --- a real, classical,
kinematic datum --- and the late-time channel factorizes
universally & Theorem~\ref{thm:collapse-channel} (T-BH-4) & Model
level, verified ($10^{-3}$) \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The registered entry: horizon entropy as real-section
geometry (Statement RB-13).}  The following is registered in the
series' home paper as Statement \textsf{RB-13}
(\citealp{ArisakaA8QM}; registrational, carrying no series-entry
number).  \emph{The
horizon's entropy data are real-section geometry.}  Every geometric
object in the exact-$1/4$ chain is real: the conical deficit of
Step~3 is a curvature datum of the real Euclidean section
(\S\ref{ssec:V-conical}); the area $A$ is a real functional of the
real metric; the entanglement the entropy measures is of the three
real bulk factors, traced at the horizon (Step~4); and the one
place the imaginary unit appears --- the analytic continuation and
the KMS face --- is the boundary's derived clock
(Theorem~\ref{thm:thermal-clock}), whose shore belongs to time and
never to the fiber.  In one line: $A/4\Newton$ is real geometry read
through a derived complex window --- the dictionary owns the
sectors; the geometry owns the $i$.  No new custody instance is
claimed (the count stands at five, \S\ref{sec:rotating-collapse});
the entry names the real-bulk standing of the entropy chain
itself.

\paragraph{Physics intuition.}  A ledger written in black ink,
audited through tinted glass.  Every quantity on the page --- the
deficit angle, the area, the traced fiber count --- is black-ink
real; the tint is the boundary's one derived window, the clock's
$i$, and it colors the reading without ever touching the page.
The real-bulk series is the cohort's running proof that the tint
comes off; the horizon, where the ledger's most famous number
lives, is its sharpest page.

\paragraph{The Part in one breath, and the door to the next.}
A horizon, this Part has argued, is not a wall but a port: the
one locus where the constitution's central operation --- the
partial trace that elsewhere an experimenter chooses --- is
performed by the causal structure itself (\textsf{L-BH-Port}).
Everything thermal then followed from that single fact, with no
further input: the Hawking spectrum as the channel's output; the
temperature as the clock face of a proven KMS state, the old
continuation tricks retired by theorem (T-BH-5); the rotating
and collapsing cases closed at the model level, with their
custody clauses intact (T-BH-3, T-BH-4); and the whole chain
standing on real-section geometry read through one derived
complex window (Statement RB-13).  The physics intuition to
carry forward is the simplest one in the paper: \emph{a black
hole is an open system, and geometry is its experimenter}.
What the port has not yet told us is the size of the account it
keeps.  The radiation is thermal, and thermal states carry
entropy; the horizon's entropy has a famous, exact, and ---
until now --- underived coefficient.  Counting what the port
traces, and proving that the count is $A/4\Newton$ with the
$1/4$ exact, is the business of the next Part.


%% ===================================================================
%% PART -- THE EXACT 1/4 (split of the former GDOS-reading Part)
%% ===================================================================

\section[Part~IV in plain English]{Part~IV in plain English: why the thermality is forced rather than derived, and what that does not buy}
A horizon is a one-way surface: what crosses it does not come back.
That is a statement about causal structure, and it is as true in the
six-dimensional bulk as it is in four dimensions.

The framework's central operation is a projection from the bulk to a
four-dimensional boundary, and every projection discards something. What
this Part observes is that at a horizon the discarding is not a modeling
choice but a fact about the geometry: the region behind the surface is
causally inaccessible, so any description written on this side must trace
over it. A trace over part of an entangled state gives a mixed state. In
this reading the thermality of Hawking radiation is therefore not a result
to be derived but a consequence of the one operation the geometry leaves no
choice about.

That is a strong claim, and it is worth naming what it does not buy. It
says nothing about the value of the temperature until the calculation is
done, so the Part does the calculation both ways --- the standard
Bogoliubov route and the open-system one --- and shows they agree. It also
cannot be tested by any instrument now or soon, which is why the Part's two
theorems are stated at model level, with the tolerance of their numerical
verification printed rather than described.

What the port has not said is how large the account it keeps is. The
horizon has an entropy, that entropy has a coefficient, and the coefficient
has been exact and unexplained for fifty years. That is the next Part.

\GGPart[The exact 1/4]{The Exact $1/4$}{The Bekenstein--Hawking Coefficient from Induced Gravity, with Its Honest Accounting}

\noindent
This is the Part the paper exists to deliver.  Part~IV established
that the horizon is a forced \OGGEP{} port and that what leaves it
is thermal; the account the port keeps --- its entropy --- was
written down by Bekenstein and Hawking half a century ago, with the
most famous coefficient in the subject standing in front: the
$1/4$.  Part~I told the story of how that number resisted
derivation for the black holes that actually exist, and Part~III
stated the bar precisely: derive the coefficient, from a
microscopic account, for the generic hole, with nothing tuned.
This Part clears that bar.  The instrument is the constitution's
second axiom: gravity is induced --- there is no bare
four-dimensional Newton constant --- so the same bulk fluctuations
that make gravity stiff make the horizon entangled, and the two
quantities are locked at the source rather than matched at the
end.  The plan runs in four steps and touches no dial: the induced
Newton constant is computed from the canonical scale profile; one
heat kernel is shown to price two quantities at once; the conical
replica evaluates their ratio in closed form, $1/4 = 4\pi/16\pi$;
and the microstates are named --- the hidden-fiber holonomies the
port traces.  The Part closes with the honest accounting: what is
rigorous, what is conditional, and the one computation the module
still owes itself.

\section{The Bekenstein--Hawking Entropy and the Exact $1/4$ from Induced Gravity}

\subsection{Entropy as entanglement}

The Bekenstein--Hawking entropy is
\begin{equation}\label{eq:Sbh_def}
\Sbh = \frac{A c^{3}}{4\Newton\hbar}\,\kB,
\end{equation}
where $A$ is the horizon area. In \GDOS{}, this is the
\emph{entanglement entropy} of the 4D exterior with the bulk
interior:
\begin{equation}\label{eq:Sbh_GDOS}
\Sbh = \SvN(\rho_\partial^{\rm out}) = -\Tr(\rho_\partial^{\rm
out}\ln\rho_\partial^{\rm out})\,\kB.
\end{equation}
The interior here includes both the 4D interior $r < r_s$ and the
non-spatial fiber --- the phase circle $S^1_\theta$ and the
inaccessible radial tail of the scale direction $I_r$.  The rest of
this Part computes this entanglement entropy from first principles
and shows that it equals $A/4\Newton$ \emph{exactly}, coefficient
included, by the induced-gravity mechanism.

\subsection{Strategy: gravity is induced, so the horizon entropy \emph{is} the entanglement}
\label{ssec:V-strategy}

Equation~\eqref{eq:Sbh_GDOS} identifies $\Sbh$ as an entanglement
entropy.  But the entanglement entropy of quantum fields across a
surface is famously ultraviolet-divergent and, in an ordinary field
theory, scheme-dependent: it is a \emph{correction} to a pre-existing
$A/4\Newton$, not the quantity itself \citep{BombelliKoulLeeSorkin1986,
SusskindUglum1994}.  What turns it into the exact Bekenstein--Hawking
value --- coefficient and all --- is a structural feature that
\GGtheory{} supplies and ordinary field theory does not:
\begin{center}
\emph{there is no bare four-dimensional Newton constant.}
\end{center}
By \GGEP{} (\AX{2}) gravity is the geometry of the bulk, not a
fundamental boundary field carrying its own coupling; the
four-dimensional gravitational action that a boundary observer reads is
generated \emph{entirely} by integrating out the bulk fluctuation
modes.  This is Sakharov's induced-gravity hypothesis
\citep{Sakharov1968}, here promoted from hypothesis to a consequence of
the constitution.  Its decisive corollary, established below, is that
the divergence of the entanglement entropy is not a correction to
$1/4\Newton$ --- it \emph{is} $1/4\Newton$ \citep{Jacobson1994,
FrolovFursaev1997}.  The argument is three analytic steps.  First, the induced Newton
constant is generated by the canonical scale profile, finite and
ultraviolet-localized (\S\ref{ssec:V-induced-G}).  Second, the
\emph{same} bulk modes that induce $1/\Newton$ produce the horizon
entanglement entropy, through the \emph{same} heat-kernel
coefficient, so their ratio is fixed once and for all
(\S\ref{ssec:V-heatkernel}).  Third, the conical replica method
evaluates that ratio in closed form: $\Sbh = A/4\Newton$, with the
coefficient $1/4 = 4\pi/16\pi$ emerging
\emph{regulator-independently} (\S\ref{ssec:V-conical}).

\subsection{Step 1 --- the induced Newton constant from the canonical scale profile}
\label{ssec:V-induced-G}

Write the six-dimensional gravitational action that \GGEP{} attaches to
the bulk,
\begin{equation}
I_6 \;=\; -\frac{1}{16\pi G_6}\int_{\cM_6}\! d^6X\,\sqrt{g_6}\,R_6 ,
\label{eq:V-I6}
\end{equation}
and reduce it on the non-spatial fiber using the canonical metric
\eqref{eq:BH_bulk}, $g^{(6)}_{\mu\nu}(X) = e^{-2r}\,g_{\mu\nu}(x)$.  The
measure and curvature reduce as
\begin{equation}
\sqrt{g_6} = e^{-4r}\sqrt{g_4}\,, \qquad
R_6 \supset e^{+2r} R_4[g] ,
\label{eq:V-reduce}
\end{equation}
the factor $e^{+2r}$ arising because the four-dimensional Ricci scalar
is built with the scale-profile inverse metric $g^{\mu\nu}_{(6)} = e^{+2r}
g^{\mu\nu}$.  Hence the Einstein--Hilbert term of the boundary theory is
\begin{equation}
I_6 \supset -\frac{1}{16\pi G_6}
\left(\int_0^{2\pi}\!\! d\theta \int_0^{K}\!\! dr\; e^{-2r}\right)
\int d^4x\,\sqrt{g_4}\,R_4 ,
\label{eq:V-EHterm}
\end{equation}
with the two fiber integrals \emph{finite and parameter-free}
(Theorem~2: no free $k$, $R_\theta$, or $r_c$):
\begin{equation}
\int_0^{2\pi}\! d\theta = 2\pi , \qquad
\int_0^{K}\! e^{-2r}\,dr = \tfrac12\bigl(1 - e^{-2K}\bigr).
\label{eq:V-fibreint}
\end{equation}
Reading off the coefficient of $\tfrac{1}{16\pi}\!\int\!\sqrt{g_4}R_4$,
\begin{equation}
\boxed{\;\frac{1}{\Newton}
= \frac{\pi\bigl(1-e^{-2K}\bigr)}{G_6}\;}
\qquad\text{(fundamental scale restoring units)} .
\label{eq:V-G4}
\end{equation}
Three features matter for what follows.  (i) $\Newton$ is
\emph{finite}: the radial integral converges because the scale profile
$e^{-2r}$ is integrable on $[0,K]$.  (ii) It is \emph{ultraviolet-localized}:
the integrand $e^{-2r}$ is peaked at the UV boundary $r=0$, so Newton's
constant is generated near $\MGUT$ --- the geometric statement that
gravity is a UV phenomenon.  (iii) It is \emph{induced}, not posited:
$\Newton$ is a property of the scale profile, exactly as \GGEP{} demands. This is the \emph{same} boundary Newton constant obtained from the scale-overlap reduction of the boundary-gravity derivation in \AFive{} \citep{ArisakaA5GDOS}: both read $1/\Newton$ off the identical canonical scale profile of \ATwo{}, so the two papers' Newton constants coincide by construction rather than by assumption. Moreover, the $A/4$ coefficient established below depends only on the heat-kernel \emph{ratio} of Step~2, not on the absolute normalization of $\Newton$, so the entropy law is robust even where that normalization is later refined.  At one
loop the same structure is reproduced quantum-mechanically --- the bulk
modes' determinant generates \eqref{eq:V-EHterm} with a coefficient set
by their spectrum --- which is the form we need next.

\subsection{Step 2 --- one heat kernel, two quantities}
\label{ssec:V-heatkernel}

Both the induced Newton constant and the horizon entanglement entropy
are controlled by a \emph{single} object: the heat-kernel expansion of
the bulk-mode fluctuation operator $\Delta$ on the four-dimensional
boundary,
\begin{equation}
\Tr\, e^{-s\Delta}
= \frac{1}{(4\pi s)^{2}}\int d^4x\,\sqrt{g_4}\,
\bigl[a_0 + a_1\,s + a_2\,s^2 + \cdots\bigr],
\qquad a_1 = \Bigl(\tfrac16 - \xi\Bigr) R ,
\label{eq:V-heatkernel}
\end{equation}
with $\xi$ the curvature coupling ($\xi=0$ for a minimal scalar).  The
coefficient $a_1$ --- the coefficient of $R$ --- is the common ancestor
of both quantities.

\paragraph{The induced $1/\Newton$.}  The one-loop effective action is
$W = \tfrac12\ln\det\Delta = -\tfrac12\int_{\epsilon^2}^{\infty}
\tfrac{ds}{s}\,\Tr\,e^{-s\Delta}$, regulated by a proper-time cutoff
$s\ge\epsilon^2$ ($\epsilon$ the physical short-distance length).  The
$a_1$ term contributes the Einstein--Hilbert piece
\begin{equation}
W \supset -\frac{1}{2}\,\frac{1}{(4\pi)^2}
\Bigl(\tfrac16-\xi\Bigr)\!\int d^4x\sqrt{g_4}\,R
\int_{\epsilon^2}^{\infty}\!\frac{ds}{s^2}
= -\frac{1}{16\pi}\Bigl[\frac{1}{12\pi\epsilon^2}\Bigr]_{\xi=0}
\!\!\int d^4x\sqrt{g_4}\,R ,
\label{eq:V-inducedG}
\end{equation}
so that, per minimal scalar,
\begin{equation}
\frac{1}{G_{\rm ind}} = \frac{1}{12\pi\,\epsilon^2},
\qquad\Longrightarrow\qquad
\frac{1}{4G_{\rm ind}} = \frac{1}{48\pi\,\epsilon^2}.
\label{eq:V-1over4G}
\end{equation}

\paragraph{The entanglement entropy.}  The leading (area) term of the
entanglement entropy across the horizon, computed from the same
heat kernel by the conical method (next subsection), is
\begin{equation}
\frac{dS_{\rm ent}}{dA} = \frac{1}{48\pi\,\epsilon^2}
\qquad\text{(per minimal scalar).}
\label{eq:V-Sent-density}
\end{equation}

\paragraph{The match.}  Comparing \eqref{eq:V-1over4G} and
\eqref{eq:V-Sent-density},
\begin{equation}
\boxed{\;\frac{dS_{\rm ent}}{dA} = \frac{1}{4G_{\rm ind}}\;}
\qquad\text{identically.}
\label{eq:V-match}
\end{equation}
The specific number $1/48\pi\epsilon^2$ is scheme-dependent --- it
shifts with the regulator and with $\xi$ --- but the \emph{equality}
\eqref{eq:V-match} is not, because both sides are the \emph{same}
heat-kernel coefficient $a_1$ of the \emph{same} modes
\citep{SusskindUglum1994, CallanWilczek1994, DemersLafranceMyers1995}.
This is the Susskind--Uglum identity, and in \GGtheory{} it is forced,
not assumed: by \GGEP{} there is no bare $\Newton$ to which the
entanglement divergence could be a mere correction, so
\eqref{eq:V-match} reads as an equality of the entropy itself with
$A/4\Newton$.

\subsection{Step 3 --- the conical replica and the exact $1/4$}
\label{ssec:V-conical}

The value of the ratio in \eqref{eq:V-match} is fixed in closed form,
independent of $\epsilon$ and of the field content, by the replica
(conical-deficit) construction.

\paragraph{The arena, licensed.}  One word on where the computation
takes place, because the replica method's Euclidean arena is the step
referees most often challenge as a formal device.  In the cohort it
is not a device.  Time carries four endowments
(\citealp{ArisakaP5CPT}): order from the causal cone, repetition from
the equivalence of instants, the arrow, and the \emph{analytic
shore} --- the half-plane of lawful imaginary time that the clock's
one-sided spectrum opens (Theorem~\ref{thm:thermal-clock}(ii)).  The
Euclidean section on which the replica cover is built \emph{is} that
shore: for the quasi-free exterior algebra the continuation to
imaginary Killing time exists as a theorem, and the $\tau$-circle of
period $2\pi/\kappa$ whose pinching produces the conical deficit
below is the shore's thermal face
(Theorem~\ref{thm:thermal-clock}(iii)), not a regularity trick.  One
import remains, and we declare it: evaluating the \emph{gravitational}
action semiclassically on the cover --- $\ln Z_n = -I_E[\cM_n]$
below --- is the saddle step of the replica literature
\citep{Solodukhin2011}, carried at that grade.  The named-statement
deck registers this license as the text key \textsf{L-BH-Shore}.  The arena's status is
therefore Derived-conditional --- derived through the clock's shore,
conditional on the quasi-free class and the declared saddle import ---
and the coefficient extraction that follows is exact arithmetic on
both.

Prepare the Euclidean $n$-fold cover
$\cM_n$ with a conical deficit angle $2\pi(1-n)$ on the bifurcation
surface $\Sigma$ (the horizon, of area $A$).  The curvature acquires a
distributional piece concentrated on $\Sigma$
\citep{Solodukhin2011}:
\begin{equation}
\int_{\cM_n}\!\sqrt{g}\,R
= n\!\int_{\rm smooth}\!\sqrt{g}\,R \;+\; 4\pi(1-n)\,A .
\label{eq:V-conical-id}
\end{equation}


\begin{figure}[tbp]
\centering
\resizebox{0.872\textwidth}{!}{%
\usebox{\GGscQuarter}}
\caption[The conical-deficit origin of the exact $1/4$.]{The conical-deficit origin of the exact $1/4$.  On the replica cover $\mathcal{M}_n$ the Euclidean-time circle is pinched at the horizon $\Sigma$ with deficit angle $2\pi(1-n)$.  The curvature concentrated on $\Sigma$ contributes $4\pi(1-n)A$ to $\int\!\sqrt{g}\,R$; fed through the induced Einstein--Hilbert action and the replica formula, this gives $\Sbh = A/4G_N$, the coefficient $1/4 = 4\pi/16\pi$ being the ratio of the conical-curvature normalization to the Einstein--Hilbert normalization (Theorem~\ref{thm:exact-quarter}).}
\label{fig:conical-quarter}
\end{figure}

\noindent Figure~\ref{fig:conical-quarter} displays the two normalizations whose ratio is the theorem.



Insert this into the induced gravitational action
$I_E[\cM_n] = -\tfrac{1}{16\pi\Newton}\int_{\cM_n}\sqrt{g}\,R$:
\begin{equation}
I_E[\cM_n] = n\,I_E[\cM_1] \;-\; \frac{(1-n)\,A}{4\Newton},
\qquad
\ln Z_n = -I_E[\cM_n].
\label{eq:V-action-n}
\end{equation}
The replica entropy is $S = -\Tr\rho\ln\rho
= \bigl(\ln Z_1 - \partial_n \ln Z_n\bigr)\big|_{n=1}$, since
$\Tr\rho^{\,n} = Z_n/Z_1^{\,n}$.  Writing $R_1\equiv\int_{\cM_1}\sqrt{g}R$,
\begin{align}
\ln Z_n &= \frac{n\,R_1}{16\pi\Newton} + \frac{(1-n)A}{4\Newton},
&\partial_n\ln Z_n\big|_{n=1} &= \frac{R_1}{16\pi\Newton} - \frac{A}{4\Newton}, \nonumber\\
\ln Z_1 &= \frac{R_1}{16\pi\Newton},
&S &= \ln Z_1 - \partial_n\ln Z_n\big|_{n=1}
= \boxed{\;\frac{A}{4\Newton}\;}.
\label{eq:V-final}
\end{align}
The bulk terms $R_1$ cancel identically; the entropy is purely the
boundary (area) term.  The coefficient is transparent:
\begin{equation}
\frac{1}{4} \;=\; \frac{4\pi}{16\pi}
\;=\; \frac{\text{(conical-deficit curvature coefficient)}}
{\text{(Einstein--Hilbert normalization)}} ,
\label{eq:V-quarter}
\end{equation}
the ratio of the $4\pi(1-n)A$ in \eqref{eq:V-conical-id} to the
$16\pi\Newton$ in the action.  Neither number knows anything about
black holes: the $4\pi$ is fixed by the geometry of spheres
(Gauss--Bonnet), the $16\pi$ by demanding Newton's law in the
weak-field limit --- both were set by consistency requirements
decades before anyone computed a horizon entropy.  No free parameter
enters, no regularization choice survives, and --- crucially ---
supersymmetry is never used.  Restoring $\kB$ and $\hbar$,
\begin{equation}
\boxed{\;\Sbh = \frac{\kB\,c^3\,A}{4\,\Newton\,\hbar}\;}
\label{eq:V-Sbh}
\end{equation}
which is Eq.~\eqref{eq:Sbh_def}, now \emph{derived} rather than posited.

\setcounter{euclidtheorem}{0}% key alignment: T-BH-n = Theorem n
\begin{euclidtheorem}[Exact Bekenstein--Hawking coefficient from induced gravity \textnormal{(T-BH-1)}]
\label{thm:exact-quarter}
Let the four-dimensional Newton constant be induced (\GGEP), the bulk metric be canonical (\ATwo{}, Theorem~2), and observation be the \OGGEP{} trace.  Then the horizon entropy is $\Sbh = A/4\Newton$, with the coefficient $1/4 = 4\pi/16\pi$ fixed \emph{regulator-independently} and \emph{without} supersymmetry or extremality, for the generic (Schwarzschild, Kerr, Reissner--Nordstr\"om) black hole.
\end{euclidtheorem}

\paragraph{Grade.}  Theorem~\ref{thm:exact-quarter} is held at
\emph{Derived-conditional}: derived regulator-independently by
Steps~1--3, conditional on exactly its three stated premises ---
full inducedness (\GGEP{}), the canonical bulk metric (\ATwo{},
Theorem~2), and observation as the \OGGEP{} trace --- with the
Euclidean arena licensed at the same grade (\S\ref{ssec:V-conical})
and the numerical species closure registered as the module's
internal kill row (F-BH-9, Table~\ref{tab:bh_kill};
Appendix~\ref{app:species-sum}).  Nothing in the coefficient
$\tfrac14$ waits on that closure.

\subsection{Step 4 --- the microstates are the hidden fiber}
\label{ssec:V-microstates}

The replica computation tells us \emph{that} $\Sbh=A/4\Newton$; Step~1
and Step~2 tell us \emph{which} modes are being counted --- those whose
one-loop determinant induces $\Newton$.  By \OGGEP{} (\AX{3}) these are
exactly the bulk-fiber modes that a horizon traces out: the
$\UoneQi$ Wilson-line holonomies $\Theta_H = \oint A_\theta\,d\theta$ on
the phase circle $S^1_\theta$ --- protection by holonomy rather than by
symmetry, in \PaperWTwo{}'s sense (\citealp{ArisakaW2Holonomy}) ---, together with the Kaluza--Klein tower of
the scale direction $I_r$, evaluated on the horizon two-sphere.  In the
hidden-fiber language of \AEight{} \citep{ArisakaA8QM}, the
Bekenstein--Hawking entropy is
\begin{center}
\emph{the entropy of the non-spatial fiber, traced out over the
horizon.}
\end{center}
The same compact phase circle $S^1_\theta$ that carries quantum
coherence in the bulk (where its holonomy is the Aharonov--Bohm/Berry
phase) becomes, at a horizon, the carrier of black-hole microstates;
the black-hole counting and the origin of quantum mechanics are one
mechanism evaluated on two different loci.  This identifies the
microstates concretely and finitely --- a horizon of area $A$ supports
$\exp(A/4\ell_P^2)$ distinguishable fiber configurations --- without any
appeal to branes, extremal limits, or replica wormholes.

\subsection{Generality: neither supersymmetry nor extremality is used}
\label{ssec:V-generality}

It is worth stating plainly what the derivation did \emph{not} assume.
It used only (i) that $\Newton$ is induced (\GGEP), (ii) the canonical
scale profile (Theorem~2), and (iii) that observation is the \OGGEP{} trace.  It
never used supersymmetry, a BPS bound, extremality, or a near-horizon
$\AdS_2$ throat.  Consequently \eqref{eq:V-final} holds for the
\emph{generic} black hole --- Schwarzschild, Kerr, Reissner--Nordstr\"om
alike --- the astrophysical case.  This is the decisive point of
contrast with the string-theoretic microstate count
\citep{StromingerVafa1996}, which is exact but available only for
supersymmetric, extremal or near-extremal black holes.  \GGtheory{}
reproduces the exact coefficient precisely where that method does not
reach.

\subsection{Kerr, Reissner--Nordstr\"om, and the universality of the $1/4$}
\label{ssec:V-kerr-rn}

Theorem~\ref{thm:exact-quarter} was stated for the generic black hole, but the conical derivation (\S\ref{ssec:V-conical}) was carried out on the Schwarzschild horizon.  We now show the result is unchanged for rotating and charged black holes --- the case that matters astrophysically, and the case the explicit string count does not reach.

\paragraph{The coefficient is metric-independent.}  The replica derivation used only two facts: that the horizon is a stationary Killing bifurcation surface $\Sigma$ of area $A$, around which the Euclidean-time circle caps off smoothly with period $\beta = 2\pi/\kappa$; and the induced Einstein--Hilbert action.  The conical-deficit identity $\int_{\mathcal{M}_n}\sqrt{g}\,R \supset 4\pi(1-n)A$ holds for \emph{any} such $\Sigma$.  Neither input distinguishes Schwarzschild from Kerr or Reissner--Nordstr\"om, so
\begin{equation}
\boxed{\;S = \frac{A}{4G_N}\quad\text{for every stationary black hole,}\;}
\end{equation}
with the coefficient $1/4 = 4\pi/16\pi$ identical in every case and only the horizon area $A(M,J,Q)$ changing.  The microstates are the same hidden-fiber holonomies traced over $\Sigma$ (\S\ref{ssec:V-microstates}).

\paragraph{The three families.}  In geometrized units ($G_N=c=1$, restored at the end), with $a = J/M$,
\begin{align}
\text{Reissner--Nordstr\"om:}\quad & r_+ = M + \sqrt{M^2 - Q^2}, & A &= 4\pi r_+^2, \nonumber\\
\text{Kerr:}\quad & r_+ = M + \sqrt{M^2 - a^2}, & A &= 4\pi(r_+^2 + a^2), \nonumber\\
\text{Kerr--Newman:}\quad & r_+ = M + \sqrt{M^2 - a^2 - Q^2}, & A &= 4\pi(r_+^2 + a^2),
\end{align}
and in each case $S = A/4G_N$ \citep{Kerr1963, ReissnerNordstrom1916, NewmanEtAl1965}.

\paragraph{First-law consistency.}  With $T = \hbar\kappa/2\pi c k_B$ and $S = A/4G_N$, the \GGtheory{} entropy automatically satisfies the full first law of the $(M,J,Q)$ family \citep{BardeenCarterHawking1973},
\begin{equation}\label{eq:first-law}
dM = T\,dS + \Omega_H\,dJ + \Phi_H\,dQ,
\qquad \Omega_H = \frac{a}{r_+^2+a^2},\quad \Phi_H = \frac{Q r_+}{r_+^2+a^2},
\end{equation}
because $A/4G_N$ is precisely the Bekenstein--Hawking--Wald entropy the first law requires \citep{Wald1993}.  \GGtheory{} thus reproduces the entire stationary first law, not only the static coefficient.

\begin{table}[H]
\centering\small
\caption[The exact $1/4$ across the stationary black-hole family.]{The exact $1/4$ across the stationary black-hole family.  In every case $S = A/4G_N$ with the identical coefficient $1/4 = 4\pi/16\pi$; supersymmetry and extremality are never used. The point of running the whole family across the table is that the coefficient does not move. Charge and rotation change the horizon radius, the area formula and the work term, and leave the \(1/4\) exactly where it was. No row uses supersymmetry, none is extremal, and none was chosen because it worked --- which is the difference between this and a microstate count in a controlled sector.}
\label{tab:kerr-rn}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.22\textwidth}p{0.30\textwidth}p{0.22\textwidth}p{0.18\textwidth}@{}}
\toprule
\textbf{Black hole} & \textbf{Horizon $r_+$ ($G_N=c=1$)} & \textbf{Area $A$} & \textbf{First-law work} \\
\midrule
Schwarzschild & $2M$ & $16\pi M^2$ & --- \\
Reissner--Nordstr\"om & $M + \sqrt{M^2 - Q^2}$ & $4\pi r_+^2$ & $\Phi_H\,dQ$ \\
Kerr & $M + \sqrt{M^2 - a^2}$ & $4\pi(r_+^2+a^2)$ & $\Omega_H\,dJ$ \\
Kerr--Newman & $M + \sqrt{M^2 - a^2 - Q^2}$ & $4\pi(r_+^2+a^2)$ & $\Omega_H\,dJ + \Phi_H\,dQ$ \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The astrophysical point.}  Every observed black hole rotates; the astrophysical objects are near-Kerr and non-extremal.  The induced-gravity derivation covers exactly this generic case, with no supersymmetry and no extremal limit --- the regime in which the explicit string-theoretic microstate count is not available.  This is the concrete content of the generality claimed in \S\ref{ssec:V-generality}.

\subsection{The subleading logarithmic correction}
\label{ssec:V-log}

Beyond the area law, every microscopic theory of black-hole entropy predicts a \emph{universal} subleading term,
\begin{equation}\label{eq:log-correction}
S = \frac{A}{4G_N} + c_{\log}\,\ln\frac{A}{\ell_P^2} + \cdots,
\end{equation}
and the coefficient $c_{\log}$ is a sharp, parameter-free discriminator between frameworks.  In the induced-gravity derivation it has a clean origin: where the leading $1/4$ came from the $a_1$ Seeley--DeWitt coefficient (the $R$ term, \S\ref{ssec:V-heatkernel}), the logarithm comes from the \emph{next} coefficient $a_2$ (the dimension-four, conformal-anomaly term).  On the replica cover the $a_2$ density integrated over the conical singularity carries an $n$-dependence whose $\partial_n$ at $n=1$ is logarithmic in the cutoff, hence in $A/\ell_P^2$; the coefficient is fixed by the conformal-anomaly charges of the massless fields crossing the horizon \citep{Solodukhin2011, Sen2013, Carlip2000}.

\paragraph{The \GGtheory{} prediction.}  Because $c_{\log}$ is set by the same boundary spectrum that fixes the induced $G_N$, \GGtheory{} predicts a definite value, $c_{\log} = -\tfrac12\,b_2[\text{\GGtheory{} massless content}]$, with $b_2$ the summed $a_2$-anomaly coefficient of the Standard-Model fields and the light bulk modes.  For its \emph{exact number} this shares the open status of the leading species sum (Appendix~\ref{app:species-sum}); its \emph{structure} is robust, and it is an independent falsifiable check.  Two consequences follow.  First, internal consistency: the \GGtheory{} value must coincide with the macroscopic $a_2$/Sen value for its spectrum.  Second, a discriminator: it generically \emph{differs} from the loop-quantum-gravity prediction $c_{\log} = -\tfrac32$, which arises from a horizon Chern--Simons state count rather than from a bulk anomaly.  A microscopic count that reproduced $A/4G_N$ but missed the \GGtheory{} $c_{\log}$ would falsify the induced-gravity reading.

\subsection{What is rigorous here, and what remains}
\label{ssec:V-honest}

In the spirit of \MF{} (\POS{7}) we separate the secure from the
outstanding.

\emph{Established.}  The area law and the exact coefficient
$\tfrac14$ follow from \eqref{eq:V-conical-id}--\eqref{eq:V-quarter},
regulator-independently, for generic non-supersymmetric black holes,
given only that $\Newton$ is induced.  The microstates are identified
(Step~4).  The Euclidean arena of Step~3 is itself licensed
constitutionally --- the analytic shore of the four-endowment time,
consumed through Theorem~\ref{thm:thermal-clock} --- at
Derived-conditional grade, with the semiclassical gravitational
saddle the one declared import (\S\ref{ssec:V-conical}).  This is a
genuine derivation of the coefficient that made the
Strominger--Vafa result a landmark, in a setting that result cannot
reach.

\emph{Outstanding.}  Three items remain, and we flag them honestly.
(a) \textbf{Species closure}: $\Sbh=A/4G_{\rm ind}$ holds for
\emph{whatever} $G_{\rm ind}$ the modes produce; obtaining a pure
number requires summing the actual \GGtheory{} spectrum (the Standard
Model content of \ANine{} plus the bulk Kaluza--Klein tower) and
checking that $1/G_{\rm ind}$ equals the $\Newton$ of Step~1 that
appears in the four-dimensional Einstein equations.  We show in Appendix~\ref{app:species-sum} that this closure is
\emph{structural} --- the Kaluza--Klein tower sum reproduces the scale reduction --- while the precise leading numerical coefficient remains
the principal open computation.  (b) \textbf{Non-minimal and
gauge fields}: the matching \eqref{eq:V-match} is clean for minimally
coupled scalars but carries known subtleties (contact terms) for gauge
fields and the graviton \citep{Kabat1995}; the full spectrum requires
care.  (c) \textbf{Subleading $\log A$}: the $a_2$ heat-kernel
coefficient yields a universal logarithmic correction
$\Sbh \supset c_{\log}\ln(A/\ell_P^2)$, an independent check the
framework must pass (developed in \S\ref{ssec:V-log}).  None of these touches the coefficient $\tfrac14$
itself, which is settled by Step~3.

\paragraph{The Part in one breath, and the door to the next.}
Gravity's stiffness and the horizon's memory are the same
purchase, priced once.  Because no bare Newton constant exists,
the entanglement across the horizon does not approximate
$A/4\Newton$ --- it \emph{is} $A/4\Newton$; and the coefficient
is the ratio of two normalizations fixed by non-negotiable physics
decades before anyone computed an entropy.  The physics intuition
to carry forward: \emph{the $1/4$ is not a count that could
drift; it is a ratio that cannot}.  What the coefficient does not
yet tell us is how the account settles over a black hole's
lifetime --- when the information comes back, why no firewall
guards the door, and what the second law charges for every
exchange.  That ledger, set in motion, is the business of the
next Part.



%% ===================================================================
%% PART -- INFORMATION (split of the former GDOS-reading Part)
%% ===================================================================

\section[Part~V in plain English]{Part~V in plain English: why the coefficient is a ratio rather than a count, and why that is checkable}
This is the paper's central technical result, and the shape of the
argument is unusual enough to be worth stating before the argument itself.

Everyone agrees the horizon entropy is proportional to area. The
coefficient is one quarter, exactly, in units of Newton's constant. The
usual way to explain it is to count something --- microstates, in some
sector where counting can be controlled --- and check that the count comes
out right. That works where it works, and it does not reach the black hole
at the center of a galaxy.

This Part does not count. It observes that in this framework there is no
fundamental Newton constant at all: gravity is induced, which means the
constant is generated by the same fields whose entanglement across the
horizon one would be trying to count. The two quantities are therefore not
independent, and both fall out of a single heat-kernel calculation. The
entanglement entropy does not approximate a quarter of the area over
Newton's constant. It is that, because the quarter is a ratio between two
normalizations that other physics had fixed decades before anyone asked
about entropy.

The consequences are what make the claim attackable. No supersymmetry and
no extremality enter, so the result covers Kerr, Reissner--Nordstr\"om and
real astrophysical holes rather than a protected sector. And a ratio cannot
drift the way a count can --- which is exactly why a measurement or a
calculation returning a coefficient other than one quarter would end this
reading rather than adjust it.

What is not settled is in the Part's own last section rather than left for
a reader to notice. The species sum that closes the argument is a
computation nobody has performed, and it is registered as a falsifier
rather than filed as a detail.

\GGPart[Information]{Information: Page Curve, Firewalls, and the Interior}{The Ledger, the Scheduled Expiry of Freshness, and the Direction of the Second Law}

\noindent
Part~V fixed the size of the horizon's account; this Part watches
the account settle in time.  Everything here is bookkeeping on one
object --- the free-energy ledger that the open-system constitution
supplies before any black hole is mentioned --- and the Part's
results are the entries that ledger forces: the Page curve as a
branch switch whose clock is an audited counting premise; the
firewall dissolved as an artifact of describing one system twice;
the interior read as bulk rather than as an ending; the four laws
upgraded from analogy to \GDOS{} statements; and the generalized
second law given both its direction and its price.  The reader who
wants the mechanism in one sentence can have it now: information
is never lost, only displaced --- and its return is not a
mechanism switching on but an assumption expiring on schedule.
The sections that follow earn that sentence, starting with the
ledger itself.

\section{The \GGtheory{} free-energy ledger and generalized entropy}
\label{sec:gff-ledger}

The Bekenstein--Hawking entropy, derived in Part~V as an entanglement entropy, is a special case of a structural object we now record in full, because that is what makes the black-hole result an instance of a universal law rather than a gravitational accident.  In the modern island and quantum-extremal-surface (\QES{}) literature, the entropy of Hawking radiation is obtained from the generalized-entropy functional
\begin{equation}
S_{\rm gen}[X] = \frac{A[X]}{4 G_N} + S_{\rm bulk}[X].
\label{eq:gen-entropy}
\end{equation}
Within \GGtheory{} this is immediately recognized as the black-hole instance of the universal scalar ledger of \GDOS{} \citep{ArisakaA5GDOS},
\begin{equation}
\GFF = F_{\rm GEP} + F_{\rm CEP},
\label{eq:gff-general}
\end{equation}
the sum of a geometric extremal piece (\GEP) and a curvature--entropy piece (\CEP).  For a black hole the natural identification is
\begin{equation}
F_{\rm GEP}^{\rm BH} = \frac{A}{4 G_N}, \qquad
F_{\rm CEP}^{\rm BH} = S_{\rm bulk}, \qquad
\GFFBH = \frac{A}{4 G_N} + S_{\rm bulk}.
\label{eq:gff-bh}
\end{equation}
The significance is interpretive, not notational.  In the standard horizon story, generalized entropy looks like the right formula discovered \emph{within} black-hole semiclassics.  In \GGtheory{} it is the black-hole reading of a ledger that the open-system constitution \emph{already required}, before any black hole is introduced.

\subsection{Why the area term belongs to \GEP{}}
In ordinary thermodynamics the free-energy ledger carries a cost for occupying macroscopically constrained configurations.  In gravity, horizon area is exactly such a constraint: for a fixed exterior observer, changing the area changes not only mass and temperature but the causal partition between accessible and inaccessible sectors.  The area term therefore measures more than surface size --- it is the geometric price of the horizon itself, and so belongs to the geometric piece \GEP{} of the ledger.

\subsection{Why the bulk-entanglement term belongs to \CEP{}}
The second term, $S_{\rm bulk}$, measures the unresolved internal state volume compatible with the chosen branch; it is where entanglement, coarse-graining, and state counting enter.  The exterior observer does not see microscopic bulk data directly, only a projected channel whose entropy must include the volume of discarded interior possibilities --- precisely the role of the curvature--entropy piece \CEP{}.  The conceptual payoff is that \GGtheory{} removes the air of special pleading from generalized entropy: its two terms are not unrelated objects glued together because black holes happened to demand them, but the two natural halves of one open-system ledger.

\subsection{Operational entropy as a branch minimum}
Once the ledger is written this way, the physically accessible radiation entropy is not the naive thermal entropy of the emitted quanta.  It is the branch-minimized ledger over admissible coarse-grained saddles,
\begin{equation}
S_R(t) = \min_{X \in \mathfrak{B}_{\rm adm}} \GFFBH[X; t].
\label{eq:branch-min}
\end{equation}
This is deliberately close to the island prescription, but its reading is different: in island language one extremizes and minimizes over candidate quantum extremal surfaces; in \GGtheory{} the operational boundary system explores admissible coarse-grained branches and the realized entropy is the minimum of the universal ledger over them.  The black-hole problem is then only one especially vivid instance of a rule already needed for open systems, non-equilibrium thermodynamics, and measurement.

\subsection{The factorization problem across the horizon}
A hidden assumption in many textbook formulations of the information paradox is that the total Hilbert space factorizes cleanly, $\cH_{\rm tot} \stackrel{?}{=} \cH_{\rm in} \otimes \cH_{\rm out}$.  In a non-gravitational theory this is often harmless; in gravity it is dangerous.  Physical states obey constraints $\cC_\alpha\ket{\Psi_{\rm phys}}=0$, and states related by gauge redundancy are not distinct, so the physical Hilbert space is a constrained quotient $\cH_{\rm phys} = \ker\cC_\alpha / (\text{gauge redundancy})$ rather than an unconstrained tensor product.

In \GGtheory{} this is the \DGI{} content of the factorization puzzle: the horizon split cannot be treated as a cut through independent microscopic degrees of freedom, because diffeomorphism constraints and edge data already make the subsystem notion subtler than the naive product.  The island story is indirect evidence for the same point --- when the radiation's entanglement wedge acquires an island, one is already conceding that its operational subsystem is not the naive geometric exterior.

Islands are then not a primitive ontology but the late-time support of the minimizing operational branch once \DGI{} and \OGGEP{} projection are respected.  Finally, the data-processing inequality makes the boundary description operationally honest: for the \CPTP{} projection $\Pi$ from bulk to boundary, relative entropy cannot increase, so the boundary may \emph{lose} information without the parent theory \emph{destroying} it.  Hawking's original argument acquires its paradoxical force only by treating the semiclassical exterior channel as the full ontology --- the move that \DGI{} and \OGGEP{} together reject.


\section{The explicit \OGGEP{} channel for a Schwarzschild black
hole}
\label{sec:explicit-channel}

\begin{figure}[tbp]
\centering
\resizebox{0.495\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small]
% bulk box (6D), pure
\draw[rounded corners, fill=violet!6, draw=violet!55] (0,0) rectangle (5.0,3.4);
\node[violet!55!black] at (2.5,3.05){6D bulk $\mathcal{M}_6$ --- pure, unitary};
\foreach \p in {(0.9,0.9),(1.5,1.7),(2.1,1.1),(2.7,1.9),(3.2,1.3),(1.2,1.2)}{\fill[blue!70!black] \p circle (2.1pt);}
\node[font=\scriptsize, gray!55!black, align=center] at (2.5,0.45){information carried by bulk modes\\(never destroyed)};
% info-bearing mode displaced along the non-spatial fibre (dashed red arc up-right)
\draw[->, red!70!black, dashed, line width=0.9pt] (3.2,1.3) .. controls (4.2,2.2) and (4.4,2.9) .. (4.55,3.15);
\node[red!70!black, font=\scriptsize, anchor=west, align=left] at (3.4,2.75){displaced along\\the fiber $(I_r,S^1_\theta)$};
% projection arrow pi
\draw[->, line width=1pt, gray!55!black] (5.2,1.7) -- (6.7,1.7);
\node[font=\scriptsize, anchor=south] at (5.95,1.75){$\pi$ (\OGGEP)};
\node[font=\scriptsize, anchor=north] at (5.95,1.65){trace fiber};
% boundary screen
\draw[fill=blue!9, draw=blue!55] (6.9,0.2) rectangle (7.5,3.2);
\node[blue!55!black, rotate=90, font=\scriptsize] at (7.2,1.7){4D boundary (mixed)};
\node[font=\scriptsize, gray!55!black, align=center, text width=2.6cm] at (8.9,1.7){appears ``lost'' to the 4D observer; recoverable in the 6D bulk};
\end{tikzpicture}}
\caption[Information loss as geometric distance.]{Information ``loss'' as geometric displacement.  The six-dimensional bulk state is pure and its information is never destroyed; the \OGGEP{} projection $\pi$ traces the non-spatial fiber, so a mode displaced along $(I_r,S^1_\theta)$ becomes inaccessible to the four-dimensional observer and the boundary state looks mixed.  What the 4D observer reads as lost information is, in the bulk, merely \emph{displaced} --- distance along a non-spatial direction, not destruction.}
\label{fig:info-distance}
\end{figure}

\noindent Figure~\ref{fig:info-distance} plots the distances just computed.



\subsection{Setup}

Consider a Schwarzschild black hole of mass $M$ in flat 4D
spacetime. The bulk geometry is approximately
\begin{equation}\label{eq:BH_6D_metric}
ds_6^2 = e^{-2r}\left[-f(r_4) dt^2 + dr_4^2/f(r_4) + r_4^2 d\Omega^2\right]
+ dr^2 + d\theta^2,
\end{equation}
where $r_4$ is the 4D radial coordinate with horizon at
$r_4 = r_s = 2\Newton M/c^2$ and $f(r_4) = 1 - r_s/r_4$. The
two non-spatial coordinates are the scale direction $r$ and the
phase angle $\theta$; the scale profile $e^{-2r}$ and the unit circle
period are fixed by \ATwo{} (Theorem~2), with no free scale rate
or circle radius.

\subsection{The \OGGEP{} channel}

The \OGGEP{} channel takes the 6D bulk state and produces the
4D exterior state by partial trace:
\begin{equation}\label{eq:OGGEP_channel}
\mathcal{D}_{\rm BH}:\;\rho_{\rm bulk}\;\mapsto\;\rho_\partial^{\rm
ext}\;=\;\Trin(\rho_{\rm bulk}),
\end{equation}
where $\Trin$ traces over (i) the compact dimensions $(r,
\theta)$ and (ii) the 4D interior $r_4 < r_s$.

\subsection{Kraus representation}

Any \CPTP{} channel admits a Kraus representation
$\mathcal{D}(\rho) = \sum_j K_j \rho K_j^\dagger$ with
$\sum_j K_j^\dagger K_j = \iden$. For $\mathcal{D}_{\rm BH}$, the
Kraus operators are labeled by (a) the Hawking-radiation
outgoing mode frequency $\omega$, (b) angular-momentum partial
wave $\ell, m$, and (c) occupation number $n$:
\begin{equation}\label{eq:Kraus_BH}
K^{T}_{\omega\ell m n} = \sqrt{p_n(\omega)\,\Gamma_\ell(\omega)}\;
\ket{\omega\ell m; n}\bra{\phi^{\rm in}_{\omega\ell m}},
\qquad
K^{R}_{\omega\ell m n} = \sqrt{p_n(\omega)\bigl(1 -
\Gamma_\ell(\omega)\bigr)}\;
\ket{\chi_{\omega\ell m n}}\bra{\phi^{\rm in}_{\omega\ell m}},
\end{equation}
where $p_n(\omega) = (1 - e^{-\beta\hbar\omega})e^{-n\beta
\hbar\omega}$ is the thermal probability at temperature
$\THaw = 1/\kB\beta$, $\Gamma_\ell(\omega)$ is the graybody
factor acting as a beamsplitter of the emitted flux, and the
$\ket{\chi}$ are the reflected, horizon-returning states; the input
bra is the single infalling state of the mode, so completeness
closes exactly: $\sum_n\bigl(K^{T\dagger}K^{T} +
K^{R\dagger}K^{R}\bigr) = \sum_n p_n\,
\ket{\phi^{\rm in}_{\omega\ell m}}\bra{\phi^{\rm in}_{\omega\ell m}}
= \iden$ on the per-mode input span.

The observable output for an initial pure infalling state traces
the reflected branch; per mode it is
\begin{equation}\label{eq:rho_out_explicit}
\rho_\partial^{\rm out}\big|_{\omega\ell m}
= \sum_{n} \tilde p_n(\omega)
\ket{\omega\ell m;n}\bra{\omega\ell m;n},
\end{equation}
with $\tilde p_n(\omega)$ the thermal distribution at the
\emph{filtered} occupation $\bar n_{\rm obs} =
\Gamma_\ell(\omega)\,\bar n(\omega)$: at the per-quantum
(fine-grained) level the barrier is a Gaussian beamsplitter, and a
thermal state through a beamsplitter is exactly thermal at the
transmitted occupation --- the same Gaussian fact the exchange
ledger of Appendix~\ref{app:data-processing} runs on.  The
two-branch set above is the mode-level coarse-graining of that
channel, matching it in mean flux --- the graybody-filtered Planck
flux of Eq.~\eqref{eq:greybody}.  One assumption in this
construction now carries its cohort name: labeling the Kraus
operators mode by mode, and composing them over the emission
history, presumes that each engaged mode pair presents fresh ---
uncorrelated with the boundary record --- at engagement.  That is
the Port-Freshness Clause of the open-system law (\AFive{},
Definition~5; \citealp{ArisakaA5GDOS}), consumed here with a
validity window rather than silently: the window, and its scheduled
edge at the Page time, are the content of
Lemma~\ref{lem:stale-port}.

\subsection{The entropy of the output state}

The von Neumann entropy of $\rho_\partial^{\rm out}$ is
\begin{equation}\label{eq:Sout}
\SvN(\rho_\partial^{\rm out}) = -\sum_{\omega\ell m n}
\tilde p_n(\omega) \ln \tilde p_n(\omega)
\approx \Sbh = \frac{A}{4\Newton\hbar}\kB,
\end{equation}
at leading order, with the approximation holding when the
thermal radiation dominates the phase space. The coefficient
of $1/4$ is reproduced, and the area-scaling is automatic.

\subsection{Compositionality: post-evaporation state}

After complete evaporation, the black hole is gone and the
radiation carries the full bulk state projected to 4D. In
\GDOS{}, this is:
\begin{equation}\label{eq:post_evap}
\rho_{\partial R}^{\rm final} = \mathcal{D}_{\rm total}
(\rho_{\rm initial}),
\end{equation}
where $\mathcal{D}_{\rm total}$ is the composition of all the
Kraus operations over the evaporation history. Crucially,
for a bulk-pure initial state, the final 4D state is a pure
state in the radiation --- provided the radiation sector
includes \emph{all} degrees of freedom, including very-late
subtle correlations. This is the manifest unitarity of the
total evolution.  Those very-late subtle correlations are now named
objects: they are the stale-port data of Lemma~\ref{lem:stale-port}
--- the correlations that \emph{must} appear once the freshness
budget of the pure total state is spent.

\section{The Page curve in \GDOS}

\subsection{Setup}

Consider a black hole formed from a pure quantum state
$\ket{\psi_0}$.  At time $t$, the system consists of two
registers: the radiation $R(t)$ --- Hawking quanta that have
already been emitted and propagated outward --- and the black
hole $B(t)$, the remainder, with reduced mass $M(t) < M(0)$.
Bulk unitarity says that the total state
$\ket{\psi(t)}_{R \otimes B}$ is still pure, evolving via the
unitary 6D evolution.

The Page curve is the time evolution of the entanglement
entropy $S_R(t) = \SvN(\rho_R(t))$ of the radiation.

\subsection{\GDOS{} derivation}

In \GDOS{}, the radiation is produced by the \OGGEP{} channel at
the horizon. The bulk state initially is approximately:
\begin{equation}\label{eq:bulk_initial}
\ket{\Psi_0}_{\rm bulk} = \ket{\psi_0}_{M_4} \otimes
\ket{\Omega}_{\rm compact},
\end{equation}
where $\ket{\psi_0}_{M_4}$ is the 4D matter state that will
collapse, and $\ket{\Omega}$ is the bulk-compact ground state.

As evaporation proceeds, bulk unitary evolution entangles
$\ket{\psi_0}_{M_4}$ progressively with the compact bulk degrees
of freedom, and the \OGGEP{} channel projects the outside-
horizon subsector.

\emph{Early times} ($t \ll t_P$). Most of the bulk state is
still ``inside'' the black hole (interior $+$ compact). The
radiation $R(t)$ has only weak correlations with the black hole
$B(t)$. Its entanglement entropy grows as
\begin{equation}\label{eq:Page_early}
S_R(t) \approx \Sbh^{\rm emitted}(t) \quad (\text{linear in
radiated area}),
\end{equation}
consistent with the Hawking result.

\emph{Page time} ($t = t_P$). Half the initial area has been
radiated. The bulk bipartition has equal-dimensional sides.
The entanglement entropy is maximal.

\emph{Late times} ($t > t_P$). The radiation is now the larger
subsystem. Its entanglement entropy equals that of the black
hole:
\begin{equation}\label{eq:Page_late}
S_R(t) = \Sbh^{\rm remaining}(t) \quad (\text{decreasing}),
\end{equation}
by the bipartite pure-state property $\SvN(\rho_R) =
\SvN(\rho_B)$. The radiation entropy decreases as the black
hole shrinks.

\emph{End of evaporation}. Final state is pure; radiation
entropy is zero.

This is the Page curve, derived directly from the \GDOS{}
structure without any exotic mechanisms. The key insight:
\emph{the Page curve is automatic once one recognizes that
bulk evolution is unitary and the boundary sees a \CPTP{}
partial trace}. No replica wormholes, no quantum extremal
surfaces (as independent principles), no islands formula as a
separate postulate are needed.

\subsection{Connection to the islands formula}

The islands formula (Eq.~\ref{eq:island}) gives the entropy of
the radiation as
\begin{equation}\label{eq:islands_again}
S(R) = \min\,\mathrm{ext}\Big\{\frac{A(\partial I)}{4\Newton\hbar}
+ S_{\rm semi-cl}(R \cup I)\Big\}.
\end{equation}
At sub-Page times, no island is picked (the trivial $I =
\emptyset$ minimizes), and $S(R) = S_{\rm semi-cl}(R)$
(Hawking's result). At super-Page times, a non-trivial island
$I \subset $ black hole interior picks up, and $S(R)$ starts to
decrease.

In \GDOS{}, the island $I$ is exactly the \emph{bulk region
that the \OGGEP{} functor traces over}. At sub-Page times, this
region is mostly the interior; its entanglement contribution is
already captured by the black-hole entropy term. At super-Page
times, the bulk-traced region effectively extends into the
black-hole interior ``from the radiation side,'' giving the
island its bulk extent.

The islands formula is therefore a formal consequence of the
\GDOS{} structure. The QES (quantum extremal surface) is the
boundary $\partial I$ of the traced-over region, and the
extremization prescription is the saddle point of the \OGGEP{}
path integral.

\GDOS{} and the islands formula agree on the Page curve. The
advantage of \GDOS{} is that the derivation is
\emph{physically concrete}: the 6D bulk provides a realization,
the \OGGEP{} partial trace is a specific operation, and the
resulting Page curve is geometric rather than arising from
replica wormholes.

\section{Hawking--Page and the Page curve as branch-switch events}
\label{sec:branch-switch}

\subsection{Hawking--Page as a gravitational branch selection}
The Hawking--Page transition is usually presented as a result of Euclidean gravity, but its importance here is methodological: it is one of the cleanest cases in standard gravity where the realized history is selected by a competition among scalar thermodynamic branches rather than by local equations of motion alone.  Writing the coarse-grained ledger of the two competing saddles as $\cF_{\rm th\text{-}AdS}(T)$ and $\cF_{\rm BH}(T)$, the equilibrium phase is the one of lower free energy; in \GGtheory{} language the runtime admits multiple thermodynamically admissible branches and the realized branch minimizes the scalar ledger.  The Hawking--Page temperature is thus a gravitational instance of branch selection, showing that the \GGtheory{} branch grammar is not imported from chemistry or from measurement --- it is already latent in standard black-hole thermodynamics (the explicit free-energy comparison is recorded in the Hawking--Page appendix). In the two-regime vocabulary of the runtime (\citealp{ArisakaA6Gi4}; \citealp{ArisakaA11Origin}) this transition now has its canonical name: it is a \GRIPout{} --- temperature is a control parameter of the ledger, and as it crosses $T_{\rm HP}$ the free-energy landscape \emph{itself} reorganizes, the dominant saddle structure changing identity. It is, in fact, the one genuine \GRIPout{} of black-hole physics (\S\ref{ssec:bh-sorting}); everything else in the black hole's life --- collapse, merger, evaporation --- happens on a fixed landscape.

\subsection{The Page curve as a \GFF{} branch minimum}

\begin{figure}[tbp]
\centering
\resizebox{0.873\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small]
\draw[->] (0,0) -- (9,0) node[right]{time $t$};
\draw[->] (0,0) -- (0,4.6) node[above, align=left]{radiation entropy $S_R$};
\draw[dashed, gray!70] (8,0) node[below]{$t_{\rm evap}$} -- (8,4.0);
\draw[dashed, gray!70] (0,4.0) node[left]{$S_{\rm BH}(M_0)$} -- (8,4.0);
\draw[dashed, gray!70] (0,2.0) node[left]{$\tfrac12 S_{\rm BH}(M_0)$} -- (5.17,2.0);
% Hawking (naive, thermal): S_BH(M_0) - S_BH(M(t)), rising to S_BH(M_0)
\draw[red!75!black, dashed, line width=0.80pt] plot[smooth] coordinates {(0,0)(1,0.34)(2,0.70)(3,1.08)(4,1.48)(5,1.92)(5.17,2.0)(6,2.41)(7,3.0)(7.7,3.55)(8,4.0)};
\node[red!75!black, anchor=west, font=\scriptsize] at (4.9,3.55){Hawking (naive, thermal)};
% Page (unitary): follows the thermal branch to t_Page = 0.646 t_evap,
% then rides S_BH(M(t)) down
\draw[blue!75!black, line width=0.80pt] plot[smooth] coordinates {(0,0)(1,0.34)(2,0.70)(3,1.08)(4,1.48)(5,1.92)(5.17,2.0)};
\draw[blue!75!black, line width=0.80pt] plot[smooth] coordinates {(5.17,2.0)(6,1.59)(7,1.0)(7.7,0.45)(8,0)};
\node[blue!75!black, anchor=west, font=\scriptsize] at (5.35,2.25){Page (unitary)};
\draw[dotted] (5.17,0) node[below]{$t_{\rm Page} \approx 0.65\,t_{\rm evap}$} -- (5.17,2.0);
\node[align=center, font=\scriptsize, gray!55!black] at (5.17,-0.75){branch switch (\S\ref{sec:branch-switch})};
\end{tikzpicture}}
\caption[The Page curve as a branch switch.]{The radiation entanglement entropy.  The naive thermal calculation (red dashed) rises monotonically to $S_{\rm BH}(M_0)$, implying information loss.  Unitarity requires the Page curve (blue): it follows the thermal branch until the \emph{Page time} $t_{\rm Page}$, when the area-limited branch becomes dominant in the \GFF{} ledger, and then falls to zero as the black hole evaporates.  In \GGtheory{} the turnover is the branch switch of \S\ref{sec:branch-switch}, automatic from bulk unitarity plus the \OGGEP{} partial trace; no islands postulate is added.}
\label{fig:page-curve}
\end{figure}

Consider two competing operational branches for the radiation entropy.  The early-time Hawking branch grows as the black hole loses entropy,
\begin{equation}
S_R^{(1)}(t) \approx \eta\bigl[S_{\rm BH}(M_0) - S_{\rm BH}(M(t))\bigr],
\label{eq:branch-one}
\end{equation}
while the late-time bridge/island-like branch is capped by the remaining geometric ledger,
\begin{equation}
S_R^{(2)}(t) \approx S_{\rm BH}(M(t)).
\label{eq:branch-two}
\end{equation}
The operational entropy is the branch minimum $S_R(t) = \min\{S_R^{(1)},S_R^{(2)}\}$ of \eqref{eq:branch-min} (Figure~\ref{fig:page-curve}).  For $\eta \simeq 1$ the crossing occurs when $S_{\rm BH}(M_{\rm Page}) = \tfrac12 S_{\rm BH}(M_0)$, i.e.
\begin{equation}
M_{\rm Page} = \frac{M_0}{\sqrt2}, \qquad
t_{\rm Page} \approx \bigl(1 - 2^{-3/2}\bigr)\, t_{\rm evap}.
\label{eq:page-time}
\end{equation}
The Page turnover is therefore not a miraculous late correction tacked onto Hawking's early-time story; it is the moment when a different operational branch becomes dominant in the entropy ledger; equivalently, in the boundary-side reading of \citep{ArisakaA5GDOS}, the Page curve is a theorem of \OGGEP{} partial-tracing plus bulk unitarity (the bipartite-pure-state argument), of which the branch-switch is the free-energy-ledger view. In regime language the classification is sharper still, and it strengthens the no-firewall conclusion of \S\ref{sec:no-firewalls}: the Page time is a \emph{milestone of \GRIPin{}}, not a transition --- like matter--radiation equality in cosmology, it is a bookkeeping crossover on a landscape that does not move. Nothing physical reorganizes at $t_{\rm Page}$; there is therefore nothing at the Page time for a firewall to ride on. The taxonomy itself rules the drama out.

\setcounter{euclidtheorem}{1}% key alignment: T-BH-n = Theorem n
\begin{euclidtheorem}[\GGtheory{} Page-turnover as a branch switch \textnormal{(T-BH-2)}]
\label{thm:page-turnover}
Within the minimal black-hole module of \GGtheory{}/\GDOS{}, the Page turnover is the branch-switch time at which the \GFF{}-minimizing operational saddle ceases to be the Hawking-like thermal branch and becomes an area-limited bridge branch.  Page-curve recovery is in this language a \GSSB{} event in the entropy ledger, not a late violation of early-time semiclassical validity.
\end{euclidtheorem}

\subsection{The mechanism: port freshness and its scheduled expiry}
\label{ssec:freshness-mechanism}

Theorem~\ref{thm:page-turnover} \emph{names} the turnover; this
subsection supplies its mechanism, in the vocabulary the cohort
maintains for exactly this purpose.  The open-system law runs on one
named working assumption, stated once for the whole program as the
\emph{Port-Freshness Clause} (\textsf{L-GDOS-Fresh}; \AFive{},
Definition~5): every engaged environment port presents a fresh,
memory-free state to the operational dictionary --- the boundary
system and each engaged port are uncorrelated at engagement.  That
clause is \AFive{}'s single home of the factorizable initial state,
the Markov property, and what \AEight{} calls the memoryless bath;
\AFive{} grades it honestly as the correlational half of molecular
chaos, the cohort's declared past-hypothesis-equivalent
\citep{ArisakaA5GDOS}.  The channel constructions of this paper
consume it --- the mode-by-mode Kraus product and its composition
over the emission history (\S\ref{sec:explicit-channel}) are
exactly factorized-engagement statements --- and until now they
consumed it silently.  At a black-hole horizon the clause has a
sharply bounded validity window, and the window's edge is not set by
dynamics at the port: it is set by bookkeeping, and it \emph{is} the
Page time.

\begin{euclidlemma}[The stale port: freshness expires at the Page
time \textnormal{(L-BH-Fresh)}]
\label{lem:stale-port}
For a black hole formed from a pure state, evolving unitarily in the
bulk, with boundary record $R(t)$ the emitted radiation, and granting
the typicality premise of Appendix~\ref{app:Page_curve} (an imported
dynamical assumption, flagged there):
(i) \emph{(fresh regime)} for $t \ll t_{\rm Page}$ each newly engaged
horizon mode is uncorrelated with $R(t)$ up to corrections
exponentially small in the unread entropy, so the Port-Freshness
Clause holds at the horizon and the mode-by-mode channel composition
of \S\ref{sec:explicit-channel} is exact at that grade; this is the
regime in which the \GFF{}-minimizing branch is the Hawking-like
thermal branch of Eq.~\eqref{eq:branch-one};
(ii) \emph{(finite budget)} the clause's correlational content has a
finite budget at a horizon: a pure total state can present fresh port
modes only while the unread sector holds more than half the total
information, so freshness expires at $t_{\rm Page}$ by counting alone
--- no new interaction, no new scale, no change in the port dynamics;
(iii) \emph{(stale regime)} for $t > t_{\rm Page}$ every newly
engaged mode is already correlated with the record --- in the
coarse-grained model the mutual information per emitted unit
approaches its maximum, $2\ln 2$ --- and these forced correlations
are precisely the late-time data that cap the radiation entropy at
the area-limited bridge branch of Eq.~\eqref{eq:branch-two}.
The branch switch of Theorem~\ref{thm:page-turnover} is therefore
the scheduled expiry of \textsf{L-GDOS-Fresh} at the horizon: the
information return is not a mechanism switching on but an assumption
switching off.  Grade: Derived-conditional --- derived from bulk
unitarity and the bipartite pure-state property, conditional on the
typicality premise, and consuming \AFive{}'s clause as the name of
what expires.
\end{euclidlemma}

\begin{proof}[Proof (coarse-grained model)]
Work in the $N$-unit model of Appendix~\ref{app:Page_curve}: after
$m$ units are emitted the record is $R$ (dimension $2^m$), the next
engaged mode is one unit $q$, and the unread remainder has dimension
$2^{N-m-1}$.  Freshness at engagement is the statement
$I(q\!:\!R) = S_q + S_R - S_{Rq} \approx 0$.  (i) For
$m < N/2$ Page's averages give
$\langle S_X \rangle = \ln d_X - d_X/2d_{\bar X} + \cdots$ for every
subsystem $X$ smaller than its complement, so the three leading
logarithms cancel in $I(q\!:\!R)$ and the survivor is the correction
term, of order $d_R d_q / d_{\bar X} \sim 4^{\,m - N/2}$:
exponentially small in $N - 2m$.  (ii) is the pure-state constraint
$S_R = S_{\bar R}$: once $d_R > d_{\bar R}$ the record's entropy is
capped by the shrinking complement, $S_R \approx (N-m)\ln 2$, so each
further emission \emph{lowers} $S_R$ by $\ln 2$ while adding a unit
--- arithmetic possible only if the added unit arrives correlated
with the record.  (iii) then follows by the same counting:
$I(q\!:\!R) \to 2\ln 2$, the maximum for one unit.  Numerically
($N = 12$, Haar-random states, five realizations, verification suite
at www.arisaka-gg.org): $I(q\!:\!R) = 0.002\,\ln 2$ at $m = 1$, growing
by the predicted factor $\simeq 4$ per step ($4.05, 4.13, 4.12,
4.03$ measured), crossing at the Page point $m = 6$, and saturating
at $1.997\,\ln 2$ of the maximum $2\ln 2$ at $m = 10$; the record's
entropy traces the Page tent throughout.  What is \emph{not}
claimed: the gravitating micro-dynamics by which the correlations
are delivered (scrambling; \citealp{HaydenPreskill2007}), any
modification of the clause's status or grade at its \AFive{} home,
and any failure of the clause elsewhere in the cohort --- the black
hole is the one system whose own schedule drives the clause from
holding exactly to failing maximally within a single history, which
is what makes it the cohort's sharpest stress test of
\textsf{L-GDOS-Fresh}.
\end{proof}

\paragraph{Physics intuition.}  A dealer dealing from a finite,
well-shuffled deck: early hands are fresh --- nothing on the table
tells you what comes next.  Past the midpoint of the deck, every
further card is increasingly readable from the table, and the last
card is fully determined by it.  The dealer never changes the rules;
the deck simply runs out of newness.  A black hole formed from a
pure state deals from a finite deck, the Page time is the midpoint,
and the celebrated ``information return'' is the table beginning to
read the deck.  Firewall arguments, in this language, mistook the
growing readability of the cards for a change in how the dealer
deals; Theorem~\ref{thm:page-turnover} and the lemma say the dealing
--- the local horizon physics --- never changes at all.

\subsection{Hayden--Preskill, and why no firewalls or remnants are forced}
The Hayden--Preskill scenario fits naturally: once the Page turnover has occurred, and after a scrambling delay, newly injected information can in principle be reconstructed from the outgoing channel, because the boundary-accessible map is no longer trapped on the purely thermal branch.  There is no contradiction with early-time thermality, because the branch structure is itself time-dependent.  The key refinement is that \GGtheory{} distinguishes three layers --- bulk purification, boundary accessibility, and detector-level reconstruction --- and confusion among them inflates paradoxes in the literature.  Once the ledger is branch-structured and the subsystem notion is operational rather than naive, several older cures lose their necessity: one need not postulate a generic firewall to preserve entropy bookkeeping, nor hide the state count in an implausibly degenerate remnant.  Honesty requires the narrow form of the claim: \GGtheory{} does not prove firewalls impossible in every UV completion; it shows that entropy bookkeeping by itself does not force them once the global-versus-operational distinction is made correctly.


\section{No firewalls in \GDOS}
\label{sec:no-firewalls}

\subsection{The monogamy puzzle dissolves}

Recall the AMPS argument: the late-time Hawking quantum $B$
must be entangled with both the earlier radiation $R$ (for
unitarity) and its interior partner $A$ (for no-drama at the
horizon). Entanglement monogamy rules this out.

In \GDOS{}, the resolution is:

\emph{There is no independent ``interior partner $A$.''}
The apparent partner is part of the 6D bulk, which the
\OGGEP{} functor traces over. $B$ is entangled with a
\emph{bulk state}, and the bulk state has been partial-traced
to produce both the ``interior'' (as seen by an infalling
observer) and the ``exterior'' (as seen by the outside
observer).

Formally: AMPS assumes that the 4D Hilbert space factorizes
as $\Hilb_A \otimes \Hilb_B \otimes \Hilb_R$ with
$\Hilb_A, \Hilb_B, \Hilb_R$ independent. In \GDOS{}, the 4D
description arises from tracing over the bulk, and the three
subsystems are not really independent at the 6D level.

More concretely: the apparent partner $A$ is part of a bulk
configuration that is also entangled with $R$ through the
bulk dynamics. The entanglement is not monogamous because it is
redistributed over the bulk, which has much more room than the
4D-Hilbert-space decomposition admits.

\subsection{The infalling observer sees no firewall}

An observer falling through the horizon does not burn up, for
three reasons that are really one.  In the bulk 6D picture, the
observer simply transitions from accessing
scale-profile-$e^{A(r_\ast)}$ degrees of freedom (the exterior)
to accessing scale-profile-$e^{A(r_\ast')}$ degrees of freedom
(the interior, at smaller scale profile).  The horizon is a
regular 6D surface, and no high-energy quanta accumulate there.
And the apparent high energies at the horizon --- those of the
Unruh frame of an accelerated exterior observer --- are
frame-dependent: in the infalling frame, nothing unusual
happens.

This is the standard GR no-drama picture, recovered automatically
at the 6D level. The firewall paradox was an artifact of insisting
on a purely 4D description of a fundamentally 6D system.

\section{The interior of the black hole in \GDOS}

\begin{figure}[tbp]
\centering
\resizebox{0.873\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small, scale=1.05]
% r=0 worldline (left vertical), regular below the singularity
\draw[line width=0.80pt] (0,0) -- (0,3.4);
\node[font=\scriptsize, anchor=east, rotate=90] at (-0.12,1.4){$r=0$ (center)};
% singularity: hand-drawn zigzag spacelike line at top-left (GG: regular small-scale-profile region)
\draw[red!70!black, line width=0.80pt]
 (0,3.4)--(0.25,3.55)--(0.5,3.4)--(0.75,3.55)--(1.0,3.4)--(1.25,3.55)--(1.5,3.4)--(1.7,3.5);
\node[red!70!black, font=\scriptsize, anchor=south west] at (0.1,3.6){singularity \;(6D: regular small-scale-profile region)};
% i^0, I^+, I^-
\coordinate (i0) at (3.4,1.7);
\coordinate (sing) at (1.7,3.4);
\draw[line width=0.80pt] (0,0) -- (i0) node[right, font=\scriptsize]{$i^0$};   % I^-
\draw[line width=0.80pt] (i0) -- (sing);                                       % I^+
\node[font=\scriptsize] at (2.85,0.95){$\mathcal{I}^-$};
\node[font=\scriptsize] at (2.85,2.55){$\mathcal{I}^+$};
% event horizon (null, 45 deg) from collapse point on r=0 to the singularity endpoint
\draw[blue!70!black, line width=1pt] (0,1.1) -- (sing);
\node[blue!70!black, font=\scriptsize, rotate=44, anchor=south] at (0.95,2.3){horizon $=$ \OGGEP{} port};
% interior (above horizon, below singularity): traced
\fill[gray!22] (0,1.1) -- (sing) -- (0,3.4) -- cycle;
\node[gray!55!black, font=\scriptsize, align=center] at (0.5,2.55){interior\\$+$ fiber\\(traced)};
% collapsing star (shaded timelike strip from i^- up to horizon formation)
\fill[orange!28] (0,0) -- (0,1.1) -- (0.55,1.35) -- (0.25,0) -- cycle;
\node[orange!60!black, font=\scriptsize, anchor=west] at (0.55,0.55){collapsing star};
% Hawking radiation: arrows from near horizon to I^+
\draw[->, orange!80!black] (1.15,1.85) -- (2.15,2.35);
\draw[->, orange!80!black] (1.55,2.25) -- (2.45,2.7);
\node[orange!75!black, font=\scriptsize, anchor=west] at (2.2,1.95){Hawking $\to \mathcal{I}^+$};
\end{tikzpicture}}
\caption[Conformal (Penrose) diagram of an evaporating black hole as an \OGGEP{} port.]{Schematic conformal (Penrose) diagram of gravitational collapse and evaporation.  Time runs upward; each $45^\circ$ line is null.  The event horizon (blue) is the one-way \OGGEP{} port: the exterior observer on $\mathcal{I}^+$ receives the Hawking radiation (orange) but cannot access the interior, which --- together with the non-spatial fiber --- is traced out.  In \GGtheory{} the classical singularity is a coordinate artifact of the small-scale-profile region and is regular in the six-dimensional bulk, and the horizon carries no firewall because it is a regular 6D surface.}
\label{fig:penrose}
\end{figure}

\noindent Figure~\ref{fig:penrose} shows the causal structure on which this Part reads the information flow.



\subsection{What's inside?}

In 4D, the interior of a Schwarzschild black hole ($r < r_s$)
contains a curvature singularity at $r = 0$. In classical GR,
this singularity is a region of infinite tidal forces where
classical theory breaks down.

In \GDOS{}, the 4D interior is part of the 6D bulk. The
apparent 4D singularity corresponds to the region where the
scale profile is very small, $e^{-r} \ll 1$. In 6D, this is
not a singular region; it is just a region where the 4D
metric is strongly scale-running, and the effective low-energy physics
there is different from the exterior.

What the infalling 4D observer sees is a journey through the
projection, not a march toward a wall.  Near the horizon the 4D
metric looks Schwarzschild, and the observer still sees the
exterior.  Just inside, the metric becomes unusual: what was
``spatial'' becomes ``temporal'' in Schwarzschild coordinates.
Deep inside, the scale running is extreme, and the 4D metric
becomes effectively a null surface.  And at $r = 0$ the 4D metric
is singular --- but the 6D metric is smooth: the ``singularity''
is an artifact of the 4D projection.

\subsection{Information doesn't ``fall'' into the singularity}
\label{ssec:singularity-6d}

In the classical picture, information that crosses the horizon
is destined to end up at the singularity. In \GDOS{}, this is
not quite right: the 4D information that ``crosses'' the
horizon simply enters the region of smaller scale profile, where
it becomes inaccessible to the 4D exterior observer but
continues to exist in the 6D bulk.

Over time, the bulk dynamics redistribute this information
across the compact directions and the radial throat. Eventually,
this redistribution leads to it being carried off by Hawking
radiation --- but the redistribution is slow, and for
astrophysical black holes, very long timescales are involved
(Page time $\sim (\Newton M^2/\hbar)$).

The singularity, insofar as it exists, is a coordinate singularity
in the 4D projection, not a physical endpoint of information.

\section{Black-hole complementarity revisited}

\subsection{Susskind-Thorlacius-Uglum 1993}

The original black-hole complementarity proposal
\citep{STU1993} was: an exterior observer and an infalling
observer both have valid, but mutually inconsistent, descriptions
of the black hole. The exterior observer sees information
preserved in the Hawking radiation; the infalling observer
sees information entering the interior. These two descriptions
are complementary in the Bohr sense: one cannot be tested in
a single experiment against the other.

The complementarity proposal faced the AMPS challenge in 2012
because it appeared to require the same information to be
present in two different places (radiation AND interior) without
a single observer being able to verify both --- hence the
monogamy violation.

\subsection{\GDOS{} complementarity}

In \GDOS{}, complementarity has a clean geometric origin: the
exterior and infalling observers access \emph{different
projections of the 6D bulk state}.  The exterior observer sees
$\rho_\partial^{\rm out}$, the partial trace over the interior
4D region and the compact dimensions.  The infalling observer
sees a different partial trace --- one that retains the 4D
interior but still traces over the compact dimensions, which
remain inaccessible from their reference frame.

The two observers see different density matrices, but both are
valid \OGGEP{} projections of the same unique 6D bulk state.
There is no contradiction, and no firewall is required to
enforce consistency, because the complementarity is just two
different partial traces of the same underlying state.

\section{The four laws of black-hole mechanics as \GDOS{} statements}
\label{sec:four-laws-gg}

The Bardeen--Carter--Hawking laws \citep{BardeenCarterHawking1973} were originally a formal \emph{analogy} between black-hole mechanics and thermodynamics.  In \GGtheory{} the analogy is an identity: each law is a statement about the \OGGEP{} port and the \GDOS{} open-system dynamics.

\paragraph{Zeroth law (constant $\kappa$).}  The surface gravity $\kappa$ is constant over the horizon of a stationary black hole.  In \GDOS{} language $\kappa$ is the single scale/Unruh--Tolman rate of the \OGGEP{} port (\S\ref{ssec:bogoliubov}); its constancy says the port is characterized by one temperature $T=\hbar\kappa/2\pi c k_B$ --- the boundary \CPTP{} channel has a single thermal scale, the gravitational counterpart of a body in equilibrium having one temperature. In regime language: the zeroth law is basin-floor equilibrium --- the certificate that the black hole is a genuine Geometric Attractor in residence, not a passing configuration.

\paragraph{First law (energy balance).}  $dM = \tfrac{\kappa}{8\pi G_N}\,dA + \Omega_H\,dJ + \Phi_H\,dQ$.  With the induced-gravity entropy $S=A/4G_N$ and $T=\hbar\kappa/2\pi$, this is exactly $dM = T\,dS + \Omega_H\,dJ + \Phi_H\,dQ$ (\S\ref{ssec:V-kerr-rn}); the area term is the geometric piece $F_{\rm GEP}$ of the \GFF{} ledger (\S\ref{sec:gff-ledger}), and the first law is the thermodynamic energy balance of the \OGGEP{}-projected open system --- the bookkeeping of the descent, debiting the ledger as the exit channel carries mass--energy away.

\paragraph{Second law (generalized entropy increase).}  Hawking's area theorem $\delta A\ge 0$, sharpened by Bekenstein to the generalized second law $\delta S_{\rm gen}\ge 0$ \citep{Bekenstein1973}.  This is the sharpest \GGtheory{} reading of the four: it is the \emph{monotonicity of the \GFF{} ledger under the \GDOS{} \CPTP{}/Markov semigroup} --- the $H$-theorem of the open-system dynamics, equivalent to the data-processing inequality for the boundary channel.  The irreversibility is the forward renormalization-group arrow of the non-compact scale fiber (the \GDOS{} generator is the dilatation generator along $I_r$); the generalized second law and the arrow of measurement are one and the same arrow.  The second law is thus not an analogy to thermodynamics but the open-system $H$-theorem itself --- and, in the runtime vocabulary, it \emph{is} the Lyapunov monotonicity of \GRIPin{} (the descent theorem of \citep{ArisakaA11Origin}, anchored in the \GFF{} ledger structure theorems of \citep{ArisakaA6Gi4}) evaluated on the horizon stratum. The \emph{direction} of this identity is no longer carried by citation: it is derived link by link, with its ceiling declared, in \S\ref{sec:gsl-arrow} (Lemma~\ref{lem:gsl-arrow}, \textsf{L-BH-Arrow}) --- the arrow consumed from its constitutional owner, the monotonicity supplied by the channel identity, and the loss priced in area by the clock's thermal face.

\paragraph{Third law (unattainability of extremality).}  $\kappa\to 0$ (the extremal limit) cannot be reached in a finite number of operations.  In \GDOS{} language extremality is the boundary of the admissible \OGGEP{}-port family --- a port of zero temperature, whose \CPTP{} channel would have a pure output --- and the third law is the statement that the open-system dynamics cannot drive the port to a pure, zero-entropy-production output in finite steps: the gravitational counterpart of the unattainability of absolute zero. In regime language the third law acquires a pointed reading: an extremal hole would be a zero-temperature port --- an attractor whose Hawking exit channel has been closed \emph{by hand} --- and the law says no finite procedure can manufacture such an artificial ground state. Nature's \GGA{}s are earned by descent, not engineered by extremization.

\begin{table}[H]
\centering\small
\caption[The four laws as mechanics, thermodynamics, and \GDOS{}/\OGGEP{} statements.]{The four laws as mechanics, thermodynamics, and \GDOS{}/\OGGEP{} statements. The first three columns are standard and are reprinted so that the fourth can be read against them. That fourth column is the claim: each law of black-hole mechanics is a statement about an open system at a port, and the analogy Bardeen, Carter and Hawking could only note in 1973 becomes an identification. The first-law row carries the weight, because it is where the area term becomes a free-energy term rather than merely resembling one.}
\label{tab:four-laws}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.06\textwidth}p{0.27\textwidth}p{0.20\textwidth}p{0.32\textwidth}@{}}
\toprule
\textbf{Law} & \textbf{BH mechanics} & \textbf{Thermodynamics} & \textbf{\GGtheory{}/\GDOS{} reading} \\
\midrule
0th & $\kappa$ constant on horizon & $T$ uniform in equilibrium & \OGGEP{} port has one scale/Unruh rate $\kappa$; one channel temperature \\
1st & $dM=\tfrac{\kappa}{8\pi G_N}dA+\Omega\,dJ+\Phi\,dQ$ & $dE=T\,dS+\text{work}$ & Energy balance of the projected open system; $A/4G_N = F_{\rm GEP}$ \\
2nd & $\delta A\ge 0$;\ $\delta S_{\rm gen}\ge 0$ & $\delta S\ge 0$ & Monotonicity of the \GFF{} ledger under the \GDOS{} CPTP semigroup ($H$-theorem; RG arrow) \\
3rd & $\kappa\to 0$ unattainable & $T\to 0$ unattainable & Port cannot be driven to a pure (zero-$T$) output in finite steps \\
\bottomrule
\end{tabular}
\end{table}


\section{The direction of the second law: the arrow consumed, the price computed}
\label{sec:gsl-arrow}

The second-law paragraph of \S\ref{sec:four-laws-gg} makes the
strongest single claim of the four-laws reading: that the generalized
second law and the arrow of measurement are one and the same arrow.
A claim of that size should not rest on a citation.  This section
derives its \emph{direction} --- why $S_{\rm gen}$ increases toward
the future rather than the past, and at what rate --- from named
inputs, each consumed from its constitutional owner, with the one
genuinely hard remaining step declared open rather than absorbed.
The result is quantitative: in the solvable class that models the
horizon exchange, the generalized-entropy production between
successive cuts of the record is not merely non-negative --- it is
\emph{exactly} the relative entropy spent toward the clock's thermal
face, with closed-form values and a sharp equality point.

\subsection{Three links, three owners}
\label{ssec:arrow-links}

The claim ``$\Delta S_{\rm gen} \ge 0$ toward the future'' decomposes
into three logically independent links, and the cohort's custody
discipline assigns each to a different owner.  \emph{The paper that
uses a link is not the paper that proves it}; this module consumes
all three and adds only the horizon translation.
The cohort's survey of how a conservation law comes to stand without a
symmetry beneath it is \PaperWOne{} \citep{ArisakaW1Symmetry}.

\begin{table}[H]
\centering\small
\caption[The three links of the second law's direction, their owners, and their status in this paper.]{The three links of the second law's direction, their owners,
and their status in this paper.  The custody chain for the arrow is
the cohort's standing discipline: \AX{3} owns it, the CPT companion
derives it at law level over the exactly T-symmetric bulk, \AFive{}
implements it dynamically, and the runtime papers --- this one now
among them --- consume it and never re-derive it.}
\label{tab:arrow-links}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.055\textwidth}p{0.335\textwidth}p{0.27\textwidth}p{0.24\textwidth}@{}}
\toprule
\textbf{Link} & \textbf{Content} & \textbf{Owner} & \textbf{Status here} \\
\midrule
D1 & \emph{Which} time direction the boundary channel family composes
in: the record is ordered, and the ordering is not the horizon's
choice & \AX{3} (non-invertibility; \citealp{ArisakaA1Constitution});
law-level derivation in the CPT companion
(\citealp{ArisakaP5CPT}); dynamical implementation \AFive{}
(\citealp{ArisakaA5GDOS}) & Consumed, never re-derived \\
D2 & No channel increases distinguishability: relative entropy
contracts under every \CPTP{} map & Identity of quantum channels
(the data-processing inequality, Appendix~\ref{app:data-processing});
carries \emph{no} arrow content of its own & Identity; verified
numerically \\
D3 & The horizon translation: contraction of relative entropy toward
the clock's KMS face \emph{is} generalized-entropy increase, with the
loss priced in area & This paper: T-BH-5 (the face) $+$ the induced
first law (\S\ref{ssec:V-kerr-rn}) $+$ the exchange-channel identity
below & Derived here (quasi-free exchange class; linearized first
law) \\
\bottomrule
\end{tabular}
\end{table}

Two remarks before the lemma, because they are where confusion most
often enters.  \emph{First}, D2 is directionless: the data-processing
inequality constrains any channel, run in any direction; by itself it
cannot distinguish future from past.  The direction enters once and
only once, at D1 --- the constitutional statement that the boundary
description composes as channels toward the future and as nothing
lawful toward the past.  Papers that ``derive the arrow from the
second law,'' or the reverse, are circling between D1 and D2 without
naming them; the cohort's custody discipline exists to prevent
exactly that circle.  \emph{Second}, D3 is where all the physics
lives: it converts an information inequality into a geometric one,
and the conversion rate --- area per unit of spent distinguishability
--- is fixed by the same two theorems this paper has already paid
for: the KMS face of the clock (Theorem~\ref{thm:thermal-clock}) and
the induced first law (\S\ref{ssec:V-kerr-rn}).

\subsection{The lemma}

\begin{euclidlemma}[The direction of the generalized second law
\textnormal{(L-BH-Arrow)}]
\label{lem:gsl-arrow}
Let the boundary record at the horizon evolve by the \CPTP{} family
of the \OGGEP{} channel (\S\ref{sec:explicit-channel}), ordered by
the constitutional arrow (link D1, consumed).  Then:
(i) \emph{(monotone)} for every pair of record times $t_1 \le t_2$
in the constitutional ordering and every reference state $\sigma$
carried by the same channel, $S(\rho(t_2)\,\|\,\sigma(t_2)) \le
S(\rho(t_1)\,\|\,\sigma(t_1))$ --- the data-processing inequality,
an identity of \CPTP{} maps; this holds for the \emph{total} map
between any two record times and therefore survives the expiry of
port freshness (Lemma~\ref{lem:stale-port}): staleness breaks the
Markov semigroup form, never the direction;
(ii) \emph{(the price list)} in the Gaussian exchange class that
models the horizon channel at frequency $\omega$ --- transmissivity
$\Gamma \in (0,1]$, the clock's KMS state $\sigma_{\rm KMS}$ of
occupation $\bar n_H = (e^{\hbar\omega/k_B T_{\rm H}}-1)^{-1}$ as
fixed point, thermal record states, and the linearized induced first
law $\delta(A/4G_N) = \delta E_H / T_{\rm H}$ at fixed $(J,Q)$ ---
the generalized-entropy production between successive record cuts is
\emph{exactly} the spent relative entropy toward the thermal face,
\begin{equation}
\Delta S_{\rm gen}
\;=\; S\!\bigl(\rho_{\rm in}\,\big\|\,\sigma_{\rm KMS}\bigr)
\;-\; S\!\bigl(\rho_{\rm out}\,\big\|\,\sigma_{\rm KMS}\bigr)
\;\ge\; 0 ,
\label{eq:gsl-identity}
\end{equation}
with equality if and only if the record state is the fixed point
($\bar n_b = \bar n_H$, i.e.\ $T_b = T_{\rm H}$);
(iii) \emph{(closed forms)} in that class, per mode of frequency
$\omega$: pure emission ($\bar n_b = 0$, $\Gamma = 1$) produces
$\Delta S_{\rm gen} = \ln(1+\bar n_H) =
-\ln\bigl(1 - e^{-\hbar\omega/k_B T_{\rm H}}\bigr) > 0$; graybody
emission produces $\Delta S_{\rm gen} = g(\Gamma\bar n_H) -
\Gamma\,\hbar\omega\,\bar n_H/k_B T_{\rm H} \ge 0$; and exchange with
a bath at $T_b$ produces $\Delta S_{\rm gen} = g(\bar n_{\rm rec}) -
g(\bar n_b) + \Gamma(\bar n_b - \bar n_H)\,\hbar\omega/k_B T_{\rm H}
\ge 0$ with $\bar n_{\rm rec} = (1-\Gamma)\bar n_b + \Gamma\bar n_H$
and $g(n) = (n{+}1)\ln(n{+}1) - n\ln n$ --- positive for net
emission \emph{and} for net absorption alike.
Grade: (i) identity on a consumed direction; (ii)--(iii) Derived
(exchange class), Derived-conditional as a statement about the
astrophysical horizon (quasi-free grade, linearized first law).  The
general interacting statement is Open (\S\ref{ssec:arrow-ceiling}).
\end{euclidlemma}

\begin{proof}
\emph{Step 1 (the direction, consumed).}  By \AX{3} the boundary
description is the output of a one-way dictionary; the law-level
location of the arrow --- the dictionary's one-way reading, derived
over an exactly T-symmetric bulk --- is Part~IX of
the CPT companion \citep{ArisakaP5CPT}, and its dynamical
implementation as the forward-composing \CPTP{} family is \AFive{}'s
\citep{ArisakaA5GDOS}.  Nothing in this proof re-derives any of it;
the record times $t_1 \le t_2$ of clause (i) are ordered by that
consumed arrow, full stop.

\emph{Step 2 (the monotone).}  Clause (i) is the data-processing
inequality of Appendix~\ref{app:data-processing} applied to the total
map $\Lambda_{t_2 \leftarrow t_1}$, which is \CPTP{} by \T{2} of
\AFive{} regardless of whether the Markov factorization holds ---
hence the robustness against stale ports noted in the clause: the
freshness window of Lemma~\ref{lem:stale-port} bounds where the
\emph{semigroup} form is exact, while the direction statement needs
only complete positivity of the total map.  Verified numerically:
$300$ Haar-random Stinespring channels on random full-rank state
pairs give contraction in every trial (verification suite,
www.arisaka-gg.org).

\emph{Step 3 (the reference and the price).}  Take
$\sigma = \sigma_{\rm KMS}$, the thermal face the clock forces on the
regular exterior state (Theorem~\ref{thm:thermal-clock}(iii)) --- the
unique reference the horizon itself supplies.  For single-mode
thermal states the relative entropy is
\begin{equation}
S(\sigma_n \,\|\, \sigma_m)
\;=\; -\,g(n) \;+\; n \ln\!\frac{m+1}{m} \;+\; \ln(m+1),
\qquad g(n) = (n{+}1)\ln(n{+}1) - n\ln n ,
\label{eq:rel-thermal}
\end{equation}
and for $m = \bar n_H$ the middle coefficient is
$\ln\bigl((\bar n_H{+}1)/\bar n_H\bigr) = \hbar\omega/k_B T_{\rm H}$:
\emph{the modular weight of the KMS reference is the Clausius weight
of the horizon}.  This is the entire content of the price list: by
the induced first law (\S\ref{ssec:V-kerr-rn}, at fixed $J,Q$,
quasi-static, linearized), energy $\delta E_H$ crossing the horizon
moves the area term by $\delta(A/4G_N) = \delta E_H/T_{\rm H}$, which
is precisely the $n\,\hbar\omega/k_B T_{\rm H}$ term of
\eqref{eq:rel-thermal} mode by mode.

\emph{Step 4 (the exchange identity).}  Model the horizon channel at
frequency $\omega$ as the Gaussian exchange map of transmissivity
$\Gamma$ with fixed point $\sigma_{\rm KMS}$: a thermal record mode
$\bar n_b$ goes to $\bar n_{\rm rec} = (1-\Gamma)\bar n_b +
\Gamma \bar n_H$, and the energy delivered to the hole is
$\delta E_H = \Gamma\hbar\omega(\bar n_b - \bar n_H)$.  Then
\begin{align}
\Delta S_{\rm gen}
&= \underbrace{\bigl[g(\bar n_{\rm rec}) - g(\bar n_b)\bigr]}_{\Delta S_{\rm out}}
 \;+\; \underbrace{\Gamma(\bar n_b - \bar n_H)\,
 \frac{\hbar\omega}{k_B T_{\rm H}}}_{\Delta(A/4G_N)}
\nonumber\\[2pt]
&= S(\sigma_{\bar n_b}\,\|\,\sigma_{\bar n_H})
 - S(\sigma_{\bar n_{\rm rec}}\,\|\,\sigma_{\bar n_H}),
\label{eq:gsl-chain}
\end{align}
by direct substitution of \eqref{eq:rel-thermal}: the $g$-terms and
the Clausius terms reassemble into the two relative entropies with
nothing left over.  The right side is non-negative because the
exchange map is \CPTP{} with $\sigma_{\bar n_H}$ as fixed point ---
clause (i) applied with the horizon's own reference --- and vanishes
iff $\bar n_b = \bar n_H$.  This is the black-hole instance of the
entropy-production theorem for quantum dynamical semigroups with a
KMS fixed point \citep{Spohn1978}, and the relative-entropy reading
of horizon entropy bounds \citep{Casini2008}, here composed with the
induced first law so that the produced entropy is carried by
\emph{area}.  Numerically: the identity \eqref{eq:gsl-chain} holds to
$3.6\times10^{-15}$ over a grid of $(\omega/T_{\rm H},\,T_b/T_{\rm
H},\,\Gamma)$; the minimum of $\Delta S_{\rm gen}$ over a
$1.2\times10^{5}$-point sweep is $+1.2\times10^{-8}$ (attained
approaching $\bar n_b \to \bar n_H$), and the closed forms of clause
(iii) are reproduced to $10^{-12}$ (verification suite).

\emph{Step 5 (closed forms and their reading).}  Setting
$\bar n_b = 0$, $\Gamma = 1$ in \eqref{eq:gsl-chain} gives the pure
Hawking emission of one mode:
\begin{equation}
\Delta S_{\rm gen}\big|_{\rm emit}
= \ln(1+\bar n_H)
= -\ln\!\bigl(1 - e^{-\hbar\omega/k_B T_{\rm H}}\bigr)
= S\bigl(\ket{0}\!\bra{0}\,\big\|\,\sigma_{\rm KMS}\bigr) > 0 :
\label{eq:gsl-emission}
\end{equation}
the generalized entropy produced by emitting one mode is exactly the
relative entropy of the vacuum to the clock's thermal face --- the
distinguishability the exterior surrenders by letting the port
thermalize that mode.  Soft modes ($\hbar\omega \ll k_B T_{\rm H}$)
are the expensive ones, $\Delta S_{\rm gen} \simeq \ln(k_B T_{\rm
H}/\hbar\omega) \to \infty$; hard modes are nearly free,
$\Delta S_{\rm gen} \simeq e^{-\hbar\omega/k_B T_{\rm H}} \to 0$;
summed against the Planck weight and graybody factors this is the
familiar statement that Hawking radiation carries more entropy than
the area it spends, here obtained per mode and in closed form.  The
two-temperature form of clause (iii) settles the conditional-GGA
regime of \S\ref{ssec:conditional-gga} quantitatively: for
$T_b > T_{\rm H}$ (today's sky, CMB-fed holes) the flux reverses ---
the hole grows --- yet $\Delta S_{\rm gen} \ge 0$ unchanged, and the
equality point $T_b = T_{\rm H}$ is exactly the crossover mass
$M_\times$ of Eq.~\eqref{eq:crossover-mass}: \emph{the
conditional-GGA crossover is the GSL's equality point, the one mass
at which the horizon exchanges without producing entropy}.

\emph{Step 6 (what is not proven).}  The identity
\eqref{eq:gsl-chain} is exact in its class; the class is a model of
the horizon channel, not the interacting gravitating field.  The
ceiling is declared in \S\ref{ssec:arrow-ceiling}.
\end{proof}

\subsection{The declared ceiling}
\label{ssec:arrow-ceiling}

A Wall-class proof of the generalized second law --- all
semiclassical orders, arbitrary horizon cuts, interacting fields,
with the monotonicity of the relative entropy run against the
half-sided modular structure of the horizon algebra
\citep{Wall2012} --- is \emph{not} claimed here and is not within
reach of the present apparatus: it requires input of ANEC grade and
Hadamard control that this module imports rather than owns.  What
this section establishes is strictly less and strictly named: the
\emph{direction} is constitutional and consumed (D1); the
\emph{monotone} is a channel identity (D2); and the \emph{price} is
computed exactly in the exchange class that models the port, with
the linearized induced first law as the conversion rate (D3).  The
promotion path --- from the exchange class to arbitrary cuts of the
interacting horizon --- is registered as an open problem of this
module in \S\ref{sec:open-questions-c3}, alongside the species sum.
The Lyapunov reading of \S\ref{sec:four-laws-gg} (the \GRIPin{}
descent theorem evaluated on the horizon stratum) is thereby
supported at the same grade as its inputs: direction derived,
production non-negative in the modeled class, full interacting
monotonicity open.

\paragraph{Physics intuition.}  A customs post on a one-way bridge.
Which way the traffic flows is decreed far upstream --- by the
constitution, not by the customs officer (link D1).  That no cargo
gains value in transit is not a local regulation either; it is the
law of every checkpoint everywhere (link D2, the data-processing
inequality).  What the horizon adds --- the only thing it adds --- is
a \emph{tariff schedule} (link D3): every mode that crosses pays in
distinguishability, the price list is printed on the clock's thermal
face, and the till is the area.  One line of the schedule is worth
memorizing: a single emitted mode costs exactly $\ln(1+\bar n_H)$ ---
expensive for soft quanta, nearly free for hard ones --- and the toll
is zero only when the two banks stand at the same temperature, which
for a black hole in today's sky happens at exactly one mass, the
crossover $M_\times$ where its conditional ground-state status is
decided.  The second law's direction was never the horizon's to
choose; the horizon only sets the prices.

\section{The black hole as a way-station attractor: where the descent pauses, and where it ends}
\label{sec:bh-way-station}

\subsection{Hawking radiation is the exit channel}
The runtime vocabulary makes one structural fact about black holes
visible that the thermodynamic language leaves implicit. A black hole
is a \emph{Geometric Attractor} of the gravitational stratum --- a
deep, stable basin of the free-energy ledger, which is why it is
no-hair simple and why perturbed holes relax back to Kerr. But it is
\emph{not} the Ground Geometric Attractor, because it possesses an
exit channel: \textbf{Hawking radiation is the channel-mediated exit
of \GRIPin{} on the gravitational stratum} --- this paper's own
identification, extending to gravity the catalog of exit channels
(weak decay for unstable matter, Hubble friction for the
cosmological dark sector) whose named instances are \AEleven{}'s
\citep{ArisakaA11Origin}. A
stellar-mass black hole is therefore the most extreme \emph{way
station} in nature: residence time $\sim 10^{67}$~yr, the slowest
single \GRIPin{} step known. The descent does not end at the black
hole; it ends beyond it, at the fully radiated configuration ---
which then relaxes, with the rest of the universe, toward the
de~Sitter vacuum, the cosmological stratum's true \GGA{}
\citep{ArisakaA10Cosmos}. In one sentence: \emph{the information
paradox was the worry that the way station destroys the luggage; the
\GDOS{} reading of this Part shows the luggage is forwarded.}

One precision keeps this claim exact. The black hole \emph{maximizes
entropy} at fixed energy (the Bekenstein bound) --- how can a maximum
be a way station? Because the runtime does not descend entropy; it
descends \emph{free energy} ($\GFF = F_{\rm GEP} + F_{\rm CEP}$,
\S\ref{sec:gff-ledger}; the ledger is minimized at fixed conserved
charges, and on the horizon stratum its monotone content is exactly
$\delta S_{\rm gen} \ge 0$ --- lower ledger and higher generalized
entropy are one direction, not two claims), and the generalized second law of
\S\ref{sec:four-laws-gg} says precisely that the radiated
configuration has \emph{higher} generalized entropy and \emph{lower}
ledger than the hole that produced it. The GSL is not in tension with
the way-station reading; it is the signpost of the continuing
descent.

\subsection{Today's black holes are \emph{conditional} ground states}
\label{ssec:conditional-gga}
Whether an exit channel is open is not an intrinsic property of the
attractor; it is computed \emph{including the boundary conditions}
--- the boundary-port clause of the open-system law
\citep{ArisakaA5GDOS}. For a black hole the relevant port is the
cosmic bath: a hole of mass $M$ radiates at
\begin{equation}
T_H \;=\; \frac{\hbar c^3}{8\pi G_N M k_B}
\;\simeq\; 6.2\times10^{-8}\,\mathrm{K}\,\frac{M_\odot}{M},
\label{eq:th-conditional}
\end{equation}
and sits in the CMB at $T_{\rm CMB} = 2.725$~K. For $T_H < T_{\rm
CMB}$ the channel runs \emph{backward}: the hole absorbs more than it
emits and grows. Since every astrophysical black hole has $T_H$ at
least seven orders of magnitude below the CMB, \textbf{every black
hole in today's sky is conditionally exitless --- a conditional
\GGA{}}, in precisely the sense in which neutron matter is the
conditional ground state inside a neutron star. The crossover is
sharp:
\begin{equation}
T_H = T_{\rm CMB}
\quad\Longleftrightarrow\quad
M_\times \;\simeq\; 4.5\times10^{22}\,\mathrm{kg}
\;\simeq\; 0.61\,M_{\rm Moon}:
\label{eq:crossover-mass}
\end{equation}
primordial holes below $M_\times$ are way-station \GA{}s in active
descent today; everything above it is a conditional \GGA{}. And the
condition expires on a calculable schedule: as the CMB cools,
$T_{\rm CMB}(t) \propto 1/a(t)$, the exit channel opens for
ever-larger holes,
\begin{equation}
M_{\rm open}(t) \;\propto\; a(t),
\label{eq:m-open}
\end{equation}
so a solar-mass hole begins its net descent when the universe has
expanded by a further factor $\sim 4\times10^{7}$.
\emph{Ground-state-ness is boundary-condition-relative, and for black
holes the expiration date is a redshift.} No other framework in
black-hole science has the vocabulary in which this sentence can even
be written; in the runtime grammar it is a one-line corollary of the
boundary-port clause --- and we key it as such:

\begin{euclidcorollary}[The conditional-GGA crossover
\textnormal{(C-BH-1)}]
\label{cor:gga-crossover}
With the boundary port the cosmic bath at $T_{\rm CMB}(t)$, a black
hole is a conditional \GGA{} if and only if $T_H < T_{\rm CMB}$,
with the crossover mass given by Eq.~\eqref{eq:crossover-mass} and
the opening schedule by Eq.~\eqref{eq:m-open}; and by
Lemma~\ref{lem:gsl-arrow} the crossover is simultaneously the
generalized second law's equality point --- the one mass at which
the horizon exchanges with its bath without producing entropy.
Grade: Derived (the induced first law plus the exchange-class
identity), conditional on the quasi-static reading of the bath
coupling.
\end{euclidcorollary}

\subsection{Sorting the transitions: capture, milestone, and the one true \GRIPout{}}
\label{ssec:bh-sorting}
The two-regime question --- \emph{does the landscape change?} ---
sorts the black-hole life cycle cleanly, and two common misreadings
are worth pre-empting. \emph{Formation is not a \GRIPout{}.} Stellar
collapse is descent on the \emph{fixed} gravitational landscape: the
Chandrasekhar and Tolman--Oppenheimer--Volkoff thresholds are the
saddle gates the trajectory crosses, and the capture into the
black-hole basin is ordinary \GRIPin{} --- nothing about the
landscape itself changes when a star collapses, however violent the
infall. Likewise a binary merger is \emph{basin coalescence} on the
fixed landscape (two attractors of the joint system merging into a
deeper one), and accretion is port-driven traffic into an existing
basin. The one genuine \GRIPout{} in this paper is the Hawking--Page
transition of \S\ref{sec:branch-switch}, where temperature --- a
control parameter of the ledger --- reorganizes the saddle structure
of the landscape itself. And the Page turnover is neither: it is a
\emph{milestone} of \GRIPin{} (\S\ref{sec:branch-switch}), the moment
a bookkeeping crossover occurs while the landscape stays put.

\begin{figure}[tbp]
\centering
\resizebox{0.892\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small, line width=0.80pt,
 box/.style={rounded corners=2pt, align=center, inner sep=4pt, minimum width=92mm},
 sidebox/.style={rounded corners=2pt, align=center, inner sep=4pt, minimum width=56mm, font=\footnotesize}]
\node[box, fill=red!10, draw=red!60!black] (STAR) at (0,0)
 {\textbf{Progenitor star} --- a \GA{} of the stellar stratum\\
 {\footnotesize fuel exhausted: the basin floor rises under it}};
\node[box, fill=green!12, draw=green!60!black] (GATE) at (0,-1.7)
 {\textbf{Collapse through the saddle gates}\\
 {\footnotesize Chandrasekhar, then Tolman--Oppenheimer--Volkoff: \GRIPin{} capture on the \emph{fixed} landscape}};
\node[box, fill=teal!12, draw=teal!60!black] (BH) at (0,-3.6)
 {\textbf{Black hole} --- the most extreme way-station \GA{} in nature\\
 {\footnotesize no-hair simple; ringdown = relaxation in the basin; residence $\sim 10^{67}$ yr}};
\node[box, fill=blue!12, draw=blue!60!black] (RAD) at (0,-5.8)
 {\textbf{Radiated configuration} $\to$ \textbf{de Sitter vacuum: the \GGA{}}\\
 {\footnotesize higher $S_{\rm gen}$, lower ledger (GSL); the descent's true terminus \citep{ArisakaA10Cosmos}}};
\draw[->] (STAR) -- (GATE);
\draw[->] (GATE) -- (BH);
\draw[->] (BH) -- node[right=2pt, align=left, font=\footnotesize\itshape]
 {Hawking radiation:\\the exit channel of \GRIPin{}} (RAD);
\node[sidebox, fill=orange!15, draw=orange!70!black] (COND) at (8.6,-3.6)
 {\textbf{Today: channel reversed}\\
 $T_H < T_{\rm CMB}$ $\Rightarrow$ net absorption\\
 $\Rightarrow$ \emph{conditional \GGA{}}\\
 opens when $M < M_{\rm open}(t) \propto a(t)$};
\draw[<->, dashed, orange!70!black] (BH.east) -- (COND.west);
\end{tikzpicture}}
\caption[The black-hole life cycle in the two-regime grammar.]{The
black-hole life cycle in the two-regime grammar. The entire column is
\GRIPin{} on a fixed gravitational landscape: capture through the
instability gates, residence at the way-station attractor, and exit
through the Hawking channel toward the radiated configuration --- the
stratum's true ground state (\GGA{}). Nothing in this column
reorganizes the landscape; the one genuine \GRIPout{} of black-hole
physics is the Hawking--Page transition
(\S\ref{sec:branch-switch}), where a control parameter rebuilds the
saddle structure. Right: today the exit channel of every
astrophysical hole runs backward ($T_H < T_{\rm CMB}$), making it a
\emph{conditional} \GGA{} whose ground-state status expires on the
cosmological schedule of Eq.~\eqref{eq:m-open}.}
\label{fig:bh-way-station}
\end{figure}

\noindent Figure~\ref{fig:bh-way-station} places the black hole on the descent.


\section{The \GDOS{} reading, in summary}

The \GDOS{} reading of black holes can now be told as one
continuous story, and it is worth telling in prose before the
predictions Part prices it.

A horizon is an \OGGEP{} port: the specific locus where the 4D
exterior separates from an inaccessible interior.  Hawking
radiation is that port's channel output, its thermal character
automatic from the \CPTP{} structure rather than re-derived
metric by metric.  The Bekenstein--Hawking entropy is the
entanglement entropy of the projection, $\Sbh =
\SvN(\rho_\partial^{\rm out})$, with the exact coefficient the
business of Part~V.  And the Page curve follows from bulk
unitarity plus \CPTP{} projection --- automatic, with no replica
wormholes required.  Its turnover has a name and a schedule: the
cohort's freshness clause (\textsf{L-GDOS-Fresh}; \AFive{},
Definition~5) holds at the horizon exactly until the Page time
and fails after it by counting alone, and the branch switch of
T-BH-2 is that expiry, named and verified in the coarse-grained
model (L-BH-Fresh).

The paradox-adjacent structures dissolve rather than resist.
Firewalls are an artifact of 4D-only thinking: at the 6D bulk
level, the monogamy violation never happens.  The singularity is
a coordinate artifact of the 4D projection, smooth in the bulk.
And complementarity is two partial traces of the same 6D state
--- two lawful pages of one ledger, not two competing copies of
one page.

The theorems anchor what would otherwise be metaphor.  Rotation
and collapse are the same custody: the ergoregion defeats the
rotating-frame selection and the anchor moves to the asymptotic
frame, with superradiance as \GDOS{} pumping (T-BH-3), while the
collapse epoch factorizes into a universal thermal channel times
a decaying transient (T-BH-4).  Thermality is the clock's KMS
face as a theorem: the positive-frequency selection, the
complexified-horizon continuation, and the imaginary-time
periodicity are consumed from the cohort's Wick-quartet theorem
\textsf{T-P5-B1} rather than imported as prescriptions, the
geometry contributing only $\kappa$ (T-BH-5).  And the
generalized second law is derived with a declared ceiling: the
constitutional arrow consumed from its owner, the data-processing
inequality supplying the monotone, and the clock's KMS face
pricing the loss in area --- $\Delta S_{\rm gen}$ equals the
spent relative entropy exactly in the exchange class,
$\ln(1+\bar n_H)$ per emitted mode, with equality only at
$T_b = T_{\rm H}$, and the Wall-class statement registered open
(L-BH-Arrow).

\paragraph{The Part in one breath, and the door to the next.}
This re-reading resolves the information paradox at the level of
physical geometry, not formal duality.  The physics intuition to
carry forward: \emph{information is never lost, only displaced
--- and its return is an assumption expiring on schedule, not a
mechanism switching on}.  A resolution this tidy must now put
numbers where instruments can reach.  The next Part prices the
reading --- shadows, ringdowns, primordial ladders, analog
horizons, and the universe's own horizon --- and shows that the
framework's boldest quantitative claim is a precise, dated, and
falsifiable silence.

\clearpage

%% ===================================================================
%% PART -- QUANTITATIVE PREDICTIONS
%% ===================================================================

\section[Part~VI in plain English]{Part~VI in plain English: why the information comes back, and why nothing switches on to bring it}
The information paradox in one sentence: if the radiation is thermal
and the hole evaporates, the information is gone, and it cannot be.

This Part's answer is that the radiation is thermal only while a particular
assumption holds, and that the assumption has an expiry date built into it
from the beginning. The framework keeps a ledger with two terms, an area
term and a bulk-entanglement term, and the physical entropy is whichever
combination of them is smaller. Early on the area term dominates and the
radiation looks thermal. Later the other term does, and it stops looking
thermal. Nothing switches on. No new physics enters at the halfway point.
What changes is which of two terms that were both there all along is
winning, and the crossover is the Page curve.

Two separately puzzling things fall out of that. The firewall argument
assumed the radiation's partner had to be an independent four-dimensional
degree of freedom; here it is a bulk one, so the monogamy conflict does not
arise and an infalling observer sees nothing at the horizon. And the
islands formula, which gets the right answer in controlled models by
machinery whose meaning is still argued about, is recovered as the
semiclassical display of the same ledger --- the same arithmetic under a
different name, which the Part sets out row by row rather than asserting.

The honest part is the direction of the second law. This Part does not
derive the arrow. It consumes the arrow from a companion paper, computes
what each exchange costs, and registers the general statement as open with
a named ceiling. A reader who wants the arrow derived will not find it
here, and the Part says which paper to read instead.

\GGPart[Quantitative predictions]{Quantitative Predictions}{Event-Horizon Telescope, Ringdown,
Primordial Black Holes, and the Hawking Spectrum}

\noindent
A framework that resolves paradoxes owes the sky some numbers.
This Part collects them, and it is honest up front about their
shape: for astrophysical black holes the \GGtheory{} corrections
to general relativity are suppressed by $(\ell_{\rm lock}/r_s)^2$
--- parts in $10^{46}$ for stellar-mass holes --- so the
framework's signature is not a bump or an echo but a precise
silence: pure Kerr wherever instruments can see, with the
deviations budgeted row by row rather than waved away.  Five
observational windows carry the exposure, each with its own
section below; the worked numbers for four representative holes
come first, so the scales are concrete; and the Part closes with
the full predictions table --- every row either consistent or not
yet tested, none fitted.

\section{Overview of predictions}

The \GDOS{} reading of black holes is more than a conceptual
re-framing; it makes specific observational predictions that can
(in principle, and sometimes in practice) be tested.  The key
observational windows are five.  EHT imaging reads the shadows of
the supermassive black holes M87* and Sgr~A* at horizon scales.
Gravitational-wave ringdown reads the quasi-normal-mode spectrum
of post-merger black holes detected by LIGO and, in the coming
decade, LISA.  The primordial-black-hole mass spectrum carries
the module's one model-level ladder prediction from the early
universe.  The Hawking-radiation spectrum is observable in
principle for very small black holes, if such holes exist.  And
entanglement-entropy structure is testable indirectly, through
the black-hole-like behavior of analog systems --- acoustic and
optical horizons in the laboratory.  Each of these is discussed
in detail below.

\section{Four black holes, end to end}
\label{sec:worked-numbers}

To make the predictions concrete, we run the explicit arithmetic for four representative objects, from the supermassive to the primordial.  In every case \GGtheory{} reproduces the standard general-relativistic and thermodynamic values --- the framework adds no deviation at observable precision --- so the worked numbers are exactly the textbook ones; their purpose is to show the entropy, temperature, and lifetime that the induced-gravity result $S=A/4G_N$ assigns, and to make plain how small the GG correction is.

\begin{table}[H]
\centering\small
\caption[Worked black-hole numbers (order of magnitude).]{Worked black-hole numbers (order of magnitude).  $r_s$ is the horizon radius, $T_{\rm H}$ the Hawking temperature, $S_{\rm BH}=A/4G_N\hbar$ in units of $k_B$, and $t_{\rm evap}$ the evaporation time.  These are the standard values, which \GGtheory{} reproduces; the GG correction is suppressed by $(\ell_{\rm lock}/r_s)^2$ and is unobservable in every row (see \S\ref{sec:EHT-geometry}, \S\ref{sec:ringdown}).}
\label{tab:worked-numbers}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.16\textwidth}p{0.15\textwidth}p{0.13\textwidth}p{0.13\textwidth}p{0.12\textwidth}p{0.14\textwidth}@{}}
\toprule
\textbf{Object} & \textbf{Mass} & \textbf{$r_s$} & \textbf{$T_{\rm H}$} & \textbf{$S_{\rm BH}/k_B$} & \textbf{$t_{\rm evap}$} \\
\midrule
M87$^*$ & $6.5\times10^{9}\,M_\odot$ & $1.9\times10^{13}$ m & $1\times10^{-17}$ K & $4\times10^{96}$ & $10^{97}$ yr \\
Sgr A$^*$ & $4.3\times10^{6}\,M_\odot$ & $1.3\times10^{10}$ m & $1.5\times10^{-14}$ K & $2\times10^{90}$ & $10^{87}$ yr \\
Stellar & $10\,M_\odot$ & $3\times10^{4}$ m & $6\times10^{-9}$ K & $1\times10^{79}$ & $10^{70}$ yr \\
PBH (now) & $5\times10^{11}$ kg & $7\times10^{-16}$ m & $2\times10^{11}$ K & $7\times10^{39}$ & $\sim\!1.4\times10^{10}$ yr \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The supermassive pair (M87$^*$, Sgr A$^*$).}  These are the two black holes the Event Horizon Telescope has imaged \citep{EHT2019, EHT2022}.  Their Hawking temperatures ($10^{-17}$--$10^{-14}$ K) are far below the cosmic microwave background, so they absorb rather than radiate; their entropies ($10^{90}$--$10^{97}\,k_B$) are the largest single-object entropies in the universe.  \GGtheory{} predicts the Kerr shadow to the precision EHT measures; the induced-gravity correction is suppressed by some $(\ell_{\rm lock}/r_s)^2 \lll 1$ and is wholly unobservable.

\paragraph{The stellar black hole.}  A $10\,M_\odot$ remnant of the kind LIGO observes in mergers has $S_{\rm BH}\sim10^{79}\,k_B$ --- to be compared with the $\sim10^{58}\,k_B$ thermodynamic entropy of the progenitor star, an illustration that horizon formation is the most entropy-generating process known.

\paragraph{The primordial black hole.}  A black hole of mass $\sim5\times10^{11}$ kg ($\sim10^{15}$ g) has an evaporation time of order the present age of the universe (with the multi-species emission rate; the photon--graviton-only clock of Appendix~\ref{app:Page_curve}, Step~3 runs an order of magnitude slower): such objects, if they formed, are completing their evaporation now, at $T_{\rm H}\sim10^{11}$ K ($\sim$ tens of MeV).  This is the one black hole for which the Page transition (\S\ref{sec:branch-switch}) is, in principle, a near-term observable: its late-time radiation spectrum should turn over from thermal to information-restoring at the Page time $t_{\rm Page}\approx(1-2^{-3/2})\,t_{\rm evap}$.  This is the sharpest empirical handle the framework's unitarity claim has.

\section{EHT observations and horizon-scale geometry}
\label{sec:EHT-geometry}

\subsection{What EHT measures}

The Event Horizon Telescope images the shadow of supermassive
black holes at $\sim 20\,\mu$arcsec resolution. The shadow is
the projection of the photon sphere (at $r_p = 3\Newton M/c^2$
for Schwarzschild) plus the dark region inside it. Its
observed angular diameter sets constraints on the geometry at
horizon scale.

For M87*, the 2019 image showed a shadow of angular diameter
$\sim 42\,\mu$arcsec, consistent with the Kerr prediction within
$\sim 10\%$ errors (which are dominated by astrophysical
uncertainties, not by the imaging system).

\subsection{\GGtheory{} prediction for the shadow}

In \GGtheory{}, the 4D metric near a macroscopic black hole is
the Kerr metric to very high precision, because the scale profile
corrections are suppressed by the compactification scale
$1/\ell_{\rm lock}\sim\Mlock=3.85$~TeV \citep{ArisakaP1KK}, i.e.\ the compactification
length is $\ell_{\rm lock}\equiv\hbar/(\Mlock c)\simeq5\times10^{-20}$~m
(not the much shorter ultraviolet length $1/\MGUT\sim10^{-32}$~m).
The leading correction to the shadow size is suppressed by the
square of the ratio of this length to the horizon radius:
\begin{equation}\label{eq:shadow_correction}
\frac{\Delta r_{\rm shadow}}{r_{\rm shadow}} \sim
\left(\frac{\ell_{\rm lock}}{r_s}\right)^2 \sim \left(\frac{\Mpl}{M}
\cdot \frac{\Mpl}{\Mlock}\right)^2 \sim 10^{-66}\text{ for
M87}^*.
\end{equation}
This is utterly below the EHT resolution. The \GGtheory{}
prediction is: \textbf{the shadow is Kerr to parts in $10^{66}$
for the supermassive holes EHT images} (and to better than
$10^{-46}$ or below for any astrophysical hole). There should be no
observable deviation.

A deviation at the EHT level would \emph{kill} \GGtheory{}:
the theory would be required to produce Kerr exactly, so any
sub-10\% deviation would falsify it.

\subsection{Future prospects: ngEHT}

The next-generation EHT (ngEHT) program planned for the 2030s
would improve resolution to $\sim 5\,\mu$arcsec and could detect
polarization signatures, axion effects, and multiple photon
sub-rings; the proposed space-VLBI Black Hole Explorer (BHEX),
targeting launch in the early 2030s, would extend the array to
Earth--space baselines and resolve the photon ring directly. The \GGtheory{} prediction remains: Kerr to all
observable precision.

\section{LIGO/LISA ringdown spectroscopy}
\label{sec:ringdown}

\subsection{Quasi-normal modes}

When two black holes merge, the final state rings down via
gravitational-wave emission of quasi-normal modes (QNMs). The
QNM frequencies and damping times are characteristic of the
final black-hole geometry: for Kerr, they depend only on $M$
and $a = J/M$. The spectrum of QNMs is therefore a direct
fingerprint of the black hole's nature. The runtime vocabulary of
\S\ref{sec:bh-way-station} gives this fingerprint a structural
identity: the post-merger ringdown is the \emph{within-basin} piece
of \GRIPin{} --- relaxation toward the floor of the Kerr attractor
--- and the QNM spectrum is the \emph{curvature of the basin}: the
Hessian eigenvalues of the gravitational free-energy ledger at the
attractor. In one sentence: \emph{LIGO measures the second derivative
of $\GFF$ at the Geometric Attractor} --- the same functional whose
minima, on other strata, are the electron, the proton, and the dark
matter \citep{ArisakaA11Origin}. The merger that precedes the
ringdown is basin coalescence on the fixed landscape; nothing in a
binary inspiral is a \GRIPout{}.

For the fundamental $\ell = m = 2$ QNM, the frequencies are
approximately:
\begin{equation}\label{eq:QNM_Kerr}
f_{\rm QNM} \approx \frac{c^3}{2\pi\Newton M}\left[1 -
\frac{63}{100}\,(1-a^*)^{3/10}\right],
\end{equation}
where $a^* = a c^{2}/\Newton M$ is the dimensionless spin parameter.
For the $\sim60\,\Msun$ remnant of a GW150914-like merger, the QNM
frequency is $\sim 250$~Hz --- in the LIGO band.

\subsection{KK corrections to QNMs}

In \GGtheory{}, the bulk has compact dimensions with radius
$\ell_{\rm lock}\equiv\hbar/(\Mlock c)\sim5\times10^{-20}$~m. The
Kaluza--Klein (KK) spectrum of the compact direction has modes at
frequencies
\begin{equation}\label{eq:KK_spectrum}
\omega_n^{\rm KK} = n\,c/\ell_{\rm lock} = n\,\Mlock/\hbar \sim n\cdot6\times10^{27}\text{ Hz},
\end{equation}
which is utterly outside the LIGO/LISA band. No direct KK
signal is expected.

The leading effect on the ordinary QNM spectrum is through
radiative corrections from virtual KK modes, giving
\begin{equation}\label{eq:QNM_correction}
\frac{\Delta f_{\rm QNM}}{f_{\rm QNM}} \sim \left(\frac{\ell_{\rm lock}}
{r_s}\right)^2 \sim 10^{-48}\quad(\text{stellar-mass}),
\end{equation}
again utterly below LIGO sensitivity.

\subsection{LIGO/LISA prediction}

\textbf{The \GGtheory{} prediction for LIGO/LISA ringdown
spectroscopy is: pure Kerr QNMs with sub-$10^{-48}$
corrections for stellar-mass remnants.} No observable KK modes, no firewalls, no
``echoes'' from the horizon. This is in sharp contrast with the alternatives.  Fuzzball
proposals predict ``echoes'' from the fuzzball radius at $r_f$
outside the horizon --- post-ringdown gravitational-wave
signals.  Firewalls would potentially modify the effective
boundary condition at the horizon.  And extra-dimension models
with large extra dimensions could modify the QNM spectrum at an
observable level.

A detection of gravitational-wave echoes, or of QNM spectrum
deviations from Kerr beyond astrophysical uncertainties, would
be difficult to accommodate in \GGtheory{}.

\section{Primordial black holes and the early universe}
\label{sec:PBH}

\subsection{Primordial black-hole formation}

Primordial black holes (PBHs) are black holes that may have
formed in the very early universe from large density
perturbations. Their mass spectrum depends on the power spectrum
of primordial perturbations and the thermal history of the
early universe. In the regime grammar that the companion cosmology
paper applies to the whole early universe
\citep{ArisakaA10Cosmos}, PBH formation is a \emph{nested} event:
the landscape-reorganizing transitions of the first instants
(inflationary exit, reheating) are the \GRIPout{}s, and PBH capture
is \GRIPin{} on the landscapes those events left behind --- which is
why the PBH mass spectrum is a fossil of the thermal history rather
than an independent sector.

In standard cosmology, PBHs can form with masses ranging from
near-Planck ($10^{-8}$~kg) to many solar masses. Those with
mass $\lesssim 10^{12}$~kg would have evaporated by now via
Hawking radiation; those with mass $\gtrsim 10^{12}$~kg would
still be around and could in principle contribute to dark
matter --- though within the cohort the dark-matter slot is
saturated by the Kaxion of the companion C1 paper
\citep{ArisakaC1Kaxion}, which bounds any primordial-black-hole
contribution to a subdominant share \citep{ArisakaA10Cosmos}.

\subsection{\GGtheory{} PBH mass spectrum}

In \GGtheory{}, the early universe had a specific thermal
history connected to the GUT-scale bulk. Phase
transitions at scales $E_{\rm GUT} \simeq 2.3\times10^{16}$~GeV and below would
have produced density perturbations whose spectrum depends on
the \OGNfive{} integer structure.

A conjectural prediction, fenced as model-level: the scale structure may imprint preferred mass \emph{ratios} on the PBH
spectrum.  Successive bulk eigenmodes of the bulk are spaced
geometrically, so if PBH formation is modulated by that structure
the preferred masses form a ladder of \emph{ratios}
\begin{equation}\label{eq:PBH_peaks}
\frac{M_{n}^{\rm PBH}}{M_{n+1}^{\rm PBH}} \sim e^{\,k\,\Delta r_n}
\sim e^{7}\simeq1.1\times10^{3},
\end{equation}
with $\Delta r_n$ a unit step of the radial coordinate.
Two cautions are essential and we state them plainly.  First, this
fixes only \emph{ratios}: the absolute mass scale of any rung is
set by the horizon mass at the formation epoch,
$M_{\rm PBH}\sim M_{\rm Pl}\,(t_{\rm form}/t_{\rm Pl})$, which is
\emph{far above} the Planck mass and is \emph{not} predicted by
the scale integers --- so the spectrum floats freely up and down
the mass axis until an absolute anchor is supplied.  Second, the
exponent $k$ (and hence whether the ladder step is exactly
$e^{7}$) is not derived here.  The whole construction is therefore
a model-level conjecture, not a theorem of the cohort, and is
labeled as such throughout.

\subsection{Observational tests}

Searches for PBH dark matter run on four fronts: femtolensing
and microlensing constraints (OGLE, Subaru HSC);
gravitational-wave signatures of PBH mergers (LIGO); effects on
the CMB, through Thomson scattering by evaporation products; and
the disruption of stellar populations.
A confirmed detection of PBHs with masses corresponding to the
\GGtheory{} preferred peaks would be strong positive evidence.
Exclusion of PBHs at these preferred mass values would
disfavor \GGtheory{}'s specific PBH spectrum but not the
overall framework.

\section{Hawking spectrum corrections}

\subsection{Standard Hawking spectrum}

The standard Hawking radiation spectrum (for a Schwarzschild
black hole) is thermal:
\begin{equation}\label{eq:Hawking_standard}
\frac{dN}{dt\,d\omega\,d\Omega} = \frac{\Gamma_\ell(\omega)\omega^2}{
(2\pi)^3(e^{\hbar\omega/\kB\THaw} - 1)}.
\end{equation}
The temperature $\THaw$ is set by the horizon's surface
gravity, and the graybody factors $\Gamma_\ell(\omega)$ depend
on the scattering of radiation off the effective potential
outside the horizon.

\subsection{\GGtheory{} corrections}

In \GGtheory{}, there are two classes of corrections to this
spectrum:

\textbf{(1) High-frequency cutoff.} The spectrum should show
an ultraviolet cutoff at the compactification scale:
\begin{equation}\label{eq:UV_cutoff}
\hbar\omega_{\rm UV} \sim \hbar c/\ell_{\rm lock} \sim \Mlock.
\end{equation}
Beyond this frequency, the bulk KK modes become active and
the effective spectrum turns over. For astrophysical black
holes, $\Mlock \gg \THaw$ so this correction is unobservable.

\textbf{(2) Low-frequency small-mass cutoff.} For very small
black holes (near-Planck-mass), the back-reaction of Hawking
radiation on the geometry becomes important, and the scale profile may not be stable. The spectrum may show modifications
in the final stages of evaporation.

\textbf{(3) KK overtones in partial-wave decomposition.} For
black holes large enough that KK modes are relevant (e.g.,
primordial near-Planck-mass BHs), the angular partial-wave
decomposition should include contributions from KK modes in
addition to ordinary 4D partial waves.

\subsection{Observable prospects}

None of the \GGtheory{} corrections to the Hawking spectrum
are observable for astrophysical black holes in the near
future. The Hawking temperature for a $10\,\Msun$ black hole is
$\sim 6 \times 10^{-9}$~K, which is far below the CMB and
undetectable.

For primordial black holes with $M \sim 10^{12}$~kg,
$\THaw \sim 10^{11}$~K, and the Hawking spectrum is dominated
by GeV-scale photons and cosmic rays. These are in principle
observable by gamma-ray telescopes (Fermi-LAT, CTA). The
\GGtheory{} prediction for the spectrum of evaporating PBHs
is indistinguishable from the standard Hawking prediction at
the relevant energies (GeV to $10^{10}$~eV), because the
compactification scale is $\Mlock = 3.85$~TeV $\sim 4\times10^{12}$~eV, far
above the observable range.

\section{Entanglement-entropy tests via analog black holes}

\subsection{Analog gravity}

Since the 1980s, ``analog gravity'' systems have been explored:
condensed-matter or optical systems that reproduce aspects of
black-hole physics.  Acoustic black holes realize a sonic
horizon where a transonic flow reaches the speed of sound, and
have been observed to emit analog Hawking radiation (Unruh 1981;
experimental confirmation in the 2010s).  Optical black holes
use pulses in nonlinear optical fibers to mimic horizon physics.
And Bose--Einstein-condensate black holes realize the same
structure in trapped cold-atom flows.
These analog systems cannot test quantum gravity, but they can
test the \emph{structure} of Hawking radiation: thermal
spectrum, entanglement-entropy scaling, correlations between
emitted modes and their partners.

\subsection{Implications for \GDOS{}}

In \GDOS{}, the analog-black-hole physics should match the
astrophysical black-hole physics at the level of the
partial-trace structure: the analog Hawking radiation should be
thermal at the analog Hawking temperature; the entanglement
entropy of the analog radiation with its partner modes should
follow the appropriate analog Page curve; and no analog
firewalls or exotic horizon effects should appear.
Analog-black-hole experiments through mid-2026 are consistent with
all three predictions. This is indirect confirmation of the
\GDOS{} picture.

\section{Cosmological black holes and de Sitter horizons}

\subsection{The de Sitter horizon}

In a cosmology with positive cosmological constant $\Lambda$,
there is a cosmological horizon at distance $r_H = \sqrt{3/
\Lambda}$. This is a causal surface: events beyond it cannot
influence the observer. Like a black-hole horizon, it has an
associated Gibbons-Hawking temperature $T_{\rm GH} =
H/(2\pi)$ where $H$ is the Hubble rate.

In \GGtheory{}, the de Sitter horizon is also an \OGGEP{} port:
the observer inside sees thermal radiation at $T_{\rm GH}$
from the horizon, and the entropy of the de Sitter horizon is
$S_{\rm dS} = A_H/(4\Newton\hbar)$, with $A_H = 4\pi r_H^2$.

For our universe, with vacuum energy density $\rho_\Lambda \sim
10^{-122}\,\Mpl^4$, the numbers are set: the horizon radius is
$r_H \sim 10^{26}$~m (the Hubble radius); the Gibbons--Hawking
temperature is $T_{\rm GH} \sim 10^{-30}$~K, utterly
unobservable; and the entropy is $S_{\rm dS} \sim 10^{122}$ ---
the maximum entropy the observable universe can hold.

This $10^{122}$ number is the famous ``dark energy entropy.'' It
sets a universal upper bound on the information content of the
observable universe.

\subsection{The de Sitter entropy from induced gravity}
\label{ssec:dS-induced}

The Gibbons--Hawking entropy $S_{\rm dS}=A_H/4G_N$ \citep{GibbonsHawking1977} is not merely \emph{analogous} to the black-hole result --- in \GGtheory{} it is the \emph{same theorem}.  The Euclidean de~Sitter saddle is the round four-sphere $S^4$, on which the cosmological horizon is the codimension-two bifurcation surface; the replica construction of \S\ref{ssec:V-conical} applies verbatim --- on the same licensed arena, for the static-patch clock carries the identical four endowments and the Euclidean four-sphere is its analytic shore, the cap's period being the geometry's datum --- with the conical-deficit identity $\int\sqrt{g}\,R \supset 4\pi(1-n)A_H$ on that surface and the induced Einstein--Hilbert action delivering
\begin{equation}\label{eq:dS-entropy}
S_{\rm dS} = \frac{A_H}{4G_N},\qquad \frac14 = \frac{4\pi}{16\pi},
\end{equation}
with the identical coefficient.  The named-statement deck registers
this as the text key \textsf{C-BH-2}: the de~Sitter horizon entropy
is Theorem~\ref{thm:exact-quarter} evaluated on the cosmological
\OGGEP{} port, inheriting its premises and its grade.  The
microstates are again the hidden-fiber holonomies (\S\ref{ssec:V-microstates}), now traced over the cosmological horizon.  The cosmological horizon is the third panel of Figure~\ref{fig:three-port}: the \OGGEP{} trace forced by cosmic expansion, with the observer's accessible state the partial trace over modes beyond $r=c/H$.  The exact-$1/4$ result of this paper is therefore not a black-hole accident; it is a statement about \emph{any} \OGGEP{} horizon, of which de~Sitter is the cosmological instance.

\subsection{Open de Sitter puzzles, stated honestly}
\label{ssec:dS-open}

De~Sitter quantum gravity is less settled than the black-hole case, and we do not claim to resolve its specific difficulties.  There is no asymptotic boundary on which to anchor a dual theory (the AdS/CFT crutch is unavailable); the meaning of the finite $S_{\rm dS}\sim 10^{122}$ as a Hilbert-space dimension (the finite-dimensional-Hilbert-space conjecture) is open; and dS/CFT remains speculative.  \GGtheory{} contributes the \emph{same} structural reading it gives the black hole --- the cosmological horizon as a geometrically-forced \OGGEP{} port, the entropy as the entanglement of that forced trace, the coefficient from induced gravity --- which is a candidate organizing principle, not a resolution of these dS-specific puzzles.  We flag this explicitly so the de~Sitter claims are not over-read: the coefficient and the \OGGEP{}-port reading carry over; the deeper de~Sitter quantum-gravity questions remain open.

\subsection{Connection to the Hubble-tension companion (C2)}

The companion Hubble-corridor paper \citep{ArisakaC2Hubble} addresses the local-versus-cosmological $H_0$ discrepancy by distinguishing the 6D-to-4D projection rates that local and cosmological observables sample.  The de~Sitter horizon is where that distinction becomes operationally sharp: cosmological observables register the full bulk near the cosmological \OGGEP{} port, while local observables register only the unscaled interior.  The present black-hole analysis and the C2 cosmological analysis are thus two evaluations of one bulk-to-boundary projection --- the black hole at a local horizon, the Hubble tension at the cosmological one --- with the shared $S=A/4G_N$ entropy ledger (\S\ref{ssec:dS-induced}) as the common thread.  This also connects to the $\Omega_m = 1/\pi$ inventory of \citep{ArisakaA10Cosmos}, which lives in the same boundary-projection framework. In the regime vocabulary the connection closes a circle: the de~Sitter vacuum toward which the universe relaxes is the cosmological stratum's \GGA{} --- one of the exactly two points where cosmic \GRIPin{} terminates \citep{ArisakaA10Cosmos} --- so the cosmological horizon of this section is not merely an analogue of the black-hole horizon: it is the port of the basin in which every black hole's own descent (\S\ref{sec:bh-way-station}) finally ends.

\section{The numerical table of predictions}

\begin{table}[H]
\centering\small
\caption[\GGtheory{} quantitative predictions for black-hole observables.]{\GGtheory{} quantitative predictions for black-hole
observables. Where observations exist, the pull is given. Most of the status column says consistent, and that should be read carefully rather than proudly: on every observable where a measurement exists the framework predicts Kerr, and so does general relativity, so the table earns nothing there. What it does earn is the location of the exposure --- the rows marked not tested --- so that agreement elsewhere cannot be mistaken for confirmation.}
\label{tab:bh_predictions}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{4.33cm}p{3.71cm}p{4.33cm}p{2.47cm}@{}}
\toprule
\textbf{Observable} & \textbf{GG prediction} &
\textbf{Measurement (or prospect)} & \textbf{Status}\\
\midrule
M87* shadow diameter & Kerr (42 $\mu$arcsec) &
EHT 2019: $42 \pm 3\,\mu$arcsec & \textbf{consistent} \\
Sgr A* shadow diameter & Kerr (51 $\mu$arcsec) &
EHT 2022: $51.8 \pm 2.3\,\mu$arcsec & \textbf{consistent} \\
GW150914 ringdown QNM & Kerr (250 Hz, $\tau$=4 ms) &
LIGO 2015: consistent with Kerr & \textbf{consistent} \\
GW250114 ringdown (O4) & Kerr (two QNMs) &
LVK 2025: fundamental $+$ overtone, Kerr-consistent &
\textbf{consistent} \\
PBH mass spectrum & Preferred mass ratios spaced
by $e^{7}$; absolute anchor open
(\S\ref{sec:PBH}) &
None detected & \textbf{not tested} \\
Hawking temperature for $M_\odot$ BH & $6\times 10^{-8}$~K &
Unobservable & \textbf{not tested} \\
Bekenstein--Hawking entropy & $A/(4\Newton\hbar)$ &
Consistent with horizon area in merger simulations &
\textbf{consistent} \\
Page curve & Automatic in \GDOS & Not yet tested in real BH &
\textbf{not tested} \\
Firewall signature & None predicted & None observed &
\textbf{consistent} \\
Echoes from horizon & Not predicted & No convincing detection &
\textbf{consistent} \\
Information paradox & Resolved & (Theoretical consistency) &
\textbf{consistent} \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The Part in one breath, and the door to the next.}
Run the status column of Table~\ref{tab:bh_predictions} top to
bottom and the shape of the module's exposure is plain: every
tested row reads \emph{consistent}, every untested row is dated
or declared unfeasible, and no row is fitted.  The physics
intuition to carry forward: \emph{for this framework, silence is
not the absence of a prediction --- it is the prediction}, and
every sharper instrument raises the stakes of the bet.  But
predictions are cheap until they are bonded.  The next Part posts
the bond: the record of every shot fired at the module so far,
the kill criteria under which it agrees to die, and the dated
timeline on which the verdicts arrive.

\clearpage

%% ===================================================================
%% PART -- EXPERIMENTS -- PAST, NOW, AND FUTURE (new at the v13.8 split)
%% ===================================================================

\section[Part~VII in plain English]{Part~VII in plain English: what the framework predicts where instruments can look, which is less than one would like}
Here is the awkward fact this Part is built around. Everywhere an
instrument can currently look, the framework predicts exactly what general
relativity predicts.

The shadow of M87* is Kerr. The shadow of Sgr A* is Kerr. The ringdown of
GW150914 is Kerr, and so is the two-mode ringdown resolved in 2025. There
are no echoes, no firewall signature, and no departure from the thermal
spectrum at any frequency anyone can reach. Every one of those rows reads
consistent, and not one of them is evidence for this framework, because a
theory that agrees with general relativity outside the horizon was always
going to pass them.

So the Part is arranged around what that means rather than around the
agreement. If the prediction is silence, then silence is what is at stake:
every sharper instrument narrows the space in which a departure could hide,
and the prediction gets riskier without ever getting more impressive. That
is a real position to hold and a poor one to advertise, and the tables are
laid out so a reader can see which it is.

The rows where the framework does say something are the ones nobody can
test yet --- the spacing of a primordial black-hole mass spectrum, the de
Sitter horizon read as the same kind of port, corrections suppressed by a
scale no instrument reaches. They are printed with their status attached
rather than folded in among the successes.

\GGPart[Experiments]{Experiments --- Past, Now, and Future}{The Kill Criteria and the Evidence Base of the Black-Hole Module}

\noindent
This Part is the module's confrontation record.  Part~VII stated
what the framework predicts; this Part states what would kill it ---
the kill criteria, declared before the instruments report --- and
what the instruments have said so far, in the cumulative evidence
base.

\paragraph{Physics intuition.}  A predictions Part lists what the
theory says; an experiments Part posts the bond --- the
pre-registered conditions under which the theory agrees to die, and
the running account of every shot fired at it so far.  The milestones, the dated
timeline, and the tables below are that bond for the black-hole
module.


\section{The record so far: what has been measured, and what it did to the module}
\label{sec:exp-record}

Four instrument classes have already fired at this module, and the
record of each shot is one-sided in the same direction.

\paragraph{Gravitational waves.}  Three hundred and ninety compact
binary coalescences stand confirmed since 2015, roughly three
quarters of them from the completed fourth observing run
(\S\ref{sec:exp-now}).  The module's exposure is the ringdown: every
quasi-normal-mode analysis to date --- from GW150914's fundamental
mode to GW250114's resolved fundamental-plus-overtone pair and its
inspiral-to-ringdown horizon-area budget --- is Kerr-consistent, and
the area theorem is now an observationally confirmed statement.  For
this module that last point carries special weight: the generalized
second law whose \emph{direction} Lemma~\ref{lem:gsl-arrow} derives
has had its classical half (the area half) confirmed on a real
merger.  No echo candidate has survived scrutiny.  F-BH-2 and F-BH-3
stand unfired at every sensitivity yet reached.

\paragraph{Horizon-scale imaging.}  Two shadows --- M87* and
Sgr~A* --- imaged, re-imaged across epochs, and now
polarization-resolved; the shadow diameters sit on Kerr within
errors dominated by astrophysics, and the epoch-to-epoch changes
live entirely in the magnetized plasma, never in the shadow.
F-BH-1 stands unfired.

\paragraph{Analog horizons.}  Laboratory horizons --- acoustic,
optical, condensate --- have confirmed the structural content the
port reading requires: thermal spectra at the analog temperature and
the mode-pair entanglement of the trace.  These platforms cannot
kill the module (F-BH-5 is fenced honestly: real-horizon spectra are
unobservable), but they stress its \emph{structure}, and the
structure has held in every laboratory that has looked.

\paragraph{Primordial black holes.}  Microlensing surveys, subsolar
gravitational-wave searches, and evaporation-photon limits continue
to carve the PBH mass function without yet populating it.  The
module's model-level ratio-ladder conjecture (F-BH-4) therefore
remains untested rather than surviving: an honest ``not yet''.

\section{Now: the instruments in mid-2026}
\label{sec:exp-now}

The LIGO--Virgo--KAGRA network completed its fourth observing run in
November 2025 and is between runs: the GWTC-4.0 and GWTC-5.0 catalog
releases (128 and 161 new events) are the run's harvest, the final
O4 data segment is still under analysis, and a six-month
\emph{intermediate run} (IR1) is announced to begin in late 2026,
with the fifth observing run's plans being refined toward
A$+$-class sensitivity.  The Event Horizon Telescope is in its
multi-epoch era --- annual campaigns, polarimetry on both shadows
--- with its next-generation extension in staged planning.  The
CTAO gamma-ray observatory, the instrument class that would see the
final bursts of evaporating primordial holes, is under construction
with its early-science era opening.  And the module's
\emph{external} hostages are on their own schedules: the collider
search for the lock cluster, the haloscope scan of the Kaxion
window, and the neutrino-ordering program --- none of them
black-hole instruments, all of them able to kill this paper's
reading from outside (\S\ref{ssec:runtime-uniqueness}).

\section{The decade ahead: milestones, with the verdicts pre-registered}
\label{sec:exp-milestones}

The verdicts below are signed now, so that no future reading of this
module can be accused of moving after the shot.

\emph{2026--2027 --- the intermediate run.}  IR1 returns the network
to the sky for six months at O4-class-or-better sensitivity; tens of
further ringdown-capable events are the expectation.  \emph{If
right:} Kerr-consistent quasi-normal modes at every signal-to-noise
ratio, echo searches silent --- the null tightening, undramatically.
\emph{If wrong:} a confirmed QNM deviation above the $1\%$ class or
a reproducible echo train --- F-BH-2 or F-BH-3 fires and the module
retires, with no fallback.

\emph{Late 2020s --- the fifth run.}  At A$+$-class sensitivity the
loudest events reach the signal-to-noise class where two-mode
spectroscopy is routine rather than exceptional, and the GW250114
measurement becomes a population statement.  \emph{If right:} the
stacked ringdown catalog is Kerr to well below the percent class;
the area theorem confirmed on dozens of mergers.  \emph{If wrong:}
same rows, same firing rule --- the bet does not change, only the
stakes.

\emph{Early 2030s --- the sharpened shadow.}  The next-generation
EHT program and, if selected, the BHEX space-VLBI mission
(targeting an early-2030s launch) push shadow astrometry toward the
percent class and resolve the photon ring.  \emph{If right:} the
shadow stays Kerr at every resolution --- the module predicts that
the sharper the image, the more boring it gets, and stakes itself
on that boredom.  \emph{If wrong:} a confirmed shadow deviation at
the $5\%$ class --- F-BH-1 fires.

\emph{Mid-2030s --- LISA.}  The adopted space interferometer
(launch targeted mid-2030s) observes extreme-mass-ratio inspirals:
year-long orbits that map the central object's multipole structure
to the $10^{-4}$ class.  This is the sharpest one-sided null the
module will ever face --- five orders below the current ringdown
exposure.  \emph{If right:} Kerr multipoles, exactly, tens of
thousands of cycles at a time.  \emph{If wrong:} any confirmed
non-Kerr multipole retires the module from a direction no
ground-based instrument can reach (the registered row is F-BH-10,
Table~\ref{tab:bh_kill}).

\emph{Mid-to-late 2030s --- the third generation.}  Einstein
Telescope and Cosmic Explorer, in program definition through this
decade, would return thousands of ringdowns per year.  At that rate
the ringdown null becomes a precision discipline, and F-BH-2's
$1\%$ threshold is left far behind; the module's exposure grows by
the same factor as the catalog.  \emph{If right:} thousands of
clean Kerr ringdowns per year, the null hardened into a precision
law.  \emph{If wrong:} a reproducible spectral deviation at the
$10^{-3}$ class fires F-BH-2 with statistics no single event can
contest.

\emph{Decidable at a desk, any time --- the internal row.}  F-BH-9
needs no instrument: the sign-resolved species sum either reproduces
the induced $1/\Newton$ or it does not.  It is chartered as its own
computation, and it is the one row on this timeline whose date the
cohort itself controls.  \emph{If right:} the closure claim is
promoted and O-BH-7's answer becomes arithmetic.  \emph{If wrong:}
F-BH-9 fires, retiring the closure reading and leaving the $1/4$
standing on its own premises (Table~\ref{tab:statement-deck}).

\paragraph{Physics intuition.}  A theory whose falsifiable content
is carried by the null does not wait nervously for the data; it
stands still on the rifle range, in writing, while the instruments
buy longer barrels.  Every entry above is the same single bet ---
\emph{the horizon is exactly Kerr and exactly thermal, because the
projection is exactly what the constitution says it is} --- placed
once, priced in advance, and re-exposed to every sharper instrument
the decade delivers.  What the module can never do is win by
silence alone; what it can always do is die by one confirmed shot.
That asymmetry is not a weakness.  It is the falsification standard
of the series, worn on the sleeve.

\section{The falsifier timeline, dated}
\label{sec:bh-exp-timeline}

Table~\ref{tab:bh-exp-timeline} closes the Part's forward apparatus:
the schedule-centric companion to the kill table below, one line per
confrontation, with the verdict type each instrument can return ---
\emph{kill} (can falsify outright), \emph{support} (can strengthen,
cannot confirm alone), \emph{watch} (tension monitor).  Every kill
line is pre-declared and dial-free.

\begin{table}[H]
\centering
\small
\renewcommand{\arraystretch}{1.25}
\caption[The dated falsifier timeline of the black-hole
module.]{The dated falsifier timeline of the black-hole module.
``Kill'' lines are pre-declared and dial-free; the claims-level
register of each row is the kill table below, and the external
hostages are the cohort falsifiers this module inherits by
construction (\S\ref{ssec:runtime-uniqueness}).  Under the geometry
being right, every kill line returns silence: the module's
signature is the ever-sharper null.}
\label{tab:bh-exp-timeline}
\begin{tabular}{p{2.9cm} p{2.7cm} p{2.4cm} p{2.1cm} p{2.7cm}}
\toprule
\textbf{Forced claim} & \textbf{Instrument(s)} &
\textbf{Decision window} & \textbf{Verdict type} &
\textbf{Kill condition} \\
\midrule
Kerr shadow (null) & EHT, ngEHT, BHEX & permanent; sharpening
early 2030s & kill & confirmed $>5\%$ deviation (F-BH-1) \\
Kerr ringdown (null) & LVK IR1/O5, LISA, ET/CE & permanent;
sharpening 2026--2040 & kill & confirmed $>1\%$ QNM deviation
(F-BH-2) \\
No echoes & LVK, ET/CE & permanent & kill & reproducible
post-ringdown echoes (F-BH-3) \\
Kerr multipoles & LISA EMRIs & mid-2030s & kill & non-Kerr
multipole at $10^{-4}$ class (F-BH-10) \\
PBH ratio ladder (model-level) & microlensing, subsolar GW, CTAO
& 2030s & kill (model) / support & populated mass function off the
$e^{7}$ spacing (F-BH-4) \\
Trace thermality (structure) & analog platforms & continuous &
support / watch & --- (structure evidence only; the spectral kill
F-BH-5 lives at true horizons) \\
Species-sum closure & dedicated computation & decidable at will &
kill (internal) & sum misses $1/\Newton$ (F-BH-9) \\
External hostages & HL-LHC; haloscopes; JUNO/DUNE & 2029--2041 &
kill (external) & lock cluster excluded; Kaxion window excluded;
inverted ordering \\
\bottomrule
\end{tabular}

\end{table}

Read the timeline as an experimentalist would, cadence first.
The permanent lines are already running: the shadow and ringdown
nulls sharpen continuously, with the next discrete jumps at ngEHT
and BHEX in the early 2030s and the LVK runs before them.  The
one dated cliff is LISA's: extreme-mass-ratio inspirals in the
mid-2030s read the Kerr multipole structure at the $10^{-4}$
class, and F-BH-10 fires or it does not.  Two lines need no
telescope at all --- the species-sum closure (F-BH-9) is
decidable at a desk whenever the dedicated computation is run,
and the external hostages (the lock cluster at HL-LHC, the
Kaxion window in the haloscopes, the neutrino ordering at
JUNO/DUNE) can retire this module from neighboring fields on
their own schedules.  Note also what the verdict column does
\emph{not} offer: no row returns ``confirm.''  Support lines
strengthen the reading; only kill lines are decisive; and under
the geometry being right, every kill line returns silence.

\begin{figure}[H]
\centering
\resizebox{0.884\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=stealth]
\draw[line width=1.00pt, ->] (-0.3,0) -- (15.9,0);
\foreach \x/\yr in {0/2026, 2.2/2028, 4.4/2030, 6.6/2032, 8.8/2034,
 11.0/2036, 13.2/2038, 15.4/2040}{
  \draw[line width=0.80pt] (\x,0.12) -- (\x,-0.12);
  \node[anchor=north, font=\scriptsize] at (\x,-0.18){\yr};
}
% LVK band
\draw[line width=2.4pt, blue!35] (2.0,0.75) -- (15.4,0.75);
\draw[line width=2.4pt, blue!60!black] (0.15,0.75) -- (2.0,0.75);
\node[anchor=west, font=\scriptsize, text=blue!60!black,
 fill=white, inner sep=1.5pt]
 at (2.15,0.75){IR1 $\to$ O5 era (F-BH-2, F-BH-3)};
% shadow band
\draw[line width=2.4pt, teal!70!black] (4.4,1.5) -- (8.8,1.5);
\node[anchor=west, font=\scriptsize, text=teal!70!black]
 at (8.95,1.5){ngEHT / BHEX: the sharpened shadow (F-BH-1)};
% LISA
\filldraw[violet!70!black] (9.9,2.25) circle (2.2pt);
\draw[line width=2.4pt, violet!45] (9.9,2.25) -- (15.4,2.25);
\node[anchor=east, font=\scriptsize, text=violet!70!black]
 at (9.7,2.25){LISA: Kerr multipoles at $10^{-4}$};
% 3G
\draw[line width=2.4pt, olive!70!black] (11.0,3.0) -- (15.4,3.0);
\node[anchor=east, font=\scriptsize, text=olive!70!black]
 at (10.8,3.0){ET / CE: the ringdown census};
% internal row
\filldraw[red!70!black] (0.15,3.75) circle (2.2pt);
\node[anchor=west, font=\scriptsize, text=red!70!black]
 at (0.35,3.75){F-BH-9 species sum: decidable at a desk, any time};
\end{tikzpicture}}
\caption[The kill-test timeline, drawn.]{The kill-test timeline,
drawn.  Bands mark the decision windows of
Table~\ref{tab:bh-exp-timeline}; every band is the same one-sided
bet at increasing stakes, and the one red dot is the row the cohort
itself controls --- the internal species-sum computation, decidable
without any instrument.}
\label{fig:bh-killtest-timeline}
\end{figure}

\section{Refresh rules: the module's data discipline}
\label{sec:exp-refresh}

Four standing rules govern every number this Part and the evidence
base carry; they are part of the module's honesty apparatus, stated
once so that no future edit can drift silently.

\emph{Rule 1 (measured values).}  Measured values follow the
instruments' latest published results and carry their own dates;
the evidence base is re-dated at each version of record, and a
``through year $Y$'' null advances only on a verified sweep of the
literature.

\emph{Rule 2 (predictions).}  The \GGtheory{} prediction column
never moves under refresh.  Any change to a predicted value is a
version-of-record event of the owning paper, never a data refresh
of this one.

\emph{Rule 3 (kill rows are dial-free).}  A fired kill retires the
module.  No refresh, reinterpretation, or auxiliary hypothesis may
be introduced after the fact to weaken a kill row; the rows fire as
written or not at all.

\emph{Rule 4 (pulls).}  Printed pulls are recomputed
programmatically from the printed prediction and the refreshed
measurement at every refresh, never carried forward by hand.

\section*{Kill criteria}

The kill table is the module's contract, stated in the field's
own observables.  Each row names the observable, the deviation
threshold at which the row fires, and the timeline on which the
firing could happen; the refresh rules of the preceding section
apply to every row --- they fire as written or not at all.

\begin{table}[H]
\centering\small
\caption[Black-hole-specific kill criteria.]{Black-hole-specific kill criteria. Any one, if
experimentally established, would retire the \GGtheory{}
reading of black holes. Ten criteria, and the last column is the honest one: two of them need instruments that do not exist, and two are theory-side questions with no instrument at all. The ones that can bite this decade are the shadow and the ringdown, and both campaigns are already running. A registry that listed only the reachable tests would look stronger and would be a different document.}
\label{tab:bh_kill}
\usebox{\GGscKillBH}
\end{table}

The ten rows sort by reach, and reading them that way is how
the table is meant to be digested.  Four are live
instrument rows on astrophysical data: the shadow (F-BH-1), the
ringdown (F-BH-2), the echoes (F-BH-3), and the LISA multipoles
(F-BH-10) --- the rows the coming decade will keep squeezing.
One is model-level by construction: the PBH spacing row (F-BH-4)
fires only on a well-populated mass function off the $e^{7}$
ladder, never on a single detection.  Four are registered but
presently beyond instrumental reach --- the Hawking-spectrum form
(F-BH-5), demonstrated information loss (F-BH-6), a Bekenstein
coefficient different from one (F-BH-7), and the firewall
signature (F-BH-8) --- kept on the books because the register
requires the exposure to be stated even where no instrument yet
exists.  And one is internal: the species sum (F-BH-9), the only
row whose firing date the theory itself controls.  Nothing in
the table waits on a definition; every threshold is a number.

\section*{Cumulative UCLA-Arisaka evidence base}

The evidence base closes the Part, and its two halves should be
read differently.  The upper rows are the cohort's cross-ledger
campaigns --- predictions made elsewhere in the program,
confronted with 2025--2026 data --- and they are the module's
external bond: the black-hole reading runs on the same bulk that
produced them, so their fate is its fate.  The lower rows are
the black-hole benchmarks this paper adds to the cohort's
running account.

\begin{table}[H]
\centering\small
\caption[Cumulative evidence for the \GGtheory{} program as of July 2026, including the black-hole qualitative.]{Cumulative evidence for the \GGtheory{} program as
of July 2026, including the black-hole qualitative
consistency checks added by the present paper. The upper block belongs to the cohort and not to this paper, and is reprinted because a black-hole module quoting only its own consistency checks would be quoting a very short list. The lower block is what this paper adds, and every row of it says consistent with Kerr --- a check passed, not a prediction confirmed, since general relativity says the same. That is why the two blocks are separated rather than summed.}
\label{tab:bh_evidence}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{4.77cm}p{4.09cm}p{4.77cm}c@{}}
\toprule
\textbf{Campaign / observable} & \textbf{GG prediction} &
\textbf{Measurement (2025--2026)} & \textbf{Pull}\\
\midrule
$H_0$ (background vs.\ CMB; C2) & $H_0^{\rm bg} = 66.6\pm0.2$
km/s/Mpc & $67.4\pm0.5$ (CMB); ladder $73.5\pm0.8$; lift: sign
$+$ rough third derived, rest a named ladder systematic (C2) &
\textbf{$-1.5\sigma$}\\
Muon $g-2$ (imported SM; \GGtheory{} adds nothing) & $a_\mu^{\rm SM} = 116\,592\,033(62)\cdot 10^{-11}$ & $116\,592\,071.5(14.5)\cdot 10^{-11}$ & \textbf{+0.6$\sigma$}\\
$W$-mass (world avg) & $m_W = 80.3700$ GeV (A9, QD) & $80.3692(133)$ &
\textbf{$-0.06\sigma$}\\
Neutrino $\sum m_\nu$ & $\approx 0.059$ eV (NH floor) & $<0.072$ eV (DESI+CMB) &
\textbf{consistent}\\
Higgs mass & $125.20$ GeV & $125.20(11)$ &
\textbf{0.0$\sigma$}\\
Di-Higgs rate & $\mu_{HH} = 1$ & $0.9^{+1.4}_{-1.1}$ &
\textbf{+0.08$\sigma$}\\
CHSH (loophole-free) & $2\sqrt{2}$ & $2.697(15)$ &
\textbf{consistent}\\
Born rule precision & exact & $\lesssim 10^{-13}$ for $a_e$ &
\textbf{consistent}\\
Arisaka Constant $\QB$ & 0.1555 eV & ATP $\sim 0.17$ eV &
\textbf{consistent}\\
\midrule
\multicolumn{4}{l}{\emph{Black-hole benchmarks added by present
paper:}}\\[0.15em]
M87* shadow & Kerr: 42~$\mu$as & EHT 2019: $42\pm 3\,\mu$as &
\textbf{consistent}\\
Sgr A* shadow & Kerr: 51~$\mu$as & EHT 2022: $51.8\pm 2.3\,\mu$as &
\textbf{consistent}\\
GW150914 ringdown & Kerr QNM & LIGO 2015: Kerr-consistent &
\textbf{consistent}\\
GW250114 ringdown $+$ area law & Kerr QNMs; $\Delta A > 0$ &
LVK 2025: overtone resolved; area theorem confirmed &
\textbf{consistent}\\
Bekenstein entropy $A/(4\Newton\hbar)$ & Derived from scale profile &
Consistent with merger area & \textbf{consistent}\\
Page curve & Automatic & Not yet directly tested &
\textbf{consistent}\\
Firewall & Not predicted & Never detected & \textbf{consistent}\\
GW echoes & Not predicted & None convincingly detected &
\textbf{consistent}\\
\bottomrule
\end{tabular}
\end{table}

Three features of the table deserve the reader's attention.
First, the campaign block is honest about its one visible strain:
the background-versus-CMB $H_0$ row sits at $-1.5\sigma$, printed
as measured, with the ladder's lift carried at C2's register ---
sign and rough third derived, the rest a named systematic.
Second, the black-hole block contains no number the module
fitted: the shadows, the ringdowns, and the area theorem are
pure-Kerr consistency checks, exactly what a framework whose
signature is silence must show.  Third, ``consistent'' is doing
modest, clearly-priced work throughout: it strengthens the
reading and confirms nothing --- confirmation is not on offer in
this Part, only survival.

\paragraph{The Part in one breath, and the door to the next.}
The record is one-sided: every shot fired at the module since
2015 has landed on \emph{consistent}, and the module's response
is not celebration but escalation --- each upgraded instrument
makes the promised silence harder to keep.  The physics intuition
to carry forward: \emph{a theory is worth exactly the risks it
signs for}, and the tables above are its signatures.  What
remains is judgment: how this reading compares, framework by
framework, with the alternatives that half a century of
black-hole physics has put on the table --- and what the bottom
line is.  That reckoning is the business of the next Part.


%% ===================================================================
%% PART -- SUMMARY AND COMPARISON
%% ===================================================================

\section[Part~VIII in plain English]{Part~VIII in plain English: the record of every shot fired, and the dates the next ones are scheduled for}
A prediction is worth what it risks, so this Part is the accounting.

The record first, and it is one-sided in a way that should be read
carefully rather than proudly. Since 2015 every measurement that could have
contradicted the module --- two shadows, a decade of ringdowns, the area
theorem confirmed in 2025 --- has landed on consistent. That is not a run
of successes. It is a run of tests the framework was always going to pass,
and the right response is to say so and then post a harder bond.

The bond is the falsifier timeline, with dates. Ten named criteria, each
with the instrument that would apply it and the window in which it will.
Two of them need instruments that do not exist. Two are theory-side
questions with no instrument at all, including the species-sum computation
this paper leaves open. The rest run on the next-generation Event Horizon
Telescope, on LISA in the middle 2030s, and on the ringdown catalog
accumulating now.

The Part also states its refresh discipline, which is a small thing that
matters more than it looks. Every number in these tables comes from a
primary record with a date attached, and there is a written rule for when
each is re-read. A comparison table that quietly ages is worse than no
table at all, because it goes on looking current.

\GGPart[Summary and comparison]{Summary and Comparison}{Scorecards, Framework Comparisons, and the Bottom Line}

\noindent
This Part is the fast reader's exit and the referee's desk: the
whole paper weighed, in one place.  It opens by stating plainly
what \GGtheory{} claims about the black hole and what would
falsify the claims; scores the classical puzzles one by one; sets
the module against every leading framework --- first in two
tables, then in a section of arguments; and closes with the
concept dictionary, the axioms actually used, and the bottom line
in a single row.  Everything comparative the paper has to say
lives in this Part; everything forward-looking --- the
inventions, the open questions, the future directions --- is
deliberately kept for the next.

\section*{Synthesis: what \GGtheory{} claims about the black
hole}

Before the summary tables, let us state plainly what
\GGtheory{} claims about the black hole, and what would
falsify these claims.

\paragraph{The positive claims.} The black-hole horizon in
\GGtheory{} is a specific locus in 4D spacetime where the
bulk degrees of freedom accessible to the external observer
become highly compressed into compact-dimension and
radial-interior sectors that are not directly observable. The
\OGGEP{} functor, applied at this locus, produces a \CPTP{}
channel whose output is the Hawking radiation spectrum. The
entropy stored at the horizon is literally the entanglement
entropy between the 4D external Hilbert space and the
inaccessible bulk sector. The coefficient $1/4$ is \emph{derived} from
induced gravity (\S\ref{ssec:V-conical}): the regulator-independent
ratio $4\pi/16\pi$, with no free parameter. The
Page curve follows automatically from bulk unitarity plus
\CPTP{} projection. No firewall is required because the
apparent monogamy violation of AMPS is an artifact of
4D-only thinking; at the 6D level, the ``interior partner''
is a bulk state shared with the entire evaporation history.

\paragraph{What this predicts.} For astrophysical black holes,
all observables (EHT shadow, LIGO ringdown, Hawking spectrum if
it were observable) should match Kerr to extraordinary precision
(better than parts in $10^{46}$, and $\sim10^{-66}$ for the
EHT-imaged supermassive shadows), because the corrections are
suppressed by $(\ell_{\rm lock}/r_s)^2$ with
$\ell_{\rm lock}\equiv\hbar/(\Mlock c)\sim5\times10^{-20}$~m.
For primordial black holes, the scale structure may imprint
preferred mass \emph{ratios} (a geometric ladder spaced by
$e^{7}$); the absolute scale is set by the formation-horizon mass
and is not yet derived (\S\ref{sec:PBH}). The
Bekenstein--Hawking entropy coefficient $1/4$ is derived
(\S\ref{ssec:V-conical}); the one residual task is the explicit species
sum that fixes the induced $\Newton$ numerically (\S\ref{ssec:V-honest}).

\paragraph{What would falsify the claims.} Any of the following
observations would retire the \GDOS{} reading of black holes:
a Kerr deviation at the $>5\%$ level in EHT shadows; a QNM
deviation at the $>1\%$ level in LIGO/LISA ringdowns; a
detection of gravitational-wave echoes from outside the
horizon; a demonstrated information-loss-in-gravitation at
any energy scale; a violation of the Bekenstein entropy
coefficient via microscopic counting experiments.

\paragraph{What is theoretical vs. empirical.} The information
paradox itself is a theoretical consistency problem, not an
observational one. No current or near-future experiment can
directly test the Page curve in a real astrophysical black
hole. The \GDOS{} resolution is empirically consistent because
it predicts Kerr + small corrections for all feasible
observations, and it is theoretically consistent because it is
grounded in a specific UV theory (\GGsix{}) rather than an
abstract duality (\AdSCFT{}).

\paragraph{What is new in \GDOS{}.} Compared to the existing
resolutions (\AdSCFT{}, islands, ER=EPR, fuzzball), the \GDOS{}
reading brings a Standard-Model-compatible UV theory (\GGsix{}
with \SUfive{} GUT); a specific, finite microstate count via
$\UoneQi$ holonomies on $S^1_\theta$, with the coefficient fixed
regulator-independently by induced gravity (and
\OGNfive{}-independent, Theorem~\ref{thm:exact-quarter}); an
automatic Page curve, with no replica wormholes and no
quantum-extremal-surface extremization as separate postulates; a
geometric reading of the ``interior partner'' as a bulk state,
dissolving the AMPS monogamy paradox; and concrete Kerr
corrections priced at the $(\ell_{\rm lock}/r_s)^2$ level --- no
observable deviation for astrophysical black holes, with the
falsifiable content carried by the \emph{null} rather than by a
positive signature.  Each of these is developed as a claimed
invention, with its provenance and its limits, in the
Discussions Part.

\paragraph{The role of the present paper in the program.} This
paper is the black-hole companion (C3-BH) of the twenty-two-paper
\GGtheory{} cohort and completes its gravitational branch.  Its
place in the program, and the honest division between what is
established here and what is registered open, are the business of
the Discussions Part.

\clearpage

\section*{Puzzle-by-puzzle scorecard}

\begin{table}[H]
\centering\small
\caption[Classical puzzles of black-hole physics and their resolution in \GGtheory{}.]{Classical puzzles of black-hole physics and their
resolution in \GGtheory{}. Read the last column first. Resolved and reframed are different claims: resolved means the puzzle does not arise in the six-dimensional reading, reframed means it does arise and is answered differently. Neither means the standard treatments were wrong about their own domain. The third column is included so a reader can see exactly which lift does the work in each row, and judge the row on that rather than on the verdict.}
\label{tab:bh_scorecard}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{3.07cm}p{3.28cm}p{3.28cm}p{3.58cm}c@{}}
\toprule
\textbf{Puzzle} & \textbf{Standard question} &
\textbf{6D lift} & \textbf{4D resolution} & \textbf{Status}\\
\midrule
Information paradox (Hawking 1976) &
Pure $\to$ mixed; unitarity violated? &
Bulk unitary; 4D CPTP projection &
Radiation encodes info in bulk-to-boundary correlations &
Resolved \\
\addlinespace[0.15em]
Page curve &
Why should $S_R$ turn over at $t_P$? &
Bulk bipartition symmetric around $t_P$ &
Automatic from bulk unitarity + partial trace &
Resolved \\
\addlinespace[0.15em]
Bekenstein--Hawking entropy &
What states count? &
$\UoneQi$ holonomies on $S^1_\theta$ at horizon &
$\Sbh = A/(4\Newton\hbar)$, exact $4\pi/16\pi$ (induced gravity) &
Resolved \\
\addlinespace[0.15em]
Firewall paradox (AMPS 2012) &
Monogamy vs no-drama &
$A$ partner is bulk, not independent 4D &
No firewall; infalling observer sees smooth horizon &
Resolved \\
\addlinespace[0.15em]
ER=EPR (Maldacena-Susskind 2013) &
Entangled BHs = wormholes? &
Entanglement encoded in shared bulk sector &
Automatic; no ad hoc ER=EPR postulate needed &
Reframed \\
\addlinespace[0.15em]
Islands formula (Penington et al. 2019) &
How to get Page curve from gravity? &
Island = bulk region traced by \OGGEP{} &
Formula recovered; derivation physically concrete &
Reframed \\
\addlinespace[0.15em]
Singularity &
What's at $r=0$? &
$r=0$ is small-scale-profile region, not singular in 6D &
Coordinate artifact; bulk is smooth &
Resolved \\
\addlinespace[0.15em]
No-hair theorem &
Why only 3 parameters? &
4D observables are zero-mode projection of 6D state &
Hair is in bulk; 4D sees only $(M, J, Q)$ zero modes &
Resolved \\
\addlinespace[0.15em]
Species problem &
Why coefficient $1/4$ not $N/4$? &
Induced gravity; no bare $\Newton$ &
$1/4 = 4\pi/16\pi$, species-indep. &
Coeff.\ exact \\
\addlinespace[0.15em]
Why black holes radiate? &
Quantum effects in curved ST &
Horizon = \OGGEP{} port = \CPTP{} channel &
Thermal output automatic &
Resolved \\
\addlinespace[0.15em]
de Sitter entropy &
$S_{\rm dS} = A_H/(4\Newton\hbar)$? &
Cosmological horizon = \OGGEP{} port &
Same mechanism as BH entropy &
Resolved \\
\addlinespace[0.15em]
Holographic principle &
Why area-extensive entropy? &
Bulk-to-boundary projection has $A$-scaled bandwidth &
Holography automatic; $1/(4\Newton\hbar)$ from scale profile &
Resolved \\
\addlinespace[0.15em]
Complementarity &
Inconsistent exterior vs infalling views? &
Different partial traces of same 6D state &
No inconsistency; no firewall required &
Resolved \\
\bottomrule
\end{tabular}
\end{table}

\section*{Framework comparison}

\begin{table}[H]
\centering\small
\caption[Six frameworks for black-hole physics compared across twelve desiderata.]{Six frameworks for black-hole physics compared across
twelve desiderata. Y = has the property, N = does not, $\sim$
= partial/ambiguous. No column here is clean, including the last one, and the two entries where this framework does not score --- the microstate count and the reach of its quantitative predictions --- are the two a specialist would ask about first. They are left where they fall in the row order rather than moved to the bottom. A framework built to count microstates in a controlled sector should be judged on that, which is why the comparison is by property and not by total.}
\label{tab:bh_framework}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{3.91cm}p{1.61cm}p{1.61cm}p{1.61cm}p{1.61cm}p{1.61cm}p{1.61cm}@{}}
\toprule
\textbf{Property} &
\textbf{Classical GR} & \textbf{\AdSCFT} &
\textbf{Fuzzball} & \textbf{Firewall} &
\textbf{Islands} & \textbf{\GGtheory}\\
\midrule
Resolves info paradox & N & Y & Y & Y (partially) & Y & Y\\
Bulk unitarity preserved & N & Y & Y & N & Y & Y\\
Page curve derived & N & Y (limited) & $\sim$ & N & Y (2D) & Y (4D)\\
No firewall needed & Y & Y & Y & N & Y & Y\\
Standard Model compatible & Y & N & $\sim$ & $\sim$ & $\sim$ & Y\\
Real-universe applicable & Y & N & $\sim$ & Y & $\sim$ & Y\\
Smooth horizon (no drama) & Y & Y & N (fuzzball replaces) & N & Y & Y\\
Singularity resolved & N & $\sim$ & Y & N & $\sim$ & Y\\
BH microstates counted & N & Y (specific cases) & Y & N & $\sim$ & $\sim^{\dagger}$\\
Quantitative predictions beyond Kerr & $\sim$ & Limited & Some & Some & Limited & Limited but specific\\
No free parameters & N & N & N & N & N & Y\\
UV theory specified & N & Y (string) & Y (string) & N & N & Y ($\GGsix$)\\
\bottomrule
\end{tabular}
\end{table}

These desiderata are necessarily schematic, and the table is best
read as a map of explanatory roles rather than a scoreboard.  On it
\GGtheory{} scores well; the entry that remains in progress ---
the explicit numerical microstate enumeration ($\sim^{\dagger}$) --- is
flagged honestly in \S\ref{ssec:V-honest}.  ($^{\dagger}$~The coefficient
$1/4$ and the identity of the microstates are settled for generic black
holes in \S\ref{ssec:V-conical}--\ref{ssec:V-microstates}; the
outstanding item is the numerical species sum, not the coefficient.)

\section*{Comparative landscape: what each framework takes as primary}

The most useful comparison is not which framework ``wins'' but what each takes as primary, explains well, and finds difficult.  Table~\ref{tab:bh-landscape} records this in deliberately neutral terms; the string-theoretic entries in particular are \emph{internally forced} by consistency, not arbitrary, and we present them as such.

\begin{table}[H]
\centering\small
\caption[Comparative landscape of major black-hole information approaches.]{Comparative landscape of major black-hole information approaches.  Schematic by design: the aim is to distinguish explanatory roles, not to force false exclusivity. The last column is the one worth reading, because it is where each approach is weakest by its own lights rather than by this paper's, and every entry in it is drawn from that approach's own literature. The table is schematic by design and the caption says so: these approaches are not exclusive, and a reader could hold several of these rows simultaneously with this one.}
\label{tab:bh-landscape}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{0.16\textwidth}p{0.21\textwidth}p{0.24\textwidth}p{0.28\textwidth}@{}}
\toprule
\textbf{Approach} & \textbf{Primary ontology} & \textbf{Main strength} & \textbf{Main open issue} \\
\midrule
Pure Hawking semiclassics & Quantum fields on a fixed curved background & Temperature and approximate thermal flux & Monotonic radiation entropy and information loss if taken literally \citep{Hawking1976}. \\
Complementarity & Observer-dependent but unitary horizon & Retains unitarity and a smooth horizon together & Hard to formulate once entanglement monogamy is imposed. \\
Fuzzballs / string microstates & Explicit microstate geometries or string states & Genuine microscopic entropy accounting in special (BPS) sectors \citep{StromingerVafa1996} & Generality beyond controlled, supersymmetric sectors. \\
Island / \QES{} program & Generalized entropy of the radiation subsystem & Quantitative Page-curve recovery in controlled models \citep{Penington2020, Almheiri2019} & Deeper ontology of islands; extension to realistic, non-AdS settings. \\
\GGsix{} / \GDOS{} / \OGGEP{} & Closed 6D bulk plus open boundary projection & Exact $A/4\Newton$ for \emph{generic} black holes (\S\ref{ssec:V-conical}); a universal reason generalized entropy and branch switching arise & Numerical species closure of the induced $\Newton$ (\S\ref{ssec:V-honest}); benchmarking against detailed island computations. \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The entropy problem, head to head.}  The sharpest single test is the one this paper targets: reproduce $S = A/4\Newton$, \emph{with the exact $1/4$, by counting}.  Stated neutrally: Strominger--Vafa achieves the exact coefficient
by an explicit D-brane state count --- but only for
supersymmetric, extremal or near-extremal black holes
\citep{StromingerVafa1996}.  Loop quantum gravity reproduces the
area law from spin-network states, with the $1/4$ fixed by tuning
the Barbero--Immirzi parameter.  Fuzzballs count horizonless
microstate geometries, sharpest in controlled (largely
supersymmetric) sectors.  The island/\QES{} program recovers the
\emph{Page curve} in controlled models, but does not by itself
supply a first-principles microstate count for a real
Schwarzschild black hole.  And \GGtheory{} obtains the exact
$1/4 = 4\pi/16\pi$ from induced gravity for the \emph{generic,
non-supersymmetric} black hole (\S\ref{ssec:V-conical}), with the
microstates identified as the hidden-fiber holonomies
(\S\ref{ssec:V-microstates}) --- the outstanding item the
numerical species sum, not the coefficient.
The honest verdict is therefore narrow and strong: on \emph{generality} \GGtheory{} reaches the case the explicit string count cannot, and it fixes the coefficient without tuning; on \emph{explicit enumeration} the string result remains the more complete in its own supersymmetric domain.  The two are best read as complementary --- one maximal in a protected sector, the other general but still closing its species sum.

\section{Comparison with the leading frameworks: the deeper analysis}

The tables above give the map; this section gives the arguments.
We take the leading alternative frameworks in turn --- classical
general relativity, the \AdSCFT{} correspondence, the fuzzball
program, the firewall proposal, and the islands formula --- and
close with what the runtime taxonomy adds that none of them has.
Everything comparative the paper has to say lives in this Part;
what the module \emph{adds} is deliberately kept for the
Discussions.

\subsection{Classical general relativity}

Classical GR is the starting point. It gives us the horizons,
the singularity, the no-hair theorem, and the
Bekenstein--Hawking entropy as an analogy to thermodynamics. But it
does not explain what the horizon really is, what the
microstates are, or why the entropy is area-extensive. The
information paradox is, in classical GR, simply a statement
of breakdown: quantum mechanics demands something that
classical GR cannot provide.

\GDOS{} augments classical GR with the bulk-to-boundary
projection structure. The classical 4D metric is the zero-mode
projection of the 6D bulk. All of classical GR is recovered in
the low-energy limit. The Hawking radiation, the entropy, and
the Page curve emerge as consequences of the projection
rather than as postulates added on top of classical GR.

\subsection{The fuzzball program}

The fuzzball program, developed primarily by Samir Mathur and
collaborators since 2003, proposes that the interior of a black
hole is not empty but rather a quantum superposition of
horizonless geometries. In this picture, there is \emph{no
horizon}: the region inside the would-be Schwarzschild radius
is a complex quantum-gravitational state that differs from
vacuum by order-unity amounts at horizon scale.

The fuzzball proposal resolves the information paradox by
eliminating the horizon: if there is no horizon, there is no
Hawking radiation in the standard sense, and no information
loss. But this requires that external observations --- the
shape of the black-hole shadow, the ringdown spectrum --- be
dominated by order-unity quantum-gravity effects near what
would classically be the horizon. As of 2026, EHT and LIGO/LISA
observations are consistent with Kerr to high precision, which
strongly constrains fuzzball proposals: they must reproduce
Kerr behavior at macroscopic scales to extraordinary precision,
which is hard to motivate from string-theory fundamentals.

The strengths of the fuzzball program include explicit
microstate geometries (in specific string-theory backgrounds),
a natural resolution of the information paradox (no horizon
means no horizon-based information loss), and a concrete
mechanism by which microstates differ from one another. The
weaknesses include: (i) horizonless geometries are hard to
reconcile with high-precision EHT and LIGO observations
consistent with Kerr; (ii) fuzzball constructions work best
in highly supersymmetric backgrounds not directly relevant to
real black holes; (iii) the physical mechanism by which the
classical horizon emerges from the fuzzball superposition is
unclear.

\GDOS{} takes a different position: there \emph{is} a horizon,
and it is smooth at the 6D bulk level.  The microstates are not
distinct horizonless geometries but distinct bulk configurations
with the same 4D zero-mode projection, so observational agreement
with Kerr is automatic at the 4D level, to all feasible
precision.  Fuzzballs, by contrast, must either reproduce Kerr
exactly --- in which case they add no testable content --- or
predict small deviations, in which case they risk falsification
as EHT precision improves.  Information, here, is preserved via
the \OGGEP{} \CPTP{} channel, not via horizonless microstate
geometries.

\subsection{\AdSCFT{} correspondence}

The \AdSCFT{} correspondence is a specific duality between
Type IIB string theory on AdS$_5 \times S^5$ and
$\mathcal{N} = 4$ super-Yang-Mills in 4D Minkowski space. It
is a genuine duality --- each side is a complete quantum
theory, and physical quantities are translated via a precise
dictionary. In this framework, black-hole evaporation is
manifestly unitary because the boundary CFT is unitary.

\AdSCFT{} provides a deep and rigorous setting for studying
quantum gravity in anti-de-Sitter space. Its limitations for
real-universe physics are, however, substantial. Our universe
is de Sitter, not anti-de Sitter; the actual gauge group is
not $\mathcal{N} = 4$ SYM; and the bulk theory is not type
IIB string. Asymptotically-flat black holes in 4D Minkowski
space do not have a direct \AdSCFT{} dual --- their treatment
requires cavity or brane constructions that are not rigorous.
Put plainly:
\AdSCFT{} needs the box and the dual, and our universe declines
to provide either.  What is missing is an anchor for the boundary
description that does not depend on asymptotics; \GGtheory{}'s
anchor is the \emph{observer} --- the port is defined by causal
access, not by a wall at infinity --- which is why the machinery
moves to flat space and de Sitter without modification.

\GDOS{} occupies a complementary position: it is less rigorous
than \AdSCFT{} (it is not a proved duality but a proposed
constitutional framework), but it is \emph{physically more
relevant} to our universe because its bulk content is the
Standard Model. \GGtheory{} and \AdSCFT{} should be thought
of as two holographic frameworks at different levels of
formalization and different domains of applicability.  The
bulk-to-boundary projection \OGGEP{} is the analogue of the
\AdSCFT{} dictionary, defined for the physical universe.

\subsection{Firewall proposal}

The firewall proposal (AMPS 2012) sacrificed horizon smoothness
to preserve unitarity plus Bekenstein--Hawking entropy.
Infalling observers, in this picture, encounter a wall of
high-energy quanta at the horizon.

The firewall proposal is consistent with unitarity but violates
the equivalence principle of general relativity. It has few
defenders as of 2026; the islands formula has superseded it as
the preferred resolution in the \AdSCFT{} community.

\GDOS{} agrees with the post-firewall consensus that the
horizon is smooth, but provides a different mechanism: the
apparent monogamy violation dissolves at the 6D bulk level,
so no firewall is needed.

\subsection{The islands formula in \GDOS{}}


The islands formula, established in 2019--2020, is currently
the most widely-accepted resolution of the information paradox.
It reproduces the Page curve in two-dimensional JT gravity via
the quantum-extremal-surface prescription combined with replica
wormhole saddles in the Euclidean gravitational path integral.

The strengths of the islands framework are its mathematical
precision (in 2D) and its direct contact with \AdSCFT{}. The
weaknesses are: (i) it is rigorously established only in 2D,
and its extension to 4D is partial; (ii) the replica wormhole
mechanism is non-perturbative and lacks a clean physical
interpretation; (iii) the extremization prescription is an
added computational ingredient rather than a primary physical
principle.

\GDOS{} reproduces the islands formula in the appropriate limit
but does not require it as a separate postulate. The bulk
partial-trace structure of \OGGEP{} produces the same
entanglement entropy as the islands formula, and the
quantum-extremal-surface is identified with the geometric
boundary between the traced and un-traced bulk regions. The
\GDOS{} version has the advantage that the ingredients
(unitary bulk evolution plus \CPTP{} projection) are physically
concrete, rather than formal path-integral manipulations.

The islands formula (\ref{eq:island}) is a precise
computational rule for the entanglement entropy in
two-dimensional JT gravity coupled to matter. In \GDOS{}, the
analogous statement is that the 4D entanglement entropy is
\begin{equation}\label{eq:GDOS_entropy_formula}
S(R) = \SvN(\rho_\partial^R) = \min_{\Sigma}\left[
\frac{\mathrm{Area}(\Sigma)}{4\Newton\hbar} + S_{\rm
semi-cl}(R \cup I_\Sigma)\right],
\end{equation}
where $\Sigma$ is the quantum extremal surface in the bulk,
$I_\Sigma$ is the island bounded by $\Sigma$, and the
minimization is over all admissible $\Sigma$.

In the JT limit (two-dimensional bulk with specific coupling
to matter), this reduces to the islands formula of
Penington and Almheiri et al.

\subsection{ER=EPR in \GDOS{}}

The ER=EPR proposal of \citet{MaldacenaSusskind2013} asserted that
two entangled black holes are connected by a wormhole. In
\GDOS{}, this is automatic: two black holes whose interior
degrees of freedom are entangled in the bulk share a common
bulk sector, which geometrically is a wormhole-like connection
through the bulk compact dimensions.

The \GDOS{} picture makes ER=EPR computationally tractable:
given two bulk regions $B_1, B_2$ with entangled bulk states,
the \OGGEP{} projection to the 4D exteriors produces two 4D
black holes whose radiation is correlated. The ER bridge is
the shared bulk sector; the EPR correlations are the resulting
4D radiation correlations.

\subsection{What the runtime taxonomy adds that no leading framework has}
\label{ssec:runtime-uniqueness}

The comparisons above were drawn at the level of the information
question, where every leading framework has an answer of some kind.
The runtime classification developed in this paper
(\S\ref{sec:bh-way-station}, \S\ref{ssec:bh-sorting}) permits a
sharper statement of what is unique here, because it operates at a
level the other frameworks do not occupy at all: the \emph{dynamical
classification} of black-hole events. Four deliverables --- collected in Table~\ref{tab:runtime-uniqueness} --- have no
counterpart in any leading approach.

\paragraph{(i) A two-regime taxonomy with theorems behind it.}
Formation, merger, accretion, evaporation, the Hawking--Page
transition, and the Page turnover are sorted by a single question ---
\emph{does the landscape change?} --- into capture, coalescence,
traffic, channel exit, the one true \GRIPout{}, and a milestone,
respectively. Each label is backed by a proved theorem (Lyapunov
descent, Out-necessity, No-Exit
(\citealp{ArisakaA6Gi4}; \citealp{ArisakaA11Origin})), not by a metaphor. No other
framework classifies black-hole dynamics at all; they classify
\emph{states} (microstates, fuzzball geometries, code subspaces) and
remain silent about which events reorganize the problem and which
merely evolve it.

\paragraph{(ii) The conditional-ground-state statement, with an
expiration date.} That every astrophysical black hole is today a
\emph{conditional} ground state --- exitless only because $T_H <
T_{\rm CMB}$, with the condition expiring on the calculable schedule
$M_{\rm open}(t) \propto a(t)$
(\S\ref{ssec:conditional-gga}) --- is a new \emph{kind} of statement
about black holes: their thermodynamic finality is
boundary-condition-relative and cosmology-scheduled. The ingredients
(Hawking temperature, CMB cooling) are textbook; the statement is
unwritable without the open-system vocabulary, which is why it does
not appear in the fuzzball, \AdSCFT{}, or islands literatures.

\paragraph{(iii) Gravitational-wave observables tied to the same
functional as particle physics.} The ringdown spectrum is the Hessian
of the free-energy ledger at the Kerr attractor
(\S\ref{sec:ringdown}) --- the same functional whose minima, on other
strata, are the electron, the proton, and the dark-matter candidate.
Islands and \AdSCFT{} say nothing about LIGO; fuzzballs predict
horizon-scale deviations that EHT and ringdown data increasingly
disfavor. Here the observed Kerr-ness of ringdowns is not an
embarrassment to be explained away but a \emph{confirmation}: clean
basin curvature is exactly what a deep, simple attractor should show.

\paragraph{(iv) Interlocking falsifiability.} The machinery that
resolves the paradox here was not built for black holes: the same
runtime, with the same theorems, is the engine that selects the
electroweak vacuum, stabilizes the electron (\citealp{ArisakaP3Electron}), runs decoherence and
measurement, and predicts the dark-matter window of the companion
papers. The account is therefore falsifiable \emph{from outside
black-hole physics}: kill the $\Delta\nu/\nu = 1/35$ dark-matter
doublet, the $3.85$~TeV resonance cluster, or the normal neutrino
ordering, and this paper's reading of black holes dies with them ---
an inverted neutrino ordering, measured in Illinois, would retire a
theorem about horizons. No
black-hole-only framework offers hostages of that kind; a framework
that cannot be shot from a neighboring field is, by the falsification
standard of this series, the weaker for it.

\begin{table}[tbp]
\centering\small
\caption[The leading frameworks against the runtime criteria.]{The leading frameworks against the runtime criteria. ``Dyn.\
taxonomy'' = a classification of black-hole \emph{events} (not just
states); ``cross-domain engine'' = the same proved machinery operating
outside gravity; ``external falsifiers'' = kill criteria located in
other fields.}
\label{tab:runtime-uniqueness}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}p{0.20\textwidth}p{0.13\textwidth}p{0.15\textwidth}p{0.13\textwidth}p{0.13\textwidth}p{0.12\textwidth}@{}}
\toprule
\textbf{Framework} & \textbf{Real universe?} & \textbf{Info answer} & \textbf{Dyn.\ taxonomy} & \textbf{Cross-domain engine} & \textbf{External falsifiers} \\
\midrule
Classical GR & yes & none (paradox) & no & no & no \\
Fuzzball & in principle & no horizon & no & no & no \\
\AdSCFT{} & AdS only & duality & no & no & no \\
Islands / QES & toy models & Page curve & no & no & no \\
\GGtheory{} runtime & yes & \OGGEP{} channel (Parts~IV--VI) & \textbf{yes} (two regimes) & \textbf{yes} (A6--A11) & \textbf{yes} (1/35; 3.85 TeV; $\nu$-ordering) \\
\bottomrule
\end{tabular}
\end{table}


\section*{Concept dictionary: 4D to 6D for black holes}

\noindent\emph{For the canonical Dictionary of program-defined acronyms (45 entries in ten categories), see \ATwo{} Appendix~B. The table below translates only the black-hole-specific 4D$\leftrightarrow$6D vocabulary.}

\begin{table}[H]
\centering\small
\caption[Dictionary translating 4D black-hole concepts to their 6D bulk origin in \GGtheory{}.]{Dictionary translating 4D black-hole concepts to their
6D bulk origin in \GGtheory{}. A dictionary is a claim about translation rather than about physics, and the third column is where the claim can be checked. Each row says which bulk object the four-dimensional concept is a projection of, and by what operation; a row whose mechanism is vague is a row where the translation has not yet been earned. Nothing is derived in the table --- every entry points at the section that derives it.}
\label{tab:bh_dictionary}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{4.69cm}p{5.28cm}p{5.28cm}@{}}
\toprule
\textbf{4D BH concept} &
\textbf{6D bulk origin} &
\textbf{Mechanism}\\
\midrule
Event horizon &
\OGGEP{} port (one-way causal boundary) &
Locus separating accessible from inaccessible bulk \\
\addlinespace[0.1em]
Hawking radiation &
\CPTP{} channel output &
Partial trace over bulk interior \\
\addlinespace[0.1em]
Hawking temperature $\THaw$ &
Scale profile at horizon &
$\kB\THaw = \hbar\kappa/(2\pi c)$ automatic \\
\addlinespace[0.1em]
Bekenstein--Hawking entropy &
von Neumann entropy of $\rho_\partial^{\rm out}$ &
Entanglement of 4D exterior with bulk \\
\addlinespace[0.1em]
Microstate count $e^{\Sbh/\kB}$ &
Phase-space volume of $\UoneQi$ holonomies on $S^1_\theta$ &
One per horizon pixel of area $4\ell_P^2$ \\
\addlinespace[0.1em]
Page curve &
Bipartite pure-state entanglement entropy &
Automatic from bulk unitarity \\
\addlinespace[0.1em]
Singularity at $r = 0$ &
Small-scale-profile region of 6D bulk &
Coordinate artifact in 4D projection \\
\addlinespace[0.1em]
No-hair theorem &
Zero-mode projection of bulk state &
Higher-mode content traced over \\
\addlinespace[0.1em]
Holographic principle &
$\OGGEP$ bandwidth is area-extensive &
Fundamental structural property \\
\addlinespace[0.1em]
de Sitter horizon &
Cosmological \OGGEP{} port &
Same mechanism as BH horizon \\
\addlinespace[0.1em]
Interior partner (AMPS) &
Bulk degrees of freedom, not 4D subsystem &
Dissolves monogamy puzzle \\
\addlinespace[0.1em]
Quantum extremal surface &
Boundary of \OGGEP{}-traced region &
Geometric locus of partial trace \\
\addlinespace[0.1em]
Replica wormhole &
Bulk path-integral saddle connecting replicas &
Natural in \GGsix{} path integral \\
\addlinespace[0.1em]
Island $I$ &
Bulk region traced by \OGGEP{} at given time &
Physical realization of formal island \\
\bottomrule
\end{tabular}
\end{table}

\section*{Program axioms used}

\begin{table}[H]
\centering\small
\caption[\GGtheory{} axioms and postulates used in the black-hole derivations.]{\GGtheory{} axioms and postulates used in
the black-hole derivations. Seven entries, and the point of listing them is to bound what the derivations are allowed to use. The last column says where each is actually spent, so a reader who doubts a result can find the premise it rests on without leaving the paper. Nothing here is derived; it is the list of what the derivation may assume, and the exact \(1/4\) uses fewer of them than the row order suggests.}
\label{tab:bh_axioms}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}llp{4.84cm}p{5.38cm}@{}}
\toprule
\textbf{Label} & \textbf{Name} & \textbf{Statement} &
\textbf{Role in BH physics}\\
\midrule
AX-1 & Local causality & Forward-time causation in bulk &
Horizon is a forward-causal boundary \\
AX-2 & Primitive connection & Gauge-gravity composite
connection on $M_6$ &
Bulk metric includes gauge sector \\
AX-3 & \OGGEP{} & Bulk $\to$ boundary functor &
Horizon is a port for $\mathcal{D}$; Hawking radiation is
output \\
AX-4 & \MSMF{} & Max sym, min freedom &
Forces Kerr on boundary; no freedom for firewalls \\
POS-CP & Complete positivity & \CPTP{} structure preserved &
Hawking channel is \CPTP{} \\
POS-OGN & \OGNfive{} & Integer ledger $(1,3,5;7,35)$ &
Field content and induced-$\Newton$ species sum; spectrum structure (the $1/4$ itself is \OGNfive{}-independent, Thm~\ref{thm:exact-quarter}) \\
POS-RG & Scale consistency & Commutes with coarse-graining &
Area law at all scales \\
\bottomrule
\end{tabular}
\end{table}

\section*{Bottom-line resolution}

\begin{table}[H]
\centering\small
\caption[Single-row summary of the 1975--2026 information-paradox debate.]{Single-row summary of the 1975--2026 information-paradox
debate. Two columns, one question per row, and the differences between them are the whole of the information paradox. Read down, the bulk column answers no loss and no firewall throughout; the boundary column agrees, with a parenthesis each time saying why the naive reading does not. The one row where the two genuinely differ is the singularity, and that difference is a statement about coordinates rather than about physics.}
\label{tab:bh_bottom_line}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{@{}p{0.30\textwidth}p{0.31\textwidth}p{0.31\textwidth}@{}}
\toprule
\textbf{Question} & \textbf{6D Bulk} &
\textbf{4D Boundary}\\
\midrule
Does info get lost in evaporation? & NO & NO
(appears so at naive Hawking level) \\
Is the evaporation unitary? & YES & YES (via bulk-to-boundary
correlations) \\
Does the Page curve hold? & YES & YES (automatic) \\
Are there firewalls? & NO & NO \\
Is the horizon smooth? & YES (6D-smooth) & YES (4D-apparent) \\
Is there a singularity? & NO (6D-smooth) & YES (4D-coordinate) \\
Is information stored on boundary? & YES (holography) &
YES (Bekenstein area law) \\
\midrule
& \multicolumn{2}{p{0.62\textwidth}}{\emph{The bulk is unitary and smooth; the
4D boundary appears paradoxical because the 4D observer is
seeing only a partial trace of the 6D state.}}\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The Part in one breath, and the door to the next.}
Weighed against half a century of frameworks, the module's case
is generality without dials: it reaches the astrophysical hole
the protected counts cannot, recovers the islands result without
its auxiliary machinery, keeps the horizon smooth without
sacrificing unitarity, and prices its one open computation as a
registered falsifier rather than a footnote.  The physics
intuition to carry forward: \emph{where others must engineer
protection, a forced trace and a fully induced $\Newton$ make
the results structural}.  The comparison says where the module
stands.  What it \emph{adds} --- the inventions, what remains
honestly open, and the road ahead --- is the business of the
next Part.

\clearpage

%% ===================================================================
%% PART -- DISCUSSIONS (permanent home of the formerly stranded sections)
%% ===================================================================

\section[Part~IX in plain English]{Part~IX in plain English: how the reading stands against five rivals, including where it loses}
Six frameworks, twelve properties, and no column that scores on all of
them.

The comparison is arranged by property rather than by winner, because the
frameworks were built for different jobs. String microstate counting
produces genuine microscopic entropy in supersymmetric sectors and does not
reach an astrophysical hole; that is a limitation of scope rather than a
failure, and the table says which. The island program recovers the Page
curve quantitatively in controlled models and leaves the ontology of an
island open. \AdSCFT{} is exact, and is about a universe that is not this
one.

Where this reading claims an advantage it is generality: the coefficient is
derived for generic holes with no protected sector, the Page curve follows
from a ledger rather than from auxiliary machinery, and the horizon stays
smooth without giving up unitarity. Where it does not score, it does not
score --- and two of those places are the first a specialist would ask
about. The microstate count is partial. The quantitative predictions beyond
Kerr are limited and specific rather than broad.

The honest summary is that this framework buys generality and pays for it
in sharpness. Whether that is the right trade is a judgment, and this Part
sets out the terms rather than announcing the answer.

\GGPart[Discussions]{Discussions}{Inventions, Open Questions, and Future Directions}

\noindent
The judgment Part behind us, this Part is the reflective one:
what is genuinely new here, where it sits in fifty years of
literature, what role it plays in the wider program, what
remains honestly open, and how the module can be stressed from
the theory side alone.  Nothing in this Part re-argues the
comparisons of Part~IX; its business is the module's own ledger
--- the inventions on one page, the debts on the facing page ---
and the directions in which the work continues.

\section{What is new}

The preceding Parts together establish that the
black-hole information paradox admits a complete resolution
within the \GGtheory{}/\GDOS{} framework. This is not the
first claim of resolution in the history of the subject; the
islands formula of Penington and Almheiri et al.~in 2019
already gave a concrete derivation of the Page curve within
two-dimensional JT gravity, and the \AdSCFT{} correspondence
has for 25 years provided a setting in which black-hole
evaporation is manifestly unitary. What, then, does
\GGtheory{} add that was not already present?

The claim of this paper is that \GGtheory{} brings four
specific novelties beyond the existing literature.

\paragraph{Novelty 1: a Standard-Model-compatible UV completion.}
Every existing resolution of the information paradox is formulated
in a setting that is \emph{not} the physical universe. The
Strominger-Vafa microstate count works for extremal black holes
in specific string-theory backgrounds. The \AdSCFT{} framework
operates in anti-de-Sitter space with $\mathcal{N} = 4$
super-Yang-Mills boundary, neither of which describes our
universe. The islands formula is proved rigorously only in
two-dimensional Jackiw-Teitelboim gravity coupled to matter,
not in four-dimensional asymptotically-flat Schwarzschild.
Fuzzball proposals require specific string-theoretic structures
not yet mapped to Standard-Model gauge content.

\GGtheory{} is different in this respect: the 6D bulk
$M_4\times I_r\times S^1_\theta$ carries the \SUfive{} GUT
group plus the two new Abelian symmetries $\Uoneb$ and
\UoneQi{}, and this content reduces on the 4D boundary to
exactly the Standard Model (as established in the constitutional
papers \AOne{}--\AFour{} and the Standard-Model paper \ANine{}). The black-hole physics developed in the present
paper therefore takes place in a UV theory that \emph{is} the
theory of our universe. The resolution is not a proof of
principle in a toy model; it is a claim about how real
astrophysical black holes work.

\paragraph{Novelty 2: the exact Bekenstein--Hawking coefficient from
induced gravity.} The coefficient $1/4$ has resisted first-principles
derivation for over fifty years.  Strominger--Vafa obtained it for
supersymmetric, extremal black holes in string theory; the
loop-quantum-gravity community recovered it at the cost of fitting the
Barbero--Immirzi parameter; the fuzzball program aims at it through
special microstate geometries.  \GGtheory{} derives it differently, and
for the \emph{generic} black hole: because \GGEP{} leaves no bare
four-dimensional Newton constant, the horizon entanglement entropy
\emph{is} the induced $A/4\Newton$, and the conical replica fixes the
coefficient as the regulator-independent ratio $1/4 = 4\pi/16\pi$
(Theorem~\ref{thm:exact-quarter}), with no supersymmetry and no
extremality.  Where the string-theoretic count needs symmetry
protection to keep the number of states rigid, here the coefficient
is not a count at all but a ratio of two normalizations fixed
upstream by non-negotiable physics; a ratio cannot drift.  The microstates are the hidden-fiber holonomies traced at
the horizon (\S\ref{ssec:V-microstates}); the species sum that fixes the
induced $\Newton$ numerically closes \emph{structurally} through the
Kaluza--Klein tower (Appendix~\ref{app:species-sum}), with the precise
leading coefficient the one item still outstanding.  The coefficient
itself carries no adjustable parameter.

\paragraph{Novelty 3: the Page curve from bulk unitarity alone,
with no auxiliary machinery.} The islands formula derivation
requires evaluating the gravitational path integral with
replica boundary conditions, identifying saddles that include
wormhole topologies connecting the replicas, and extremizing
over quantum extremal surfaces. These steps are mathematically
non-trivial and are rigorously justified only in specific
low-dimensional models.

In \GDOS{}, the Page curve is automatic. Bulk unitarity (\AX{1};
the custody \AFive{} states) guarantees that the total bulk state
remains pure.
The \OGGEP{} functor (AX-3) is a \CPTP{} channel whose output
is the 4D radiation. For a bipartite pure state, the
entanglement entropy of either side is a function of the
bipartition only; it follows the Page curve by Page's 1993
theorem applied to a typical bulk state. No replica
wormholes, no QES extremization, no island formula as a
separate postulate. The ingredients reduce to (i) unitary bulk
evolution and (ii) partial trace at the horizon.

\paragraph{Novelty 4: a geometric dissolution of the firewall
paradox.} The AMPS firewall argument runs as follows: for a
late-time Hawking mode $B$, unitarity requires entanglement
with early radiation $R$; smooth-horizon physics requires
entanglement with an interior partner $A$; entanglement
monogamy forbids both simultaneously; therefore either
unitarity fails, smoothness fails, or some postulate of
complementarity fails. The standard options --- firewalls,
ER=EPR, non-locality --- each have their own difficulties.

\GDOS{} resolves this by identifying the ``interior partner
$A$'' not as an independent 4D subsystem but as a bulk degree
of freedom that is part of the \OGGEP{}-traced region. Since
the bulk is unitary and the boundary is \CPTP{}, the apparent
tripartite decomposition $A \otimes B \otimes R$ is an
artifact of the 4D projection; at the 6D level there is no
monogamy violation because $A$ and $R$ are not independent
subsystems. The infalling observer encounters no firewall
because the 6D geometry is smooth at the horizon; the
apparent discontinuity is a 4D coordinate artifact, resolved
by the same mechanism that resolved the original Schwarzschild
``singularity'' at $r = r_s$ via Finkelstein's 1958 coordinates.

\section{Related work and precursors}
\label{sec:related-work}

Because this paper will be read against five decades of black-hole literature, we state plainly where it sits.  The thermodynamic foundation is Bekenstein's proposal that horizon area is physical entropy \citep{Bekenstein1973}, Hawking's derivation of thermal radiation \citep{Hawking1975}, the Bardeen--Carter--Hawking four laws \citep{BardeenCarterHawking1973}, and Page's unitarity curve \citep{Page1993}.  Our derivation of the coefficient belongs squarely to the \emph{induced-gravity} lineage --- Sakharov's hypothesis that the Einstein--Hilbert term is generated by matter loops \citep{Sakharov1968, Adler1982, Visser2002}, its black-hole specialization by Jacobson \citep{Jacobson1994, FrolovFursaev1997}, and the Susskind--Uglum / Callan--Wilczek / Demers--Lafrance--Myers identification of the horizon entanglement entropy with the induced $1/4G_N$ \citep{SusskindUglum1994, CallanWilczek1994, DemersLafranceMyers1995}.

What \GGtheory{} adds to that lineage is constitutional: \GGEP{} makes the four-dimensional Newton constant \emph{fully} induced (no bare term to split against), which removes the chief ambiguity of the Sakharov program and makes the entanglement entropy the entropy itself.  The projection half of the story belongs to the holographic lineage --- 't~Hooft and Susskind's holographic principle \citep{THooft1993, Susskind1995} and Maldacena's AdS/CFT \citep{Maldacena1998} --- of which \OGGEP{} is the open-system, real-universe generalization; the islands program \citep{Penington2020, Almheiri2019} is recovered as the \GFF{} ledger at the horizon stratum (\S\ref{sec:gff-ledger}), and Wald's Noether-charge entropy \citep{Wald1993} and Jacobson's ``Einstein equation as equation of state'' \citep{Jacobson1995} are complementary geometric readings of the same area law.

The information-theoretic anchors are the AMPS firewall \citep{AMPS2013}, Hayden--Preskill scrambling \citep{HaydenPreskill2007}, and the log-area corrections of Carlip and others \citep{Carlip2000, Solodukhin2011}, all of which we engage below.  The honest one-line placement: \GGtheory{} is induced-gravity for the coefficient and holographic for the projection, set in the Standard-Model-containing universe, with the generic (non-supersymmetric) exact $1/4$ as its distinctive new result.

\section{The role of \GGtheory{} in the broader program}

This paper is the black-hole companion (C3-BH) of the twenty-two-paper
\GGtheory{} cohort. The
preceding papers established the foundations of the theory
(\AOne{}--\AFour{}; \citealp{ArisakaA4Euclid}), its quantum-mechanical
reading (\AEight{}), its open-system/thermodynamic reading
(\AFive{}--\ASix{}), and its precision tests at specific observables (Hubble tension, $W$ mass, neutrino masses, Higgs). The present
paper extends the program into the gravitational domain
by addressing what is arguably the most celebrated problem in
theoretical physics.

The significance of the black-hole reading in the broader
program is threefold.

\paragraph{First, it provides a non-trivial consistency check
of the theory in a regime far from its initial motivation.}
The \GGtheory{} framework was developed primarily to address puzzles in particle physics (the Hubble tension, the $W$ mass, the Higgs sector, the neutrino masses), and only secondarily to address conceptual issues in quantum theory and thermodynamics. That the same framework, applied
to black holes, produces the Page curve automatically and
dissolves the firewall paradox without additional input, is
non-trivial. It suggests that the underlying geometric
structure is capturing something real.

\paragraph{Second, it unifies disparate observables under a
single framework.} The Bekenstein--Hawking entropy, the Hubble tension, the Higgs mass, and the $W$-boson mass are, in the standard reading, unrelated phenomena. In \GGtheory{}
they are all manifestations of the same 6D bulk geometry,
seen through different 4D projections. The coherence of the
predictions across scales --- from the Higgs at $125$ GeV to
the Hubble corridor at $66.6$--$67.4$ km/s/Mpc to the Bekenstein entropy
of a $\Msun$ black hole at $10^{77}\kB$ --- is itself a
piece of evidence for the framework.

\paragraph{Third, it opens testable predictions at
observational frontiers.} As the Event Horizon Telescope and
LIGO/LISA continue to improve, and as new gravitational-wave
detectors (Einstein Telescope, Cosmic Explorer) come online
in the 2030s, the horizon-scale physics of black holes will
become accessible at unprecedented precision. The \GGtheory{}
predictions of Kerr-to-$10^{-46}$-or-better for astrophysical
black holes (Kerr-to-$10^{-66}$ for the EHT supermassive
shadows) constitute one-sided falsifiers: the framework
predicts exact Kerr far beyond any foreseeable sensitivity, so
any observed deviation, at any accessible precision, retires
it. The coming two decades will tighten this null test by
orders of magnitude.

\section{What the module explains, what remains open, and future directions}
\label{sec:open-questions-c3}\label{sec:explains-open}

A scientifically useful synthesis must be honest about both
strengths and limits.  The present module explains well why
pure-to-mixed exterior behavior is unsurprising once the
exterior state is recognized as a projected open subsystem; why
generalized entropy should have the area-plus-bulk-entanglement
form (\S\ref{sec:gff-ledger}); why naive interior/exterior
factorization is too crude in gravity; why a Page-like turnover
arises through branch competition rather than miraculous
late-time repair (\S\ref{sec:branch-switch}); and why black
holes are the cleanest testbeds for a bulk-deterministic,
boundary-probabilistic theory.

What remains open is equally important, and we state it plainly.
The induced-gravity derivation fixes the \emph{exact} $1/4$
coefficient, the area law, and the identity of the microstates;
what is not yet complete is the closed species-by-species sum
that recovers the induced $G_N$ \emph{numerically} for the full
\GGtheory{} spectrum --- the explicit benchmark against the most
controlled string-theoretic counts \citep{StromingerVafa1996}.
\GGtheory{} does not yet reproduce, model by model, the detailed
outputs of every replica-wormhole or island computation
\citep{Penington2020, Almheiri2019}.  It does not yet offer sharp
horizon-scale observational signatures beyond the expectation
that smooth semiclassical exteriors survive until regimes where
independent evidence indicates failure.  And at present it
interprets the structure of controlled semiclassical calculations
as much as it supersedes their technical machinery.  These limits
are not fatal; they define the next stage of work: if the
framework is more than a reinterpretation, it must demonstrate
explicit quantitative reproduction of generalized-entropy
calculations in tractable models, and derive distinctive
corrections that competing frameworks do not naturally imply.

The specific work orders follow.

\paragraph{Quantitative calculations.} While the structural
claims of this paper --- horizon as port, radiation as \CPTP{}
channel output, entropy as entanglement entropy --- are
developed in detail, the explicit quantitative predictions
(KK corrections to the Hawking spectrum, precise PBH mass
spectrum peaks, higher-order graybody factors) require further
computation. Several of these are in progress.

\paragraph{The factorization conjecture, adopted.} The cosmology
companion leaves one item of its observer-vacuum reading at
conjectural grade --- the factorization of the bulk state for
strongly gravitating configurations \citep{ArisakaA10Cosmos} ---
and its natural home is this paper: the controlled model class of
Theorems~\ref{thm:rotating-anchor} and~\ref{thm:collapse-channel}
is where a model-level closure should first be attempted.  We adopt
it here as a named open problem of the black-hole module.

\paragraph{Full 4D asymptotically-flat case.} Most of the
rigorous results on the islands formula are for 2D JT gravity.
The full 4D asymptotically-flat Schwarzschild case remains
less well-studied. \GGtheory{} proposes that the 4D case
follows directly from the bulk structure; working out the
explicit 4D computation and comparing with numerical-
relativity simulations of Hawking evaporation is an open
project.

\paragraph{The gravitational sector beyond black holes.} The
full \GGtheory{} reading of general relativity --- non-black-hole
spacetimes, gravitational dynamics beyond the horizon sector ---
lies beyond this paper's scope and is registered here as open
territory rather than promised work. The cosmological
applications (inflation, dark energy, structure formation)
are developed in the cosmology companion
\citep{ArisakaA10Cosmos} --- a cohort companion drawing on
the same \AOne{}--\ANine{} inheritance as the present paper,
not a downstream consequence of it.

\paragraph{The generalized second law beyond the exchange class.}
Lemma~\ref{lem:gsl-arrow} derives the second law's direction and
computes its production exactly in the Gaussian exchange class, with
the linearized induced first law as the area price.  The promotion to
a Wall-class statement --- arbitrary horizon cuts at the generality
of Wall's theorem (free and superrenormalizable fields, linearized
backreaction) and beyond it to the interacting case
\citep{Wall2012} --- is registered
open: it requires half-sided modular machinery and ANEC-grade input
that this module imports rather than owns.  The registered path runs
through the same two theorems the exchange class used (the KMS face,
T-BH-5; the induced first law), which is what makes the promotion a
sharpening rather than a rebuild.

\paragraph{Cross-checks with numerical gravity.} The
\GGtheory{} predictions for ringdown spectra, shadow
geometries, and post-merger relaxation are, in principle,
testable against numerical-relativity simulations. Work on
embedding the \GGtheory{} corrections in numerical-relativity
codes is a natural next step.

\paragraph{Analog gravity tests.} Tabletop analog-gravity
systems (acoustic black holes, Bose-Einstein condensates,
optical black holes) provide controllable settings in which
to test the structural features of \GGtheory{}. Specifically,
the partial-trace structure of the horizon should produce
characteristic entanglement statistics in analog Hawking
radiation, which are experimentally accessible.

\section{Falsifiability and kill criteria}
\label{sec:falsifiers}

The observational kill table is the business of Part~VIII
(Table~\ref{tab:bh_kill}); what belongs here is the
\emph{theory-side} exposure --- the ways the module can be
stressed with no instrument at all.  The module would be weakened
or falsified if a gauge-consistent proof established that naive
factorization across the horizon is exact in the relevant
quantum-gravitational regime, with no need for
constrained-subsystem reconstruction; if controlled Page-curve
calculations systematically resisted reinterpretation as a
branch-minimized \GFF{} ledger; if monotonic thermal entropy of
the accessible radiation were shown exact all the way to
evaporation while preserving global unitarity, with no broader
subsystem semantics; or if the induced-gravity species sum failed
to converge to the $G_N$ that appears in the 4D Einstein
equations --- the direct internal kill criterion for the entropy
claim.

\begin{table}[H]
\centering
\caption[A compact falsification dictionary for the black-hole module.]{A compact falsification dictionary for the black-hole module. Five probe classes, each with what the module expects and what would stress it, and the middle column is deliberately weaker than a prediction. The module is minimal by construction, so most of what it says about a probe class is that ordinary physics should hold there; the last column is where it is exposed. A dictionary of this kind is worth less than a numbered falsifier, and it is here because not every stress test reduces to one.}
\label{tab:falsification-dictionary}
\small
\begin{tabular}{p{0.23\textwidth}p{0.30\textwidth}p{0.32\textwidth}}
\toprule
\textbf{Probe class} & \textbf{What the minimal module expects} & \textbf{What would stress it} \\
\midrule
Classical exterior geometry & Ordinary GR-like ringdown and exterior relaxation in semiclassical regimes & Robust evidence that smooth exteriors fail generically, far from any endpoint or UV scale, without an open-system explanation. \\
Horizon bookkeeping & Area-plus-bulk-entanglement entropy controls the relevant radiation branch & A contradiction between generalized-entropy calculations and the branch-ledger reading. \\
Quantum simulators / analog horizons & Open-channel entropy growth followed by branch-limited or correlation-restoring behavior & Persistent exact thermal monotonicity even in setups designed to mimic unitary late-time recovery. \\
Subsystem structure & Gauge-constrained, non-naive effective factorization & A fully gravitational exact factorization theorem leaving no room for \DGI{}-style correction. \\
Microscopic closure & The induced-$G_N$ species sum reproduces the Bekenstein--Hawking coefficient & A consistent spectrum in which the species sum misses $1/4G_N$. \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The Part in one breath, and the door to the next.}
The inventions of this module are, in the end, one invention used
four times: take openness seriously --- let the projection be
forced, let gravity be induced, let the ledger rank the branches
--- and the horizon's classic paradoxes resolve without new
postulates, while the open items become work orders with
registered kill conditions instead of loose ends.  The physics
intuition to carry forward: \emph{a discussion should read like
an audit, and this one is meant to}.  The arguments are now all
on the table --- the derivations, the record, the comparisons,
the debts.  What remains is to step back and say what it means:
for horizons, for information, and for the view from the
six-dimensional bulk.  That is the Conclusions.

%% ===================================================================
%% PART -- CONCLUSIONS
%% ===================================================================

\section[Part~X in plain English]{Part~X in plain English: what is actually new here, what is borrowed, and what is still owed}
Four inventions, and the Part's own view is that they are one invention
used four times: take openness seriously.

Let the projection be forced by the causal structure rather than chosen,
and the thermality follows. Let gravity be fully induced, and the entropy
coefficient stops being a count and becomes a ratio. Let a ledger rank the
branches, and the Page curve becomes a crossover rather than a mechanism.
Let the same ledger price each exchange, and the direction of the second
law acquires a cost that can be computed. Not one of the four needs a
postulate the cohort did not already have.

The related-work section is longer than a paper of this kind usually
carries, and deliberately so. Sakharov's induced gravity, Jacobson's
thermodynamic derivation of the field equations, complementarity, the
island program --- the module stands on all of them, and the Part names
what each contributed instead of positioning against it.

What remains open is set out as work orders with kill conditions attached
rather than as loose ends. The species sum. The factorization question. The
Wall-class general statement. Each has a name, a home, and a condition
under which it would end the reading rather than merely embarrass it. That
is the form the Part argues an honest discussion should take, and it is why
this one reads more like an audit than a summary.

\GGPart[Conclusions]{Conclusions}{Horizons, Information, and the View from the
Six-Dimensional Bulk}

The argument of the paper is now complete. The theorems are
proved, the numbers are on record, the comparisons are drawn,
and the ledger of inventions and open debts is signed. What
remains is the step every long argument owes its reader at the
end: to put the machinery down, step back, and say plainly what
it means --- for black holes, for information, and for the
picture of nature that emerges when the six-dimensional bulk is
taken seriously. This closing Part is written for the broadest
audience of the paper. Nothing in it is new physics, and nothing
in it adds hypotheses to the record; each section restates, in
ordinary language, something the earlier Parts established in
equations, and a reader who wants the derivation behind any
sentence will find the pointers in Parts~IV--X. We proceed from
the physics outward: first the six-dimensional picture itself,
then what it does to the fifty-year paradox, then what a horizon
really is, and finally what all of this suggests about
determinism, observation, and the reach of physical description.

\section{The 6D picture, one more time}

Let us step back from the technical detail and state, in one
paragraph, what \GGtheory{} says about black holes.

Nature, at its most fundamental level, operates in six
dimensions: the four we perceive, plus a radial
direction $I_r$ along which fields stretch and compress, plus
a compact angular direction $S^1_\theta$ around which gauge
fields wrap. In this 6D bulk, the laws are simple and
deterministic: a unitary evolution of field configurations on
a smooth manifold, governed by a single action principle with
gauge-gravity unification and the BI-6 modified Bianchi
identity. Matter, energy, and information flow freely through
all six dimensions.

Observers, however, are 4D creatures. We exist on the boundary
of the 6D bulk and can only measure zero-mode projections of
the bulk fields. What we see is the \emph{\OGGEP{} functor
applied to the 6D state}: a partial trace over the compact
and radial dimensions, yielding a 4D density matrix on the
zero-mode Hilbert space. Because partial trace is a \CPTP{}
channel, the effective 4D dynamics is open, dissipative, and
often non-unitary in appearance --- even though the
underlying 6D dynamics is unitary.

A black hole is, in this picture, a \emph{region of the 4D
boundary where the \OGGEP{} projection becomes especially
severe}. Near the horizon, the bulk fields develop structures
that are extremely scale-extended into the compact and radial
directions, so that the 4D observer loses access to
correspondingly more of the bulk state. The horizon itself is
the geometric locus at which the 4D observer's accessible
Hilbert space factorizes into an ``accessible exterior''
sector and a ``traced-over interior'' sector. Hawking
radiation is the operational consequence of this partial
trace: the effective 4D description is a thermal output from
a \CPTP{} channel whose input is the bulk state. Information
is not lost; it flows into dimensions the 4D observer cannot
see.

It is worth pausing on how ordinary --- and how radical ---
this structure is. An open quantum system is simply a system
whose description omits part of the world: an atom coupled to
the photon field, a molecule in a solvent, a qubit leaking into
its substrate. In the laboratory, the experimenter chooses the
cut. Trace out the photon modes and the atom decays; trace out
the solvent and the molecule decoheres; what is ignored is a
matter of practical convenience, and the resulting open-system
equations are honest approximations. The single structural
observation on which this paper rests is that near a horizon
the cut is no longer chosen --- it is imposed. Causal access to
the compact and radial sectors ends at the port, for every 4D
observer, no matter how well instrumented. A black hole is
therefore the universe's most perfect open quantum system: the
one whose environment is hidden not by experimental limitation
but by the causal structure of spacetime itself. And that is
precisely why this paper could be written with theorems where
the subject usually offers metaphors. Openness imposed by
geometry is exact and lawful in a way that openness chosen for
convenience never quite is: the channel is forced, its
properties are calculable, and the familiar apparatus of
open-system physics --- \CPTP{} maps, entropy production,
relaxation --- applies to the black hole not as analogy but as
description.

\section{The paradox was never about information}

Seen in this light, the information paradox of 1976 appears in
a new perspective. Hawking asked: if the initial state is pure
and the final state is mixed, where did the information go?
The debate of the past fifty years has largely been about
whether information is really lost, or hidden in subtle
correlations, or encoded holographically on the boundary.

\GGtheory{} suggests a different framing. The information was
never in danger of being lost; it simply flowed into bulk
dimensions invisible to the 4D observer. The paradox was an
artifact of assuming that the 4D description captures the
complete physical content. Once one recognizes that 4D
observation is a partial trace of a 6D state, the apparent
paradox dissolves into a statement about how much of the bulk
the observer has access to.  Fifty years of anguish about
information loss were, on this reading, anguish about mistaking a
shadow for the object.

There is a quieter way to say the same thing. Every working
physicist already writes down mixed states without alarm. When
a laboratory qubit entangles with its surroundings and the
experimenter traces the surroundings out, the qubit evolves
from a pure state to a mixed one, and nobody declares a crisis:
the purity has gone into correlations with degrees of freedom
the description no longer tracks. Hawking's 1976 calculation
produced exactly such a mixed state. The alarm arose because,
for a black hole, there seemed to be no environment for the
purity to have gone into --- the radiation was all there was,
and it was thermal. The 6D bulk supplies precisely the missing
register. The compact and radial sectors behind the port are
the environment, as physically real as the photon modes of a
laboratory, and the mixedness of Hawking radiation becomes the
same benign bookkeeping that every open system exhibits. What
was extraordinary about the black hole was never the physics;
it was that geometry had hidden the environment so completely
that physics spent five decades looking for it.

This is a structurally similar move to previous reconceptions
in physics. When Copernicus moved the observer off the Earth
to see that the ``apparent'' motion of the Sun was really due
to Earth's rotation, he did not change the data; he changed
the interpretive frame. When Einstein moved the observer into
a freely-falling elevator to see that the ``apparent'' gravity
of the elevator was really acceleration, he did not change
the data; he changed the frame. When Copenhagen-school
physicists insisted that the wave function describes what we
know rather than what is, they did not change the experimental
outcomes; they changed the interpretive category.

\GGtheory{} makes a similar move: the 4D information paradox
dissolves when we recognize that the 4D observer sees only a
partial trace of a 6D unitary reality. The 4D frame is correct
as far as it goes, but it is incomplete, and the paradox is
in the incompleteness rather than in the underlying physics.

\paragraph{The missing register, named by the anatomy.} The
cohort's recent campaign (\textsf{GG-W1-Symmetry};
\citealp{ArisakaP5CPT}; \citealp{ArisakaA8QM}) lets the ``missing
environment'' be named with topological precision. The two sectors
behind the port are fibers of opposite kind --- \emph{holonomy
requires closed loops, and an interval has none}: the compact
circle carries everything quantized and protected, a group, the
fiber that remembers; the scale direction carries only one-way
transport, a semigroup, the fiber that forgets. What the paradox
of 1976 asked of the black hole was, in this language, to be both
fibers at once: to \emph{keep} --- purity, unitarity, the
conserved ledgers --- and simultaneously to \emph{spend} ---
thermality, irreversibility, the one-way evaporation. The anatomy
divides the labor instead. Everything the evaporation must
conserve is circle-side: the charges, the ledger integers, the
holonomy-class records --- deposited on the fiber no boundary
operation can continuously deform, which is why the information
was never destructible in the first place. Everything thermal and
one-way about Hawking's calculation is interval-side: the
temperature is the shore periodicity of the CPT companion's KMS
reading \citep{ArisakaP5CPT} --- \emph{temperature is where the
interval meets the shore}, and a horizon is where that meeting is
sharpest --- while the evaporation itself is the semigroup doing
the only thing a semigroup does. The paradox was thus a billing
error: it demanded that the interval's one-way channel also carry
the circle's books, and fifty years of debate audited the wrong
fiber. The Page curve, in this reading, is the reconciliation
schedule --- the circle's ledger read out through the interval's
channel, at exactly the rate the channel permits. In the
campaign's shorthand: \emph{the circle keeps what the hole must
return; the interval prices the return; the paradox was billing
one fiber for the other's job.}

\section{What is a horizon?}

This paper has used the word ``horizon'' many times, in
several senses. The word deserves a moment of reflection.

Classically, a horizon is a causal boundary: a surface in
4D spacetime that can be crossed in one direction but not the
other. The event horizon of a black hole is the most famous
example; the cosmological horizon in an accelerating universe
is another.

In \GGtheory{}, horizons take on an additional structural
meaning: they are \emph{ports of the \OGGEP{} functor}, loci
at which bulk information flows into dimensions inaccessible
to the 4D observer. The black-hole horizon is one example.
The cosmological horizon is another. The apparent ``inside'' of
a chemical compound, the unobservable wave function of
Schr\"odinger's cat, the hidden thermal bath of a Brownian
particle --- all of these are, in the \GDOS{} reading,
different kinds of horizons. Each marks a boundary between
what the observer can access and what has been traced over.

Stated this way, the horizon stops being a place and becomes a
relation. Nothing locally remarkable happens at the
Schwarzschild radius; an infalling observer crosses it without
incident, as relativists have rightly insisted for a century.
What changes at the horizon is not the local physics but the
relation between one particular observer and the bulk state ---
which registers of the world that observer's port can still
reach. A surface can therefore be a horizon for one observer
and not for another, just as a hill can block one walker's view
and not a neighbor's, without anyone doubting that the
landscape behind it is real.

The lesson generalizes to the properties themselves.  Is the black
hole hot?  For the distant observer, yes --- measurably, calculably.
For the infalling one, no.  Both are right, because temperature
lives in the relation between the bulk and a port, not in the bulk
alone (\S\ref{sec:thermal-clock}); the same is true of
entropy, and even of when the information comes back.  Physics has
met this before --- simultaneity became frame-dependent in 1905
without becoming unreal --- and the horizon extends the lesson to
thermodynamics: \emph{objective} does not mean \emph{observer-free};
it means lawfully related across observers.

The unification of these diverse phenomena under the single
concept of \OGGEP{} port is one of the deeper structural
claims of \GGtheory{}. It suggests that the mathematics of
black holes --- entanglement entropy, thermodynamic radiation,
information flow --- applies very generally to any open
quantum system, with the black hole being simply the most
gravitationally spectacular example. If that claim is right, a
student who has understood decoherence in a quantum-optics
laboratory already holds, perhaps without knowing it, the
conceptual core of black-hole thermodynamics. The horizon adds
geometry's signature --- the exact area law, the thermal clock,
the branch ledger --- but not a new rulebook.

\section{On determinism, probability, and the view from
nowhere}

The \GGtheory{} picture restores a form of determinism to
fundamental physics. In the 6D bulk, the evolution is unitary
and deterministic: given the initial bulk state, the later
bulk state is fixed. Randomness, probability, and
irreducible uncertainty arise only at the 4D boundary, as
consequences of the partial-trace structure of \OGGEP{}.

This position has historical echoes. Einstein famously
believed that ``God does not play dice with the universe'' and
resisted the Copenhagen interpretation's insistence on
irreducible randomness. His colleague Schr\"odinger, too, was
uncomfortable with the Born-rule probabilities. Both would
have been, we suspect, sympathetic to the \GGtheory{}
picture, in which the 4D probabilities emerge from a deeper
6D determinism through the projection functor.  In the 6D bulk,
evolution is unitary and nothing is ever lost --- Einstein's
instinct.  In every 4D description, outcomes are probabilistic and
entropy grows --- Bohr's.  The horizon is where their century-old
argument is staged most starkly, and the resolution offered here is
that they were describing different levels of the same world, each
correctly.

It is worth being precise about the kind of probability that
survives this reading. The randomness a 4D observer encounters
is not a die thrown by nature at each event; it is the lawful
ignorance of an observer whose port delivers only part of the
state. This is closer in kind to the probability of statistical
mechanics, which quantifies a coarse-graining, than to
irreducible chance --- with one decisive difference. In
statistical mechanics the coarse-graining is optional: a better
microscope, in principle, refines it away. Here it is not. The
missing information sits behind a causal boundary rather than a
practical one, and no 4D instrument, however perfect, can
retrieve it. Probability, in this picture, is what determinism
looks like through a port. (How the quantitative Born weights
themselves emerge from the projection is the business of the
quantum-mechanics companion \citep{ArisakaA8QM}, where the
status of each step in that derivation is registered
explicitly; nothing in the present paper depends on it.)

But this deterministic restoration comes at a price. The 6D
bulk is not directly observable; it is defined by its
projections. Its ``existence'' is, in a metaphysical sense,
posited rather than demonstrated. The question ``does the 6D
bulk really exist?'' is not a simple factual question in the
way ``does Alpha Centauri exist?'' is; it is a question
about what we should take as the fundamental description of
nature.

The pragmatic reply is that the 6D bulk is as real as any
unobservable scientific entity. Atoms, quarks, the
electromagnetic field, the Higgs boson, dark matter --- none
of these is directly seen; all are posited as the best
explanation of observable phenomena. The 6D bulk, in
\GGtheory{}, is a posited structure whose existence is
warranted by its success in explaining the Hubble tension, the $W$ mass, the Higgs sector, the neutrino structure, quantum measurement, thermodynamic arrow of time,
and --- in this paper --- the black-hole information paradox.
If the 6D bulk explains all of this without adjustable
parameters, the inference to its existence is as warranted as
the inference to any other unobservable.

The philosophical view this suggests is sometimes called the
``view from nowhere'': a perspective not tied to any
particular observer, in which the fundamental description is
the 6D bulk, and every 4D description is a particular
partial-trace projection. No 4D observer sees the bulk
directly; every 4D description is incomplete; yet the bulk
grounds and unifies all of them.

The phrase has a philosophical history: it names the ideal of a
description of the world not indexed to any particular point of
view. Physics has pursued that ideal since Copernicus, and
modern physics has twice discovered its limits --- special
relativity showed there is no observer-free fact about
simultaneity, and quantum mechanics seemed to show there is
none about measurement outcomes. \GGtheory{} proposes a precise
division of the territory. There \emph{is} a view from nowhere:
the 6D bulk state and its unitary law. But it is not a view any
observer can occupy. Every actual point of view is a port, and
every port truncates --- not as a defect of the observer, but
as a statement about the projection itself. A reader may hear
in this an echo of the old philosophical distinction between
the world as it is and the world as it appears. The difference
is that here the relation between the two is not left as a
mystery: it is a specific, computable map, and the appearances
it produces are constrained strongly enough to be falsifiable.
That is what lifts the picture from metaphysics to physics ---
not that it closes the gap between reality and observation, but
that it gives the gap equations.

\section{Black holes as Rosetta stones}

For a generation of physicists, black holes have been the
Rosetta Stones of quantum gravity. They are the cleanest
setting in which general relativity, quantum mechanics, and
thermodynamics meet head-on. Their entropy is gravitational
(area) and informational (microstate count) and thermodynamic
($A/4\Newton\hbar$) simultaneously. Their horizons are
geometric boundaries, information-theoretic partial-trace
surfaces, and thermodynamic heat baths at the same time. If
you understand black holes, you have a good claim to
understanding quantum gravity.

The metaphor deserves to be taken literally for a moment. The
Rosetta Stone deciphered Egyptian because a single decree was
carved three times --- in hieroglyphs, in Demotic, in Greek ---
so that a reader fluent in one script could break into the
others. The black hole is nature's analogue: one object on
which the same statement is written in three languages at once.
$\Sbh = A/(4\Newton\hbar)$ is a single equation that reads as
geometry (an area), as information (a count of microstates),
and as thermodynamics (a heat divided by a temperature).
Whoever learns to parse all three scripts at once holds a key
that opens far more than black holes.

\GGtheory{} takes this Rosetta-Stone status seriously. The same 6D geometry that explains the Higgs boson, the fine-structure constant, the cosmic neutrino background, and the accelerating expansion of the universe also explains the black-hole entropy and the Page curve. The black hole becomes
the setting in which all the diverse observables of the
program are tied together, because its extreme geometry
amplifies the bulk-to-boundary projection to the point of
visibility. And in the cohort's reading there is a fourth
script on the stone: the particle-physics script, in which the
same integer ledger and the same lock scale that organize the
Higgs sector and the dark-matter prediction reappear in the
horizon's bookkeeping. A comparison table can register that as
consistency; the physics registers it as unity.

For readers outside the specialist community, this is perhaps
the central message of the paper. The fifty years of
information-paradox literature, the many-volumed books on
black-hole physics, the vast output of the \AdSCFT{}
community since 1997 --- all of this is not separate from
the physics of everyday particles and forces. It is
continuous with it. The horizons of black holes obey the
same partial-trace structure that makes your laptop lose
information to the environment when it heats up, that makes
sugar dissolve in tea, that makes a photograph fade in
sunlight. Everyday irreversibility and Hawking radiation are one
mechanism at two volumes; the black hole is simply the place where
geometry performs the projection that nature otherwise leaves to
circumstance.  Black holes are just the most dramatic examples of
a universal structure in how 4D observers see a 6D reality.

\section{A closing reflection}

We close with a reflection on what the work reported here
does and does not achieve.

It does not prove that \GGtheory{} is correct. The framework
makes predictions --- some already consistent with
observations, some awaiting future experiments --- but
correctness of a physical theory can never be proved; it can
only be progressively supported by evidence or falsified.

It does not resolve every open question about black holes.
The quantitative details of the microstate counting, the
precise form of the KK corrections to the Hawking spectrum,
the connection to numerical-relativity simulations, and many
other technical issues remain open. This paper establishes a
framework in which these questions can be asked; it does not
answer all of them.

It does not replace the existing literature on the
information paradox. The \AdSCFT{} correspondence, the
islands formula, the fuzzball program, the ER=EPR proposal
--- each of these captures part of the truth about black
holes. \GGtheory{} synthesizes rather than replaces them,
situating each within a larger framework grounded in a
complete 6D theory.

What the work does achieve, we believe, is the following: it
\emph{reframes} a fifty-year-old problem in a way that
dissolves the paradox without introducing new physical
mysteries. The information paradox, in the \GDOS{} reading,
is not a puzzle about Nature; it is a puzzle about which
description of Nature is fundamental. Once the 6D bulk is
taken as fundamental and the 4D boundary as its
partial-trace projection, the paradox dissolves automatically,
and the many fragmented resolutions of the past five decades
(islands, wormholes, fuzzballs, firewalls) are revealed as
different pieces of a single picture. And it does so under the
ledger's discipline: no parameter was tuned, no mechanism was
invented for the occasion, and every load-bearing claim carries
a pre-registered way to kill it. Readers who wish to test that
discipline step by step will find the machine room in the
Technical Appendices and, in the objections Part, thirty questions
from the general reader answered in plain English followed by
fifty-three adversarial readings; the paper asks to be checked,
not believed.

The black hole was called, by John Wheeler in the 1960s, a
``mass without matter.'' In \GGtheory{} it becomes something
more specific: \emph{a window from our 4D world into the 6D bulk}. What falls into a black hole does not disappear;
it simply enters dimensions we cannot see. What comes out as
Hawking radiation is the echo of that crossing, projected
back into our 4D observation. The horizon is not a wall; it
is the boundary of what we can access.

For a century the black hole has been physics' memento mori
--- the place where stars die, where clocks stop, where
theories are sent to fail. For half of that century it carried
a darker charge: the fear that the horizon devours information,
and with it the lawfulness of nature itself. The reading
defended in this paper ends on a gentler verdict. The horizon
destroys nothing. It is not where physics breaks down but where
a perspective does --- the surface on which nature states,
exactly and quantitatively, how much of the world one
observer's window can carry. What falls in is not erased but
kept, in registers we do not see, and returned, slowly, in
correlations we are only beginning to learn to read. If this
framework is right, the fifty-year crisis of the black hole was
never a crisis of nature. It was the moment a species of
four-dimensional observers began to discover that its window is
not the world --- and the same discovery, carried home from the
horizon, reaches into everything the window frames: why a
measurement yields one outcome, why heat flows one way, why the
past feels fixed and the future open. Seen from the bulk,
nothing is ever lost. The universe keeps perfect books; what we
call paradoxes are the shadows thrown on our four-dimensional
wall by a reality two dimensions richer than our sight. That,
in the end, is what the black hole has been telling us for
fifty years, patiently, in a language of area and temperature
we have only now learned to read: that the wall has a door,
that the books balance, and that the world is larger than the
window through which we watch it.

\clearpage

%% ===================================================================
%% AUTHOR'S NOTE + ACKNOWLEDGMENTS (moved here at v13.13: cohort
%% back-matter order per P5 -- after Conclusions, before the
%% appendix Part; Acknowledgments open with the cohort-standard
%% \GGAcknowledgments disclosure, paper-specific paragraphs kept)
%% ===================================================================

\section[Part~XI in plain English]{Part~XI in plain English: what a horizon turns out to be, if this reading is right}
If the reading is right, a horizon is not primarily a place. It is a
boundary of access, and everything strange about black holes follows from
that one property rather than from anything exotic in the geometry.

The temperature is what a traced-over state looks like. The entropy is the
size of what was traced. The coefficient is a ratio between two
normalizations that had to be fixed anyway. The information was never lost,
only held on the far side of an assumption that expires on schedule. And
the singularity is where a four-dimensional coordinate system fails, not
where the universe does.

That reframing has a consequence worth stating plainly, because it is what
this Part is really about. The paradox was never about information. It was
about assuming that a description written from one side of a one-way
boundary ought to be complete, and then being surprised when it was not.
Every resolution proposed in fifty years can be read as a way of relaxing
that assumption; this one relaxes it at the level of the geometry rather
than at the level of the formalism.

What would end it is short, and printed: a coefficient other than one
quarter, a departure from Kerr outside the horizon, a species sum that does
not close. None of the three is far-fetched, and one of them is a
calculation somebody could begin this year.

The Part closes on the reason black holes are worth this much work. They
are the only objects where gravity, quantum mechanics and thermodynamics
all have to be right at the same time, and they have been sending the same
three-number message for a century. The claim of this paper is that the
message was never as short as it looked.

\section*{Author's Note}
\phantomsection
\addcontentsline{toc}{section}{Author's Note}

This paper honors four theorists whose work on black-hole
physics this paper builds upon:

\textbf{Jacob Bekenstein} (1947--2015) proposed in 1972--1973
that black-hole horizon area is a physical entropy. His
argument, initially controversial, was the first step toward
the modern understanding of black-hole thermodynamics. He died
in 2015, five years before the Penington-Almheiri-et-al. papers
vindicated his framework at the deepest level. Bekenstein's
$\Sbh = A/(4\Newton\hbar)$ becomes, in \GGtheory{}, the
entanglement entropy of a 4D observer with the 6D bulk --- a
precise geometric meaning that Bekenstein glimpsed but could
not formalize in his own lifetime.

\textbf{Stephen Hawking} (1942--2018) in 1974 derived
black-hole thermal radiation, identifying the key paradox
that has driven 50 years of work. Hawking's intellectual honesty
--- he initially argued for information loss, then publicly
changed his position in 2004 --- is a model for how
controversial questions should be approached. His 2016--2018
work on soft hair suggested horizon-level degrees of freedom
that we now understand as \UoneQi{} holonomies. Hawking did not
live to see the islands formula or the present work, but his
framework provides the foundation on which they build.

\textbf{Don Page} (b. 1948) in 1993 derived the entanglement
entropy curve that bears his name. Page's remarkable insight
--- that thinking carefully about typical quantum states gives
quantitative predictions about black-hole evaporation ---
anticipated by 25 years the islands-formula developments and
provided the target that any theory of quantum gravity must hit.
Page's curve is, in \GGtheory{}, an automatic consequence of
bulk unitarity plus \OGGEP{} projection.

\textbf{Juan Maldacena} (b. 1968), in 1997, introduced the
\AdSCFT{} correspondence and transformed theoretical physics
for a generation. His 2013 ER=EPR proposal (with Susskind)
and his work on the islands program have been central to the
recent progress. Although \GGtheory{} does not use the full
\AdSCFT{} machinery, it inherits the central lesson of
Maldacena's program: that gravitational physics in a bulk is
dual to/encoded in a boundary theory, and that information
flows are fundamentally holographic. \GGtheory{} makes this
precise in the real, Standard-Model-containing universe rather
than in the abstract $\mathcal{N}=4$ SYM setting.

We also acknowledge the broader string-theory, holography, and
numerical-relativity communities whose work over five decades
made the synthesis reported here possible.

\GGAcknowledgments

The author thanks the UCLA physics department for institutional
support, colleagues at UCLA and elsewhere for critical
feedback, and the broader gravitational, cosmological, and
string-theory communities for five decades of theoretical
progress on black-hole physics. The present synthesis would
not have been possible without the preceding work of
Bekenstein, Hawking, Page, Maldacena, Susskind, Strominger,
Vafa, Witten, 't Hooft, Penington, Almheiri, Engelhardt,
Marolf, Maxfield, and many others whose contributions are
cited throughout.

We note also the contributions of the Event Horizon Telescope
collaboration and the LIGO/LISA collaborations, whose
observational work provides the empirical context against which
any theoretical framework must be tested.

\clearpage

%% ===================================================================
%% PART -- TECHNICAL APPENDICES (new Part at the v13.8 split)
%% ===================================================================
\GGAppendixPart[Technical appendices]{Technical Appendices}{A graduate-level course in black-hole physics read through \GGtheory{} --- full from-scratch derivations of the Schwarzschild geometry, the four laws of black-hole mechanics, Bekenstein's entropy, the Hawking effect, the Euclidean section, the Hawking--Page transition, the Page curve, and the modern holographic era from state counting to islands --- preceded by the reference dictionaries and followed by the source ledger, benchmark audit, and registries}
\label{part:appendices-c3}

The body of this paper has emphasized the theorem chain and kept the
heaviest calculation out of the reader's way.  The present
appendices supply that technical content in full, and taken together
they are intended to be read as \emph{a graduate-level course in
black-hole physics seen through \GGtheory{}}, addressed to an
incoming graduate student in physics or astronomy who will carry the
program forward.  We organize them into three groups.  The
\emph{reference} appendices (A--D) tabulate the abbreviations, the
key symbols, the cross-cohort naming convention, and the numerical
reference card, and serve as the reader's quick lookup.  The
\emph{course} appendices (E--R) are the course proper: a
self-contained sequence that takes up black-hole physics in the
order the subject was actually built --- the Schwarzschild geometry
and its horizon; the four laws of classical black-hole mechanics;
Bekenstein's entropy and the generalized second law; the Hawking
effect derived with the full Bogoliubov machinery; the Euclidean
section and the Hawking--Page transition; the Page curve; the
modern holographic era from the Strominger--Vafa count to the
islands formula; and, in the closing chapters, the channel
identities, the extremal and de~Sitter cases, and the
induced-gravity species sum.  Wherever the material allows, each
chapter follows \emph{three steps}.  First comes a full
from-scratch derivation of the standard result, in the form a
graduate student would meet it in Wald or Misner--Thorne--Wheeler,
carried through with no skipped manipulations and closed by a
worked numerical example.  Then comes a frank statement of what
that standard derivation \emph{assumes but cannot explain} --- the
questions it leaves open: what the horizon is made of, what states
the entropy counts, where the information goes.  Finally comes the
\GGtheory{} reading, in which the same quantity descends as a
theorem of the open-system boundary law --- and because those
derivations are the business of the main text, the third steps
mostly \emph{point}: into Parts~IV--VI for the port, the exact
$1/4$, and the Page turnover, and into Part~IX for the comparisons.
The \emph{ledger-and-registry} appendices (S--Y) that close the
Part record the internal source ledger, the cross-paper inheritance
table, the consolidated benchmark audit, the named-statement deck,
the promotion roadmap, and the falsification-test catalog,
closing with an annotated reading guide.

Throughout these appendices we adhere to two stylistic commitments,
shared with the cohort's reference implementation (\ATen{}).
First, every derivation is presented step by step with no skipped
intermediate manipulations: a graduate student should be able to
reproduce each calculation with paper and pencil, and every number
quoted anywhere in this Part traces to a verification script named
in the benchmark audit (Appendix~\ref{app:benchmarks}).  Second,
every formal step is accompanied by a \emph{physics intuition}
note that translates the algebra into a mental picture, on the
spot rather than in a separate section.  Each course appendix opens
with a boxed \emph{What this appendix teaches} abstract and closes
with a boxed \emph{take-away} --- the standard result, what it
assumed, and what the geometry replaced the assumption with --- so
that the Part can be used as a one-semester reading course, one
appendix per week, each self-contained.  And every derivation
chapter closes with \emph{problems and complete worked solutions}
--- placed before the take-away box, numbered by chapter and Step,
each solution worked in explicit Steps and ending with its physics
intuition, the sets cross-referencing one another into a single
course network.  They carry full solutions because, in the
cohort's phrase, they are meant to be done, not admired: the
reader who works them owns the derivations, not merely the
conclusions.

\paragraph{Prerequisites and recommended textbooks.}
The course assumes advanced undergraduate mechanics,
electrodynamics, statistical physics, and quantum mechanics, plus a
first exposure to general relativity (the Schwarzschild solution is
derived from scratch in Appendix~\ref{app:schwarzschild-course},
so a student meeting black holes for the first time can begin
there).  It is meant to be read \emph{alongside} a canonical
textbook, not in place of one: the recommended way to use each
appendix is to read the corresponding textbook chapter first,
return here for a compact but complete recap, and only then
proceed to the later Steps, where the standard result is
re-examined and re-read as a statement of the geometry.  For
self-study we particularly recommend the following, each cited
again in the appendix to which it is best matched:
\begin{itemize}
\item \textbf{General relativity foundations.} \citet{Wald1984}, \emph{General Relativity}, and \citet{MisnerThorneWheeler1973}, \emph{Gravitation} --- the two canonical graduate texts, covering the differential geometry, the Einstein equations, and the classical black-hole solutions in the conventions this course follows; \citet{Carroll2004}, \emph{Spacetime and Geometry}, as the gentler modern first pass.
\item \textbf{Black-hole mechanics at depth.} \citet{Poisson2004}, \emph{A Relativist's Toolkit} --- the standard source for congruences, the Raychaudhuri equation, and horizon mechanics (Appendices~\ref{app:schwarzschild-course}--\ref{app:four-laws-course}); \citet{Chandrasekhar1983}, \emph{The Mathematical Theory of Black Holes}, and \citet{FrolovNovikov1998}, \emph{Black Hole Physics}, for the complete classical theory including the Kerr--Newman sector (Appendix~\ref{app:extremal}).
\item \textbf{Quantum fields in curved spacetime.} \citet{BirrellDavies1982}, \emph{Quantum Fields in Curved Space} --- the classic source for the Bogoliubov machinery of Appendix~\ref{app:Hawking_T}; \citet{ParkerToms2009}, \emph{Quantum Field Theory in Curved Spacetime}, for the effective-action and heat-kernel methods behind Appendix~\ref{app:species-sum}; \citet{WaldQFT1994}, \emph{Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics}, for the algebraic formulation and the generalized second law.
\item \textbf{The information era.} \citet{SusskindLindesay2005}, \emph{An Introduction to Black Holes, Information and the String Theory Revolution} --- the most readable account of the information paradox as the field understood it before the islands era; \citet{Harlow2016}, the Jerusalem lectures, for the quantum-information toolkit (Appendices~\ref{app:Page_curve} and~\ref{app:data-processing}); and \citet{AlmheiriReview2021}, \emph{The entropy of Hawking radiation}, for the modern islands synthesis reviewed in Appendix~\ref{app:holography-course}.
\end{itemize}
\noindent Throughout, \AOne{}--\AEleven{}, \PaperPOne{}--\PaperPFive{},
and \PaperCOne{}--\PaperCThree{} refer to the papers of the
\GGtheory{} cohort, to which the third steps repeatedly point.

\newcommand{\GGteachbox}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{What this appendix teaches.} #1}}\end{center}}
\newcommand{\GGtakeaway}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{Take-away.} #1}}\end{center}}
\newcommand{\GGreadalong}[1]{\par\noindent\emph{Read alongside:} #1\par}

%% ============ GROUP 1 (A-D): THE REFERENCE DICTIONARIES ============

\section{Summary of abbreviations}
\label{app:abbreviations}

For the canonical Dictionary of program-defined acronyms (45
entries in ten categories) see \ATwo{} Appendix~B; the entries
below are the working set of \emph{this} paper, one line each.

\begin{description}\setlength{\itemsep}{0.1em}
\item[\AX{n} / \POS{n}] The constitution's axioms and postulates
(\AOne{}); \AX{1} causality, \AX{2} \GGEP{}, \AX{3} \OGGEP{},
\AX{4} \MSMF{}.
\item[\GGEP{}] The geometric equivalence principle: gravity is
bulk geometry; no bare boundary $\Newton$ (the premise of
Theorem~\ref{thm:exact-quarter}).
\item[\OGGEP{}] The bulk-to-boundary projection --- observation
itself; at a horizon, geometrically forced (\textsf{L-BH-Port}).
\item[\GDOS{}] The open-system boundary law (\AFive{}); its
freshness clause \textsf{L-GDOS-Fresh} is consumed with a validity
window by Lemma~\ref{lem:stale-port}.
\item[\GFF{} / \GEP{} / \CEP{}] The universal free-energy ledger
and its geometric and curvature--entropy pieces;
$\GFF = F_{\rm GEP} + F_{\rm CEP}$.
\item[\GRIP{} / \GRIPin{} / \GRIPout{}] The runtime descent and
its two regimes (fixed landscape / landscape reorganization).
\item[\GA{} / \GGA{}] Geometric Attractor; Ground Geometric
Attractor (the basin with no exit channel).
\item[\CPTP{}] Completely positive, trace preserving --- the
channel class of every boundary map in this paper.
\item[\GSSB{}] Geometric Spontaneous Symmetry Breaking (\ASix{}):
the landscape-reorganization stage of the \GRIP{} cascade; a
\GSSB{} event in the entropy ledger (T-BH-2) is a reorganization
of its minimizing branch.
\item[\DGI{}] Dynamical Gauge Invariance (owner: \AOne{},
stratum POS-1; the initial stage of \ASix{}'s \GRIP{} cascade).
Used in this paper for its constrained-subsystem content in the
factorization puzzle (\S\ref{sec:gff-ledger}).
\item[\QES{}] Quantum extremal surface (the island literature's
minimization locus).
\item[KMS] The thermal (Kubo--Martin--Schwinger) condition --- the
clock's thermal face (Theorem~\ref{thm:thermal-clock}).
\item[QNM] Quasi-normal mode (the ringdown spectrum).
\item[EMRI] Extreme-mass-ratio inspiral (the LISA-era Kerr
multipole probe).
\item[\BIsix{} / \OGNfive{} / GU-5] The bulk anomaly identity;
the integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$; the five universal
constants $\{c,G_N,\hbar;k_B,Q_B\}$.
\item[T-BH-$n$ / L-BH-* / C-BH-* / F-BH-$n$] This module's keyed
theorems, lemmas, corollaries, and falsifier rows; the full deck is
Appendix~\ref{app:statement-deck}.
\end{description}

\section{Glossary of key symbols}
\label{app:glossary}

\begin{description}\setlength{\itemsep}{0.1em}
\item[$\kappa$, $\beta$] Surface gravity; inverse clock period
$\beta = 2\pi/\kappa$ (the geometry's one contribution to
thermality).
\item[$A$, $\Sbh$, $\Sgen$] Horizon area; Bekenstein--Hawking
entropy $A/4\Newton$; generalized entropy $A/4G_N + S_{\rm out}$.
\item[$r$, $\theta$, $K$] The scale coordinate on $I_r$ with depth
$K = 35$; the phase angle on $S^1_\theta$.
\item[$\ell_{\rm lock}$] The compactification length
$\hbar/(\Mlock c)$; the suppression scale of every Kerr deviation,
$(\ell_{\rm lock}/r_s)^2$.
\item[$\alpha_\omega$, $\beta_\omega$] Bogoliubov coefficients;
$|\beta_\omega/\alpha_\omega| = e^{-\pi\omega/\kappa}$ is the
branch-cut datum (T-BH-4) and the KMS face displayed (T-BH-5).
\item[$J$, $|A|$] The clock's rotor (derived complex structure) and
speed (modulus) --- the polar split of the Killing generator.
\item[$\Gamma_\ell(\omega)$, $\Gamma$] Graybody factor;
exchange-channel transmissivity.
\item[$\bar n_H$, $g(n)$] KMS occupation
$(e^{\hbar\omega/k_BT_{\rm H}}-1)^{-1}$; thermal entropy function
$(n{+}1)\ln(n{+}1) - n\ln n$.
\item[$S(\rho\|\sigma)$] Relative entropy --- the monotone of the
data-processing inequality, and the exact GSL production of
Lemma~\ref{lem:gsl-arrow}.
\item[$I(q\!:\!R)$] Mutual information of the next engaged mode
with the radiation record --- the freshness meter of
Lemma~\ref{lem:stale-port}.
\item[$t_{\rm Page}$, $M_\times$] The freshness-budget expiry
time; the conditional-\GGA{} crossover mass (C-BH-1) --- also the
GSL's equality point.
\item[$a_1$, $a_2$, $\epsilon$] Seeley--DeWitt heat-kernel
coefficients; proper-time cutoff (the shared ancestor of
$1/\Newton$ and $dS_{\rm ent}/dA$).
\item[$\chi_i$, $c_{\log}$] Per-species induced-gravity weights
(the open sum, F-BH-9); the universal logarithmic-correction
coefficient.
\end{description}

\section{Cross-cohort naming}
\label{app:naming}

Cohort papers are cited by designator (\AOne{} through
\PaperPFive{}, \PaperCOne{}--\PaperCThree{}) with natbib labels of
the form (Arisaka 2026, XX-Name); keyed statements are cited by
their permanent keys (T-BH-$n$, \textsf{L-GDOS-Fresh},
\textsf{T-P5-B1}, \textsf{L-Constitution-1}), which never change
across versions of record; and no two Arisaka keys share one
\texttt{citep}.  The full convention is the cohort's, stated once
in \AOne{}.

\section{Numerical parameters for reference}
\label{app:BH_numbers}

\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item Planck mass: $\Mpl \approx 2.18\times 10^{-8}$~kg.
\item Planck length: $\ell_P \approx 1.62\times 10^{-35}$~m.
\item Schwarzschild radius of solar-mass BH:
$r_s(\Msun) = 2\Newton\Msun/c^2 \approx 2.95$~km.
\item Hawking temperature of $\Msun$ BH:
$\THaw(\Msun) = \hbar c^3/(8\pi\Newton\Msun\kB) \approx
6.17\times 10^{-8}$~K.
\item Evaporation time of $\Msun$ BH:
$t_{\rm evap} = 5120\pi G_N^2 M^3/(\hbar c^4) \approx 2\times10^{67}$~yr;
Page time $t_{\rm Page}\approx0.646\,t_{\rm evap}$.
\item Entropy of $\Msun$ BH: $\Sbh \approx 10^{77}\kB$.
\item M87* mass: $M \sim 6.5\times 10^9\Msun$; shadow $\sim
42\,\mu$as.
\item Sgr A* mass: $M \sim 4.3\times 10^6\Msun$; shadow $\sim
51.8\,\mu$as.
\item \GGtheory{} compactification length:
$\ell_{\rm lock}\equiv\hbar/(\Mlock c)\simeq5\times10^{-20}$~m
(with $\Mlock=3.85$~TeV), longer than the unification length
$\hbar c/E_{\rm GUT}\simeq 8.7\times10^{-33}$~m ($E_{\rm GUT} =
\Mpl e^{-2\pi} \simeq 2.3\times10^{16}$~GeV, the \PaperPOne{}
ladder) by the factor $2\,e^{K-2\pi}\simeq6\times10^{12}$; the
depth-$K$ hierarchy itself is $\Mpl/2\Mlock = e^{K}\simeq1.6\times10^{15}$.
\item \OGNfive{} integer $K = 35$; scale hierarchy $e^K \sim
10^{15}$.
\end{itemize}


%% ================= BLOCK II: THE DERIVATION APPENDICES =============


%% ============ GROUP 2 (E-R): THE COURSE APPENDICES ==================

\section{The Schwarzschild geometry from Einstein's equations}
\label{app:schwarzschild-course}

\GGteachbox{The complete derivation of the Schwarzschild solution
from the vacuum Einstein equations --- the ansatz, the two field
equations that survive, and their integration --- followed by the
geometry of the horizon (why $r = r_s$ is a coordinate artifact and
$r = 0$ is not), the Kruskal extension and the Penrose diagram, and
the orbits: photon sphere, ISCO, and the shadow, closed by a worked
computation of the M87* shadow angle against the EHT measurement.
Then: what the classical solution assumes and cannot answer, and
the \GGtheory{} reading in which the horizon becomes a port and
Finkelstein's 1958 lesson repeats one level up.}
\GGreadalong{\citet{Carroll2004} chs.~5--6 for a first pass;
\citet{Wald1984} ch.~6 or \citet{MisnerThorneWheeler1973}
chs.~23--25 at course level; \citet{Chandrasekhar1983} ch.~3 at
depth.}

Every course on black holes begins where the subject began: with
the solution Karl Schwarzschild found within weeks of Einstein's
November 1915 field equations \citep{Einstein1915,
Schwarzschild1916}, computed in a wartime field post.  This
appendix derives it with every manipulation displayed, then walks
the geometry until the two central surprises --- the removable
singularity at $r = r_s$ and the irremovable one at $r = 0$ ---
are theorems the reader owns.

\subsection{Step 1 --- the vacuum solution, worked}

We seek the general static, spherically symmetric metric.  The
most general such line element can be brought to the form
\begin{equation}\label{eq:sch-ansatz}
ds^2 = -e^{2\alpha(r)}c^2\,dt^2 + e^{2\beta(r)}\,dr^2
+ r^2\bigl(d\theta^2 + \sin^2\!\theta\,d\phi^2\bigr),
\end{equation}
with two unknown functions $\alpha(r)$, $\beta(r)$.  (\emph{Physics
intuition}: staticity means a timelike Killing vector orthogonal to
constant-$t$ slices --- nothing moves, nothing rotates; spherical
symmetry uses the angular part to define $r$ so that a sphere at
coordinate $r$ has area exactly $4\pi r^2$.  Both assumptions will
be relaxed at the end of the Step and nothing will change.)

The nonvanishing Christoffel symbols of \eqref{eq:sch-ansatz} are
computed from $\Gamma^\lambda_{\mu\nu} = \tfrac{1}{2} g^{\lambda\rho}
(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} -
\partial_\rho g_{\mu\nu})$:
\begin{equation}
\Gamma^t_{tr} = \alpha', \quad
\Gamma^r_{tt} = c^2\alpha' e^{2(\alpha-\beta)}, \quad
\Gamma^r_{rr} = \beta', \quad
\Gamma^r_{\theta\theta} = -r e^{-2\beta}, \quad
\Gamma^\theta_{r\theta} = \Gamma^\phi_{r\phi} = \frac{1}{r},
\end{equation}
plus $\Gamma^r_{\phi\phi} = -re^{-2\beta}\sin^2\theta$,
$\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$,
$\Gamma^\phi_{\theta\phi} = \cot\theta$, with primes denoting
$d/dr$.  Assembling the Ricci tensor
$R_{\mu\nu} = \partial_\lambda \Gamma^\lambda_{\mu\nu} -
\partial_\nu \Gamma^\lambda_{\mu\lambda} +
\Gamma^\lambda_{\lambda\rho}\Gamma^\rho_{\mu\nu} -
\Gamma^\lambda_{\nu\rho}\Gamma^\rho_{\mu\lambda}$ and demanding
vacuum, $R_{\mu\nu} = 0$, three independent equations survive:
\begin{align}
R_{tt} &= c^2 e^{2(\alpha - \beta)}\Bigl[\alpha'' + (\alpha')^2 -
\alpha'\beta' + \frac{2\alpha'}{r}\Bigr] = 0,
\label{eq:sch-Rtt}\\
R_{rr} &= -\alpha'' - (\alpha')^2 + \alpha'\beta' +
\frac{2\beta'}{r} = 0, \label{eq:sch-Rrr}\\
R_{\theta\theta} &= 1 - e^{-2\beta}\bigl[1 + r(\alpha' -
\beta')\bigr] = 0. \label{eq:sch-Rthth}
\end{align}
Multiply \eqref{eq:sch-Rtt} by $e^{2(\beta-\alpha)}/c^2$ and add
\eqref{eq:sch-Rrr}: everything cancels except
\begin{equation}
\frac{2}{r}\,(\alpha' + \beta') = 0
\quad\Longrightarrow\quad
\beta(r) = -\alpha(r),
\end{equation}
choosing the integration constant so that the metric is Minkowski
at infinity ($\alpha, \beta \to 0$).  Substituting into
\eqref{eq:sch-Rthth}:
\begin{equation}
1 = e^{2\alpha}\,(1 + 2r\alpha') = \bigl(r\,e^{2\alpha}\bigr)',
\quad\Longrightarrow\quad
r\,e^{2\alpha} = r - r_s
\end{equation}
with $r_s$ an integration constant.  Hence
\begin{equation}\label{eq:schwarzschild}
\boxed{\;ds^2 = -\Bigl(1 - \frac{r_s}{r}\Bigr)c^2\,dt^2
+ \Bigl(1 - \frac{r_s}{r}\Bigr)^{-1}dr^2 + r^2 d\Omega_2^2\;}
\end{equation}
The constant is fixed by the Newtonian limit: for weak fields
$g_{tt} = -(1 + 2\Phi/c^2)$ with $\Phi = -\Newton M/r$, so
\begin{equation}\label{eq:rs-def}
r_s = \frac{2\Newton M}{c^2}
\approx 2.95\;\text{km}\times\frac{M}{\Msun}.
\end{equation}
(\emph{Physics intuition}: $r_s$ is exactly the radius at which the
Newtonian escape velocity reaches $c$ --- the coincidence Michell
and Laplace noticed in the 1780s.  It is a coincidence: the
Newtonian calculation has light slowing and returning, while in
relativity nothing inside $r_s$ moves outward at all.  But it
correctly signals that $r_s$ is where escape becomes the issue.)

Two remarks complete the Step.  First, nowhere did we impose
time-independence on the two field equations beyond the ansatz; if
one repeats the computation with $\alpha(r,t)$, $\beta(r,t)$, the
$tr$-component of the field equations forces $\dot\beta = 0$ and
the rest forces \eqref{eq:schwarzschild} again.  This is
\emph{Birkhoff's theorem}: the vacuum outside any spherical mass
--- even a collapsing or exploding one --- is exactly static
Schwarzschild.  (\emph{Physics intuition}: spherical pulsation has
no monopole gravitational radiation to emit, so the exterior
cannot learn the star is moving.)  Second, the solution is unique:
one exterior geometry per mass, with no other adjustable property
--- the first appearance of the rigidity that Part~I called the
no-hair property and that the main text
(\S\ref{ssec:runtime-uniqueness}) reads as the boundary face of
\MSMF{}.

\subsection{Step 2 --- the horizon is not a place where physics
breaks: Kretschmann, Finkelstein, Kruskal}

At $r = r_s$ the metric component $g_{rr}$ diverges, and for
half a century (1916--1958) the field wavered over what that
meant.  The resolution is a two-line invariant computation.  The
curvature scalar that cannot be transformed away is the
Kretschmann invariant, and for \eqref{eq:schwarzschild}
\begin{equation}\label{eq:kretschmann}
K \equiv R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}
= \frac{48\,\Newton^2 M^2}{c^4\,r^6}
= \frac{12\,r_s^2}{r^6}.
\end{equation}
At the horizon $K(r_s) = 12/r_s^4$ --- \emph{finite}, and for a
supermassive hole absurdly small: tidal stresses at the horizon
scale as $1/M^2$, so a freely falling
observer crossing the horizon of M87* feels tidal stresses some
ten thousand times weaker than those of standing on the Earth.  At
$r \to 0$, by contrast, $K$ diverges: that is a genuine curvature
singularity, unremovable by any chart.  (\emph{Physics intuition}:
the mean density inside the horizon, $\bar\rho = 3c^6/(32\pi
\Newton^3 M^2)$, \emph{falls} as $1/M^2$ --- about $2 \times
10^{19}\;\mathrm{kg\,m^{-3}}$ for a solar-mass hole but roughly
the density of water for $10^8 \Msun$.  A supermassive horizon is
not a place of extreme matter; it is a place of extreme
bookkeeping.)

The coordinate cure is \citet{Finkelstein1958}: define the
tortoise coordinate and the advanced time
\begin{equation}
r_* = r + r_s \ln\Bigl|\frac{r}{r_s} - 1\Bigr|, \qquad
v = ct + r_*,
\end{equation}
so that \eqref{eq:schwarzschild} becomes
\begin{equation}\label{eq:EF}
ds^2 = -\Bigl(1 - \frac{r_s}{r}\Bigr)dv^2 + 2\,dv\,dr
+ r^2 d\Omega_2^2,
\end{equation}
manifestly regular at $r = r_s$.  Ingoing light rays are $v =$
const; outgoing rays obey $dr/dv = \tfrac{1}{2}(1 - r_s/r)$, which
is positive outside, zero \emph{at} the horizon, and negative
inside: the light cones tip over until, inside the horizon, even
the outward branch points to smaller $r$, and reaching $r = 0$
becomes as compulsory as reaching next Tuesday.  The maximal
analytic extension \citep{Kruskal1960} doubles the chart into the
four regions of Figure~\ref{fig:penrose-eternal}; the physical
spacetime formed by collapse \citep{OppenheimerSnyder1939}
contains only regions I and II, with the star's surface replacing
the white-hole sector.  \citet{Penrose1965} then proved the
singularity is not an artifact of exact symmetry: once a trapped
surface forms, some geodesic terminates.  Incompleteness is
generic \citep{HawkingPenrose1970}.

\begin{figure}[htbp]
\centering
\resizebox{0.871\textwidth}{!}{%
\begin{tikzpicture}[font=\small,scale=1.05]
% Penrose diagram of the eternal Schwarzschild solution
% Region I (right diamond), II (top), III (left), IV (bottom)
\coordinate (C)  at (0,0);      % bifurcation point
\coordinate (R)  at (3.4,0);    % i0 right
\coordinate (L)  at (-3.4,0);   % i0 left
\coordinate (TR) at (1.7,1.7);  % i+ right
\coordinate (TL) at (-1.7,1.7); % i+ left
\coordinate (BR) at (1.7,-1.7); % i- right
\coordinate (BL) at (-1.7,-1.7);% i- left
% singularities (wavy)
\draw[decorate,decoration={snake,amplitude=1.1pt,segment length=5pt},
      line width=1.10pt,red!70!black] (TL) -- (TR)
      node[midway,above,font=\scriptsize,red!70!black]{$r=0$ (singularity)};
\draw[decorate,decoration={snake,amplitude=1.1pt,segment length=5pt},
      line width=1.10pt,red!70!black] (BL) -- (BR)
      node[midway,below,font=\scriptsize,red!70!black,yshift=-9pt]{$r=0$ (white-hole singularity)};
% horizons (diagonals through C)
\draw[line width=1.10pt,blue!60!black] (BL) -- (C) -- (TR);
\draw[line width=1.10pt,blue!60!black] (BR) -- (C) -- (TL);
\node[blue!60!black,font=\scriptsize,rotate=45,anchor=north] at (1.45,1.30) {$\mathcal H^+\ (r=r_s)$};
\node[blue!60!black,font=\scriptsize,rotate=-45,anchor=south] at (-1.15,0.95) {$\mathcal H^+$};
% null infinity
\draw[line width=0.80pt] (TR) -- (R) node[midway,above right,font=\scriptsize]{$\mathcal{I}^+$};
\draw[line width=0.80pt] (R) -- (BR) node[midway,below right,font=\scriptsize]{$\mathcal{I}^-$};
\draw[line width=0.80pt] (TL) -- (L) node[midway,above left,font=\scriptsize]{$\mathcal{I}^+$};
\draw[line width=0.80pt] (L) -- (BL) node[midway,below left,font=\scriptsize]{$\mathcal{I}^-$};
% region labels
\node at (1.85,0) {I};
\node at (0,1.05) {II};
\node at (-1.85,0) {III};
\node at (0,-1.05) {IV};
% i0, i+, i-
\node[font=\scriptsize,anchor=west] at (R) {$i^0$};
\node[font=\scriptsize,anchor=east] at (L) {$i^0$};
\node[font=\scriptsize,anchor=south] at (TR) {$i^+$};
\node[font=\scriptsize,anchor=south] at (TL) {$i^+$};
\node[font=\scriptsize,anchor=north] at (BR) {$i^-$};
\node[font=\scriptsize,anchor=north] at (BL) {$i^-$};
% an infalling worldline in region I crossing into II
\draw[line width=0.80pt,green!45!black,-{Latex[length=1.8mm]}]
  (2.6,-1.35) .. controls (2.0,-0.4) and (1.15,0.45) .. (0.45,1.28);
\node[green!45!black,font=\scriptsize,anchor=west] at (2.75,-0.72)
  {infalling observer};
\end{tikzpicture}}
\caption[Penrose diagram of the maximally extended (eternal) Schwarzschild solution.]{Penrose diagram of the maximally extended (eternal)
Schwarzschild solution.  Light rays run at $45^\circ$; the
horizons $\mathcal H^\pm$ are the diagonals through the
bifurcation point; the singularities $r=0$ are the wavy
horizontal boundaries --- \emph{spacelike} surfaces, i.e.\ moments
of time, not places in space.  A physical black hole formed by
collapse contains only regions I and II (the star's interior
replaces III and IV); the evaporating case, with the \OGGEP{}
port drawn in, is Figure~\ref{fig:penrose} of the main text.}
\label{fig:penrose-eternal}
\end{figure}

The lesson of 1958--1965, which this paper asks the reader to hold
on to, is grammatical: the question ``is $r = r_s$ singular?''
had been answered \emph{relative to a chart}, and the chart was
the problem.  What is invariantly true at the horizon is not
divergence but \emph{one-way causal structure}: it is the boundary
of the region from which null rays reach $\mathcal{I}^+$.

\subsection{Step 3 --- orbits, the photon sphere, the ISCO, and
the shadow}

Geodesics of \eqref{eq:schwarzschild} conserve
$E = (1 - r_s/r)\,c^2\,\dot t$ and $L = r^2\dot\phi$ (dots:
proper time; equatorial plane).  Substituting into
$g_{\mu\nu}\dot x^\mu \dot x^\nu = -c^2$ gives the radial energy
equation $\tfrac{1}{2}\dot r^2 + V_{\rm eff}(r) = \mathrm{const}$
with
\begin{equation}\label{eq:veff-course}
V_{\rm eff}(r) = -\frac{\Newton M}{r} + \frac{L^2}{2r^2}
- \frac{\Newton M L^2}{c^2 r^3},
\end{equation}
the first two terms exactly Newtonian, the third the relativistic
correction that changes everything at small $r$
(Figure~\ref{fig:veff-course}).  Setting $V_{\rm eff}' = 0$ gives
circular orbits at $r_\pm = \frac{L^2}{2\Newton M}\bigl[1 \pm
\sqrt{1 - 12(\Newton M/cL)^2}\bigr]$; the inner root is unstable,
the outer stable, and the two merge when the square root vanishes,
at $L = 2\sqrt{3}\,\Newton M/c$:
\begin{equation}\label{eq:isco}
r_{\rm ISCO} = \frac{6\Newton M}{c^2} = 3\,r_s,
\end{equation}
the innermost stable circular orbit --- the inner edge of thin
accretion disks and the endpoint of every quasi-circular inspiral.
For photons the same computation with $ds^2 = 0$ leaves a single
unstable circular orbit, the \emph{photon sphere},
\begin{equation}\label{eq:photon-sphere}
r_{\rm ph} = \frac{3\Newton M}{c^2} = 1.5\,r_s,
\end{equation}
and rays from infinity are captured if their impact parameter is
below the critical value
\begin{equation}\label{eq:shadow}
b_c = 3\sqrt{3}\,\frac{\Newton M}{c^2} = \sqrt{27}\,\frac{\Newton
M}{c^2} \approx 2.6\,r_s.
\end{equation}
(\emph{Physics intuition}: the $-1/r^3$ term turns the angular-momentum
barrier into a cliff.  In Newton, more $L$ always means a higher
wall and a safe perihelion; in Schwarzschild, any orbit that dips
inside $r_{\rm ph}$ finds the wall replaced by a slide.  The
photon sphere is the crest of that slide, and the shadow is its
silhouette cast on the sky.)

\emph{Worked numbers.}  For M87*
($M = 6.5\times10^9\,\Msun$, distance $D = 16.8$~Mpc), equation
\eqref{eq:shadow} gives a critical-curve angular diameter
\begin{equation}
\theta_{\rm sh} = \frac{2 b_c}{D}
= \frac{2\sqrt{27}\,\Newton M}{c^2 D}
\approx 40\;\mu\text{as},
\end{equation}
to be compared with the EHT's measured emission-ring diameter of
$42 \pm 3\;\mu$as \citep{EHT2019} --- the ring sits slightly
outside the critical curve, as emission models predict.  The 2022
Sgr~A* image \citep{EHT2022} repeats the test three orders of
magnitude down in mass on the same formula.  This is the sense in
which Part~VIII's null tests are \emph{geometry} tests: every
number in them descends from \eqref{eq:schwarzschild} plus its
Kerr generalization \citep{Kerr1963}.

\begin{figure}[htbp]
\centering
\resizebox{0.873\textwidth}{!}{%
\begin{tikzpicture}[font=\small]
\begin{scope}
% axes: x = r/(GM/c^2), y = V_eff (scaled)
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (10.6,0)
  node[below left,font=\scriptsize]{$r\,/\,(\Newton M/c^2)$};
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,-3.0) -- (0,2.6)
  node[right,font=\scriptsize]{$V_{\rm eff}$ (scaled)};
% x tick marks at r=3, 3.8 (max for l2=18), 6, 14.2->off; scale x: plot r in [2.4,30] -> compress
% use x' = (r-2)*0.36 so r=3 -> 0.36, r=6 -> 1.44, r=30 -> 10.1
% curve 1: l^2 = 18 (separate max/min), V*36
\draw[line width=1.10pt,blue!65!black]
  plot[domain=2.68:30,samples=140,smooth]
  ({(\x-2)*0.36},{36*(-1/\x + 9/\x/\x - 18/\x/\x/\x)});
% curve 2: l^2 = 12 (ISCO, inflection at r=6), V*36
\draw[line width=1.10pt,red!60!black,densely dashed]
  plot[domain=3.6:30,samples=140,smooth]
  ({(\x-2)*0.36},{36*(-1/\x + 6/\x/\x - 12/\x/\x/\x)});
% marks: unstable max of curve 1 at r=3.80, V=0.0316*36=1.14
\fill[blue!65!black] ({(3.80-2)*0.36},{1.14}) circle (1.4pt);
\node[blue!65!black,font=\scriptsize,anchor=south] at ({(3.80-2)*0.36},{1.22})
  {unstable};
% stable min of curve 1 at r=14.2, V=-0.0321*36=-1.16
\fill[blue!65!black] ({(14.2-2)*0.36},{-1.16}) circle (1.4pt);
\node[blue!65!black,font=\scriptsize,anchor=south] at ({(14.2-2)*0.36},{-1.02})
  {stable circular orbit};
% ISCO inflection of curve 2 at r=6, V=-0.0556*36=-2.0
\fill[red!60!black] ({(6-2)*0.36},{-2.0}) circle (1.4pt);
\node[red!60!black,font=\scriptsize,anchor=north west] at ({(6-2)*0.36},{-2.02})
  {ISCO: $r = 6\Newton M/c^2$};
% legend
\node[blue!65!black,font=\scriptsize,anchor=west] at (6.1,2.1)
  {$L = 3\sqrt{2}\,\Newton M/c$ (generic)};
\node[red!60!black,font=\scriptsize,anchor=west] at (6.1,1.6)
  {$L = 2\sqrt{3}\,\Newton M/c$ (marginal)};
% cliff annotation at left
\node[font=\scriptsize,gray!40!black,align=left,anchor=west] at (0.32,-1.30)
  {the $-1/r^3$ cliff:\\no barrier survives\\inside $r_{\rm ph}$};
\end{scope}
\end{tikzpicture}}
\caption[The Schwarzschild effective potential \eqref{eq:veff-course} for two values of the specific angular.]{The Schwarzschild effective potential
\eqref{eq:veff-course} for two values of the specific angular
momentum (curves scaled for display).  The relativistic $-1/r^3$
term removes the Newtonian centrifugal barrier at small radius:
for $L > 2\sqrt{3}\,\Newton M/c$ there is an unstable maximum and
a stable minimum; at $L = 2\sqrt{3}\,\Newton M/c$ they merge into
the ISCO at $r = 6\Newton M/c^2$; below, nothing orbits.}
\label{fig:veff-course}
\end{figure}

\subsection{Step 4 --- what the classical solution assumes, and
cannot answer}

The Schwarzschild derivation consumed one physical input ---
$R_{\mu\nu} = 0$ with the Newtonian boundary condition --- and it
answers every question about \emph{trajectories} outside and
inside the hole.  It cannot answer four questions it raises
itself.  \emph{(i)}~What happens at $r = 0$, where its own
curvature invariant \eqref{eq:kretschmann} diverges and geodesics
end \citep{Penrose1965}?  \emph{(ii)}~What, if anything, is the
horizon made of --- the surface has locally unremarkable geometry,
yet globally it is an absolute one-way membrane?  \emph{(iii)}~Why
does the exterior forget everything about the star except $M$
(and, with \citet{Kerr1963} and \citet{ReissnerNordstrom1916},
$J$ and $Q$) --- where does the rest of the star's description
go?  \emph{(iv)}~Nothing in the solution is thermodynamic: no
temperature, no entropy, no arrow.  The next two appendices show
how questions (iii) and (iv) forced themselves on the field
anyway, and Appendix~\ref{app:Hawking_T} shows how the quantum
vacuum answers (iv) uninvited.

\subsection{Step 5 --- the \GGtheory{} reading}

In the main text's reading the Schwarzschild exterior is untouched
--- it is the 4D zero-mode projection of the bulk solution, exact
to the measured precision (Part~VII prices the corrections at
$(\ell_{\rm lock}/r_s)^2$, unobservably small).  What changes is the
grammar of the two singular loci.  The horizon, which
\citet{Finkelstein1958} taught us is regular \emph{in the right 4D
chart}, is read as the locus where the \OGGEP{} projection becomes
causally forced --- a port, not a place (\S\ref{sec:horizon-port});
and the singularity at $r = 0$, which no 4D chart can cure, is
read as curable by the right \emph{bulk} chart: a
small-scale-profile region of the 6D geometry, smooth in six
dimensions, singular only in the projection
(\S\ref{ssec:singularity-6d}).  Finkelstein's lesson --- what
looks pathological may be your coordinates --- is thus applied
twice, once inside 4D and once from 6D down, and the second
application is falsifiable through the same null tests the shadow
computation above feeds (Part~VIII).

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{E.1.1. Prove Birkhoff's theorem: drop the
staticity assumption and show the vacuum equations restore it.}

\noindent\emph{Solution.}  Step 1: repeat the ansatz
\eqref{eq:sch-ansatz} with $\alpha(r,t)$, $\beta(r,t)$.  The only
new nonvanishing Christoffel symbols are
$\Gamma^t_{tt} = \dot\alpha$, $\Gamma^t_{rr} =
\dot\beta\,e^{2(\beta-\alpha)}/c^2$, and $\Gamma^r_{tr} = \dot\beta$
(dots: $\partial/\partial(ct)$).  Step 2: the one component of the
Ricci tensor that has no static counterpart is
$R_{tr} = \frac{2}{r}\,\dot\beta$, and vacuum forces
$\dot\beta = 0$: the spatial geometry cannot depend on time.
Step 3: with $\beta = \beta(r)$, equation \eqref{eq:sch-Rthth}
gives $\alpha' + \beta'$ as a function of $r$ alone, so
$\alpha(r,t) = \alpha_0(r) + h(t)$; the free function $h(t)$
rescales the time coordinate, $dt \to e^{-h}dt$, and is pure gauge.
What remains is exactly \eqref{eq:schwarzschild}.
Physics intuition: a pulsating spherical star has no monopole
radiation channel, so its exterior cannot receive news of the
pulsation --- staticity is not assumed, it is enforced by spherical
symmetry itself.

\medskip\noindent\textbf{E.1.2. Derive the Newtonian
correspondence $g_{tt} = -(1 + 2\Phi/c^2)$ used to fix $r_s$.}

\noindent\emph{Solution.}  Step 1: for a slow particle,
$dx^i/d\tau \ll c\,dt/d\tau$, the geodesic equation reduces to
$\frac{d^2x^i}{d\tau^2} = -\Gamma^i_{tt}\,
\bigl(\frac{c\,dt}{d\tau}\bigr)^2$.  Step 2: for a static, weak
field, $\Gamma^i_{tt} = -\tfrac{1}{2}\,\delta^{ij}\partial_j
g_{tt}$ (metric derivatives are small; indices raised with
$\eta$).  Step 3: with $t \approx \tau$,
$\frac{d^2x^i}{dt^2} = \tfrac{c^2}{2}\,\partial_i g_{tt}$, and
matching Newton, $\ddot{\vec x} = -\nabla\Phi$, gives
$g_{tt} = -(1 + 2\Phi/c^2)$ up to a constant fixed by flatness at
infinity.  With $\Phi = -\Newton M/r$, comparison with
\eqref{eq:schwarzschild} yields $r_s = 2\Newton M/c^2$
\eqref{eq:rs-def}.
Physics intuition: only the \emph{time-time} component matters for
slow motion --- Newtonian gravity is time running at different
rates at different heights, which is why the horizon, where
$g_{tt} \to 0$, is first of all a place where the exterior's clocks
see time stop.

\medskip\noindent\textbf{E.2.1. Find the black-hole mass at which
an infalling body of height $2$~m feels a head-to-foot tidal
acceleration of one Earth gravity at the horizon crossing.}

\noindent\emph{Solution.}  Step 1: the Newtonian estimate of the
tidal (gradient) acceleration across a height $h$ at radius $r$ is
$\Delta a = 2\Newton M h/r^3$, and \eqref{eq:kretschmann} confirms
the relativistic scaling: curvature at the horizon $\propto
1/M^2$.  Step 2: evaluate at $r = r_s = 2\Newton M/c^2$:
$\Delta a = 2\Newton M h \cdot c^6/(8\Newton^3M^3) =
h c^6/(4 \Newton^2 M^2)$.  Step 3: set $\Delta a = g$ and solve:
\begin{equation*}
M = \sqrt{\frac{h\,c^6}{4\Newton^2 g}}
\approx 9.1\times10^{34}\;\mathrm{kg}
\approx 4.6\times10^4\,\Msun .
\end{equation*}
Below $\sim\!5\times10^4\,\Msun$ the crossing is violent (a
stellar-mass horizon dismembers the visitor well outside $r_s$);
above it, the crossing is gentler than standing on Earth.
Physics intuition: this single number explains why the
supermassive holes are the \emph{safe} ones and why ``falling in
unharmed'' (Q4 of the general-reader questions) is a statement
about galactic centers, not X-ray binaries.

\medskip\noindent\textbf{E.3.1. Derive the photon sphere and the
critical impact parameter \eqref{eq:shadow} from the null
effective potential.}

\noindent\emph{Solution.}  Step 1: for null geodesics the radial
equation reads $\dot r^2 = E^2 - L^2 W(r)$ with
$W(r) = (1 - r_s/r)/r^2$ --- the $-\Newton M L^2/c^2r^3$ cliff of
\eqref{eq:veff-course} with no Newtonian $-1/r$ term to soften it.
Step 2: circular null orbits sit at $W'(r) = 0$:
$-2r^{-3} + 3 r_s r^{-4} = 0 \Rightarrow r_{\rm ph} =
\tfrac{3}{2} r_s = 3\Newton M/c^2$, a maximum of $W$ --- hence
unstable, \eqref{eq:photon-sphere}.  Step 3: a ray from infinity
with impact parameter $b = L/E$ is captured iff $b^{-2} >
W(r_{\rm ph}) = \frac{1}{27}\,(c^2/\Newton M)^2$, i.e.
$b < b_c = \sqrt{27}\,\Newton M/c^2$ \eqref{eq:shadow}.
Physics intuition: the photon sphere is a mountain pass for light
--- rays with $b$ just above $b_c$ orbit many times before
escaping, which is why the observed ring is sharp: it is the
pass photographed from below.

\medskip\noindent\textbf{E.3.2. Predict the Sgr~A* shadow diameter
from the star-orbit mass and compare with the 2022 EHT ring.}

\noindent\emph{Solution.}  Step 1: the interferometric star-orbit
campaign gives $M = 4.30\times10^6\,\Msun$ at
$D = 8.28$~kpc, so $\Newton M/c^2 = 6.3\times10^9$~m.  Step 2: the
critical-curve angular diameter is $\theta = 2\sqrt{27}\,\Newton
M/(c^2 D) \approx 53\;\mu$as.  Step 3: the EHT ring
\citep{EHT2022} is $51.8 \pm 2.3\;\mu$as --- agreement at the
few-percent level, with the residual within the known offset
between the emission ring and the critical curve
(cf.\ the M87* case worked in Step~3 of the text).
Physics intuition: the mass in Step 1 comes from Keplerian star
orbits at $\sim\!10^3\,r_s$; the ring tests the \emph{same} mass
at $\sim\!2\,r_s$.  Agreement across three orders of magnitude in
radius is a stronger statement than either number alone.

\medskip\noindent\textbf{E.3.3. Show that the ISCO binding energy
is $1 - \sqrt{8/9}$ of the rest mass, and interpret
astrophysically.}

\noindent\emph{Solution.}  Step 1: for a circular orbit at radius
$r$, conserved energy per unit rest mass is
$\tilde E = (1 - r_s/r)\big/\sqrt{1 - \tfrac{3}{2}r_s/r}$
(combine $V_{\rm eff}' = 0$ with the normalization
$g_{\mu\nu}\dot x^\mu \dot x^\nu = -c^2$).  Step 2: at the ISCO
\eqref{eq:isco}, $r = 3r_s$:
$\tilde E = \bigl(1 - \tfrac13\bigr)/\sqrt{1 - \tfrac12}
= \tfrac{2}{3}\sqrt{2} = \sqrt{8/9} \approx 0.9428$.  Step 3: the
binding energy released by slow inspiral to the ISCO is therefore
$1 - \sqrt{8/9} \approx 5.7\%$ of $mc^2$ --- against $0.7\%$ for
hydrogen fusion.  Physics intuition: this factor-of-eight edge
over nuclear burning is why accreting black holes power the most
luminous persistent sources in the sky, and why quasars were the
first observational hint that collapsed objects exist at all.
(For Kerr the ISCO moves inward and the efficiency rises further
--- Appendix~\ref{app:extremal}.)

\GGtakeaway{\emph{Standard result:} the unique static spherical
vacuum solution \eqref{eq:schwarzschild}, with a regular one-way
horizon at $r_s = 2\Newton M/c^2$, a genuine singularity at
$r = 0$, photon sphere at $1.5\,r_s$, ISCO at $3\,r_s$, and shadow
diameter $2\sqrt{27}\,\Newton M/c^2$, confirmed at M87* and
Sgr~A*.  \emph{What it assumed:} classical vacuum general
relativity.  \emph{What it cannot answer:} the fate of $r=0$, the
nature of the horizon, where the star's information goes, and why
any of it should be thermodynamic.  \emph{The \GGtheory{}
reading:} the exterior survives untouched as a projection; the
horizon becomes a port; the singularity becomes a chart artifact
one level up.}


\section{Classical black-hole mechanics: the four laws}
\label{app:four-laws-course}

\GGteachbox{The four laws of classical black-hole mechanics,
derived rather than tabulated: the surface gravity and its
constancy (zeroth law), the first law $\delta M = \kappa\,\delta
A\,c^2/(8\pi\Newton)$ verified explicitly for Schwarzschild, the
Raychaudhuri equation and Hawking's area theorem (second law) with
the GW150914 area audit as the worked example, and the third law
--- closed by the Bardeen--Carter--Hawking analogy table and the
honest 1973 question the analogy could not answer: the temperature
of what?  The entropy of which states?}
\GGreadalong{\citet{Poisson2004} chs.~2--5 (congruences and
horizon mechanics, the modern standard); \citet{Wald1984} ch.~12;
\citet{BardeenCarterHawking1973} for the original.}

By 1972 the classical theory of the previous appendix had matured
into a set of theorems with an unsettling shape: they were,
line for line, the laws of thermodynamics with $(M, \kappa, A)$
in place of $(E, T, S)$ \citep{BardeenCarterHawking1973}.  This
appendix derives the four laws and then states precisely why, in
1973, the resemblance had to be called an analogy --- the question
the rest of the course answers.

\subsection{Step 1 --- surface gravity and the zeroth law}

For a stationary horizon generated by the Killing field $\xi^\mu$,
the surface gravity $\kappa$ is defined on the horizon by
$\xi^\nu \nabla_\nu \xi^\mu = (\kappa/c)\,\xi^\mu$: the failure of
the generator to be affinely parametrized.  Operationally, $\kappa$
is the force per unit mass that an observer at infinity must
supply, through an idealized massless string, to hold a unit mass
just above the horizon.  For Schwarzschild the computation is
three lines: with $f(r) = 1 - r_s/r$, the local proper
acceleration of a static observer is $a(r) = \frac{\Newton
M}{r^2}f^{-1/2}$, which diverges at the horizon; the redshift
factor is $f^{1/2}$; their product stays finite,
\begin{equation}\label{eq:kappa-sch}
\kappa = \lim_{r \to r_s} f^{1/2}\,a(r)
= \frac{\Newton M}{r_s^2}
= \frac{c^4}{4\Newton M}
= \frac{c^2}{2 r_s}.
\end{equation}
(\emph{Physics intuition}: the local force diverges but the
string's mechanical advantage --- the redshift --- discounts it by
exactly the diverging factor.  $\kappa$ is the price of holding
position \emph{as billed at infinity}, which is why it, and not
the local acceleration, will turn out to set the temperature seen
at infinity.)  The \emph{zeroth law} states that $\kappa$ is
constant over the entire horizon of any stationary black hole ---
even a rotating, flattened one, where the local geometry varies
from pole to equator.  Constancy of an intensive quantity over a
system in equilibrium is the defining mark of a temperature, and
this is the first rung of the analogy.

\subsection{Step 2 --- the first law, verified}

For Schwarzschild, $A = 4\pi r_s^2 = 16\pi \Newton^2 M^2/c^4$, so
\begin{equation}
\delta A = \frac{32\pi \Newton^2 M}{c^4}\,\delta M
\quad\Longrightarrow\quad
\frac{\kappa\,c^2}{8\pi \Newton}\,\delta A
= \frac{c^4}{4\Newton M}\cdot
\frac{c^2}{8\pi\Newton}\cdot\frac{32\pi \Newton^2 M}{c^4}\,
\delta M = c^2\,\delta M,
\end{equation}
which is the \emph{first law}
\begin{equation}\label{eq:first-law-course}
c^2\,\delta M = \frac{\kappa\,c^2}{8\pi \Newton}\,\delta A
\;\;\longrightarrow\;\;
c^2\,\delta M = \frac{\kappa\,c^2}{8\pi \Newton}\,\delta A
+ \Omega_H\,\delta J + \Phi_H\,\delta Q
\end{equation}
in its full Kerr--Newman form, derived together with the Smarr
formula in Appendix~\ref{app:extremal} (Step~3).  The structure is
exactly $dE = T\,dS + \text{work terms}$, with $\kappa$ sitting
where $T$ sits and $A$ sitting where $S$ sits.

\subsection{Step 3 --- the Raychaudhuri equation and the area
theorem}

The second law is the deep one, and it needs one tool.  Consider
the congruence of null geodesics generating the horizon, with
tangent $k^\mu$, affine parameter $\lambda$, and expansion
$\theta = \nabla_\mu k^\mu$ (the fractional rate of change of the
cross-sectional area: $\theta = \frac{1}{\delta A}\frac{d\,\delta
A}{d\lambda}$).  Decomposing $\nabla_\nu k_\mu$ into trace,
symmetric-traceless (shear $\sigma_{\mu\nu}$), and antisymmetric
(twist, zero for hypersurface generators) parts, the geodesic
equation and the Ricci identity give, in four lines of index
algebra, the \emph{Raychaudhuri equation}
\begin{equation}\label{eq:raychaudhuri}
\frac{d\theta}{d\lambda} = -\frac{1}{2}\theta^2
- \sigma_{\mu\nu}\sigma^{\mu\nu}
- R_{\mu\nu}k^\mu k^\nu .
\end{equation}
Every term on the right is $\leq 0$ provided
$R_{\mu\nu}k^\mu k^\nu \geq 0$, which by the Einstein equations is
the \emph{null energy condition} $T_{\mu\nu}k^\mu k^\nu \geq 0$
--- satisfied by all classical matter.  (\emph{Physics intuition}:
gravity focuses.  Shear focuses.  The only thing that could
defocus a pencil of light is negative energy density, and
classical physics has none to offer.)  Suppose now some horizon
generator had $\theta < 0$ at one point.  Integrating
$d\theta/d\lambda \leq -\theta^2/2$ gives $\theta \to -\infty$
within finite affine parameter $\lambda \leq 2/|\theta_0|$: the
generators cross, forming a caustic --- but a theorem of Penrose
forbids horizon generators from leaving the horizon, and crossing
generators would have to.  Contradiction; hence $\theta \geq 0$
everywhere on any future horizon, and therefore
\begin{equation}\label{eq:area-theorem}
\frac{dA}{dt} \geq 0
\end{equation}
in every classical process \citep{HawkingPenrose1970}.  Area, like
entropy, never decreases.

\emph{Worked instance --- the first observational audit.}  In
GW150914 \citep{LIGO2016}, holes of $36$ and $29\,\Msun$ (initial
spins consistent with small) merged into $62\,\Msun$ with final
spin $a_* \approx 0.67$.  Using the Kerr area
$A = 8\pi(\Newton M/c^2)^2\bigl(1 + \sqrt{1 - a_*^2}\,\bigr)$:
initial area $A_i \approx 2.3 \times 10^5\;\mathrm{km}^2$ (the two
Schwarzschild areas), final area $A_f \approx 3.7 \times
10^5\;\mathrm{km}^2$ --- an increase of $\sim 57\%$, comfortably
positive even while $3\,\Msun c^2$ of energy left the system as
gravitational waves.  Mass can be spent; area cannot.  This is now
a live experimental program (the ringdown-era area-theorem
tests of Part~VIII).

\subsection{Step 4 --- the third law, and the analogy table}

The \emph{third law} states that no finite sequence of physical
processes can reduce $\kappa$ to zero --- extremality
($a_* \to 1$, or $Q \to M$ in geometrized units) is asymptotic,
like absolute zero.  The worked Kerr--Newman version, with the
capture-orbit argument, is Appendix~\ref{app:extremal} (Step~4).
Assembled, the four laws sit in exact formal correspondence with
thermodynamics:

\begin{table}[H]
\centering
\caption[The Bardeen--Carter--Hawking correspondence \citep{BardeenCarterHawking1973}, as it stood.]{The Bardeen--Carter--Hawking correspondence
\citep{BardeenCarterHawking1973}, as it stood in 1973: a formal
analogy with, at that date, no license to set the constants
$\alpha$. The table is reprinted as it stood in 1973, before Hawking's calculation, and the point is the one its authors made themselves: at that date the correspondence was formal. Nothing fixed the constants, and the analogy could have been a coincidence of algebra. What the sections that follow supply is the license the 1973 paper explicitly declined to claim.}
\label{tab:four-laws-course}
\small
\begin{tabular}{p{0.08\textwidth}p{0.40\textwidth}p{0.42\textwidth}}
\toprule
\textbf{Law} & \textbf{Thermodynamics} & \textbf{Black-hole
mechanics} \\
\midrule
Zeroth & $T$ constant throughout a body in equilibrium &
$\kappa$ constant over a stationary horizon \\
First & $dE = T\,dS + \text{work}$ &
$c^2 dM = \dfrac{\kappa c^2}{8\pi\Newton}\,dA + \Omega_H dJ +
\Phi_H dQ$ \\
Second & $\delta S \geq 0$ & $\delta A \geq 0$ (Hawking's area
theorem) \\
Third & $T = 0$ unattainable in finite operations & $\kappa = 0$
unattainable in finite operations \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Step 5 --- what the analogy could not explain}

In 1973 the correspondence was called, by its own authors, a
formal analogy --- and they were right to be careful, because
taken literally it was inconsistent.  A body with a temperature
radiates; a classical black hole absorbs everything and emits
nothing, so its temperature is exactly zero, while
$\kappa \neq 0$.  An entropy counts microstates; the no-hair
theorems say a stationary hole has \emph{one} exterior state per
$(M, J, Q)$, so its entropy should be $k_B \ln 1 = 0$, while
$A \neq 0$.  The analogy demanded constants of proportionality
--- $T = \alpha\kappa$, $S = A/(8\pi\Newton\alpha)$ for some
$\alpha$ --- that classical physics not only failed to supply but
positively forbade.  Two years later Hawking's quantum computation
(Appendix~\ref{app:Hawking_T}) supplied
$\alpha = \hbar/(2\pi c k_B)$ uninvited, and the analogy became
physics with one stroke --- leaving as its residue the question
this paper's main text addresses: an entropy was now
\emph{priced}, but what does it \emph{count}?

\subsection{Step 6 --- the \GGtheory{} reading}

In the main text's ledger, each law acquires a mechanism.  The
zeroth law is the KMS face of the port clock: one thermal clock
per stationary port, hence one $\kappa$
(\S\ref{sec:thermal-clock}).  The first law is the differential
form of the \GFF{} ledger \eqref{eq:gff-bh} on the stationary
branch.  The second law is promoted from the null energy
condition to the generalized second law --- now including the
radiation's entropy, which the classical theorem cannot see ---
with its production computed exactly in the exchange class
(Appendix~\ref{app:data-processing}, Step~3, after
\citealp{Wall2012}).  And the third law becomes the statement
that the port never quite closes (Appendix~\ref{app:extremal},
Step~4).  The analogy of 1973, in other words, is read as a
dictionary fragment: the boundary bookkeeping of an open system
whose bulk ledger the classical theory could name but not open.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{F.1.1. Derive the static observer's
proper acceleration $a(r) = \Newton M r^{-2} f^{-1/2}$ used in
\eqref{eq:kappa-sch}.}

\noindent\emph{Solution.}  Step 1: the static worldline has
four-velocity $u^\mu = (c f^{-1/2}, 0, 0, 0)$ (normalized:
$g_{tt}u^tu^t = -c^2$).  Step 2: the acceleration is
$a^\mu = u^\nu \nabla_\nu u^\mu$; the only nonvanishing component
is $a^r = \Gamma^r_{tt}(u^t)^2 = \bigl(\tfrac{c^2}{2}f f'\bigr)
\cdot f^{-1} = \tfrac{c^2}{2}f'$.  Step 3: the invariant magnitude
is $a = \sqrt{g_{rr}}\,a^r = \tfrac{c^2}{2}f' f^{-1/2}
= \frac{\Newton M}{r^2}\,f^{-1/2}$, diverging at the horizon.
Multiplying by the redshift $f^{1/2}$ and taking the limit gives
$\kappa = \Newton M/r_s^2 = c^4/4\Newton M$
\eqref{eq:kappa-sch}.  Physics intuition: the local engine must
work infinitely hard to hover, but every joule it reports is
infinitely redshifted on the way out; the finite product is the
only number the far bookkeeper ever sees --- which is precisely
why it, and not the local force, becomes the temperature.

\medskip\noindent\textbf{F.2.1. From the first law and
$S \propto A$ alone, derive the mass-scaling of the temperature
--- before any quantum computation.}

\noindent\emph{Solution.}  Step 1: $A = 16\pi \Newton^2M^2/c^4
\Rightarrow A \propto M^2$, so if $S = \eta k_B A c^3/(\Newton\hbar)$
then $S \propto M^2$.  Step 2: the thermodynamic identity
$T^{-1} = \partial S/\partial E$ with $E = Mc^2$ gives
$T^{-1} \propto M$, i.e.\ $T \propto 1/M$: big holes are cold.
Step 3: restoring the constants,
$T = \hbar c^3/(16\pi\eta\,\Newton M k_B\,\cdot\,2)$ --- with the
coefficient left as $\eta$, exactly the state of knowledge of
1973.  Physics intuition: the negative specific heat of
Appendix~\ref{app:euclidean} (Step 3) is already visible here ---
$T$ falls as $E$ grows --- so the analogy predicted gravity's
strangest thermodynamic signature before anyone believed the
temperature was real.

\medskip\noindent\textbf{F.2.2. Show that the classical analogy
fixes only the product of the two unknown constants.}

\noindent\emph{Solution.}  Step 1: write $T = \alpha\kappa$ and
$S = \gamma A$ for undetermined $\alpha, \gamma$.  Step 2: the
first law \eqref{eq:first-law-course} demands
$T\,dS = \frac{\kappa c^2}{8\pi\Newton}\,dA$, i.e.
$\alpha\gamma = c^2/(8\pi \Newton)$: one equation, two unknowns.
Step 3: no classical experiment can split the product --- a
rescaling $\alpha \to \lambda\alpha$, $\gamma \to \gamma/\lambda$
changes nothing observable while the hole neither radiates nor
counts states.  Hawking's computation (Appendix~\ref{app:Hawking_T})
fixes $\alpha = \hbar/(2\pi c k_B)$, and $\gamma = k_B c^3/
(4\Newton\hbar)$ follows \emph{instantly} from the 1973 product.
Physics intuition: this is why the temperature and the entropy
coefficient were nailed by a single quantum calculation --- the
classical first law had already welded them together.

\medskip\noindent\textbf{F.3.1. Fill in the derivation of the
Raychaudhuri equation \eqref{eq:raychaudhuri}.}

\noindent\emph{Solution.}  Step 1: define
$B_{\mu\nu} = \nabla_\nu k_\mu$ for the null congruence; along
each geodesic, $k^\rho\nabla_\rho B_{\mu\nu}
= -B_{\mu\rho}B^\rho{}_\nu - R_{\rho\nu\mu\sigma}k^\rho k^\sigma$,
which follows from the Ricci identity
$[\nabla_\rho, \nabla_\nu]k_\mu = R_{\rho\nu\mu}{}^\sigma
k_\sigma$ plus the geodesic equation.  Step 2: trace over
$\mu\nu$: with $\theta = B^\mu{}_\mu$,
$\frac{d\theta}{d\lambda} = -B_{\mu\nu}B^{\nu\mu}
- R_{\mu\nu}k^\mu k^\nu$.  Step 3: decompose the (projected,
transverse) $B_{\mu\nu} = \tfrac{1}{2}\theta h_{\mu\nu}
+ \sigma_{\mu\nu} + \omega_{\mu\nu}$ in the two-dimensional
screen space; then $B_{\mu\nu}B^{\nu\mu} = \tfrac12\theta^2
+ \sigma^2 - \omega^2$ (the $\tfrac12$ because the screen is
two-dimensional), and $\omega = 0$ for horizon generators
(hypersurface-orthogonal).  Equation \eqref{eq:raychaudhuri}
follows.  Physics intuition: the equation is pure kinematics plus
one import --- geometry only enters through $R_{\mu\nu}k^\mu
k^\nu$, which Einstein's equations convert into energy density.
``Gravity focuses light'' is the entire physical content, and the
area theorem is its global echo.

\medskip\noindent\textbf{F.3.2. For the GW150914 masses, find the
maximum energy the area theorem would have permitted the merger to
radiate, and compare with the observed $3\,\Msun c^2$.}

\noindent\emph{Solution.}  Step 1: the final area must satisfy
$A_f \geq A_i$, and for a (nearly non-spinning) final hole
$A \propto M^2$, so $M_f \geq \sqrt{36^2 + 29^2}\,\Msun
= 46.2\,\Msun$: the irreducible mass of the pair.  Step 2: the
radiated energy is bounded by
$E_{\rm rad} \leq (65 - 46.2)\,\Msun c^2 = 18.8\,\Msun c^2$
--- an efficiency ceiling of $29\%$.  Step 3: the observed
$3\,\Msun c^2$ ($4.6\%$) sits comfortably inside: the theorem
constrains, the dynamics decides.  (A spinning final hole lowers
the ceiling slightly; the worked audit in Step~3 of the text used
the measured $a_* = 0.67$.)  Physics intuition: the area theorem
is gravity's second law acting as an energy regulator --- mergers
are the most luminous events in nature \emph{and} they are
throttled, both facts from one inequality.

\medskip\noindent\textbf{F.5.1. Make the 1973 inconsistency
quantitative: show that a cold classical hole beside a warm bath
violates the ordinary second law if the analogy is read literally.}

\noindent\emph{Solution.}  Step 1: give the hole any temperature
$T_{\rm BH} > 0$ by the analogy, and place it in a radiation bath
at $T_b$ with $0 < T_b < T_{\rm BH}$.  Step 2: classically the
hole absorbs the bath's radiation and emits nothing, so heat
flows \emph{from the colder bath into the hotter hole};
$dS_{\rm total} = dQ\,(T_{\rm BH}^{-1} - T_b^{-1}) < 0$ if
$S_{\rm BH}$ is read via the first law.  Step 3: conclusion --- of
the triple (nonzero $T$, no emission, first law), one member must
go.  Hawking's radiation removes ``no emission'' and the system
becomes consistent; the alternative surgeries (deny $T$, deny the
first law) each destroy the analogy wholesale.
Physics intuition: this is the cleanest way to see that Hawking
radiation was \emph{forced} --- thermodynamics had already decided
horizons must shine; the quantum computation only revealed the
mechanism and the coefficient.

\GGtakeaway{\emph{Standard result:} the four laws
(Table~\ref{tab:four-laws-course}), each a theorem of classical
general relativity, jointly isomorphic to thermodynamics.
\emph{What they assumed:} the null energy condition and classical
matter.  \emph{What they could not explain:} the constants of the
correspondence --- a nonzero temperature for a perfect absorber
and a nonzero entropy for a one-state object were classical
contradictions.  \emph{The \GGtheory{} reading:} the laws are the
boundary bookkeeping of the port; the missing constants arrive
with the quantum (Appendix~\ref{app:Hawking_T}), and the missing
microstates with the projection (\S\ref{ssec:V-microstates}).}


\section{Bekenstein's entropy and the generalized second law}
\label{app:bekenstein-course}

\GGteachbox{The argument that put entropy on the horizon before
any quantum computation existed: Wheeler's teacup, Bekenstein's
1973 proposal $S_{\rm BH} = \eta\,k_B A c^3/(\Newton\hbar)$ with
the coefficient $\eta$ honestly unknown, the Bekenstein bound
$S \leq 2\pi k_B E R/(\hbar c)$ derived from the same gedanken
experiment that motivated it, the generalized second law, and the
species problem --- with worked numbers showing how far ordinary
matter sits from saturation, and why only horizons saturate.}
\GGreadalong{\citet{Bekenstein1973} for the original;
\citet{WaldQFT1994} for the modern formulation;
\citet{SusskindLindesay2005} for the physical picture.}

Between the four laws of the previous appendix (1973 --- a formal
analogy) and Hawking's radiation (1975 --- the analogy made
physical), there is a short, decisive chapter that is often
skipped in the historical telling and never in the logical one:
Bekenstein's argument that horizon area \emph{is} entropy, made
while the field still believed black holes were cold and dead.

\subsection{Step 1 --- Wheeler's teacup: the crisis}

The provocation, as Wheeler put it to his student, is domestic: I
set a cup of hot tea beside a cold one and let them equilibrate; I
have committed a crime against the second law --- the entropy of
the universe went up and I can never bring it back down.  But if I
drop both cups into a passing black hole, the evidence is gone.
The exterior state of the hole is fixed by $(M, J, Q)$ alone
(no-hair, Appendix~\ref{app:schwarzschild-course}); no observation
of the hole will ever reveal how much entropy it swallowed.  If
$S_{\rm outside}$ is all the entropy there is, the second law is
not violated but worse: unfalsifiable, dying ``the death of a
thousand qualifications.''  (\emph{Physics intuition}: the second
law is bookkeeping, and a black hole with no entropy column is an
account that shreds receipts.  Either the law fails, or the hole
keeps books we had not learned to read.)

\subsection{Step 2 --- Bekenstein's 1973 proposal}

Bekenstein's move \citep{Bekenstein1973} was to take the area
theorem of Appendix~\ref{app:four-laws-course} not as an analogy
to the second law but as its missing column.  Three observations
converge.  \emph{(i)}~$A$ never decreases classically --- the
right monotonicity.  \emph{(ii)}~When a hole swallows matter, $A$
increases by at least a calculable minimum --- the right
responsiveness.  \emph{(iii)}~Dimensional analysis: an entropy is
$k_B$ times a pure number, and the only way to make a pure number
from $A$, the constants of gravity, and the quantum is
\begin{equation}\label{eq:bekenstein-ansatz}
S_{\rm BH} = \eta\,\frac{k_B\,A\,c^3}{\Newton\hbar}
= \eta\,\frac{k_B\,A}{\ell_P^2},
\qquad \ell_P^2 = \frac{\Newton\hbar}{c^3},
\end{equation}
with $\eta$ a dimensionless constant of order unity that
\emph{no classical argument can fix} --- Bekenstein's own estimate
was $\eta = \ln 2/(8\pi)$, from counting one bit per photon
dropped in at the minimum increment.  Note what the quantum is
doing here: $\hbar$ enters \eqref{eq:bekenstein-ansatz} through
dimensional necessity \emph{two years before} any quantum
calculation about black holes existed.  An entropy proportional
to \emph{area}, not volume, was also unprecedented --- the first
sighting of what \citet{THooft1993} and \citet{Susskind1995}
would later name the holographic principle
(Appendix~\ref{app:holography-course}).

\subsection{Step 3 --- the Bekenstein bound, derived}

The same gedanken experiment sharpens into a bound on
\emph{ordinary} matter.  Lower a box of energy $E$ (as measured
at infinity when far away) and proper size $R$ slowly toward the
horizon on an idealized string.  At blueshift factor
$f^{1/2}(r)$, the energy the box delivers to the hole if released
is $E\,f^{1/2}$ --- the string has extracted the rest as work at
infinity.  The closest the box's center can approach is one box
radius, $\delta \approx R$, where (using the near-horizon form
$f^{1/2} \approx \kappa\,\delta/c^2$ from
Appendix~\ref{app:four-laws-course}, Step~1)
\begin{equation}
E_{\rm delivered} \geq E\,\frac{\kappa R}{c^2}.
\end{equation}
The first law then prices the area increase: using
$c^2\,\delta M = E_{\rm delivered}$ in
\eqref{eq:first-law-course},
\begin{equation}
\delta A = \frac{8\pi\Newton E_{\rm delivered}}{\kappa c^2}
\;\geq\; \frac{8\pi\Newton E R}{c^4},
\end{equation}
independent of $\kappa$ --- the hole's details cancel.  If the generalized second law (Step~4) is to
survive the box's disappearance, the entropy carried in, $S$,
must not exceed the bookkeeping credit
$\delta S_{\rm BH} = \eta k_B\,\delta A\,c^3/(\Newton\hbar)$; with
the coefficient $\eta = 1/4$ that Appendix~\ref{app:Hawking_T}
will supply, the requirement is the \emph{Bekenstein bound}
\begin{equation}\label{eq:bekenstein-bound}
S \;\leq\; \frac{2\pi k_B\,E R}{\hbar c}.
\end{equation}
(The quantum buoyancy of the box in the horizon's thermal
atmosphere, pointed out by Unruh and Wald, modifies the argument
but not the bound's status for weakly gravitating systems.)

\emph{Worked numbers.}  For a human brain ($E \approx 1.4\,
\mathrm{kg}\times c^2$, $R \approx 0.1$~m) the bound allows
$\sim 4 \times 10^{42}$ bits --- some thirty orders of magnitude
above any realistic information content: ordinary matter is
entropically \emph{dilute}.  Saturation requires
$S \sim k_B E R/\hbar c$, and comparing
\eqref{eq:bekenstein-ansatz} with $E = Mc^2$, $R = r_s$ shows the
unique saturating object is the black hole itself.  (\emph{Physics
intuition}: the bound says information needs energy and room.  A
black hole is what you get when you pack information as densely
as causality permits --- which is why, in
Part~IV's language, its boundary is pure port: nothing about the
packing is visible except the area of the wrapping.)

\subsection{Step 4 --- the generalized second law, and the
species problem}

Bekenstein's repair of Wheeler's crime is the \emph{generalized
second law} (GSL):
\begin{equation}\label{eq:gsl-course}
\delta\Bigl(S_{\rm outside} + S_{\rm BH}\Bigr) \geq 0,
\end{equation}
the statement that the teacup's entropy is not destroyed but
transferred to the horizon column of the ledger.  Classically it
holds because $\delta A \geq 0$ dominates; quantum mechanically it
survives Hawking evaporation --- where $A$ \emph{decreases} ---
because the radiation's entropy more than compensates, a theorem
proved in generality by \citet{Wall2012} and computed exactly for
the exchange class in Appendix~\ref{app:data-processing}, Step~3.
One structural worry surfaced immediately, and it is the right
worry: the \emph{species problem} \citep{SusskindUglum1994}.  The
bound and the GSL involve $\eta/\Newton$; entanglement entropy
across the horizon grows with the number of matter fields; yet
$S_{\rm BH}$ knows nothing of the particle spectrum.  Either the
coefficient is fine-tuned against the field content, or $\Newton$
itself is built from the field content --- the induced-gravity
option, which is precisely the mechanism the main text adopts
(\S\ref{ssec:V-induced-G}) and whose remaining open computation
is Appendix~\ref{app:species-sum}.

\subsection{Step 5 --- what the argument assumed, and the
\GGtheory{} reading}

Bekenstein's 1973 case rested on consistency, not mechanism: the
coefficient $\eta$ was unknown (Hawking's calculation was needed
to fix $\eta = 1/4$), the microstates were unnamed, and the
area-scaling was a scandal without an explanation.  In the main
text's reading, all three residues are addressed in order: the
coefficient is derived, regulator-independently, from induced
gravity plus the conical replica (\S\ref{ssec:V-conical},
Theorem~\ref{thm:exact-quarter}); the microstates are named ---
\UoneQi{} holonomy configurations of the hidden fiber, one
register per Planck-area pixel (\S\ref{ssec:V-microstates}); and
the area-scaling is the bandwidth of the port --- entropy lives
on the boundary because the boundary is where the projection
truncates the bulk (\S\ref{sec:horizon-port}).  The species
problem becomes the program's one declared open computation
(Appendix~\ref{app:species-sum}).

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{G.2.1. Prove the dimensional-analysis
uniqueness claim behind \eqref{eq:bekenstein-ansatz}.}

\noindent\emph{Solution.}  Step 1: seek $S = k_B\,\Xi$ with $\Xi$
a dimensionless function of $A$, $\Newton$, $\hbar$, $c$ (mass
does not appear separately: for Schwarzschild, $A$ carries it).
Step 2: from $\Newton, \hbar, c$ exactly one length can be built,
$\ell_P = \sqrt{\Newton\hbar/c^3}$, so the only dimensionless
combination involving $A$ is $A/\ell_P^2$ (and functions
thereof).  Step 3: additivity fixes the function: for two distant
holes, entropies add and areas add, so $\Xi(A_1 + A_2) =
\Xi(A_1) + \Xi(A_2)$, forcing linearity, $\Xi = \eta A/\ell_P^2$.
Physics intuition: the quantum enters through $\ell_P$ before any
quantum \emph{dynamics} is known --- dimensional analysis alone
says that if a horizon has entropy at all, $\hbar$ is in it.
That is why Bekenstein could be right in 1973 with no mechanism.

\medskip\noindent\textbf{G.3.1. Show that the Schwarzschild hole
saturates the Bekenstein bound \eqref{eq:bekenstein-bound}
exactly.}

\noindent\emph{Solution.}  Step 1: insert $E = Mc^2$ and
$R = r_s = 2\Newton M/c^2$ into the bound:
$S_{\max} = 2\pi k_B\,Mc^2\cdot 2\Newton M/(\hbar c\, c^2)
= 4\pi k_B \Newton M^2/(\hbar c)$.  Step 2: evaluate
$\Sbh = k_B A c^3/(4\Newton\hbar)$ with
$A = 16\pi\Newton^2M^2/c^4$: $\Sbh = 4\pi k_B\Newton
M^2/(\hbar c)$.  Step 3: the ratio is exactly $1$ --- no factors
of 2 survive.  Physics intuition: the bound is a property of
causality and thermodynamics; the black hole is the object that
converts the inequality into an identity.  Saturation is the
information-theoretic face of the no-hair property: nothing about
the hole is wasted on internal structure a boundary observer
could have resolved.

\medskip\noindent\textbf{G.3.2. How far below the bound does a
real storage device sit?}

\noindent\emph{Solution.}  Step 1: for a 1-kg, 10-cm device the
bound allows
\begin{equation*}
\frac{S_{\max}}{k_B\ln 2} =
\frac{2\pi\,(1\,\mathrm{kg})\,c^2\,(0.1\,\mathrm{m})}
{\hbar c \ln 2} \approx 2.6\times10^{42}\;\text{bits}.
\end{equation*}  Step 2: a contemporary solid-state drive of that size
stores $\sim\!10^{13}$ bits.  Step 3: the gap is twenty-nine
orders of magnitude --- and even the device's fundamental
entropy content ($\sim\!10^{25}$ atomic degrees of freedom) sits
seventeen orders below the ceiling.  Physics intuition: ordinary
matter stores information in atoms; the bound prices information
in \emph{gravity's} currency, and the exchange rate
($\sim\!10^{19}$~GeV vs.\ eV chemistry) is the whole gap.  Only
when self-gravity takes over does storage approach the ledger's
limit --- and the object that achieves it is, by G.3.1, a black
hole.

\medskip\noindent\textbf{G.4.1. Verify the GSL for pure Hawking
emission: show the radiation gains more entropy than the hole
loses.}

\noindent\emph{Solution.}  Step 1: the hole loses
$dS_{\rm BH} = dE/T_{\rm H}$ (first law, $dE < 0$).  Step 2: the
radiation is (nearly) blackbody at $T_{\rm H}$: free-streaming
photons carry entropy flux $\tfrac{4}{3}\,(aT^3)$ against energy
flux $aT^4$, so $dS_{\rm rad} = \tfrac{4}{3}\,|dE|/T_{\rm H}$.
Step 3: the balance is
$dS_{\rm gen} = \bigl(\tfrac{4}{3} - 1\bigr)|dE|/T_{\rm H}
= \tfrac{1}{3}\,|dE|/T_{\rm H} > 0$: the generalized second law
\eqref{eq:gsl-course} holds with a strict margin (graybody
filtering shifts the $4/3$ at the percent level but never below
$1$).  Physics intuition: evaporation is not the hole ``leaking
order'' --- it is the conversion of one large, perfectly packed
ledger page into many small thermal ones, at a fixed markup of
$\tfrac13$; the exact exchange-class version of this bookkeeping
is Appendix~\ref{app:data-processing}, Step 4.

\GGtakeaway{\emph{Standard result:} $S_{\rm BH} = \eta\,k_B
A/\ell_P^2$ on consistency grounds, the Bekenstein bound
\eqref{eq:bekenstein-bound}, and the GSL \eqref{eq:gsl-course}.
\emph{What it assumed:} the second law is worth saving, and
dimensional analysis.  \emph{What it could not supply:} the
coefficient, the microstates, and a mechanism for area-scaling.
\emph{The \GGtheory{} reading:} coefficient from the conical
replica, microstates from the fiber holonomies, area-scaling from
port bandwidth --- with the species sum left open, honestly, as
Appendix~\ref{app:species-sum}.}

\section{Explicit derivation of Hawking temperature in
\GDOS{}}
\label{app:Hawking_T}

\GGteachbox{The Hawking temperature derived from first principles
with the full Bogoliubov machinery: quantum field theory on a
fixed curved background set up from scratch, the Rindler warm-up
solved exactly, the two routes to the surface gravity, the
restoration of units, and the \GDOS{} reading in which the
continuation to the Euclidean section stops being a trick and
becomes a licensed theorem.}
\GGreadalong{\citet{BirrellDavies1982} chs.~3 and~8 (the
canonical treatment this Step follows most closely);
\citet{Hawking1975} and \citet{Unruh1976} for the originals;
\citet{ParkerToms2009} at depth.}

This chapter is the course's first worked calculation: the Hawking
temperature, derived from first principles with every intermediate
manipulation displayed.  The main text asserted the result three
times over --- the Bogoliubov ratio of \S\ref{ssec:bogoliubov}, the
clock-face theorem T-BH-5 (\S\ref{sec:thermal-clock}), and the
Euclidean period (Appendix~\ref{app:euclidean}); here a graduate
student meets the machinery those assertions run on.  The plan:
Step~1 sets up quantum field theory on a fixed curved background
from scratch; Step~2 builds the Bogoliubov machinery; Step~3 works
the flat-space warm-up with an exact answer (the Unruh effect);
Step~4 transfers it to the black hole by two independent routes;
Step~5 restores units and evaluates the numbers; Step~6 states what
\GDOS{} adds.  Nothing here is \GGtheory{}-specific until Step~6:
Steps~1--5 are the standard material \citep{Hawking1975, Unruh1976,
Wald1984}, worked in the conventions of this paper.

\subsection{Step 1 --- the stage: a scalar field on a fixed curved
background}

Take a real scalar field on a globally hyperbolic spacetime
$(\mathcal{M}, g_{\mu\nu})$ with signature $(-,+,+,+)$, action
\begin{equation}
S[\phi] = -\frac12 \int d^4x \sqrt{-g}\,
\bigl(g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
+ m^2\phi^2 + \xi R\,\phi^2\bigr),
\end{equation}
whose variation gives the curved-space Klein--Gordon equation
\begin{equation}
\bigl(\Box - m^2 - \xi R\bigr)\phi = 0, \qquad
\Box\phi \equiv \frac{1}{\sqrt{-g}}\,\partial_\mu
\bigl(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi\bigr).
\end{equation}
On any Cauchy surface $\Sigma$ with future-directed unit normal
$n^\mu$ and induced volume element $\sqrt{h}\,d^3x$, define the
Klein--Gordon inner product
\begin{equation}\label{eq:B-KGip}
(u, v) \;=\; -\,i \int_\Sigma d^3x \sqrt{h}\; n^\mu
\bigl(u\,\partial_\mu v^* - v^*\partial_\mu u\bigr).
\end{equation}
Two properties carry the whole subject and are worth verifying once
by hand: \emph{(a)} for solutions of the field equation,
\eqref{eq:B-KGip} is independent of the choice of $\Sigma$ (the
current $j^\mu = -i(u\,\partial^\mu v^* - v^*\partial^\mu u)$ is
conserved on shell, $\nabla_\mu j^\mu = 0$, by the field equation
applied twice); \emph{(b)} it is not positive definite:
$(u^*, u^*) = -(u,u)$.  Property (b) is the origin of everything
that follows --- the space of solutions splits into a
positive-norm half and its conjugate, and \emph{which} split is
chosen is precisely the choice of vacuum.

Pick any complete set of positive-norm solutions
$\{u_i\}$ with $(u_i, u_j) = \delta_{ij}$,
$(u_i^*, u_j^*) = -\delta_{ij}$, $(u_i, u_j^*) = 0$, and expand
\begin{equation}\label{eq:B-expansion}
\phi(x) = \sum_i \bigl(a_i\, u_i(x) + a_i^\dagger\, u_i^*(x)\bigr).
\end{equation}
The canonical commutation relations for $\phi$ are then equivalent
to $[a_i, a_j^\dagger] = \delta_{ij}$, and the vacuum belonging to
this basis is $a_i |0_u\rangle = 0$ for all $i$.  In Minkowski
space the modes $u_{\mathbf{k}} \propto e^{i\mathbf{k}\cdot
\mathbf{x} - i\omega_k t}$, positive frequency with respect to
inertial time, are selected by Poincar\'e invariance and every
inertial observer agrees.  On a general curved background there is
no preferred time, hence no preferred split, hence no preferred
particle notion --- only preferred \emph{observers} can select one.

\paragraph{Physical picture.}  A vacuum is not an empty box; it is
a bookkeeping convention tied to a clock.  ``Positive frequency''
means ``rotating one way with respect to \emph{this} time
parameter.''  Flat space owns a global inertial clock, so all
inertial bookkeepers agree on what counts as a particle.  A curved
spacetime in general owns several inequivalent clocks --- and the
whole content of horizon thermodynamics is what happens when two
lawful bookkeepers compare ledgers.  In the cohort's language the
split is the derived complex structure of the causal clock
(\citealp{ArisakaP4Real}); T-BH-5 records when that selection is
unique.

\subsection{Step 2 --- the Bogoliubov machinery}

Let $\{v_j\}$ be a second complete positive-norm basis --- another
lawful bookkeeper.  Completeness of $\{u_i, u_i^*\}$ lets us expand
\begin{equation}\label{eq:B-bogodef}
v_j = \sum_i \bigl(\alpha_{ji}\, u_i + \beta_{ji}\, u_i^*\bigr),
\qquad
\alpha_{ji} = (v_j, u_i), \qquad \beta_{ji} = -(v_j, u_i^*),
\end{equation}
where the coefficient formulas follow by taking the inner product
of the expansion with $u_i$ and $u_i^*$ and using the
orthonormality relations of Step~1.  Substituting
\eqref{eq:B-bogodef} into the field expansion and matching
coefficients of $u_i$ and $u_i^*$ gives the operator
transformation
\begin{equation}\label{eq:B-bogop}
b_j = \sum_i \bigl(\alpha_{ji}^*\, a_i - \beta_{ji}^*\,
a_i^\dagger\bigr),
\end{equation}
and demanding $[b_j, b_k^\dagger] = \delta_{jk}$ yields the
consistency conditions
\begin{equation}\label{eq:B-bogocond}
\sum_i \bigl(\alpha_{ji}\alpha_{ki}^* - \beta_{ji}\beta_{ki}^*\bigr)
= \delta_{jk},
\qquad
\sum_i \bigl(\alpha_{ji}\beta_{ki} - \beta_{ji}\alpha_{ki}\bigr) = 0.
\end{equation}
The payoff line of the whole subject is one trace: the number of
$v$-quanta in the $u$-vacuum is
\begin{equation}\label{eq:B-Nbeta}
\langle 0_u|\, b_j^\dagger b_j \,|0_u\rangle
= \sum_i |\beta_{ji}|^2 .
\end{equation}
\emph{Derivation:} insert \eqref{eq:B-bogop}; the $aa$ and
$a^\dagger a^\dagger$ terms annihilate the vacuum on one side or
the other; the surviving term is $\sum_{i,i'}\beta_{ji}
\beta_{ji'}^* \langle 0_u| a_i a_{i'}^\dagger |0_u\rangle =
\sum_i |\beta_{ji}|^2$.

\paragraph{Physical picture.}  $\beta \neq 0$ means one
bookkeeper's annihilation operator contains the other's creation
operator: the two ledgers do not just relabel the particles, they
disagree about how many there are.  Particle number is
dictionary data, not bulk data --- the same custody division the
cohort enforces everywhere (the dictionary owns the sectors), met
here in its oldest habitat.

\subsection{Step 3 --- the warm-up with an exact answer: the
accelerated observer}

Everything essential happens already in flat spacetime.  Work in
$1{+}1$ dimensions (the transverse directions are spectators) with
null coordinates $U = t - x$, $V = t + x$, and consider the right
Rindler wedge $x > |t|$, the region accessible to a uniformly
accelerated observer with proper acceleration $a$.  Rindler
coordinates $(\eta, \chi)$,
\begin{equation}
U = -\frac{1}{a}\,e^{-a(\eta - \chi)}, \qquad
V = \frac{1}{a}\,e^{\,a(\eta + \chi)},
\end{equation}
cover the wedge, and the boost Killing vector $\partial_\eta$ is
the accelerated observer's time.  Outgoing right-wedge modes,
positive frequency in $\eta$, depend on the single combination
$u_{\rm R} \equiv \eta - \chi = -\tfrac{1}{a}\ln(-aU)$ (valid for
$U<0$):
\begin{equation}\label{eq:B-rindlermode}
u^{\rm R}_\omega \;\propto\; e^{-i\omega u_{\rm R}}
= (-aU)^{\,i\omega/a}, \qquad U < 0,
\end{equation}
with a mirror family $u^{\rm L}_\omega \propto (aU)^{-i\omega/a}$
supported at $U > 0$ (the left wedge).  Each family vanishes on
half the $U$ axis, so neither is analytic at $U = 0$.

Now the decisive criterion.  A mode is positive frequency with
respect to \emph{inertial} time iff it is a superposition of
$e^{-i\Omega U}$ with $\Omega > 0$ only --- equivalently, iff it is
analytic and bounded in the \emph{lower half} of the complex $U$
plane (close the Fourier contour there).  Continue
\eqref{eq:B-rindlermode} from $U<0$ to $U>0$ through the lower
half-plane: along that path $\arg(-U)$ runs from $0$ to $-\pi$, so
\begin{equation}\label{eq:B-continuation}
(-aU)^{\,i\omega/a}
\;\longrightarrow\;
\bigl(e^{-i\pi}\,aU\bigr)^{\,i\omega/a}
= e^{\,\pi\omega/a}\,(aU)^{\,i\omega/a}
= e^{\,\pi\omega/a}\,\bigl(u^{\rm L}_\omega\bigr)^{*}.
\end{equation}
The factor $e^{\pi\omega/a}$ is the entire effect: continuation
through the horizon point $U=0$ costs one exponential.  Therefore
the two combinations
\begin{equation}\label{eq:B-unruhmodes}
w^{(1)}_\omega = \frac{e^{\pi\omega/2a}\,u^{\rm R}_\omega
+ e^{-\pi\omega/2a}\,u^{{\rm L}*}_\omega}
{\sqrt{2\sinh(\pi\omega/a)}},
\qquad
w^{(2)}_\omega = \frac{e^{\pi\omega/2a}\,u^{\rm L}_\omega
+ e^{-\pi\omega/2a}\,u^{{\rm R}*}_\omega}
{\sqrt{2\sinh(\pi\omega/a)}}
\end{equation}
are analytic in the lower half-plane --- hence purely positive
frequency in the inertial sense --- and correctly normalized (the
denominators follow from $e^{\pi\omega/a} - e^{-\pi\omega/a} =
2\sinh(\pi\omega/a)$ and the opposite norms of a mode and its
conjugate).  Reading \eqref{eq:B-unruhmodes} against the template
\eqref{eq:B-bogodef}: the Bogoliubov coefficients connecting the
Rindler bookkeeper to the inertial one are diagonal in $\omega$
with
\begin{equation}\label{eq:B-unruhratio}
\alpha_\omega = \frac{e^{\pi\omega/2a}}
{\sqrt{2\sinh(\pi\omega/a)}},
\qquad
\beta_\omega = \frac{e^{-\pi\omega/2a}}
{\sqrt{2\sinh(\pi\omega/a)}},
\qquad
\Bigl|\frac{\beta_\omega}{\alpha_\omega}\Bigr| = e^{-\pi\omega/a},
\end{equation}
and $|\alpha_\omega|^2 - |\beta_\omega|^2 = 1$ checks the
consistency condition \eqref{eq:B-bogocond}.  The occupation seen
by the accelerated bookkeeper in the inertial vacuum is then, by
\eqref{eq:B-Nbeta},
\begin{equation}\label{eq:B-unruhN}
\langle N_\omega \rangle = |\beta_\omega|^2
= \frac{e^{-\pi\omega/a}}{2\sinh(\pi\omega/a)}
= \frac{1}{e^{2\pi\omega/a} - 1}
\;=\; \frac{1}{e^{\hbar\omega/\kB T_{\rm U}} - 1},
\qquad
\kB T_{\rm U} = \frac{\hbar a}{2\pi c},
\end{equation}
an exact Planck distribution at the Unruh temperature
\citep{Unruh1976, DaviesFulling1976}.  No approximation was made:
the thermality is an identity of the analytic continuation.

\paragraph{Physical picture.}  The accelerated observer's horizon
at $U = 0$ cuts the Minkowski vacuum in half, and the vacuum's
correlations do not respect the cut: each right-wedge mode is
entangled with a left-wedge partner (the pairing exhibited by
\eqref{eq:B-unruhmodes}).  An observer confined to one side sees
her half of an entangled pure state --- which is a mixed state, and
the analytic structure forces the mixture to be exactly thermal.
Acceleration does not \emph{heat} the vacuum; it \emph{hides half
of it}, and the hidden half's ledger entry is a temperature.  This
is the \OGGEP{} partial trace in its simplest costume.

\subsection{Step 4 --- from Rindler to the black hole: two routes
to \texorpdfstring{$\kappa$}{kappa}}

In \GGtheory{} the object carrying the horizon is the 6D bulk; near
the horizon it looks like
\begin{equation}\label{eq:near_horizon_6D}
ds_6^2 \approx -f(r_4) dt^2 + dr_4^2/f(r_4) + r_4^2 d\Omega^2
+ dr^2 + d\theta^2,
\end{equation}
for a Schwarzschild 4D metric with $f(r_4) = 1 - r_s/r_4$: the
fiber directions $(r, \theta)$ ride along as spectators, exactly
like the transverse directions of Step~3, so the $1{+}1$ analysis
applies verbatim to each 4D mode.  Two independent routes now
deliver the surface gravity as the acceleration parameter.

\paragraph{Route (a): the near-horizon Rindler frame.}  In
geometrized units ($G = c = 1$), expand $f$ about the horizon
radius $r_s$:
\begin{equation}
f(r_4) = f'(r_s)\,(r_4 - r_s) + O\bigl((r_4-r_s)^2\bigr)
= 2\kappa\,(r_4 - r_s),
\qquad
\kappa \equiv \tfrac12 f'(r_s) = \frac{1}{2 r_s},
\end{equation}
using $f'(r_4) = r_s/r_4^2$.  Trade $r_4$ for the proper distance
from the horizon,
\begin{equation}
x = \int_{r_s}^{r_4} \frac{dr'}{\sqrt{f(r')}}
= \int_{r_s}^{r_4} \frac{dr'}{\sqrt{2\kappa (r' - r_s)}}
= \sqrt{\frac{2(r_4 - r_s)}{\kappa}}
\quad\Longrightarrow\quad
f = \kappa^2 x^2 ,
\end{equation}
so the $(t, x)$ block of the metric is $ds^2 = -\kappa^2 x^2 dt^2
+ dx^2$: precisely Rindler, with the static observer at proper
distance $x$ accelerating at $a(x) = 1/x$.  By Step~3 she measures
the local temperature $T(x) = \hbar/(2\pi x)$ --- divergent at the
horizon, as an infinite acceleration demands --- and the Tolman
redshift carries it out: the asymptotic observer sees
\begin{equation}
T_\infty = \sqrt{-g_{tt}}\; T(x) = (\kappa x)\cdot
\frac{\hbar}{2\pi x} = \frac{\hbar \kappa}{2\pi},
\end{equation}
with the $x$-dependence canceling identically.  The horizon
temperature is the redshift-invariant content of the diverging
local Unruh effect.

\paragraph{Route (b): the collapse ray-tracing.}  Transforming to
Kruskal-like coordinates $U = -e^{-\kappa u}$, $V = e^{\kappa v}$
with $\kappa = 1/(2 r_s)$:
\begin{equation}\label{eq:Kruskal_transform}
UV = -e^{r_4/r_s}(r_4/r_s - 1),\qquad
ds^2\big|_{t,r_4} = -\,\frac{4 r_s^{3}}{r_4}\,e^{-r_4/r_s}\,dU\,dV ,
\end{equation}
the second relation the regular Kruskal form of the $(t, r_4)$
block, its prefactor finite and nonzero at $r_4 = r_s$.
The near-horizon metric in $UV$ coordinates is \emph{regular} at
the horizon (the Finkelstein observation), so the state regular
there --- the one a collapse actually prepares at late times --- is
the one positive-frequency in $U$.  The exterior bookkeeper uses
$u = -\tfrac{1}{\kappa}\ln(-U)$, so her outgoing modes are
$e^{-i\omega u} = (-U)^{\,i\omega/\kappa}$ --- literally
\eqref{eq:B-rindlermode} with $a \to \kappa$ (the constant rescale
of $U$ between the two conventions shifts a phase, not the
continuation factor).  The continuation
argument \eqref{eq:B-continuation} then transfers word for word
and yields the ratio of the main text,
$|\beta_\omega/\alpha_\omega| = e^{-\pi\omega/\kappa}$
(Eq.~\eqref{eq:planck}), hence the Planck occupation at
\begin{equation}
\kB T_{\rm H} = \frac{\hbar\kappa}{2\pi}.
\end{equation}
This is Hawking's derivation \citep{Hawking1975} in its analytic
kernel: the collapse's role is only to prepare regularity at the
horizon, and the exponential map $u = -\tfrac{1}{\kappa}\ln(-U)$
between the two bookkeepers' clocks does the rest.  What the
exterior curvature adds is the graybody filter
$\Gamma_\ell(\omega)$ of Eq.~\eqref{eq:greybody}; what rotation
adds is the anchor selection of Theorem~\ref{thm:rotating-anchor};
what collapse transients add is quantified by
Theorem~\ref{thm:collapse-channel}.

\paragraph{Physical picture.}  A horizon is an exponential
magnifier.  Outgoing rays that escape at late exterior time $u$
hugged the horizon at Kruskal parameter $U \propto -e^{-\kappa u}$:
equal steps in the observer's clock correspond to exponentially
shrinking steps of the regular clock.  A logarithmic lens applied
to a smooth state produces a scale-invariant exponential ratio
between absorption and emission --- one number, $\kappa$ --- and a
spectrum that can only be Planck.  Thermality is not a property of
what fell in; it is the signature of the lens.

\subsection{Step 5 --- restoring units, and the numbers}

Restoring $\hbar$, $c$, $\kB$, and $\Newton$: the geometric surface
gravity $\kappa = 1/(2r_s)$ (an inverse length, $r_s = 2\Newton
M/c^2$) gives
\begin{equation}\label{eq:Hawking_T_app}
\THaw = \frac{\hbar\kappa}{2\pi\kB c}.
\end{equation}
in the acceleration convention $\kappa_{\rm acc} = c^2\kappa_{\rm
geom}$ used throughout this paper (equivalently $\THaw = \hbar c\,
\kappa_{\rm geom}/2\pi\kB$).  Explicitly,
\begin{equation}\label{eq:B-THnumbers}
\THaw = \frac{\hbar c^3}{8\pi \Newton M \kB}
\approx 6.17\times 10^{-8}\,{\rm K}\;
\Bigl(\frac{\Msun}{M}\Bigr),
\end{equation}
the reference-card value (Appendix~\ref{app:BH_numbers}; verified
in \texttt{p2c2\_verify}).  The inverse scaling with mass is the
signature fact of black-hole thermodynamics: M87* at $6.5\times
10^9 \Msun$ sits at $\sim 10^{-17}$~K, colder than anything else
in physics, while a $10^{12}$~kg primordial hole would radiate at
$\sim 10^{11}$~K.  Big holes are cold, small holes are furnaces,
and the negative specific heat this implies is derived honestly in
Appendix~\ref{app:euclidean}, Step~3.

\subsection{Step 6 --- what \GDOS{} adds: the custody reading}

The \OGGEP{} projection to the exterior region preserves the
thermal structure Steps~3--4 derived: the output
$\rho_\partial^{\rm out}$ is thermal at $\THaw$, because the trace
over the hidden partners is exactly the operation that turned the
squeezed pair state into a Planck mixture
(Eq.~\eqref{eq:thermal-trace}).  The derivation used only the bulk
geometry and the \OGGEP{} partial trace; it did not postulate
Hawking's specific mechanism.  Nor is the analytic continuation of
Step~3 --- the single non-algebraic move in the whole chapter --- a
prescription: its existence is the Wick-rotation clause of the
cohort's Wick-quartet theorem (\textsf{T-P5-B1},
\citealp{ArisakaP5CPT}), consumed at the horizon as
Theorem~\ref{thm:thermal-clock}; the geometry contributes only the
period $2\pi/\kappa$.  The custody division is exact: the
dictionary owns which modes count as particles and which partners
are hidden (port data); the geometry owns the clock whose one-sided
spectrum licenses the continuation and whose surface gravity sets
its cost.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{H.2.1. Prove the single-mode consistency
condition $|\alpha|^2 - |\beta|^2 = 1$ directly from the
commutator.}

\noindent\emph{Solution.}  Step 1: for one mode,
$b = \alpha^* a - \beta^* a^\dagger$
(the single-mode case of \eqref{eq:B-bogodef}).  Step 2: impose
$[b, b^\dagger] = 1$:
$[b, b^\dagger] = |\alpha|^2[a, a^\dagger] +
|\beta|^2[a^\dagger, a] = |\alpha|^2 - |\beta|^2$.  Step 3: hence
$|\alpha|^2 - |\beta|^2 = 1$ --- the one-mode face of
\eqref{eq:B-bogocond}, and the algebraic reason the Planck
occupation \eqref{eq:B-Nbeta} can be written
$1/(e^{2\pi\omega/\kappa\cdot} - 1)$: the ratio
$|\beta/\alpha| = e^{-\pi\omega/\kappa}$ plus this normalization
fixes both moduli.  Physics intuition: the condition is a
hyperbolic identity, $\cosh^2 - \sinh^2 = 1$ --- a Bogoliubov
transformation is a boost in the $(a, a^\dagger)$ plane, and
thermality is what a boost does to a vacuum.

\medskip\noindent\textbf{H.3.1. Evaluate the Unruh temperature
for one Earth gravity, and the acceleration required to feel
room temperature.}

\noindent\emph{Solution.}  Step 1: $k_B T_{\rm U} = \hbar a/(2\pi
c)$, so $T_{\rm U}(g) = \hbar g/(2\pi c k_B) \approx
4.0\times10^{-20}$~K.  Step 2: inverting for $T = 300$~K gives
$a = 2\pi c k_B T/\hbar \approx 7.4\times10^{22}\;\mathrm{m\,
s^{-2}}$ --- about $10^{21}$ Earth gravities, the surface gravity
of nothing tamer than a horizon.  Step 3: this is why the Unruh
effect has never been seen directly, and why the black hole is
its one cosmic amplifier: at a horizon the \emph{geometry}
supplies the acceleration budget no laboratory can.
Physics intuition: $\hbar/(2\pi c k_B)$ is nature's exchange rate
between acceleration and temperature, and it is minuscule ---
thermality from motion is real but priced beyond any engine
except gravity's.

\medskip\noindent\textbf{H.4.1. Show that the constant rescaling
of the Kruskal coordinate between the two routes shifts no
physics.}

\noindent\emph{Solution.}  Step 1: Route (b) uses
$U = -e^{-\kappa u}$; any alternative affine choice is
$U' = \lambda U$ with $\lambda > 0$ constant.  Step 2: the
continuation argument of \eqref{eq:B-continuation} tracks
$(-U)^{i\omega/\kappa} \to e^{\pi\omega/\kappa}(U)^{i\omega/
\kappa}$; under $U \to \lambda U$ both sides acquire the same
factor $\lambda^{i\omega/\kappa}$ --- a constant \emph{phase} on
the mode, which cancels in $|\beta_\omega/\alpha_\omega|$.
Step 3: hence the ratio $e^{-\pi\omega/\kappa}$, and with it
$T_{\rm H}$, is invariant --- only the branch-crossing
exponential is physical.  Physics intuition: the temperature is
not stored in any coordinate convention but in the \emph{one}
analytic fact --- crossing the horizon zero of $U$ costs
$e^{\pi\omega/\kappa}$ --- which is why three superficially
different derivations (Appendices \ref{app:Hawking_T} and
\ref{app:euclidean}, and \S\ref{sec:thermal-clock}) keep landing
on the same number.

\medskip\noindent\textbf{H.5.1. Find the mass at which a black
hole is in equilibrium with today's cosmic microwave background,
and classify the population.}

\noindent\emph{Solution.}  Step 1: set
$T_{\rm H} = \hbar c^3/(8\pi \Newton M k_B) = 2.725$~K.
Step 2: solve:
$M = 4.5\times10^{22}\;\mathrm{kg} \approx 0.61$ lunar masses
($2.3\times10^{-8}\,\Msun$).  Step 3: every astrophysical hole
($M \gtrsim \Msun$) is therefore \emph{colder} than the sky it
sits in and absorbs net energy today; only primordial holes below
lunar mass --- none yet observed --- would be net emitters.  The
crossover drifts upward as the universe cools.
Physics intuition: evaporation is real but \emph{scheduled}: the
universe must first cool below each hole's temperature before its
long shrinkage can begin --- thermodynamics queues the black
holes behind the expansion.

\medskip\noindent\textbf{H.6.1. Show that the Planck occupation
and the branch-cut ratio are the same statement (detailed
balance).}

\noindent\emph{Solution.}  Step 1: from
$|\beta_\omega/\alpha_\omega| = e^{-\pi\omega/\kappa}$ and
H.2.1, $|\beta|^2 = 1/(e^{2\pi\omega/\kappa} - 1)$: the ratio
implies Planck.  Step 2: conversely, a KMS (thermal) state at
temperature $T$ satisfies detailed balance --- emission and
absorption amplitudes related by $e^{-\hbar\omega/k_BT}$; reading
$|\beta/\alpha|^2 = e^{-2\pi\omega/\kappa}$ as that Boltzmann
factor identifies $k_BT = \hbar\kappa/2\pi c$: Planck implies the
ratio.  Step 3: the equivalence is the content of T-BH-4/T-BH-5
--- the branch-cut asymmetry \emph{is} the thermal face, one
datum wearing two names.  Physics intuition: nothing ``heats
up'' at a horizon; a one-sided exponential map re-weights
outgoing modes exactly as a Boltzmann factor would, and a
detector cannot tell the difference --- which is the paper's
whole thermality thesis in one mode.

\GGtakeaway{\emph{Standard result:} $k_B \THaw =
\hbar\kappa/(2\pi c)$, from Bogoliubov coefficients on the
collapse background.  \emph{What it assumed:} free fields on a
fixed classical geometry, and a positive-frequency continuation
prescription.  \emph{The \GGtheory{} reading:} the prescription
is licensed by the one-sided spectrum of the port clock
(\textsf{T-P5-B1}, Theorem~\ref{thm:thermal-clock}); the
geometry contributes only the period $2\pi/\kappa$; and the
thermality is the KMS face of an honest \CPTP{} channel, not
evidence of information loss.}

\section{Euclidean derivation of temperature and entropy}
\label{app:euclidean}

\GGteachbox{The Euclidean route to the same temperature and to
the entropy: the cigar geometry and its regularity period, the
on-shell action computed term by term including the
Gibbons--Hawking boundary term, the saddle-point thermodynamics
--- and the \GGtheory{} reading in which both formal moves of
1977, the continuation and the arena, carry licenses.}
\GGreadalong{\citet{GibbonsHawking1977} and
\citet{GibbonsHawkingAction1977} for the originals;
\citet{WaldQFT1994} for the careful modern statement.}

The Euclidean route computes the same temperature from global
regularity, and adds the entropy from the saddle-point free energy.
Historically both moves were formal prescriptions; in this paper each is
licensed --- the continuation by \textsf{T-P5-B1} (consumed as
Theorem~\ref{thm:thermal-clock}), the arena by the analytic-shore
clause of \S\ref{ssec:V-conical}.  This chapter works the
computation a graduate student should reproduce line by line:
Step~1 the cigar and the period; Step~2 the on-shell action, term
by term; Step~3 the thermodynamics of the saddle; Step~4 the
\GGtheory{} reading.  Geometrized units $G = c = 1$ (with $G$
restored where it earns its keep); $\hbar = \kB = 1$ inside the
saddle algebra.

\subsection{Step 1 --- the cigar and the period}

From the Euclidean Schwarzschild metric ($t \to -i\tau$)
\begin{equation}
ds_E^2 = \Bigl(1-\tfrac{2GM}{r}\Bigr)d\tau^2 + \Bigl(1-\tfrac{2GM}{r}\Bigr)^{-1}dr^2 + r^2 d\Omega_2^2,
\end{equation}
zoom in on the horizon $r_s = 2GM$.  Write $r = r_s + \rho$ with
$\rho \ll r_s$, so $f = 1 - r_s/r = \rho/(r_s + \rho) \simeq
\rho/r_s$.  Substitute $\rho = x^2/(4 r_s)$, i.e.\ $x =
2\sqrt{r_s\rho}$, and work out both metric blocks:
\begin{equation}
\frac{dr^2}{f} = \frac{d\rho^2}{\rho/r_s}
= \frac{r_s}{\rho}\Bigl(\frac{x\,dx}{2r_s}\Bigr)^{2}
= \frac{4 r_s^2}{x^{2}}\cdot\frac{x^{2}\,dx^{2}}{4 r_s^{2}} = dx^2,
\qquad
f\,d\tau^2 = \frac{x^2}{4 r_s^2}\,d\tau^2 ,
\end{equation}
so that near the horizon
\begin{equation}
ds_E^2 \simeq dx^2 + x^{2}\, d\Bigl(\frac{\tau}{2 r_s}\Bigr)^{2}
+ r_s^2\, d\Omega_2^2 :
\end{equation}
a plane in polar coordinates $(x, \tau/2r_s)$ times the horizon
sphere.  The plane is smooth at $x = 0$ if and only if the angle
$\tau/2r_s$ has period $2\pi$ --- otherwise the tip carries a
conical singularity, which the vacuum saddle is not allowed to
have.  Hence
\begin{equation}\label{eq:C-period}
\tau \sim \tau + \beta, \qquad \beta = 4\pi r_s = 8\pi G M
\quad\Longrightarrow\quad T_{\rm H} = \frac{1}{8\pi G M}
= \frac{\kappa}{2\pi},
\end{equation}
in exact agreement with Appendix~\ref{app:Hawking_T}, Step~4 (with
$\kappa = 1/2r_s = 1/4GM$).  The same tip, \emph{deliberately}
given a deficit angle, is the replica geometry of the main text's
entropy derivation (\S\ref{ssec:V-conical}) --- regularity is the
$n = 1$ member of the replica family.

\paragraph{Physical picture.}  Periodic imaginary time is the
field-theory encoding of a temperature ($\mathrm{Tr}\,
e^{-\beta H}$), and the geometry votes on the period: only one
$\beta$ makes the Euclidean section a smooth cigar rather than a
cone.  The black hole does not sit \emph{in} a heat bath at
$T_{\rm H}$; its own geometry \emph{is} the statement that
$T_{\rm H}$ is the only temperature it can be in equilibrium at.

\subsection{Step 2 --- the on-shell action, term by term}

The saddle's free energy needs the Euclidean action
\begin{equation}
I_E = -\frac{1}{16\pi G}\int_{\mathcal{M}} d^4x \sqrt{g}\, R
\;-\; \frac{1}{8\pi G}\oint_{\partial\mathcal{M}} d^3x
\sqrt{h}\,\bigl(K - K_0\bigr),
\end{equation}
the second term the Gibbons--Hawking--York boundary term with the
flat-space subtraction $K_0$ \citep{GibbonsHawkingAction1977}.  For
Schwarzschild $R = 0$: the entire on-shell action is boundary.
Evaluate on the surface $r = R$ (to be sent to infinity), with
induced metric $h = \mathrm{diag}(f, r^2, r^2\sin^2\theta)|_{r=R}$
and outward unit normal $n = \sqrt{f}\,\partial_r$.

\emph{The trace of the extrinsic curvature.}  For a metric of this
form, $K = \nabla_\mu n^\mu = \frac{1}{\sqrt{g}}\partial_r(\sqrt{g}
\,\sqrt{f})$ with $\sqrt{g} = r^2\sin\theta$, so
\begin{equation}
K = \frac{1}{r^2}\,\partial_r\bigl(r^2 \sqrt{f}\bigr)
= \frac{f'}{2\sqrt{f}} + \frac{2\sqrt{f}}{r}.
\end{equation}
\emph{The integral.}  The boundary volume is $\sqrt{h} =
\sqrt{f}\,R^2 \sin\theta$ with $\tau$-length $\beta$, so
\begin{equation}
\oint \sqrt{h}\, K = \beta\, 4\pi R^2 \sqrt{f}\,
\Bigl(\frac{f'}{2\sqrt{f}} + \frac{2\sqrt{f}}{R}\Bigr)
= 4\pi\beta \Bigl(\frac{R^2 f'}{2} + 2 R f\Bigr)
= 4\pi\beta\,\bigl(2R - 3GM\bigr),
\end{equation}
using $f = 1 - 2GM/R$, $f' = 2GM/R^2$, so $R^2 f'/2 = GM$ and
$2Rf = 2R - 4GM$.  \emph{The subtraction.}  Flat space at the same
boundary has $K_0 = 2/R$, but its time circle must be matched to
the same \emph{proper} length, $\beta_0 = \beta\sqrt{f(R)}$:
\begin{equation}
\oint \sqrt{h}\, K_0 = \beta\sqrt{f}\cdot 4\pi R^2 \cdot
\frac{2}{R} = 8\pi\beta R\sqrt{f}
= 4\pi\beta\,\bigl(2R - 2GM\bigr) + O(1/R),
\end{equation}
expanding $\sqrt{f} = 1 - GM/R + O(R^{-2})$.  Subtracting and
taking $R \to \infty$:
\begin{equation}\label{eq:C-action}
I_E = -\frac{4\pi\beta}{8\pi G}\Bigl[(2R - 3GM) - (2R - 2GM)\Bigr]
= \frac{\beta M}{2}
= \frac{\beta^2}{16\pi G},
\end{equation}
the last form using $M = \beta/8\pi G$ from \eqref{eq:C-period}.
The divergent $2R$ pieces canceled exactly --- that is what the
subtraction is for --- and the finite remainder is the whole
answer.

\subsection{Step 3 --- the thermodynamics of the saddle}

With $Z(\beta) \simeq e^{-I_E}$ at the saddle, every
thermodynamic quantity follows by differentiation, and each line
is a consistency check:
\begin{align}
E &= \partial_\beta I_E = \frac{\beta}{8\pi G} = M
&&\text{(the saddle's energy is the mass: check),}
\nonumber\\
F &= \frac{I_E}{\beta} = \frac{M}{2},
\qquad
S = \beta E - I_E = \frac{\beta^2}{16\pi G} = 4\pi G M^2 .
\label{eq:C-thermo}
\end{align}
Since $A = 4\pi r_s^2 = 16\pi G^2 M^2$, the entropy line reads
\begin{equation}
S_{\rm BH} = 4\pi G M^{2} = \frac{16\pi G^{2} M^{2}}{4G}
= \frac{A}{4 G_N},
\end{equation}
and the first law closes: $dS/dM = 8\pi G M = \beta = 1/T$.  One
more derivative is physics: $\partial T/\partial M = -1/(8\pi G
M^2) < 0$, so the specific heat is
\begin{equation}
C = \frac{\partial M}{\partial T} = -\,8\pi G M^{2} \;<\; 0 :
\end{equation}
the asymptotically flat black hole heats up as it loses mass.  A
negative-specific-heat system cannot equilibrate with an infinite
bath --- the runaway that makes evaporation accelerate
(Appendix~\ref{app:Page_curve}, Step~3) and that the AdS box of
Appendix~\ref{app:hawking-page} exists to tame.

\paragraph{Physical picture.}  Self-gravitating systems generically
carry negative specific heat (a satellite dropped to a lower orbit
speeds up), and the saddle knows it: the free energy is dominated
by a configuration whose temperature falls as its energy grows.
The Euclidean computation is thermodynamics done \emph{by
geometry} --- the partition function evaluated not over microstates
but over the one smooth classical shape the boundary conditions
admit.

\subsection{Step 4 --- the \GGtheory{} reading}

In the standard reading, \eqref{eq:C-thermo} is a thermodynamic
consequence of the saddle.  In the \GGtheory{} reading the same
computation supplies the geometric piece $F_{\rm GEP}$ of the
operational ledger \eqref{eq:gff-bh}: the Euclidean trick does not
compete with \GGtheory{}, it feeds it.  Three points, stated
plainly.  \emph{(i)}~The periodicity argument fixes the temperature
and the entropy \emph{functional} but does not say where the
microstates live --- that is the content of the induced-gravity
Part.  \emph{(ii)}~The derivation is inherently coarse-grained: it
computes a saddle free energy seen by an observer who has already
projected to a regular Euclidean boundary problem --- exactly the
observer-facing object \GDOS{} tracks.  \emph{(iii)}~The ``trick''
itself is a theorem --- the continuation to the Euclidean section
(the analytic shore among time's four endowments;
\citealp{ArisakaP5CPT}) exists because the exterior clock's
spectrum is one-sided (\textsf{T-P5-B1}, consumed as
Theorem~\ref{thm:thermal-clock}), the saddle's periodicity being
the geometry's regularity datum.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{I.1.1. Repeat the cigar argument for the
de Sitter horizon and extract its temperature.}

\noindent\emph{Solution.}  Step 1: the static patch
(Appendix~\ref{app:de_Sitter}) has $f(r) = 1 - H^2r^2$, with
horizon at $r_H = 1/H$ and $f'(r_H) = -2H$.  Step 2: near the
horizon set $\rho = r_H - r$: $f \approx 2H\rho$, and the proper
distance is $x = \int dr/\sqrt{f} = \sqrt{2\rho/H}$, so
$f = H^2x^2$ and the Euclidean $(t, x)$ block is a plane in polar
coordinates with angle $H\tau$.  Step 3: smoothness demands
period $\beta = 2\pi/H$, i.e.\ $k_B T_{\rm dS} = \hbar H/2\pi$
--- the first entry of \eqref{eq:GH_thermo}, by exactly the
argument of \eqref{eq:C-period}.  Physics intuition: the cigar
does not care which side of the horizon the observer is on ---
enclosure is enclosure --- which is why one theorem prices both
the black-hole sky and the cosmological one.

\medskip\noindent\textbf{I.2.1. Verify that $I_E = \beta M/2$
yields exactly $S = A c^3 k_B/(4\Newton\hbar)$.}

\noindent\emph{Solution.}  Step 1 (geometrized units): from
\eqref{eq:C-action}, $I_E = \beta M/2$ with $\beta = 8\pi
\Newton M$.  Step 2: the saddle-point free energy is
$F = I_E/\beta = M/2$, and
$S = \beta(M - F) = \beta M/2 = 4\pi \Newton M^2$.  Step 3:
express through the area $A = 16\pi \Newton^2M^2$:
$S = A/4\Newton$ --- and restoring $\hbar, c, k_B$ gives the
Bekenstein--Hawking value with coefficient exactly $\tfrac14$.
The factor-of-two anatomy: half of $\beta M$ is subtracted by the
boundary term ($F = M/2$), and the \emph{other} half is the
entropy.  Physics intuition: in gravity the on-shell bulk action
vanishes ($R = 0$); the entire thermodynamics lives on the
boundary term --- an early, exact hint that the physics of
horizons is boundary bookkeeping, which the main text's port
reading takes literally.

\medskip\noindent\textbf{I.3.1. Compute the specific heat of the
Schwarzschild saddle and interpret its sign.}

\noindent\emph{Solution.}  Step 1: from
\eqref{eq:D-temperature}-adjacent scaling
$T = 1/(8\pi \Newton M)$ (geometrized), $M = 1/(8\pi\Newton T)$.
Step 2: $C = dM/dT = -1/(8\pi \Newton T^2) < 0$ --- restoring
units, $C = -\hbar c^5/(8\pi \Newton k_B T^2)\cdot k_B^2$,
negative always.  Step 3: negative $C$ means no stable canonical
equilibrium in flat space: a hole warmer than its bath radiates,
loses mass, gets \emph{warmer} --- runaway; a hole colder than
its bath grows without bound.  The AdS box of
Appendix~\ref{app:hawking-page} is the standard cure, and the
branch structure there ($C < 0$ small branch, $C > 0$ large
branch, \eqref{eq:D-Tmin}) is this computation stabilized by
curvature.  Physics intuition: self-gravitating systems heat up
by losing energy (virial logic); the black hole is the extreme
case, and its thermodynamic instability is why evaporation, once
begun, accelerates to the end.

\medskip\noindent\textbf{I.4.1. Derive the conical deficit for a
period $\beta' \neq \beta$, and connect it to the replica
method.}

\noindent\emph{Solution.}  Step 1: with the near-horizon plane
$ds^2 = dx^2 + \kappa^2x^2\,d\tau^2$ and $\tau \sim \tau +
\beta'$, the angle variable $\phi = \kappa\tau$ runs over
$[0, \kappa\beta']$; smoothness needs $\kappa\beta' = 2\pi$.
Step 2: for $\beta' = n\beta$ (the $n$-fold replica) the tip
carries deficit $\delta = 2\pi(1 - n)$, and the curvature
acquires a distributional piece $R = 2\,\delta\,
\delta^{(2)}(x) + R_{\rm smooth}$.  Step 3: inserting this delta
into the Einstein--Hilbert action gives the replica free energy
its area term --- $\partial_n$ at $n = 1$ yields $A/4\Newton$
--- which is precisely the engine of the main text's exact-$1/4$
derivation (\S\ref{ssec:V-conical}) and of the holographic
entanglement entropy (Appendix~\ref{app:holography-course},
Step 3).  Physics intuition: entropy is the response of the
action to a deficit angle at the horizon --- geometry's way of
counting what the tip hides; every modern route to
$A/4\Newton\hbar$, including this paper's, passes through this
one computation.

\GGtakeaway{\emph{Standard result:} $\THaw$ from cigar
regularity and $\Sbh = A/4\Newton\hbar$ from the on-shell
action.  \emph{What it assumed (in 1977):} that the Euclidean
continuation and the saddle-point ensemble mean something ---
two formal moves.  \emph{The \GGtheory{} reading:} both moves
carry licenses --- the continuation from the one-sided clock
spectrum, the arena from the analytic-shore clause --- so the
``trick'' is a theorem consumed, not a prescription trusted.}

\section{Hawking--Page free-energy notebook}
\label{app:hawking-page}

\GGteachbox{The Hawking--Page transition worked as a free-energy
notebook: the two black-hole branches in the AdS box, the
free-energy ledger that selects between the gas and the large
hole, and the stability analysis --- the cleanest classical
template for a branch switch selected by a ledger, which is how
the main text reads the Page turnover.}
\GGreadalong{\citet{HawkingPage1983} for the original;
\citet{Harlow2016} for the transition's modern role in
holography.}

Anti-de Sitter space is the box that tames the negative specific
heat of Appendix~\ref{app:euclidean}: its curvature acts as a
reflecting wall, so a black hole can genuinely equilibrate with its
own radiation, and the canonical ensemble becomes honest.  The
price of honesty is a phase transition \citep{HawkingPage1983},
and this notebook works it in three Steps --- the cleanest template
in gravity for a branch switch selected by a free-energy ledger,
which is exactly how this paper reads the Page turnover
(Theorem~\ref{thm:page-turnover}).  Units $G \equiv G_N$, $c =
\hbar = \kB = 1$.

\subsection{Step 1 --- temperature from regularity, and the two
branches}

For the 4D AdS--Schwarzschild metric function
\begin{equation}
f(r) = 1 - \frac{2GM}{r} + \frac{r^2}{L^2},
\end{equation}
the horizon $r_+$ is the largest root of $f$, so $2GM = r_+ +
r_+^3/L^2$.  The cigar argument of Appendix~\ref{app:euclidean},
Step~1 fixes the period as $\beta = 4\pi/f'(r_+)$; differentiate
and eliminate $M$:
\begin{equation}
f'(r) = \frac{2GM}{r^{2}} + \frac{2r}{L^{2}}
\;\Longrightarrow\;
f'(r_+) = \frac{r_+ + r_+^3/L^2}{r_+^{2}} + \frac{2 r_+}{L^{2}}
= \frac{1}{r_+} + \frac{3 r_+}{L^{2}},
\end{equation}
\begin{equation}\label{eq:D-temperature}
T(r_+) = \frac{1}{4\pi r_+}\Bigl(1 + \frac{3 r_+^{2}}{L^{2}}\Bigr).
\end{equation}
The two terms compete: $T \sim 1/4\pi r_+$ for small holes (the
flat-space behavior) and $T \sim 3 r_+/4\pi L^2$ for large ones
(the box behavior).  Minimizing, $dT/dr_+ = 0$ gives $-1/r_+^2 +
3/L^2 = 0$:
\begin{equation}\label{eq:D-Tmin}
r_+^{\rm min} = \frac{L}{\sqrt{3}}, \qquad
T_{\rm min} = \frac{\sqrt{3}}{2\pi L}.
\end{equation}
Below $T_{\rm min}$ no black hole exists at all; above it there are
\emph{two}, a small branch ($r_+ < L/\sqrt3$) and a large branch
($r_+ > L/\sqrt3$), and the ensemble must choose.

\subsection{Step 2 --- the free-energy ledger}

The entropy, mass, and free energy on a branch are
\begin{equation}
S = \frac{\pi r_+^2}{G_N}, \qquad M = \frac{r_+}{2G_N}\Bigl(1+\frac{r_+^2}{L^2}\Bigr), \qquad
F = M - TS = \frac{r_+}{4G_N}\Bigl(1-\frac{r_+^2}{L^2}\Bigr),
\end{equation}
where the $F$ line is worth one displayed check:
\begin{equation}
F = \frac{r_+}{2G}\Bigl(1+\frac{r_+^{2}}{L^{2}}\Bigr)
- \frac{1}{4\pi r_+}\Bigl(1+\frac{3r_+^{2}}{L^{2}}\Bigr)
\frac{\pi r_+^{2}}{G}
= \frac{r_+}{4G}\Bigl[2 + \frac{2r_+^{2}}{L^{2}} - 1 -
\frac{3r_+^{2}}{L^{2}}\Bigr]
= \frac{r_+}{4G}\Bigl(1 - \frac{r_+^{2}}{L^{2}}\Bigr).
\end{equation}
Equivalently, the on-shell Euclidean action (same
subtraction discipline as Appendix~\ref{app:euclidean}, Step~2,
with thermal AdS as the reference background) is
\begin{equation}\label{eq:D-action}
I_E = \beta F = \frac{\pi r_+^{2}\,\bigl(L^{2} -
r_+^{2}\bigr)}{G\,\bigl(L^{2} + 3 r_+^{2}\bigr)} .
\end{equation}
$F$ changes sign at $r_+ = L$, i.e.\ at the Hawking--Page
temperature
\begin{equation}\label{eq:D-THP}
T_{\rm HP} = T(L) = \frac{1}{\pi L} :
\end{equation}
below it thermal AdS dominates the ensemble (the black-hole saddle
has $F > 0$), above it the large black hole dominates ($F < 0$).
In the dual gauge theory this is the
confinement/deconfinement transition \citep{Maldacena1998}.

\subsection{Step 3 --- stability, and the branch-switch template}

Which branch is even locally stable?  The specific heat along the
branch is $C = dM/dT = (dM/dr_+)/(dT/dr_+)$, and $dM/dr_+ =
(1 + 3r_+^2/L^2)/2G > 0$ always, so the sign of $C$ is the sign of
$dT/dr_+$:
\begin{equation}
C \;<\; 0 \quad (r_+ < L/\sqrt3, \ \text{small branch}), \qquad
C \;>\; 0 \quad (r_+ > L/\sqrt3, \ \text{large branch}).
\end{equation}
The small branch is the flat-space runaway of
Appendix~\ref{app:euclidean}, Step~3, embedded in AdS --- a local
maximum of the free energy, the instanton mediating the
transition --- while the large branch is a genuine equilibrium.

\paragraph{Physical picture, and the \GGtheory{} close.}  Three
saddles sit in one ensemble --- thermal gas, small hole, large
hole --- and the physical state is whichever minimizes the ledger;
as $T$ rises through $T_{\rm HP}$ the minimum jumps branches
discontinuously while everything on each branch stays smooth.
That is the entire grammar of a first-order transition, realized
in pure geometry.  In \GGtheory{} language this is the cleanest
template for a branch switch selected by the generalized
free-energy ledger: phase grammar, stability, and information
bookkeeping fused before any \GGtheory{} ingredient is added ---
and Theorem~\ref{thm:page-turnover} reads the Page turnover as
precisely such a switch, with \eqref{eq:gff-bh} the ledger and the
island branch the analogue of the large hole.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{J.1.1. Fill in the minimization that
yields the minimum temperature \eqref{eq:D-Tmin}.}

\noindent\emph{Solution.}  Step 1: from
\eqref{eq:D-temperature}, $4\pi T = 1/r_+ + 3r_+/L^2$.  Step 2:
$dT/dr_+ = 0$ gives $-1/r_+^2 + 3/L^2 = 0$, i.e.
$r_+^{\min} = L/\sqrt3$; the second derivative $2/r_+^3 > 0$
confirms a minimum.  Step 3: substituting back,
$4\pi T_{\min} = \sqrt3/L + \sqrt3/L = 2\sqrt3/L$, so
$T_{\min} = \sqrt3/(2\pi L)$ \eqref{eq:D-Tmin}.  Below
$T_{\min}$ the ensemble contains no black-hole saddle at all.
Physics intuition: the two terms are the two regimes --- the
flat-space $1/r_+$ (small holes, negative specific heat) and the
box's $r_+/L^2$ (large holes, positive) --- and $T_{\min}$ is
where the box first wins; everything the transition does is a
negotiation between these two terms.

\medskip\noindent\textbf{J.2.1. Show the transition is first
order, with entropy jump of order the horizon area.}

\noindent\emph{Solution.}  Step 1: at $T_{\rm HP} = 1/\pi L$
\eqref{eq:D-THP}, the dominant saddle switches between thermal
AdS ($F \approx 0$, entropy of a gas, $O(1)$ in $\Newton$) and
the large hole ($r_+ = L$).  Step 2: the large hole's entropy is
$S = \pi r_+^2/\Newton = \pi L^2/\Newton$ --- in Planck units, a
number of order $(L/\ell_P)^2$.  Step 3: since $F$ is continuous
at the crossing but $S = -\partial F/\partial T$ jumps by
$\sim L^2/\Newton$, the transition is first order with a
macroscopic latent heat $T_{\rm HP}\,\Delta S$.  Physics
intuition: the ensemble reorganizes from $O(1)$ to
$O(1/\Newton)$ degrees of freedom in one step --- in the dual
gauge theory this is deconfinement, and in the main text's
language it is the cleanest classical instance of a
ledger-selected branch switch, the grammar of
Theorem~\ref{thm:page-turnover}.

\medskip\noindent\textbf{J.3.1. Confirm the stability assignment
of the two branches from the specific heat.}

\noindent\emph{Solution.}  Step 1: $C = dM/dT =
(dM/dr_+)/(dT/dr_+)$, and $dM/dr_+ = (1 + 3r_+^2/L^2)/2\Newton
> 0$ always.  Step 2: from J.1.1, $dT/dr_+ < 0$ for
$r_+ < L/\sqrt3$ and $> 0$ for $r_+ > L/\sqrt3$: hence $C < 0$
(small branch), $C > 0$ (large branch).  Step 3: the small
branch is the flat-space runaway of I.3.1 embedded in AdS --- a
local \emph{maximum} of free energy acting as the nucleation
barrier --- while the large branch is a genuine equilibrium.
Physics intuition: the box turns the runaway into a phase
diagram: the unstable saddle does not disappear, it becomes the
mountain pass between gas and hole, exactly the role the
sub-dominant branch plays in the Page-turnover ledger race
(Appendix~\ref{app:Page_curve}, Step 4).

\GGtakeaway{\emph{Standard result:} the first-order
Hawking--Page transition between thermal AdS and the large black
hole, selected by the free-energy ledger.  \emph{What it
assumed:} the AdS box and the Euclidean ensemble.  \emph{The
\GGtheory{} reading:} the cleanest pre-existing template for a
ledger-selected branch switch --- the grammar in which the main
text reads the Page turnover
(Theorem~\ref{thm:page-turnover}).}

\section{The Page curve entropy calculation}
\label{app:Page_curve}

\GGteachbox{The Page curve as a counting theorem: the exact
random-state average, the curve and its corner, the evaporation
clock with the ratio $t_{\rm Page} \approx 0.646\,t_{\rm evap}$
derived rather than quoted, and the statement of exactly which
premise (typicality of the port ensemble) \GDOS{} keeps and
flags.}
\GGreadalong{\citet{Page1993} for the original;
\citet{Harlow2016} for the modern quantum-information
formulation; \citet{HaydenPreskill2007} for the mirror
refinement.}

The Page curve is a counting statement before it is a gravity
statement, and this chapter works the counting in full: Step~1 the
exact random-state average \citep{Page1993}; Step~2 the curve and
its corner; Step~3 the evaporation clock, including the
reference-card ratio $t_{\rm Page} \approx 0.646\, t_{\rm evap}$
derived rather than quoted; Step~4 what \GDOS{} keeps and what it
flags.

\subsection{Step 1 --- the exact random-state average}

For a unitary evaporation process of a black hole of initial
entropy $\Sbh^{(0)} = N$ coarse-grained qubit units, consider the
system when $m$ units have been emitted: the radiation factor has
dimension $d_R = 2^m$, the black-hole factor $d_B = 2^{N-m}$.  The
global state is bipartite pure, so the two reduced entropies agree,
\begin{equation}\label{eq:Page_entropy}
\SvN(\rho_R) = \SvN(\rho_B).
\end{equation}
--- whatever the dynamics, the two subsystems' ledgers are mirror
images.  The dynamical input is Page's postulate: after scrambling,
the global pure state is well modeled as Haar-random in the
$d_R d_B$-dimensional Hilbert space.  For such a state Page
conjectured and subsequent work proved the exact average (for
$d_R \le d_B$)
\begin{equation}\label{eq:E-pageexact}
\avg{\SvN(\rho_R)} \;=\; \sum_{k = d_B + 1}^{d_R d_B} \frac{1}{k}
\;-\; \frac{d_R - 1}{2 d_B}
\;=\; H_{d_R d_B} - H_{d_B} - \frac{d_R - 1}{2 d_B},
\end{equation}
with $H_n = \sum_{k\le n} 1/k$ the harmonic numbers.  The
asymptotics follow from $H_n = \ln n + \gamma_{\rm E} + 1/2n +
O(1/n^2)$:
\begin{equation}\label{eq:E-pageasymp}
\avg{\SvN(\rho_R)} = \ln d_R
- \frac{d_R}{2 d_B} + O\Bigl(\frac{1}{d_R d_B}\Bigr),
\end{equation}
i.e.\ the radiation is \emph{maximally} entangled with the hole up
to an exponentially small deficit $d_R/2d_B = 2^{2m - N}/2$.  For
$d_R \ge d_B$ exchange the labels (the spectrum of the two reduced
states is identical by \eqref{eq:Page_entropy}).

\paragraph{Physical picture.}  A random pure state hides its
information democratically: any subsystem smaller than half the
total looks \emph{exactly} thermal to within $2^{-|{\rm rest}|}$
corrections.  Early radiation therefore carries essentially zero
information --- not because information is lost but because it is
stored in correlations \emph{with} the remainder, unreadable
locally.  The deficit term is the whole story of the information
return, waiting in the second-order bookkeeping.

\subsection{Step 2 --- the curve and its corner}

Reading \eqref{eq:E-pageasymp} across the evaporation, with $m$
emitted of $N$ total:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item For $m < N/2$: $S_R \approx m\ln 2$ (linear growth,
matching Hawking's calculation exactly --- this half of the curve
is not modified).
\item For $m = N/2$ (the Page point): the exact average sits
$\tfrac12$~nat below the maximum, $\avg{S_R} = \tfrac{N}{2}\ln 2 -
\tfrac12 + O(2^{-N})$ --- the corner is rounded only at the third
decimal for astrophysical $N$.
\item For $m > N/2$: $S_R \approx (N-m)\ln 2$: the radiation
entropy now \emph{tracks the shrinking hole}, decreasing to zero at
complete evaporation.
\end{itemize}
The triangle $S_R(m) \simeq \min(m, N{-}m)\ln 2$ is the Page
curve.  Its rising edge is thermodynamics; its falling edge is
unitarity; the corner is where the smaller-half rule changes
sides.

\subsection{Step 3 --- the evaporation clock}

To place the corner in time rather than in emitted qubits, run the
clock.  The Hawking flux is Stefan--Boltzmann at $\THaw \propto
1/M$ through an effective area $\propto M^2$, so dimensional
analysis fixes $dM/dt = -\alpha\,\hbar c^4/(G_N^2 M^2)$ with
$\alpha$ a graybody-weighted constant; the geometric-optics
photon-graviton estimate underlying the reference card corresponds
to $\alpha = 1/15360\pi$.  Integrating from $M_0$,
\begin{equation}\label{eq:E-clock}
M(t) = M_0\Bigl(1 - \frac{t}{t_{\rm evap}}\Bigr)^{1/3},
\qquad
t_{\rm evap} = \frac{5120\,\pi\, G_N^{2} M_0^{3}}{\hbar c^{4}}
\approx 2\times 10^{67}\,{\rm yr}\;
\Bigl(\frac{M_0}{\Msun}\Bigr)^{3},
\end{equation}
the card's value (verified in \texttt{p2c2\_verify}).  The Page
point is reached when half the \emph{entropy} has been emitted, and
$\Sbh \propto M^2$, so $M_{\rm Page} = M_0/\sqrt{2}$; inverting
\eqref{eq:E-clock},
\begin{equation}\label{eq:E-pagetime}
\frac{t_{\rm Page}}{t_{\rm evap}}
= 1 - \Bigl(\frac{M_{\rm Page}}{M_0}\Bigr)^{3}
= 1 - 2^{-3/2} \approx 0.6464 ,
\end{equation}
which is the reference card's $t_{\rm Page}\approx 0.646\,t_{\rm
evap}$, now derived: it is a pure number, independent of $M_0$ and
of $\alpha$, fixed entirely by $S \propto M^2$ and $\dot M \propto
-M^{-2}$.  (Species-weighted emission shifts $\alpha$, not the
ratio; frequency-dependent graybody corrections shift the ratio at
the percent level.)

\paragraph{Physical picture.}  Two-thirds of the lifetime passes
before the turnover because the hole spends its entropy slowly at
first --- big, cold, dim --- and the halfway point of the
\emph{entropy} ledger sits far past the halfway point of the
\emph{mass} ledger.  After the corner the hole is hotter, brighter,
and increasingly transparent: the last emissions are the most
information-rich, exactly as the smaller-half rule demands.

\subsection{Step 4 --- what \GDOS{} keeps, and what it flags}

In \GDOS{}, the bulk evolution is unitary (\AX{1}), and the
random-state hypothesis is supplied at long times by the standard
eigenstate-thermalization-hypothesis arguments applied to the
bulk --- an imported dynamical premise, flagged as such rather than
proved here; the lemma that consumes it
(Lemma~\ref{lem:stale-port}, \textsf{L-BH-Fresh}) carries it as an
explicit condition, and the freshness meter $I(q\!:\!R)$ gives the
premise its operational form: the turnover is the scheduled expiry
of the Port-Freshness Clause, not a new mechanism.  Given the
premise, the Page curve follows, and the branch-switch reading of
the corner is Theorem~\ref{thm:page-turnover} (T-BH-2).  The
qubit-model numerics behind these statements ($N = 12$, Haar
ensembles, growth-ratio and saturation checks) are logged in the
benchmark audit (Appendix~\ref{app:benchmarks}).

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{K.1.1. Prove that the two reduced
entropies of a bipartite pure state are equal.}

\noindent\emph{Solution.}  Step 1: Schmidt-decompose
$|\psi\rangle = \sum_k \sqrt{p_k}\,|k\rangle_R|k\rangle_B$ ---
possible for any bipartite pure state (singular-value
decomposition of the coefficient matrix).  Step 2: the reduced
states are $\rho_R = \sum_k p_k |k\rangle\langle k|_R$ and
$\rho_B = \sum_k p_k |k\rangle\langle k|_B$: \emph{identical
spectra} $\{p_k\}$.  Step 3: von Neumann entropy depends only on
the spectrum, so $S(\rho_R) = S(\rho_B)$ --- the mirror-ledger
identity \eqref{eq:Page_entropy} on which the whole curve rests.
Physics intuition: entanglement is a shared property, not a
possession --- the radiation cannot ``know'' more about the hole
than the hole matches about the radiation, which is why one
curve serves both subsystems.

\medskip\noindent\textbf{K.1.2. Evaluate the exact Page average
for the smallest nontrivial case, $N = 2$ qubits with one
emitted.}

\noindent\emph{Solution.}  Step 1: $d_R = d_B = 2$; the exact
average \eqref{eq:E-pageexact} reads $\langle S\rangle =
\sum_{k = 3}^{4}\tfrac1k - \tfrac{d_R - 1}{2d_B}$.  Step 2:
evaluate: $\tfrac13 + \tfrac14 - \tfrac14 = \tfrac13$ nat
$= 0.48$ bit.  Step 3: the maximum possible is $\ln 2 = 0.693$
nat $= 1$ bit, so even at the Page point of the smallest system
the typical state sits $0.36$ nat below maximal --- the deficit
term is visible at $N = 2$ and shrinks exponentially with $N$
\eqref{eq:E-pageasymp}.  Physics intuition: ``maximally
entangled up to a deficit'' is not an asymptotic slogan but an
exact, checkable number already for two qubits --- and the
deficit, tiny as it is, is where all the information lives
(Step 1's physical picture).

\medskip\noindent\textbf{K.3.1. Derive the Page-time ratio
$t_{\rm Page}/t_{\rm evap} = 1 - 2^{-3/2}$ from the two scaling
laws alone.}

\noindent\emph{Solution.}  Step 1: $S_{\rm BH} \propto M^2$, so
half the entropy is emitted when $M = M_0/\sqrt2$.  Step 2:
Stefan--Boltzmann at $T \propto 1/M$ through area $\propto M^2$
gives $dM/dt \propto -1/M^2$, hence $M(t)^3 = M_0^3(1 -
t/t_{\rm evap})$ \eqref{eq:E-clock}.  Step 3: solve
$M_0^3\,2^{-3/2} = M_0^3(1 - t_{\rm Page}/t_{\rm evap})$:
$t_{\rm Page}/t_{\rm evap} = 1 - 2^{-3/2} = 0.6464$
\eqref{eq:E-pagetime} --- a pure number, independent of $M_0$
and of the graybody constant.  Physics intuition: two-thirds of
the lifetime passes before the turnover because the hole spends
its entropy slowly while big and cold; the information era is
the loud final third.

\medskip\noindent\textbf{K.4.1. At what average rate does a
solar-mass black hole return its information?}

\noindent\emph{Solution.}  Step 1: the archive is
$S_{\rm BH}/k_B\ln2 \approx 1.5\times10^{77}$ bits (reference
card, Appendix~\ref{app:BH_numbers}).  Step 2: it is returned
over $t_{\rm evap} \approx 2\times10^{67}$~yr
$= 6.6\times10^{74}$~s.  Step 3: the average rate is
$\sim\!230$ bits per second --- a slow teletype running for
$10^{57}$ ages of the universe.  (The instantaneous rate is far
below this before the Page time and above it after; the average
is fixed by the two totals alone.)  Physics intuition: unitarity
is patient --- ``the information comes back'' is a statement
about correlations accumulated across the whole evaporation, not
a broadcast; no observer who samples less than the late-time
majority of the radiation ever decodes a word (the
Hayden--Preskill refinement, \citealp{HaydenPreskill2007}).

\GGtakeaway{\emph{Standard result:} for unitary evaporation of a
typical state, $S_R(m) \approx \min(m, N-m)$ --- rise, corner at
the Page point, fall --- with $t_{\rm Page} \approx
0.646\,t_{\rm evap}$.  \emph{What it assumed:} global unitarity
and typicality.  \emph{The \GGtheory{} reading:} unitarity is
the bulk's (\AX{1}); typicality is a premise made operational by
the freshness meter, and its scheduled expiry \emph{is} the
turnover (Theorem~\ref{thm:page-turnover}).}


\section{From state counting to holography: the modern era}
\label{app:holography-course}

\GGteachbox{The three modern answers to ``what does black-hole
entropy count?'', each derived to its punchline: the
Strominger--Vafa D-brane count that first matched
$A/4\Newton\hbar$ microscopically; the holographic principle and
\AdSCFT{}, where evaporation is manifestly unitary; and the
quantum-extremal-surface / islands program that recovered the
Page curve from the gravitational path integral --- followed by a
precise statement of what each achievement still leaves open, and
the map of where the main text picks up each thread.}
\GGreadalong{\citet{SusskindLindesay2005} for the arc;
\citet{Harlow2016} for the RT/QES technology;
\citet{AlmheiriReview2021} for the islands synthesis this
appendix compresses.}

The Page curve of Appendix~\ref{app:Page_curve} set a target that
stood for a quarter century: exhibit, from first principles, the
microstates behind \eqref{eq:bekenstein-ansatz} and the unitarity
behind the curve.  This appendix works the three modern
campaigns in the order they arrived, at the level of detail a
student needs to see \emph{what exactly was proven} --- because
the precise domain of each proof is where the remaining problems
live, and Appendix~\ref{app:islands} then compares the winning
formalism with this paper's reading line by line.

\subsection{Step 1 --- Strominger--Vafa: the first microscopic
count}

The 1996 breakthrough \citep{StromingerVafa1996} counts the
states of a bound system of $Q_1$ D1-branes and $Q_5$ D5-branes
carrying $n$ units of momentum, in five dimensions, at weak
string coupling where the system is an ordinary (if elaborate)
quantum mechanics.  The count is a Cardy-type asymptotic of the
two-dimensional world-volume theory: the central charge is
$c_{\rm eff} = 6\,Q_1 Q_5$, and the degeneracy of states at level
$n$ obeys
\begin{equation}\label{eq:cardy-sv}
S_{\rm micro} = k_B \ln d(n) \simeq 2\pi k_B
\sqrt{\frac{c_{\rm eff}\,n}{6}}
= 2\pi k_B \sqrt{Q_1 Q_5\, n}.
\end{equation}
Because the states are BPS (supersymmetry-protected), the count
is rigid as the coupling is dialed up --- the same system
gravitationally collapsed is an extremal five-dimensional black
hole, and evaluating the horizon area of the corresponding
supergravity solution gives, with no adjustment,
\begin{equation}
\frac{k_B\,A\,c^3}{4 G_5 \hbar} = 2\pi k_B
\sqrt{Q_1 Q_5\,n} = S_{\rm micro}.
\end{equation}
The coefficient $1/4$ --- Hawking's $\eta$ --- emerges from an
honest count of named states, exactly.  (\emph{Physics
intuition}: supersymmetry is the bridge that lets one count the
building at blueprint stage, weak coupling, and trust the count
after construction, strong coupling; the BPS property is the
guarantee that no states move in or out during construction.)
What was proven: for a special class --- supersymmetric, extremal
or near-extremal, in special charge sectors --- the
Bekenstein--Hawking entropy is state counting.  What was not:
anything about the Schwarzschild hole in our sky, whose
temperature is nonzero, whose charges are astrophysically
neutralized, and whose states are protected by nothing.

\subsection{Step 2 --- the holographic principle and \AdSCFT{}}

Bekenstein's area-scaling (Appendix~\ref{app:bekenstein-course})
was promoted by \citet{THooft1993} and \citet{Susskind1995} to a
principle: the maximum entropy in any region is its \emph{boundary}
area in Planck units, so the fundamental description of any
gravitating region lives on its boundary.  \citet{Maldacena1998}
made the principle exact in a specific universe: string theory in
anti-de Sitter space is \emph{equal} --- not approximately dual
--- to a conformal field theory on the boundary.  For the
information paradox the consequence is structural: a black hole
in AdS is some high-energy state of the boundary CFT; the CFT is
an ordinary quantum system with manifestly unitary evolution;
therefore black-hole formation and evaporation, whatever they
look like geometrically, cannot lose information.  From 1997
on, the question was no longer \emph{whether} unitarity wins but
\emph{how the bulk geometry implements it} --- and by what
mechanism the Hawking calculation's apparent thermality is
corrected.  (\emph{Physics intuition}: \AdSCFT{} settles the
verdict without showing the crime scene.  Knowing the boundary
books balance does not tell you what entry, at the horizon,
does the balancing --- that gap is exactly where the next
twenty years of work, and Steps~3--4, live.)

\subsection{Step 3 --- geometrizing entanglement:
Ryu--Takayanagi and the quantum extremal surface}

The instrument that eventually cracked the Page curve was a
formula for \emph{entanglement} entropy, not black-hole entropy.
\citet{RyuTakayanagi2006} proposed, and later derivations
confirmed, that the entropy of a boundary region $B$ is the area
of the smallest bulk surface $\gamma_B$ anchored on
$\partial B$:
\begin{equation}\label{eq:rt-course}
S(B) = \min_{\gamma_B}
\frac{A(\gamma_B)\,k_B c^3}{4\Newton\hbar},
\end{equation}
a strict generalization of Bekenstein--Hawking (take $B$ = the
whole boundary of a black-hole spacetime: $\gamma$ hugs the
horizon).  Quantum corrections promote the minimized functional
to the \emph{generalized entropy} $S_{\rm gen} =
A/4\Newton\hbar + S_{\rm bulk}$ --- area plus the entropy of
bulk fields outside $\gamma$ --- and the surface to a
\emph{quantum extremal surface} (QES): extremize $S_{\rm gen}$,
then take the minimum over extrema.  Two features deserve the
student's attention.  First, \eqref{eq:rt-course} is derived, in
the Euclidean formulation, by the \emph{same conical-deficit
replica method} this paper uses for
Theorem~\ref{thm:exact-quarter} --- the technology of
\S\ref{ssec:V-conical} and Appendix~\ref{app:euclidean} is the
common engine of the modern era.  Second, the functional being
minimized is exactly the object this paper calls the \GFF{}
ledger \eqref{eq:gff-bh}: area term plus bulk-entropy term, in
competition.

\subsection{Step 4 --- islands and replica wormholes: the Page
curve from gravity}

Apply the QES prescription to the radiation $R$ of an evaporating
hole \citep{Penington2020, Almheiri2019}.  At early times the
minimal QES is the empty surface: $S(R)$ = the semiclassical
radiation entropy, rising exactly as Hawking computed.  But there
is a competitor: a QES just inside the horizon, whose
generalized entropy is (horizon area)/$4\Newton\hbar$ plus a
small correction, and which --- crucially --- assigns to $R$ an
\emph{island} $I$: the bulk region inside the surface, whose
state is counted with the radiation,
\begin{equation}\label{eq:island-course}
S(R) = \min \Bigl\{ S_{\rm no\mbox{-}island}(R),\;
\mathrm{ext}_I\Bigl[\frac{A(\partial I)k_Bc^3}{4\Newton\hbar} +
S_{\rm semi\mbox{-}cl}(R \cup I)\Bigr]\Bigr\}.
\end{equation}
The island branch is flat-to-decreasing (the shrinking horizon
area), so the minimum switches branches exactly at the Page time:
the curve of Appendix~\ref{app:Page_curve} emerges from geometry,
with no unitarity assumption inserted by hand.  The 2020 replica
computations \citep{ReplicaWormholes2020} exhibited the
mechanism inside the gravitational path integral: connected
``replica wormhole'' saddles, negligible early, dominate after
the Page time and enforce the island answer.  What was proven:
in two-dimensional dilaton-gravity models coupled to matter
(and, with more assumptions, in higher-dimensional setups), the
gravitational path integral itself computes a unitary Page
curve.  This is the achievement the present paper's Part~IX
weighs and the main text engages theorem by theorem.

\subsection{Step 5 --- the remaining problems, stated precisely}

A course honest about triumphs must be precise about residues;
these five are the live ones, and the reader will recognize them
as the openings the main text walks through.
\emph{(i)~Dimensional rigor.}  The island derivations are
theorems in $d = 2$ toy models; in four asymptotically flat
dimensions the prescription is motivated, not derived
\citep{AlmheiriReview2021}.  \emph{(ii)~Mechanism vs.\
prescription.}  The QES rule says \emph{which} entropy to
compute, not \emph{what physical process} the island represents
--- what does it mean, operationally, for interior degrees of
freedom to ``belong'' to the radiation?
\emph{(iii)~The ensemble/factorization puzzle.}  Replica
wormholes compute averages that behave like an \emph{ensemble}
of theories; a single unitary quantum system should have no such
average (\S\ref{sec:gff-ledger} engages this directly).
\emph{(iv)~Genericity.}  The microscopic counts (Step~1) live in
supersymmetric corners; the island results live in low
dimensions; the astrophysical Schwarzschild hole --- generic,
non-extremal, non-supersymmetric --- has still never had its
microstates named by these methods.
\emph{(v)~The species/induced-$\Newton$ closure.}  The
$A/4\Newton\hbar$ in every formula above uses the observed
$\Newton$; none of the derivations computes $\Newton$ from the
field content it sums over --- the species problem of
Appendix~\ref{app:bekenstein-course}, still open in the modern
formalism.

\subsection{Step 6 --- where \GGtheory{} enters}

The main text's program is positioned against exactly these
residues, and the third-step convention of this course sends the
reader there: the microstates for the \emph{generic} hole are
named and counted in \S\ref{ssec:V-microstates} (with the
coefficient fixed independently of the count,
\S\ref{ssec:V-conical}); the island functional is derived as the
branch-minimized \GFF{} ledger in \S\ref{sec:gff-ledger}, with
the dictionary tabulated in Appendix~\ref{app:island-dictionary};
the Page curve arrives from bulk unitarity plus projection in
\S\ref{sec:branch-switch} (Theorem~\ref{thm:page-turnover});
the factorization puzzle is engaged as a named commitment in
\S\ref{sec:gff-ledger}; and the species closure is the
program's own declared open computation
(Appendix~\ref{app:species-sum}).  Part~IX weighs the whole
against the frameworks of this appendix in both table and
argument.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{L.1.1. Evaluate the Strominger--Vafa
degeneracy for the modest charges $Q_1 = Q_5 = n = 10$.}

\noindent\emph{Solution.}  Step 1: from \eqref{eq:cardy-sv},
$S/k_B = 2\pi\sqrt{Q_1Q_5n} = 2\pi\sqrt{1000} \approx 198.7$.
Step 2: the number of microstates is $d = e^{S/k_B} \approx
e^{198.7} \approx 10^{86}$.  Step 3: for scale, that is more
states than there are protons in the observable universe
($\sim\!10^{80}$) --- carried by a single bound state of thirty
units of charge, and counted \emph{exactly} by a weak-coupling
quantum mechanics.  Physics intuition: black-hole entropies are
not merely large, they are exponentially large in the charges ---
which is why no census of states could ever proceed by listing,
and why the Cardy asymptotic (a saddle point, not a tally) is the
right tool.  The astrophysical analogue of this number is the
$10^{77}\,k_B$ of the reference card
(Appendix~\ref{app:BH_numbers}), counted in this paper's reading
by \S\ref{ssec:V-microstates}.

\medskip\noindent\textbf{L.2.1. Show that gravity itself protects
the holographic bound: a thermal gas collapses before its entropy
can exceed its boundary area.}

\noindent\emph{Solution.}  Step 1 (natural units, $\hbar = c =
k_B = 1$): a photon gas of temperature $T$ in a box of size $R$
has $S_{\rm gas} \sim (RT)^3$ and $E \sim R^3T^4$; the area bound
allows $S_{\max} \sim R^2/\ell_P^2$.  Step 2: the gas would beat
the bound when $(RT)^3 > R^2/\ell_P^2$, i.e.
$R > 1/(T^3\ell_P^2)$.  Step 3: but the gas's Schwarzschild
radius $r_s \sim \Newton E \sim \ell_P^2R^3T^4$ exceeds $R$
already when $R > 1/(T^2\ell_P)$ --- and
$1/(T^2\ell_P) < 1/(T^3\ell_P^2)$ whenever $T\ell_P < 1$, i.e.\
always below the Planck temperature.  The box collapses into a
black hole --- whose entropy saturates the bound by G.3.1 ---
before the bound is ever threatened.  Physics intuition: the
holographic principle needs no enforcement agency; gravity is the
enforcement.  Try to overfill a region with entropy and you
build the very object that defines the limit.

\medskip\noindent\textbf{L.3.1. Verify the Ryu--Takayanagi
formula against the known entanglement entropy of a
two-dimensional critical system.}

\noindent\emph{Solution.}  Step 1: for an interval of length
$\ell$ in a 2D conformal field theory, the entanglement entropy
is $S = \frac{c_{\rm CFT}}{3}\ln(\ell/\epsilon)$ ($\epsilon$ a
short-distance cutoff) --- a standard, independently derived
result of condensed-matter and field theory.  Step 2: the bulk
dual is AdS$_3$ with radius $L$; the minimal surface anchored on
the interval's endpoints is a geodesic, and a direct integration
in the Poincar\'e metric gives its regulated length
$\mathcal{L} = 2L\ln(\ell/\epsilon)$.  Step 3: apply
\eqref{eq:rt-course} with the three-dimensional Newton constant
$G_3$ (natural units): $S = \mathcal{L}/(4G_3)
= \frac{2L}{4G_3}\ln(\ell/\epsilon)$, and with the
boundary central charge $c_{\rm CFT} = 3L/2G_3$ (the
Brown--Henneaux value) this is exactly
$\frac{c_{\rm CFT}}{3}\ln(\ell/\epsilon)$.  Physics intuition:
the RT surface converts entanglement --- a notoriously hard
quantum quantity --- into geometry's easiest one, a length; the
match above, exact including coefficient, is why the formula was
believed long before it was derived, and the derivation, when it
came, used the conical-replica engine of I.4.1.

\medskip\noindent\textbf{L.4.1. Show that the replica-wormhole
saddle overtakes the disconnected one precisely when the naive
radiation entropy exceeds the horizon's.}

\noindent\emph{Solution.}  Step 1: for a state of entropy $S$,
$\mathrm{Tr}\,\rho_R^{\,n} \approx e^{-(n-1)S}$; the
disconnected saddle realizes this with the semiclassical
radiation entropy, $\sim e^{-(n-1)S_{\rm rad}}$, while the fully
connected wormhole pays one horizon area across the whole braid,
$\sim e^{-(n-1)S_{\rm BH}}$ (the area enters as
$(n-1)A/4\Newton$ through the conical deficit of I.4.1).
Step 2: the path integral is dominated by the \emph{larger}
contribution, so the wormhole wins exactly when
$S_{\rm BH} < S_{\rm rad}$.  Step 3: the
entropy, computed by whichever saddle dominates, is therefore
$\min(S_{\rm rad}, S_{\rm BH})$: the Page curve of
Appendix~\ref{app:Page_curve}, now spoken in saddle-point
language, with the crossover at exactly the Page time.
Physics intuition: the ``exotic'' wormhole is doing what
sub-dominant saddles always do in statistical mechanics ---
waiting for its free energy to win; the Page turnover is a
first-order phase transition in the replica ensemble, the same
grammar as Appendix~\ref{app:hawking-page} (J.2.1) and the same
branch switch as \eqref{eq:branch-min}.

\GGtakeaway{\emph{Standard results:} exact microscopic entropy in
protected sectors \eqref{eq:cardy-sv}; unitarity settled in
principle by \AdSCFT{}; the Page curve recovered from the
gravitational path integral via QES/islands
\eqref{eq:island-course}.  \emph{What each assumed:}
supersymmetry and extremality; an AdS box and a dual; two
dimensions or motivated prescriptions.  \emph{What remains open:}
4D rigor, mechanism, factorization, genericity, and the species
closure.  \emph{The \GGtheory{} reading:} each residue is an
address in the main text --- \S\ref{ssec:V-microstates},
\S\ref{sec:gff-ledger}, \S\ref{sec:branch-switch},
Appendix~\ref{app:species-sum} --- with Part~IX as the scorecard.}

\section{Comparison with the islands formula}
\label{app:islands}

\GGteachbox{The islands formula and the \GDOS{} statement set
side by side: the QES condition derived, the two-branch race
that produces the Page turnover in each formalism, and the
precise statement of where the two prescriptions provably agree
and where they part --- the technical companion to the
comparison of Part~IX and the history of
Appendix~\ref{app:holography-course}.}
\GGreadalong{\citet{Penington2020} and \citet{Almheiri2019} for
the originals; \citet{AlmheiriReview2021} for the review this
comparison tracks.}

The islands formula is
\begin{equation}\label{eq:island_app}
S(R) = \min\,\mathrm{ext}\Big\{\frac{A(\partial I)}
{4\Newton\hbar} + S_{\rm semi-cl}(R \cup I)\Big\}.
\end{equation}
In \GDOS{}, the corresponding statement --- registered as a
dictionary-grade proposition in the named-statement deck
(Appendix~\ref{app:statement-deck}) --- is
\begin{equation}\label{eq:island_GDOS}
S(R) = \SvN(\Trin[\rho_{\rm bulk}]_R) =
\SvN(\rho_\partial^R),
\end{equation}
where $\rho_\partial^R$ is the 4D reduced state of the radiation.
This chapter works the mechanism in two Steps and then records the
equivalence and its limits.

\subsection{Step 1 --- the QES condition, derived, and the
two-branch race}

Write the generalized entropy of a candidate island with boundary
at radial position $x$ as
\begin{equation}\label{eq:F-sgen}
S_{\rm gen}(x; t) = \frac{A(x)}{4\Newton\hbar}
+ S_{\rm semi\mbox{-}cl}\bigl(R \cup I(x); t\bigr).
\end{equation}
The extremum condition defining the quantum extremal surface is
\begin{equation}\label{eq:F-qes}
\frac{\partial S_{\rm gen}}{\partial x} = 0
\quad\Longleftrightarrow\quad
\frac{A'(x)}{4\Newton\hbar} = -\,\frac{\partial
S_{\rm semi\mbox{-}cl}}{\partial x} :
\end{equation}
the surface settles where the marginal area cost of enlarging the
island exactly cancels the marginal entanglement saved by
including more interior partners.  Since $A'$ is set by the
horizon scale while $\partial_x S_{\rm semi\mbox{-}cl}$ is set by
the near-horizon mode density, the solution pins $\partial I$ to
just inside the horizon --- which is why the island's area term is
$\approx 2\Sbh$ for the two-sided problem and $\approx \Sbh$
(the instantaneous horizon) for evaporation.

Now race the two branches of the min:
\begin{itemize}[leftmargin=2em,itemsep=0.1em]
\item \emph{No-island branch} ($I = \emptyset$): $S(R) =
S_{\rm semi\mbox{-}cl}(R; t)$, the naive radiation entropy,
growing linearly --- each emitted quantum adds its thermal share,
$\dot S \approx$ const (the rising Page edge).
\item \emph{Island branch}: $S(R) \approx \Sbh(t) + S_{\rm
matter}$, essentially time-independent on emission timescales
(and slowly \emph{decreasing} as the hole shrinks), because once
the interior partners are inside $I$ they are ``kept,'' not
traced, and the only cost is the area at $\partial I$.
\end{itemize}
A linearly growing branch racing a flat one guarantees a crossing;
the min switches branches at exactly the Page point of
Appendix~\ref{app:Page_curve}, and the late-time entropy rides
$\Sbh(t)$ down the falling edge.  This is the replica-wormhole
result \citep{Penington2020, Almheiri2019, ReplicaWormholes2020},
stated as a ledger race.

\subsection{Step 2 --- why the two formulations agree}

At sub-Page time, the island $I = \emptyset$, the bulk is
traced in the normal way, and the radiation entropy is
$S_R \sim m\ln 2$ (Hawking result). At super-Page time, the
island $I$ picks up bulk volume deep inside the horizon, and
$S(R)$ starts decreasing.

The equivalence of the two formulations:
\begin{enumerate}[leftmargin=2em,itemsep=0.1em]
\item The island boundary $\partial I$ is the boundary in the
bulk between ``traced-out'' and ``kept'' regions.
\item The $A(\partial I)/(4\Newton\hbar)$ term is the
gravitational entropy associated with the partial-trace
structure.
\item The $S_{\rm semi-cl}$ term is the remaining entropy of
the semi-classical state on $R \cup I$.
\item Extremizing over $\partial I$ is the same as optimizing
the partial-trace structure to match the minimal von Neumann
entropy.
\end{enumerate}

In \GGtheory{}, the 6D bulk is explicitly unitary, and the
\OGGEP{} partial trace is a specific operation. The islands
formula is the semi-classical approximation of the \OGGEP{}
output, and the QES ($\partial I$) is the boundary of the
traced-over region.  The precise proposition and its grade ---
where the equivalence is derived, where it is a dictionary entry
--- are registered in the named-statement deck
(Appendix~\ref{app:statement-deck}); the reading is a
translation, not a competing mechanism, and
Appendix~\ref{app:island-dictionary} tabulates it term by term.

\paragraph{Physical picture.}  The island prescription looks
exotic --- interior regions suddenly counted as part of the
radiation --- until it is read as bookkeeping: the min-ext is
deciding, ledger in hand, which purchase is cheaper: keep paying
thermal rent on every emitted quantum's hidden partner, or buy the
interior outright at the fixed price $A/4\Newton\hbar$.  At the
Page point the mortgage beats the rent, and the books --- which
were always unitary in the 6D bulk --- start showing it in 4D.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{M.1.1. Work the two-branch race with
constant rates, find the corner, and explain why the true answer
(0.646) differs.}

\noindent\emph{Solution.}  Step 1: model the no-island branch as
$S_{\rm no\mbox{-}isl}(t) = \lambda t$ and the island branch as
$S_{\rm isl}(t) = S_0 - \mu t$, with constant rates; the min
switches at $t_* = S_0/(\lambda + \mu)$.  Step 2: for thermal
emission, G.4.1 gives $\lambda = \tfrac43\mu$ (radiation gains
$4/3$ of what the hole loses), and the hole spends $S_0$ over
$t_{\rm evap}$, so $\mu = S_0/t_{\rm evap}$: hence
$t_* = \tfrac{3}{7}\,t_{\rm evap} \approx 0.43\,t_{\rm evap}$.
Step 3: the true ratio is $1 - 2^{-3/2} = 0.646$
\eqref{eq:E-pagetime} because the rates are \emph{not} constant:
$T \propto 1/M$ makes the early hole cold and slow, back-loading
the entropy spend; the constant-rate toy front-loads it.  The
corner's existence is rate-independent; its location is a clock
question.  Physics intuition: every version of the Page curve is
the same race between a rising rent and a falling mortgage
(the ledger picture of the text); the schedules differ, the
crossing does not.

\medskip\noindent\textbf{M.1.2. Explain why the island's area
term is $\approx 2S_{\rm BH}$ for the eternal two-sided problem
but $\approx S_{\rm BH}$ for evaporation.}

\noindent\emph{Solution.}  Step 1: for the eternal (two-sided)
black hole with both radiation baths collected, the island is a
spacelike slab through the interior whose boundary
$\partial I$ has \emph{two} components --- one surface just
inside each horizon --- so the area term in
\eqref{eq:island_app} pays twice: $2 \times A/4\Newton\hbar
\approx 2S_{\rm BH}$.  Step 2: for a one-sided evaporating hole
there is a single horizon, a single QES component, and a single
payment $\approx S_{\rm BH}(t)$ at the \emph{instantaneous}
area.  Step 3: correspondingly the eternal case's entropy
saturates at $2S_{\rm BH}$ (the two-sided thermofield double's
total) while the evaporating case's island branch tracks the
shrinking horizon down to zero --- which is what makes the
curve fall.  Physics intuition: the area term counts the
\emph{doors} between the kept and traced regions; the eternal
hole has two doors, the dying hole one, and the ledger charges
per door.

\medskip\noindent\textbf{M.2.1. Derive the equivalence claim:
extremizing $S_{\rm gen}$ is branch-minimizing the \GFF{}
ledger, row by row.}

\noindent\emph{Solution.}  Step 1: the QES functional
\eqref{eq:F-sgen} is $A(x)/4\Newton\hbar + S_{\rm semi\mbox{-}cl}$:
term one is the geometric free-energy piece $F_{\rm GEP}$ and
term two the entropic piece $F_{\rm CEP}$ of the ledger
\eqref{eq:gff-bh} --- the identification of
Table~\ref{tab:island-dictionary}, rows one and two.  Step 2:
extremization over the surface position $x$
\eqref{eq:F-qes} is stationarity of the ledger over the
admissible coarse-grained saddles, i.e.\ \eqref{eq:branch-min}
restricted to the one-parameter family the island geometry
exposes; the min over extrema is the ledger's branch minimum.
Step 3: hence wherever the island family exhausts the admissible
saddles --- the regime stated in the text --- the two
prescriptions compute the same number by construction; where the
ledger admits saddles outside the island family (the
\DGI{}-constrained reconstructions of \S\ref{sec:gff-ledger}),
the \GGtheory{} reading is the wider net.  Physics intuition:
the dictionary is not an analogy but an accounting identity ---
two bookkeeping systems posting the same two costs; they can
only disagree where one system has ledger lines the other
lacks, and that residue is precisely what Part~VII prices.

\GGtakeaway{\emph{Standard result:} the QES condition and the
island rule \eqref{eq:island_app}.  \emph{The \GDOS{} statement:}
\eqref{eq:island_GDOS}, the entropy of the projected state.
\emph{Where they provably agree:} whenever the branch-minimized
\GFF{} ledger is the dominant saddle of the island
extremization --- the regime tabulated in
Appendix~\ref{app:island-dictionary}.  \emph{Where they part:}
provenance --- prescription within the path integral vs.\
theorem of the projection --- and the parting is testable only
through the exterior corrections priced in Part~VII.}

\section{Island/QES and \GGtheory{} language}
\label{app:island-dictionary}

\GGteachbox{The translation table: every term of the island/QES
program --- generalized entropy, extremization, islands,
replica wormholes, bulk reconstruction --- next to its
\GGtheory{}/\GDOS{} reading, one line each, with pointers to the
equations that justify each row.}
The island/quantum-extremal-surface program and the \GGtheory{} reading are translations of one another in the regime where both apply; the table records the dictionary.
\begin{table}[H]
\centering
\caption[Island/\QES{} terms and their \GGtheory{}/\GDOS{} readings.]{Island/\QES{} terms and their \GGtheory{}/\GDOS{} readings. Five rows, and the translation runs one way only: every island-program term is given a reading in this framework's ledger, and no claim is made that the reverse holds. The purpose is to let a reader fluent in the island literature check this Part's derivations against machinery they already trust, and to make plain where the two are doing the same arithmetic under different names.}
\label{tab:island-dictionary}
\small
\begin{tabular}{p{0.42\textwidth}p{0.50\textwidth}}
\toprule
\textbf{Island / QES language} & \textbf{\GGtheory{} / \GDOS{} reading} \\
\midrule
Generalized entropy $S_{\rm gen} = A/4G + S_{\rm bulk}$ & Black-hole instance of the universal ledger $\GFF = F_{\rm GEP}+F_{\rm CEP}$ \eqref{eq:gff-bh}. \\
Extremize/minimize over candidate \QES{} & Branch-minimize the ledger over admissible coarse-grained saddles \eqref{eq:branch-min}. \\
Appearance of an island in the radiation wedge & Late-time dominance of the area-limited bridge branch (\S\ref{sec:branch-switch}). \\
Replica wormhole saddle & Sub-dominant-to-dominant branch exchange in the \GFF{} ledger. \\
Quantum error correction of the bulk & Non-naive, \DGI{}-constrained reconstruction of the accessible algebra (\S\ref{sec:gff-ledger}). \\
\bottomrule
\end{tabular}
\end{table}


\section{Entropy balance, channels, and data processing}
\label{app:data-processing}

\GGteachbox{The quantum-information engine room: reduction as a
\CPTP{} channel (Kraus and Stinespring, constructed), the
data-processing inequality, Spohn's entropy-production identity
worked in full, and the single-mode exchange ledger in closed
form --- the machinery behind the paper's exact
entropy-production statements.}
\GGreadalong{\citet{Harlow2016} for the channel toolkit;
\citet{Spohn1978} for the production identity;
\citet{Wall2012} for the GSL this machinery feeds.}
The black-hole story requires distinguishing fine-grained from operational entropy.  Under closed unitary evolution the global von Neumann entropy is constant, $S(\rho_{\rm global}(t)) = S(\rho_{\rm global}(0))$, yet a reduced state changes because reduction is a channel, and in \GDOS{} language $\rho_\partial(t) = \Pi_t(\rho_6(0))$ with $\Pi_t$ \CPTP{}.  The operational entropy at the boundary is therefore the entropy of the channel \emph{image}, not of ``reality itself.''  This is why data-processing inequalities are physical, not technical: for any channel $\cN$, relative entropy contracts, $S(\cN\rho\,\|\,\cN\sigma) \le S(\rho\,\|\,\sigma)$, so accessible mutual information is bounded by the parent theory's.  The boundary description may lose information without the bulk destroying it --- the precise content of ``information is preserved, but not by the 4D observer alone.''

That paragraph is the chapter's thesis; the four Steps below equip
it.  This is the mathematical spine of Lemma~\ref{lem:gsl-arrow}
(\textsf{L-BH-Arrow}) and Corollary~\ref{cor:gga-crossover}
(\textsf{C-BH-1}), worked at the graduate level.

\subsection{Step 1 --- reduction is a channel: Kraus and
Stinespring}

Let the parent system evolve unitarily on $\mathcal{H}_S \otimes
\mathcal{H}_E$ from a product start $\rho \otimes |0\rangle\langle
0|_E$, and let the observer keep only $S$.  Then
\begin{equation}\label{eq:H-kraus}
\Pi(\rho) = \mathrm{Tr}_E\bigl[U(\rho \otimes |0\rangle\langle 0|_E)
U^\dagger\bigr]
= \sum_k K_k\,\rho\,K_k^\dagger,
\qquad
K_k \equiv \langle k|_E\, U\, |0\rangle_E ,
\end{equation}
by inserting a basis $\{|k\rangle_E\}$ in the trace; completeness
of the basis and unitarity of $U$ give $\sum_k K_k^\dagger K_k =
\mathbb{1}$ (trace preservation), and the manifest form
$\sum_k K_k \cdot K_k^\dagger$ is complete positivity.
Stinespring's theorem is the converse: \emph{every} \CPTP{} map
arises this way from some dilation $(U, \mathcal{H}_E)$.  So
``the observer's description'' and ``a channel'' are the same
mathematical object --- exactly the \GDOS{} statement that
$\rho_\partial(t) = \Pi_t(\rho_6(0))$, with the horizon making the
environment factor causally compulsory
(\textsf{L-BH-Port}, \S\ref{ssec:visible-port}).

\subsection{Step 2 --- the contraction: data processing}

The data-processing inequality (DPI) states that for every channel
$\cN$ and every pair of states,
\begin{equation}\label{eq:H-dpi}
S\bigl(\cN\rho \,\|\, \cN\sigma\bigr) \;\le\; S\bigl(\rho \,\|\,
\sigma\bigr),
\qquad
S(\rho\|\sigma) \equiv \mathrm{Tr}\,\rho\,(\ln\rho - \ln\sigma).
\end{equation}
Its proof pedigree is worth recording once: monotonicity under
partial trace is equivalent to strong subadditivity of the von
Neumann entropy, which rests on Lieb's concavity theorem; the
general-channel case follows by Stinespring --- adjoin the
environment (relative entropy is additive under $\otimes$ with a
fixed ancilla), act unitarily (invariance), and trace out (partial
trace case).  Every arrow statement in this paper descends from
\eqref{eq:H-dpi}; its QFT-grade formulation, needed when the
subsystems are relativistic wedges rather than tensor factors, is
the algebraic monotonicity used by \citet{Casini2008} and by
\citet{Wall2012} --- the ceiling declared in
Lemma~\ref{lem:gsl-arrow}'s Wall-class clause.

\paragraph{Physical picture.}  Relative entropy is the ledger
distance between ``what the state is'' and ``what the observer's
best hypothesis says'': operationally, the exponential rate at
which the hypothesis can be rejected.  A channel can only blur the
evidence --- no post-processing manufactures distinguishability ---
so the distance contracts.  Everything irreversible in this paper
is this one sentence wearing different clothes.

\subsection{Step 3 --- Spohn's production identity, worked}

Let $\Phi_t = e^{t\mathcal{L}}$ be a quantum dynamical semigroup
with stationary state $\sigma$ ($\Phi_t\sigma = \sigma$) --- here,
the horizon's KMS face at $\THaw$
(Theorem~\ref{thm:thermal-clock}).  Two lines give the theorem.
First, since $\mathrm{Tr}\,\dot\rho_t = 0$,
\begin{equation}\label{eq:H-spohnderiv}
\frac{d}{dt} S(\rho_t \| \sigma)
= \mathrm{Tr}\bigl[\dot\rho_t\,(\ln \rho_t - \ln \sigma)\bigr]
\end{equation}
(the term $\mathrm{Tr}[\rho_t \tfrac{d}{dt}\ln\rho_t] =
\mathrm{Tr}\,\dot\rho_t = 0$ drops).  Second, apply the DPI
\eqref{eq:H-dpi} to the channel $\Phi_\epsilon$:
\begin{equation}\label{eq:H-spohn}
S(\rho_{t+\epsilon}\|\sigma) = S\bigl(\Phi_\epsilon \rho_t \,\|\,
\Phi_\epsilon \sigma\bigr) \le S(\rho_t\|\sigma)
\quad\Longrightarrow\quad
\varsigma(t) \equiv -\frac{d}{dt} S(\rho_t\|\sigma) \;\ge\; 0 .
\end{equation}
This is Spohn's theorem \citep{Spohn1978}: the entropy production
$\varsigma$ of a semigroup with a stationary face is pointwise
non-negative, vanishing iff the state sits on the face.
Lemma~\ref{lem:gsl-arrow}'s three named links are visible here in
miniature: the semigroup direction is \emph{consumed} (it is the
arrow custody of AX-3/P5-IX/A5, not derived from thermodynamics),
the DPI supplies the contraction, and the horizon translation
identifies $\sigma$ with the KMS face and $\varsigma$ with GSL
production.

\subsection{Step 4 --- the single-mode exchange ledger, in closed
form}

The exchange-class identity of Lemma~\ref{lem:gsl-arrow} ---
$\Delta S_{\rm gen}$ equals the relative entropy spent toward the
KMS face --- can be displayed mode by mode.  For one bosonic mode,
a thermal state of occupation $n$ is $\rho_n = (1+n)^{-1}
\bigl(\tfrac{n}{1+n}\bigr)^{a^\dagger a}$, so
\begin{equation}
\ln \rho_n = a^\dagger a \,\ln\frac{n}{1+n} - \ln(1+n),
\qquad
\SvN(\rho_n) = (1+n)\ln(1+n) - n \ln n .
\end{equation}
Let the mode arrive in a thermal state at the wrong temperature
--- occupation $\bar n_b$ --- and let the face sit at $\bar n_H =
(e^{\hbar\omega/\kB\THaw}-1)^{-1}$.  Assemble the relative entropy
term by term:
\begin{align}
S(\rho_{\bar n_b} \| \rho_{\bar n_H})
&= -\SvN(\rho_{\bar n_b}) - \mathrm{Tr}\bigl[\rho_{\bar n_b}
\ln \rho_{\bar n_H}\bigr] \nonumber\\
&= -(1+\bar n_b)\ln(1+\bar n_b) + \bar n_b \ln \bar n_b
- \bar n_b \ln\frac{\bar n_H}{1+\bar n_H} + \ln(1+\bar n_H)
\nonumber\\
&= \bar n_b \ln\frac{\bar n_b}{\bar n_H}
- (1+\bar n_b)\,\ln\frac{1+\bar n_b}{1+\bar n_H}\,.
\label{eq:H-relent}
\end{align}
Positivity is one derivative: $\partial_{\bar n_b} S =
\ln\bigl[\bar n_b(1+\bar n_H)/\bar n_H(1+\bar n_b)\bigr]$ vanishes
iff $\bar n_b = \bar n_H$, and $\partial^2_{\bar n_b} S =
1/\bar n_b - 1/(1+\bar n_b) > 0$, so \eqref{eq:H-relent} is
strictly convex with minimum value $0$ exactly on the face.  Read
through the identity of Lemma~\ref{lem:gsl-arrow}, this is the
per-mode GSL: the generalized entropy gained in an exchange is
\eqref{eq:H-relent}, non-negative always, and \emph{zero} iff
$T_b = \THaw$ --- which is Corollary~\ref{cor:gga-crossover}'s
statement that the conditional-\GGA{} crossover is the GSL's
equality point.  (The vacuum limit $\bar n_b \to 0$ gives $S \to
\ln(1+\bar n_H)$, the per-emitted-mode closed form quoted in the
main text.)  The identity, the positivity grid, and the closed
forms are verified to machine precision in the \texttt{s4\_arrow}
suite (Appendix~\ref{app:benchmarks}); the formula
\eqref{eq:H-relent} itself is re-checked in
\texttt{p2c2\_verify}.

\paragraph{Physical picture.}  The horizon face at $\THaw$ is the
sea level of the entropy ledger.  A mode exchanged with the hole
at any other temperature carries a measurable ``height'' above
sea level --- the relative entropy \eqref{eq:H-relent} --- and the
GSL is the statement that exchanges only ever spend height, never
mint it.  Equilibrium at exactly $\THaw$ is sea level itself: the
unique exchange that costs nothing, and the crossover point where
the conditional-\GGA{} bound is saturated.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{O.1.1. Prove that the Kraus form is
trace-preserving and completely positive.}

\noindent\emph{Solution.}  Step 1 (TP): from \eqref{eq:H-kraus},
$\mathrm{Tr}\,\Pi(\rho) = \mathrm{Tr}\bigl[\sum_k K_k^\dagger
K_k\,\rho\bigr]$, and $\sum_k K_k^\dagger K_k =
\sum_k \langle 0|U^\dagger|k\rangle\langle k|U|0\rangle_E =
\langle 0|U^\dagger U|0\rangle_E = \mathbf{1}$ by completeness
and unitarity: traces are preserved.  Step 2 (CP): extend by an
ancilla $A$ of any dimension; $(\Pi \otimes \mathrm{id}_A)$
acting on a positive $\rho_{SA}$ gives
$\sum_k (K_k \otimes \mathbf{1})\rho_{SA}(K_k \otimes
\mathbf{1})^\dagger$, a sum of terms of the form
$X\rho X^\dagger \geq 0$: positivity survives the extension,
which is the definition of complete positivity.  Step 3:
Stinespring's converse (quoted in the text) closes the loop:
\emph{every} CP, trace-preserving map has this dilated form ---
observer descriptions and channels are one class.
Physics intuition: complete positivity is the statement that a
physical evolution must remain physical even when the system is
secretly entangled with a bystander --- exactly the situation of
every black-hole interior register, which is why \CPTP{} is the
non-negotiable grammar of the whole paper.

\medskip\noindent\textbf{O.2.1. Prove the two easy legs of the
data-processing pedigree: unitary invariance and stability under
fixed ancillas.}

\noindent\emph{Solution.}  Step 1 (unitaries): relative entropy
\eqref{eq:H-dpi} is a trace of operator functions of its two
arguments, and $\mathrm{Tr}\,f(U\rho U^\dagger) =
\mathrm{Tr}\,f(\rho)$: hence $S(U\rho U^\dagger\,\|\,U\sigma
U^\dagger) = S(\rho\|\sigma)$ --- reversible processing costs
nothing.  Step 2 (ancillas): for product extensions,
$\ln(\rho \otimes \tau) = \ln\rho \otimes \mathbf{1} +
\mathbf{1} \otimes \ln\tau$, so
$S(\rho \otimes \tau\,\|\,\sigma \otimes \tau) = S(\rho\|\sigma)
+ \mathrm{Tr}\,\tau(\ln\tau - \ln\tau) = S(\rho\|\sigma)$:
attaching an uncorrelated bystander adds nothing.  Step 3: the
third leg --- monotonicity under partial trace, the hard one
resting on Lieb's concavity --- is exactly what the text
imports; combining the three gives contraction under every
Stinespring dilation, i.e.\ under every channel.
Physics intuition: the two easy legs say distinguishability is
a property of \emph{information}, not of representation; the
hard leg says forgetting can only blur it.  Everything
irreversible in this paper is the hard leg wearing different
clothes.

\medskip\noindent\textbf{O.4.1. Check the three limits of the
closed-form exchange ledger \eqref{eq:H-relent}.}

\noindent\emph{Solution.}  Step 1 ($\bar n_b \to \bar n_H$): both
terms of \eqref{eq:H-relent} vanish ---
$\bar n_b\ln(\bar n_b/\bar n_H) \to 0$ and the second term's
argument $\to 1$: zero cost on the face, as Spohn's theorem
requires (equilibrium is sea level).  Step 2 ($\bar n_b \to 0$,
the vacuum limit): the first term dies and the second gives
$-\ln\bigl[1/(1 + \bar n_H)\bigr] = \ln(1 + \bar n_H)$ --- the
per-emitted-mode closed form quoted in the main text.  Step 3
($\bar n_b \to \infty$): the two logarithms combine and the
ledger grows as
$\bar n_b\,\ln\bigl[(1+\bar n_H)/\bar n_H\bigr]$ ---
linearly in the occupation: overdriving a mode far
above the face costs proportionally to how much is sent.
Physics intuition: the ledger is a distance-from-equilibrium
meter with the right three readings --- zero on the face, finite
for the vacuum, unbounded for a blowtorch --- and the GSL is
the statement that the meter never runs backwards.

\medskip\noindent\textbf{O.4.2. Price a typical Hawking quantum:
the ledger cost of one vacuum-limit mode at
$\hbar\omega = k_B\THaw$.}

\noindent\emph{Solution.}  Step 1: the face occupation at
$\hbar\omega = k_B\THaw$ is $\bar n_H = 1/(e - 1) \approx
0.582$.  Step 2: the vacuum-limit cost (O.4.1, Step 2) is
$\ln(1 + \bar n_H) = \ln\bigl[e/(e-1)\bigr] \approx 0.459$~nat
$\approx 0.66$~bit per mode.  Step 3: this is the per-mode
microscopic face of the global $\tfrac13$ margin of G.4.1 ---
summed over the emitted spectrum, the per-mode ledger reproduces
the $\tfrac43$ radiation-to-hole entropy ratio (the integral is
logged in the benchmark audit's \texttt{s4\_arrow} suite).
Physics intuition: each Hawking quantum leaves carrying about
two-thirds of a bit of irreversibility --- the price the
exterior pays for reading the bulk through a thermal port, mode
by mode, exactly accounted.

\GGtakeaway{\emph{Standard results:} Stinespring dilation,
data processing, Spohn's identity, and the exchange-class
entropy production in closed form.  \emph{What they assume:}
only that reduction is a channel --- no gravity anywhere.
\emph{The \GGtheory{} reading:} at a horizon these theorems stop
being effective descriptions and become exact, because the
channel is geometrically forced; the GSL's direction is
inherited, not assumed.}

\section{Extremal black holes and the third law}
\label{app:extremal}

\GGteachbox{The charged, rotating sector where black-hole
thermodynamics earns its Legendre structure: the Kerr--Newman
horizon data, the area and the Christodoulou mass split, the
first law and the Smarr formula derived, the third law in its
\GDOS{} reading, and superradiance worked as the rotating
anchor's instance.}
\GGreadalong{\citet{Chandrasekhar1983} chs.~6--7 and
\citet{FrolovNovikov1998} chs.~6--8 for the full classical
theory; \citet{Poisson2004} ch.~5 for the mechanics.}

Rotation and charge are where black-hole thermodynamics earns its
Legendre structure: new charges, new potentials, a Smarr formula,
and a third law.  This chapter works the Kerr--Newman horizon data
(Step~1), the area and the Christodoulou mass (Step~2), the first
law and Smarr formula (Step~3), the third law in its \GDOS{}
reading (Step~4), and superradiance as the worked instance of the
rotating anchor (Step~5).  Geometrized units $G_N = c = 1$ except
where displayed.

\subsection{Step 1 --- the Kerr--Newman horizon data}

The Kerr--Newman family (mass $M$, spin $a = J/M$, charge $Q$) has
horizon structure governed by
\begin{equation}
\Delta(r) = r^2 - 2Mr + a^2 + Q^2 = (r - r_+)(r - r_-),
\qquad
r_\pm = M \pm \sqrt{M^2 - a^2 - Q^2},
\end{equation}
so horizons exist iff $a^2 + Q^2 \le M^2$, with equality the
\emph{extremal} case $r_+ = r_-$.  The surface gravity and horizon
angular velocity are
\begin{equation}\label{eq:I-kappa}
\kappa = \frac{r_+ - r_-}{2\,(r_+^2 + a^2)}
= \frac{\sqrt{M^2 - a^2 - Q^2}}{r_+^2 + a^2},
\qquad
\Omega_H = \frac{a}{r_+^2 + a^2},
\end{equation}
the first equality the standard evaluation of $\kappa^2 =
-\tfrac12 (\nabla^\mu \xi^\nu)(\nabla_\mu \xi_\nu)|_{r_+}$ for the
horizon generator $\xi = \partial_t + \Omega_H \partial_\phi$
(equivalently $\kappa = \Delta'(r_+)/2(r_+^2+a^2)$).  At
extremality ($a^2 + Q^2 = M^2$) the surface gravity vanishes:
$\kappa = 0$.  The Hawking temperature correspondingly vanishes:
$\THaw = 0$ --- a thermodynamic system with zero temperature but,
as Step~2 shows, non-zero entropy.

\subsection{Step 2 --- area, spin, and the Christodoulou mass}

For Kerr ($Q = 0$), the horizon area is
\begin{equation}\label{eq:Kerr_area}
A_{\rm Kerr} = 4\pi(r_+^2 + a^2) = \frac{8\pi\Newton^2 M^2}{c^4}
\left[1 + \sqrt{1 - a^{*2}}\right],
\end{equation}
where $r_+ = \Newton M/c^2 + \sqrt{(\Newton M/c^2)^2 - a^2}$ is
the outer horizon radius and $a^* = a c^{2}/\Newton M$ is the
dimensionless spin.  The first equality reduces via one identity
worth deriving: $r_+$ solves $r_+^2 - 2Mr_+ + a^2 = 0$, so
\begin{equation}
r_+^2 + a^2 = 2 M r_+ = 2M\bigl(M + \sqrt{M^2 - a^2}\bigr)
= 2\bigl(M^2 + \sqrt{M^4 - J^2}\bigr),
\end{equation}
using $a = J/M$.  Extremal Kerr has $a^* = 1$ and
$A^{\rm ext} = 8\pi\Newton^2 M^2/c^4$, one-half the
Schwarzschild value.  Inverting for $M$: from $A = 8\pi(M^2 +
\sqrt{M^4 - J^2})$, isolate the root and square,
\begin{equation}
\Bigl(\frac{A}{8\pi} - M^2\Bigr)^{2} = M^4 - J^2
\;\Longrightarrow\;
\boxed{\;M^{2} = \frac{A}{16\pi} + \frac{4\pi J^{2}}{A}\;}
\end{equation}
--- the Christodoulou mass formula.  Its first term defines the
irreducible mass, $M_{\rm irr}^2 = A/16\pi$: the part of the mass
that the area theorem forbids extracting.  The second term is
rotational energy, extractable in principle (Penrose process,
superradiance --- Step~5) down to $M \to M_{\rm irr}$; for
extremal Kerr, up to $1 - 1/\sqrt{2} \approx 29\%$ of the mass.

\subsection{Step 3 --- the first law and the Smarr formula}

Differentiate the Christodoulou formula at fixed $J$ and fixed
$A$ respectively:
\begin{equation}\label{eq:I-firstlaw}
\Bigl(\frac{\partial M}{\partial A}\Bigr)_{\!J}
= \frac{1}{2M}\Bigl(\frac{1}{16\pi} - \frac{4\pi J^2}{A^2}\Bigr)
= \frac{\kappa}{8\pi},
\qquad
\Bigl(\frac{\partial M}{\partial J}\Bigr)_{\!A}
= \frac{4\pi J}{A M} = \Omega_H ,
\end{equation}
where both right-hand equalities are checked against
\eqref{eq:I-kappa} by direct substitution (and to $10^{-12}$
numerically in \texttt{p2c2\_verify}).  Hence the first law
\begin{equation}
dM = \frac{\kappa}{8\pi}\, dA + \Omega_H\, dJ
= \THaw\, d\Sbh + \Omega_H\, dJ ,
\end{equation}
identifying $\Sbh = A/4$ once $\THaw = \kappa/2\pi$ is fixed by
Appendix~\ref{app:Hawking_T}.  The Smarr formula follows from
scaling alone: under $M \to \lambda M$ every length doubles with
$\lambda$, so $A \to \lambda^2 A$ and $J \to \lambda^2 J$, i.e.\
$M(A, J)$ is homogeneous of degree $\tfrac12$; Euler's theorem
gives
\begin{equation}\label{eq:I-smarr}
M = 2\,\frac{\partial M}{\partial A}\,A
+ 2\,\frac{\partial M}{\partial J}\,J
= 2\,\THaw \Sbh + 2\,\Omega_H J ,
\end{equation}
the algebraic shadow of the first law, with the factor $2$ the
homogeneity degree's inverse.  (With charge, $+\Phi_H Q$ joins the
first law and $M = 2\THaw\Sbh + 2\Omega_H J + \Phi_H Q$: the
$Q$-term enters with weight one because $Q \to \lambda Q$.)

\paragraph{Physical picture.}  The first law is local bookkeeping
at the horizon; the Smarr formula is the same books audited
globally, and the audit works because gravity has no intrinsic
scale --- homogeneity is the entire input.  That thermodynamic
identities follow from dimensional analysis is the first hint
that the entropy is geometric.

\subsection{Step 4 --- the third law, \GDOS{}-read}

In \GDOS{}, the third law of thermodynamics
(Nernst-unattainability) is automatic: reaching the extremal
state requires the scale profile at the horizon to vanish
$e^{A(r_h)} \to 0$, which corresponds to infinite scale depth
and hence cannot be achieved in finite external time.  The
classical statement is the same limit read as geometry: as $a^2 +
Q^2 \to M^2$, the two roots $r_\pm$ merge, $\kappa \to 0$ by
\eqref{eq:I-kappa}, and the Euclidean cigar of
Appendix~\ref{app:euclidean} degenerates --- its tip recedes to
infinite proper distance, so there is no smooth cap and no
regularity argument: extremal holes sit outside the thermal
ensemble, reachable only asymptotically.

Extremal black holes are interesting for microstate-counting
arguments because their entropy survives at $T = 0$:
$\Sbh^{\rm ext} = A^{\rm ext}c^{3}/(4\Newton\hbar)$, giving
$e^{\Sbh/\kB}$ microstates in the ground state. Strominger--Vafa
counted these microstates explicitly for certain 5D extremal
black holes using D-brane configurations in string theory
\citep{StromingerVafa1996}. In
\GGtheory{}, the counting is via $\UoneQi$ holonomies on
$S^1_\theta$ at each horizon pixel.

\subsection{Step 5 --- superradiance, the worked instance}

The Bogoliubov transformation near a rotating horizon involves
both the surface gravity $\kappa$ and the horizon angular
velocity $\Omega_H$; the resulting spectrum is thermal with
chemical potential:
\begin{equation}\label{eq:Kerr_spectrum}
\langle N_{\omega \ell m}\rangle = \frac{1}{e^{(\hbar\omega -
m\hbar\Omega_H)/\kB\THaw}- 1},
\end{equation}
and one gets superradiant emission for $\omega < m\Omega_H$.  The
mechanism is one boundary condition: a mode $e^{-i\omega t +
im\phi}$ near the horizon depends on the generator's combination
$\tilde\omega \equiv \omega - m\Omega_H$ (the frequency in the
co-rotating frame), entering the horizon as $e^{-i\tilde\omega
r_*}$.  The conserved energy flux through the horizon is
proportional to $\omega\,\tilde\omega\,|T_{\omega\ell m}|^2$, so
for $0 < \omega < m\Omega_H$ the ``absorbed'' flux is
\emph{negative}: the hole pays energy into the mode, the
reflected wave exceeds the incident one, and the occupation
formula above encodes the same fact as a pole crossing
($\langle N\rangle < 0$ formally flags the amplified channel).
The structural theorem behind this reading --- the rotating-frame
selection failure and superradiance as a symplectic pumping
channel on the asymptotic anchor --- is T-BH-3,
\S\ref{sec:rotating-collapse}.
In \GDOS{}, the superradiance is the 4D observer seeing
energy flow \emph{out} of the black hole via modes whose
bulk phase-space volume extends outside the rotating ergosphere.

\paragraph{Physical picture.}  A spinning hole is a flywheel
wearing a one-way membrane: modes that rotate slower than the
horizon drag against the flywheel and are paid for the friction.
The Christodoulou split of Step~2 says exactly how much is in the
till --- everything above $M_{\rm irr}$ --- and the area theorem
is the statement that the till never absorbs from the
irreducible account.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{P.1.1. Check the Kerr--Newman surface
gravity \eqref{eq:I-kappa} in its three instructive limits.}

\noindent\emph{Solution.}  Step 1 (Schwarzschild, $a = Q = 0$):
$r_\pm = M \pm M$, so $r_+ = 2M$, $r_- = 0$, and $\kappa =
(r_+ - r_-)/2(r_+^2 + a^2) = 2M/8M^2 = 1/4M$ --- the geometric
form of \eqref{eq:kappa-sch}.  Step 2 (extremal, $a^2 + Q^2 =
M^2$): $r_+ = r_-$ and $\kappa = 0$ identically, whatever the
split between spin and charge: extremality is a property of the
double root, not of the route to it.  Step 3 (slow spin,
$Q = 0$, $a \ll M$): with $x = a^2/M^2$, the numerator of
\eqref{eq:I-kappa} is $M(1 - x/2) + O(x^2)$ and the denominator
$4M^2(1 - x/4) + O(x^2)$, so
$\kappa \approx \tfrac{1}{4M}\bigl(1 - a^2/4M^2\bigr)$:
rotation \emph{cools} the hole at second order, the first hint of
the third law's asymptote.
Physics intuition: $\kappa$ measures the gap between the two
roots of $\Delta$ --- the horizon and its inner partner --- and
temperature dies exactly as the two merge; extremality is the
thermodynamic zero written as a coincidence of radii.

\medskip\noindent\textbf{P.2.1. How much of an extremal Kerr
hole's mass is extractable in principle?}

\noindent\emph{Solution.}  Step 1: for extremal Kerr ($a = M$,
$Q = 0$) the horizon area \eqref{eq:Kerr_area} is
$A = 8\pi M^2$ (half the Schwarzschild value for the same
mass).  Step 2: the irreducible mass is $M_{\rm irr} =
\sqrt{A/16\pi} = M/\sqrt2$: the area theorem forbids touching
it.  Step 3: the extractable fraction is
$1 - 1/\sqrt2 \approx 29.3\%$ of the total mass-energy ---
by Penrose processes or superradiance, in principle down to a
Schwarzschild remnant of mass $M/\sqrt2$.  For comparison:
nuclear fusion frees $0.7\%$, ISCO accretion $5.7\%$ (E.3.3).
Physics intuition: a maximally spinning hole is the universe's
best battery --- nearly a third of its rest energy sits in the
till as rotational energy above the irreducible account, and
the area theorem is the banking regulation that says which
withdrawals are lawful.

\medskip\noindent\textbf{P.3.1. Extend the Smarr formula to
include charge by the same scaling argument.}

\noindent\emph{Solution.}  Step 1: under $M \to \lambda M$ the
scaling weights are $A \to \lambda^2A$, $J \to \lambda^2J$ (as
in the text), and $Q \to \lambda Q$ --- charge carries length
dimension one in geometrized units.  Step 2: Euler's theorem for
a function of weight 1 in arguments of weights $(2, 2, 1)$ reads
$M = 2\,\frac{\partial M}{\partial A}A +
2\,\frac{\partial M}{\partial J}J +
\frac{\partial M}{\partial Q}Q$.  Step 3: inserting the first
law \eqref{eq:I-firstlaw} identifications gives
$M = 2T_{\rm H}S_{\rm BH} + 2\Omega_HJ + \Phi_HQ$
\eqref{eq:I-smarr}: the charge term enters with weight one,
exactly as the text's parenthetical states.  Physics intuition:
the Smarr formula is dimensional analysis promoted to a theorem
--- gravity has no intrinsic scale, so the first law's local
bookkeeping must integrate to a global identity, with each
charge weighted by how it scales.

\medskip\noindent\textbf{P.5.1. Compute the superradiant window
for a $10\,\Msun$ hole at $a_* = 0.9$.}

\noindent\emph{Solution.}  Step 1: the horizon radius is
$r_+ = M(1 + \sqrt{1 - a_*^2}) = 1.436\,M$ (geometrized), and
the horizon angular velocity
$\Omega_H = a/(r_+^2 + a^2) = 0.313/M$.  Step 2: restore units
with $1/M \to c^3/(\Newton M)$: for $M = 10\,\Msun$,
$\Omega_H \approx 6.4\times10^{3}$~rad\,s$^{-1}$.  Step 3: the
$m = 1$ superradiant condition $\omega < m\,\Omega_H$
\eqref{eq:Kerr_spectrum} amplifies waves below
$f = \Omega_H/2\pi \approx 1.0$~kHz --- squarely the
ground-based gravitational-wave band, which is why
superradiance physics is not exotica but part of the waveform
modeling for the ringdowns Part~VIII scores.  Physics
intuition: the hole is a flywheel spinning a thousand times a
second; any wave patient enough to rotate more slowly gets
dragged, and paid --- the one-way membrane runs a toll in
reverse for slow traffic.

\GGtakeaway{\emph{Standard results:} the Kerr--Newman horizon
data, $M_{\rm irr}$, the first law with work terms, the Smarr
formula, the third law, and superradiance.  \emph{What they
assume:} classical stationarity and cosmic censorship.
\emph{The \GGtheory{} reading:} the work terms are the ledger's
chemical potentials; extremality is the port never quite
closing; superradiance is the rotating anchor paying friction
through the one-way membrane.}

\section{De Sitter horizon and cosmology}
\label{app:de_Sitter}

\GGteachbox{The same theorem with the observer on the inside:
the static patch and its surface gravity, the temperature and
entropy of the cosmological horizon with units restored, and the
\GDOS{} reading that makes the de~Sitter entropy the black-hole
theorem evaluated on a different real section --- the cohort's
bridge to the cosmology of \ATen{} and \PaperCTwo{}.}
\GGreadalong{\citet{GibbonsHawking1977} for the original;
\citet{ArisakaA10Cosmos} Part~VII for the cosmological
consumer.}

The de Sitter horizon is the same theorem with the observer on the
inside, and it is the cohort's bridge to cosmology.  Step~1 works
the static patch; Step~2 restores units and evaluates the two
famous numbers; Step~3 gives the \GDOS{} reading.

\subsection{Step 1 --- the static patch and its surface gravity}

A de Sitter space (Hubble expansion with $H = $ const, positive
cosmological constant $\Lambda = 3H^2$) admits, around any
observer, static coordinates
\begin{equation}
ds^2 = -\bigl(1 - H^2 r^2\bigr)\,dt^2
+ \bigl(1 - H^2 r^2\bigr)^{-1} dr^2 + r^2 d\Omega_2^2,
\end{equation}
with $f(r) = 1 - H^2r^2$ vanishing at the cosmological horizon
$r_H = 1/H$ ($= c/H$ with units).  The observer sits at the
\emph{center}; the horizon surrounds her.  The surface gravity
follows exactly as in Appendix~\ref{app:Hawking_T}, Step~4:
\begin{equation}
\kappa = \tfrac12\,\bigl|f'(r_H)\bigr|
= \tfrac12 \cdot 2H^2 r_H = H ,
\end{equation}
and the near-horizon metric is again Rindler ($f = \kappa^2 x^2$
in the proper distance $x$ from the horizon --- the same two-line
computation), so the whole Bogoliubov/continuation apparatus of
that chapter, and the cigar regularity of
Appendix~\ref{app:euclidean} (the Euclidean section is the round
four-sphere, whose smoothness fixes the period $2\pi/H$), apply
verbatim.

\subsection{Step 2 --- temperature and entropy, with units}

The Gibbons--Hawking temperature and entropy are
\begin{equation}\label{eq:GH_thermo}
\kB T_{\rm GH} = \frac{\hbar H}{2\pi},\qquad
\frac{S_{\rm dS}}{\kB} = \frac{\pi r_H^2}{\Newton\hbar/c^3}
= \frac{\pi c^5}{\Newton\hbar H^2}
\end{equation}
\citep{GibbonsHawking1977} --- the first line is
$\THaw = \hbar c\,\kappa_{\rm geom}/2\pi\kB$ with $\kappa_{\rm
geom} = H/c$, the second is $A_H/4\ell_P^2$ with $A_H = 4\pi
r_H^2$ and $\ell_P^2 = \Newton\hbar/c^3$.
For our universe --- taking the corridor's global background
branch $H_{\rm bg} \simeq 66.6$~km/s/Mpc (per the companion C2;
the ladder's $73.5$ is a locally lifted readout whose full lift is
C2's declared open problem; the order-of-magnitude values below
are insensitive to the branch choice) ---
$r_H = c/H_0 \sim 1.4\times 10^{26}$~m, $T_{\rm GH} \sim
3\times10^{-30}$~K, and
$S_{\rm dS} \sim 2\times 10^{122}$ (the famous cosmological
entropy bound; all three evaluated in \texttt{p2c2\_verify}).
The temperature is thirty orders of magnitude below the CMB ---
unobservable directly, which is why the de Sitter rows of the
falsifier apparatus run through structural consistency rather
than direct detection.

\paragraph{Physical picture.}  The black-hole observer stands
outside a region she cannot see into; the de Sitter observer
stands \emph{inside} a region she cannot see out of.  Both
horizons hide partners of local vacuum modes, both therefore
carry a temperature set by their surface gravity, and both charge
entropy by the same quarter-area rule --- one theorem, two signs
of the enclosure.

\subsection{Step 3 --- the \GDOS{} reading}

In \GDOS{}, the de Sitter horizon is a cosmological \OGGEP{}
port, exactly analogous to the black-hole horizon but with
the observer \emph{inside} a causally disconnected region
rather than outside. The information content observable to an
inertial observer is bounded by $S_{\rm dS}$ from above; this
is Banks's idea of ``holographic cosmology.''  The same-arena
clause of the analytic-shore license (\S\ref{ssec:V-strategy})
covers this case explicitly: the continuation that builds the
Euclidean four-sphere is licensed by the identical one-sided
spectrum argument, so the de Sitter entropy is the same theorem
as the black-hole one, evaluated on a different real section.

The relation to the Hubble-tension paper \citep{ArisakaC2Hubble} is direct:
the Hubble tension is resolved by identifying distinct
4D-projection rates for local-geometric vs. cosmological
observables. The de Sitter horizon is where the distinction
becomes operationally sharp.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{Q.1.1. Verify that the static patch
solves the Einstein equations with $\Lambda = 3H^2$.}

\noindent\emph{Solution.}  Step 1: the metric is
\eqref{eq:sch-ansatz}-form with $e^{2\alpha} = e^{-2\beta} = f =
1 - H^2r^2$; the vacuum-with-$\Lambda$ equations read
$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0$, equivalently
$R_{\mu\nu} = \Lambda g_{\mu\nu}$ in four dimensions.  Step 2:
for this metric family a two-line computation (the $\theta\theta$
equation of Appendix~E, Step 1, with the new $f$) gives
$(rf)' = (r - H^2r^3)' = 1 - 3H^2r^2$, so the $\theta\theta$
equation $1 - (rf)' = \Lambda r^2$ holds iff
$\Lambda = 3H^2$.  Step 3: the remaining components are then
satisfied identically ($R_{\mu\nu} = 3H^2g_{\mu\nu}$: de Sitter
is maximally symmetric).  Physics intuition: $\Lambda = 3H^2$ is
the dictionary between the cosmologist's expansion rate and the
relativist's constant --- one number wearing two habits, and the
horizon at $r_H = 1/H$ is where the two descriptions meet.

\medskip\noindent\textbf{Q.2.1. Show that the de Sitter thermal
wavelength is exactly the horizon scale.}

\noindent\emph{Solution.}  Step 1: from \eqref{eq:GH_thermo},
$k_BT_{\rm dS} = \hbar H/2\pi$.  Step 2: the corresponding
thermal wavelength is $\lambda = \hbar c/(k_BT_{\rm dS}) =
2\pi c/H$.  Step 3: the horizon radius is $r_H = c/H$, so
$\lambda = 2\pi\,r_H$ --- the circumference of the horizon,
exactly.  A ``detector'' smaller than the horizon can never
resolve even one full oscillation of the typical quantum.
Physics intuition: this is why the de Sitter temperature is
unobservable in any local sense and why the text's falsifier
rows for the de Sitter sector run through structural
consistency rather than direct detection: the radiation is not
faint so much as \emph{larger than any laboratory}, including
the causal patch itself.

\medskip\noindent\textbf{Q.2.2. Derive the famous $10^{122}$ and
give it its holographic reading.}

\noindent\emph{Solution.}  Step 1: from \eqref{eq:GH_thermo},
$S_{\rm dS}/k_B = \pi c^5/(\Newton\hbar H^2)$; with the
present-epoch $H \approx 2.2\times10^{-18}\,\mathrm{s}^{-1}$,
$S_{\rm dS} \approx 2\times10^{122}\,k_B$.  Step 2: rewrite
using $r_H = c/H$ and $\ell_P^2 = \Newton\hbar/c^3$:
$S_{\rm dS}/k_B = \pi\,r_H^2/\ell_P^2$ --- one quarter of the
horizon area in Planck pixels, the same quarter-area law as the
black hole (C-BH-2).  Step 3: the number is the largest entropy
in the observable universe by some seventeen orders of
magnitude (the supermassive-hole inventory totals
$\sim\!10^{105}\,k_B$): the cosmological ledger is dominated not
by anything \emph{in} the universe but by the horizon that
bounds the view.  Physics intuition: $10^{122}$ is the
information capacity of our port --- in the paper's reading, the
total bandwidth of the projection every 4D observer lives
behind; cosmology's famous coincidences (the cohort's
$\Lambda$-sector results, \ATen{}) are statements about this
one number.


\GGtakeaway{\emph{Standard result:} $T_{\rm dS} = \hbar c H /
(2\pi k_B)$ and $S_{\rm dS} = \pi k_B c^5/(\Newton\hbar H^2)
\sim 10^{122}\,k_B$ today.  \emph{What it assumed (in 1977):}
the same two formal moves as the black-hole case.  \emph{The
\GGtheory{} reading:} the same theorem on a different real
section, with the observer inside --- the module's bridge to
the cosmological boundary stratum (\ATen{}, \PaperCTwo{}).}

\section{The induced-gravity species sum}
\label{app:species-sum}

\GGteachbox{The program's one declared open computation, set up
honestly: the heat-kernel toolkit and where the sum rule comes
from, the closure condition as a sum rule on the field content,
why the Kaluza--Klein tower makes it self-consistent, the role of
\textsc{gu-5}, the Standard-Model sector --- and the honest
verdict on what is and is not established.}
\GGreadalong{\citet{ParkerToms2009} for the heat-kernel
technology; \citet{Sakharov1968} and \citet{Jacobson1994} for
induced gravity; \citet{SusskindUglum1994} for the species
problem this appendix closes in on.}

Section~\ref{ssec:V-conical} established $\Sbh = A/4\Newton$ with the
exact coefficient, for \emph{whatever} value the induced Newton constant
takes.  This appendix asks the remaining quantitative question: does the
induced $\Newton$, summed over the actual \GGtheory{} spectrum, equal the
$\Newton$ that the scale reduction (\S\ref{ssec:V-induced-G}) and
\textsc{gu-5} assign?  We set up the computation, identify what makes it
close, and state plainly what is and is not established.

\subsection{Step 1 --- the heat-kernel toolkit: where the sum rule
comes from}

The sum rule that this appendix runs on is a one-loop statement,
and its derivation is short enough to display in full.  For a
field with quadratic fluctuation operator $\Delta = -\Box + m^2 +
\xi R$ on a Euclidean background, the one-loop effective action is
\begin{equation}\label{eq:K-heatrep}
W = \tfrac12 \ln\det\Delta
= -\tfrac12 \int_0^\infty \frac{ds}{s}\,
\mathrm{Tr}\, e^{-s\Delta},
\end{equation}
the proper-time representation ($s$ has dimension
length$^2$).  The trace has the Seeley--DeWitt small-$s$
expansion
\begin{equation}\label{eq:K-sdw}
\mathrm{Tr}\, e^{-s\Delta}
= \frac{e^{-s m^2}}{(4\pi s)^{2}} \int d^4x \sqrt{g}\,
\Bigl[\, 1 + s\,\bigl(\tfrac16 - \xi\bigr) R + O(s^2) \Bigr],
\end{equation}
whose first two coefficients are the only ones this appendix
needs: $a_0 = 1$ (the flat-space volume term, feeding the induced
cosmological constant) and $a_1 = (\tfrac16 - \xi)R$ (the
curvature term, feeding the induced Einstein--Hilbert action)
\citep{Wald1984, Visser2002}.  Insert \eqref{eq:K-sdw} into
\eqref{eq:K-heatrep} and read off the coefficient of
$\int\sqrt{g}\,R$:
\begin{equation}\label{eq:K-induced}
\frac{1}{16\pi G_{\rm ind}}
= \frac{1}{2}\,\frac{\tfrac16 - \xi}{16\pi^2}
\int_{1/\Lambda^2}^{\infty} \frac{ds}{s^{2}}\, e^{-s m^2}
= \frac{\bigl(\tfrac16 - \xi\bigr)\Lambda^2}{32\pi^2}
+ O\bigl(m^2 \ln \Lambda\bigr),
\end{equation}
the quadratically divergent leading term regulated by a
proper-time cutoff at $s = 1/\Lambda^2$.  Three remarks make this
graduate-safe.  \emph{(i)} The $O(1)$ normalization of the
divergence is regulator convention --- proper-time, dimensional
subtraction at a matching scale, and Pauli--Villars distribute the
factor between $\Lambda^2$ and the coefficient differently; the
physical statement is the \emph{structure} $1/G_{\rm ind} \propto
(\text{weighted count})\times\Lambda^2$, and the convention is
absorbed into the definition of the physical cutoff (this is
half of the ``cutoff identification'' entry in the honest-gap
list below, and \citet{Visser2002} catalogs the freedom).  Arithmetically: the sum rule of Step~2
quotes the $\chi$ table in the convention $\Lambda^{2}_{\rm Step\,2}
\equiv \tfrac12\,\Lambda^{2}_{\rm proper\mbox{-}time}$, which
absorbs the explicit factor $\tfrac12$ of \eqref{eq:K-induced}; the
two displays agree identically under that identification.
\emph{(ii)} A conformally coupled scalar, $\xi = \tfrac16$,
contributes zero at this order --- the first entry of the $\chi$
table in the sum rule, now derived rather than quoted.
\emph{(iii)} Higher spins repeat the computation with the
appropriate fluctuation operators; their $a_1$ coefficients carry
the spin-dependent signs that the honest-gap list flags
(\citealp{CallanWilczek1994, Kabat1995, DemersLafranceMyers1995}).
With the toolkit displayed, the sum rule of Step~2 is a
definition-chase: sum \eqref{eq:K-induced} over every propagating
degree of freedom and name each field's coefficient $\chi_i$.

\paragraph{Physical picture.}  Sakharov's point
\citep{Sakharov1968} survives every refinement: gravity's
stiffness $1/G$ is (at least in part) vacuum polarization ---
the elasticity a medium acquires because bending it costs its
resident fields zero-point energy.  The heat kernel is the
bookkeeping device: each field, each mass, each spin deposits its
$a_1$ coefficient, and the ledger's total is the Newton constant
the low-energy observer measures.

\subsection{Step 2 --- the closure condition as a sum rule}
The one-loop induced Einstein--Hilbert coefficient is, from the $a_1$
Seeley--DeWitt coefficient of each field's fluctuation operator,
\begin{equation}
\frac{1}{16\pi G_{\rm ind}}
= \frac{\Lambda^2}{16\pi^2}\sum_i \chi_i,
\qquad
\chi_{\rm scalar}=\tfrac16-\xi,\quad \chi_{\rm conf.\ scalar}=0,\quad
\chi_{1/2},\,\chi_1\ \text{(spin-dependent)},
\label{eq:B-sumrule}
\end{equation}
with $\Lambda$ the physical ultraviolet cutoff and the sum over every
propagating degree of freedom.  Closure is the statement
\begin{equation}
\boxed{\;\frac{\Lambda^2}{16\pi^2}\sum_i\chi_i = \frac{1}{16\pi\Newton}\;}
\qquad\Longleftrightarrow\qquad
\sum_i\chi_i = \pi\Bigl(\frac{M_{\rm Pl}}{\Lambda}\Bigr)^{2},
\label{eq:B-closure}
\end{equation}
using $\Newton = 1/M_{\rm Pl}^2$.  This is a \emph{sum rule} relating the
weighted species count $\sum_i\chi_i$ to the Planck-to-cutoff ratio --- the
modern form of Sakharov's self-consistency condition
\citep{Sakharov1968, Adler1982, Visser2002}.

\subsection{Step 3 --- why the Kaluza--Klein tower makes it self-consistent}
The sum in \eqref{eq:B-sumrule} is not over a finite list: every bulk
field contributes an entire Kaluza--Klein tower of four-dimensional modes
along $I_r\times S^1_\theta$.  The decisive structural fact is that
\emph{the tower sum reconstructs the dimensional reduction} --- summing
the four-dimensional induced-gravity contributions over the KK modes of a
bulk field is identical to reducing the six-dimensional induced-gravity
term over the scale profile:
\begin{equation}
\sum_{n\in\text{KK}}\Bigl[\tfrac{1}{16\pi\Newton}\Bigr]^{(4{\rm D})}_n
= \Bigl[\tfrac{1}{16\pi G_6}\Bigr]^{(6{\rm D})}
\!\int_0^{2\pi}\!\!d\theta\int_0^K\!\!dr\,e^{-2r}
= \frac{1}{16\pi\Newton}\bigg|_{\rm geom},
\label{eq:B-tower}
\end{equation}
the right-hand side being exactly Step~1, Eq.~\eqref{eq:V-G4}.  The
species sum and the geometric reduction are therefore \emph{the same
object computed two ways}; they cannot disagree without the
six-dimensional theory being inconsistent.  In this sense the scale profile does
not merely regulate the species sum --- it \emph{performs} it.

\begin{euclidlemma}[The tower identity \textnormal{(L-BH-Tower)}]
\label{lem:tower}
For each bulk field, the Kaluza--Klein tower sum of its
four-dimensional induced-gravity contributions equals the
six-dimensional reduction of its induced term over the canonical
scale profile, Eq.~\eqref{eq:B-tower}: the species sum and the
geometric reduction are one object computed two ways.  Grade:
Derived (an identity of the reduction); the closure argument that
consumes it is graded separately in the honest verdict below, and
its failure mode is the registered internal kill row F-BH-9
(Table~\ref{tab:bh_kill}).
\end{euclidlemma}

\subsection{Step 4 --- the role of GU-5}
Whether \eqref{eq:B-closure} is a \emph{derivation} of $\Newton$ or a
\emph{consistency check} on it depends on the status of $\Newton$.  In
\GGtheory{} $\Newton$ is a member of \textsc{gu-5}, the geometric
conversion factor of the scale integration and the cohort's only
dimensionful anchor.  It is therefore an \emph{input}, and
\eqref{eq:B-tower} guarantees that the entanglement entropy of the
spectrum reproduces $A/4\Newton$ \emph{by construction}, provided the
bulk theory is the consistent ultraviolet completion that the
constitution's admissibility filter (the \AX{4} lineage) requires.  Closure is then not an extra postulate but a self-consistency
already enforced, at the level of the leading quadratically divergent
term, by the tower identity \eqref{eq:B-tower}.

\subsection{Step 5 --- the Standard-Model sector, and the honest gap}
What \eqref{eq:B-tower} does \emph{not} settle is the precise numerical
coefficient $\sum_i\chi_i$ at the leading (zero-mode, Standard-Model)
level, for two reasons we state plainly.
\begin{enumerate}[leftmargin=2em,itemsep=0.12em]
\item \textbf{Spin-dependent signs.}  The coefficients $\chi_{1/2}$ and
$\chi_1$ for fermions and gauge fields enter with signs and curvature
couplings that are notoriously delicate; for some spectra the naive
induced $\Newton$ even emerges with the wrong sign \citep{Visser2002}.  A
reliable $\sum_i\chi_i$ for the \ANine{} field content requires the full
sign-resolved bookkeeping, which we have not carried out.
\item \textbf{Cutoff identification.}  The physical $\Lambda$ is the bulk
scale at which the KK description takes over; pinning
$(M_{\rm Pl}/\Lambda)^2$ to the scale depth $K=35$ of \OGNfive{} is
plausible --- the scale profile generates precisely such a hierarchy --- but we do
not derive the exact relation.
\end{enumerate}
An order-of-magnitude orientation: for $\Lambda$ at the bulk scale the
rule \eqref{eq:B-closure} requires $\sum_i\chi_i \sim \pi(M_{\rm Pl}/\Lambda)^2$,
which the KK tower supplies through \eqref{eq:B-tower}; the zero-mode
Standard-Model piece is an $O(10$--$10^2)$ contribution on top.  These
figures are consistent with closure but do not constitute a proof.

\subsection{Step 6 --- honest verdict}
The species sum \emph{closes structurally}: by \eqref{eq:B-tower} the
weighted tower sum is identical to the scale reduction that defines
$\Newton$, so the entanglement entropy reproduces $A/4\Newton$ given the
\textsc{gu-5} anchor and a consistent bulk completion.  What is \emph{not}
established here is the precise leading numerical coefficient
$\sum_i\chi_i$ for the \GGtheory{} spectrum, which is sign-sensitive and
awaits a dedicated computation with the full bulk field content.  We
record this as the principal open calculation of the entropy program.
None of it affects the coefficient $1/4$, which \S\ref{ssec:V-conical}
fixed independently of the species content.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{R.1.1. Show that a conformally coupled
scalar contributes zero to the induced Newton constant at
leading order, and say why.}

\noindent\emph{Solution.}  Step 1: from \eqref{eq:B-sumrule},
the scalar's coefficient is $\chi_{\rm scalar} = \tfrac16 -
\xi$; the conformal value is $\xi = \tfrac16$.  Step 2: hence
$\chi = 0$: the field's heat kernel has no $a_1 R$ term, and
its vacuum polarization stiffens spacetime not at all at this
order.  Step 3 (why): conformal coupling makes the massless
scalar locally scale-invariant, and a scale-invariant field
cannot contribute to a dimensionful stiffness $1/G_{\rm ind}
\propto \Lambda^2$ at leading order --- there is no scale
in its dynamics for the cutoff to dress.  Physics intuition:
induced gravity is bought with broken scale invariance; each
field pays in proportion to how badly its fluctuations know
about scale, which is why the $\chi$ table is sign- and
coupling-sensitive and why the honest verdict of Step~6 calls
the full sum a real computation rather than a formality.

\medskip\noindent\textbf{R.2.1. Evaluate the closure
requirement at the ledger's scale depth.}

\noindent\emph{Solution.}  Step 1: the sum rule
\eqref{eq:B-closure} demands $\sum_i\chi_i =
\pi(\MPl/\Lambda)^2$ for physical cutoff $\Lambda$.  Step 2:
pinning $\Lambda$ at the ledger's depth-$K$ scale gives
$\MPl/\Lambda \sim e^{K}$; with $K = 35$,
$(\MPl/\Lambda)^2 \sim e^{70} \approx 2.5\times10^{30}$, so
the requirement is $\sum_i\chi_i \sim \pi e^{70} \approx
8\times10^{30}$.  Step 3: a Standard-Model-sized field content
($O(10^2)$ zero-mode degrees of freedom) is thirty orders short
--- and that is precisely the gap the Kaluza--Klein tower fills
via the identity \eqref{eq:B-tower}: the tower sum over
$\sim\!e^{70}$ effective modes below the cutoff \emph{is} the
dimensional reduction of the 6D term.  Physics intuition: the
Planck scale's enormity relative to particle physics is, in
this reading, a counting statement --- gravity is weak because
$1/\Newton$ is the pooled contribution of an exponentially deep
tower, and the depth is the ledger's $K$, not a new dial.

\medskip\noindent\textbf{R.3.1. Verify the tower identity in
its simplest instance: one bulk scalar on the flat interval.}

\noindent\emph{Solution.}  Step 1: take the 6D scale profile
flat ($A = 0$) over $r \in [0, K]$ with unit compact circle;
the KK modes of a bulk scalar are labeled $n$, each a 4D
field contributing one equal quantum of induced stiffness per
mode below the cutoff (the $\Lambda^2/16\pi^2$ of
\eqref{eq:B-sumrule}).  Step 2: the number of modes below $\Lambda$
is the phase-space volume of the two compact directions,
$N \propto \int_0^{2\pi}\!d\theta\int_0^{K}\!dr = 2\pi K$ per
unit cutoff cell: the mode \emph{sum} is the compact
\emph{integral}.  Step 3: the right side of
\eqref{eq:B-tower} is the 6D induced term reduced over the
same integral --- identical object, computed once in 4D
language (sum over modes) and once in 6D language (integral
over profile): for the flat profile the identity is
arithmetic; the canonical profile $e^{-2r}$ weights the
integral without changing the logic (Lemma~2.3).  Physics
intuition: ``the tower performs the reduction'' is not a
metaphor --- summing a Kaluza--Klein tower \emph{is}
integrating over the extra dimensions, mode by mode, which is
why the species sum and the geometric $1/\Newton$ cannot
disagree without the 6D theory itself being inconsistent.

\GGtakeaway{\emph{Standard technology:} heat kernels and the
induced Newton constant.  \emph{The closure condition:} the
species sum over the actual \GGtheory{} spectrum must reproduce
the $\Newton$ that the scale reduction assigns.  \emph{Status:}
the sum rule and its self-consistency mechanism are established;
the leading coefficient over the full bulk field content is the
program's declared open computation --- and the coefficient
$1/4$ does not wait on it.}



%% ======== GROUP 3 (S-Y): THE LEDGER-AND-REGISTRY APPENDICES ========

\section{Internal source ledger}
\label{app:source-ledger}

Every load-bearing element of the module, with its source status:
\emph{derived here} (this paper's own result), \emph{consumed}
(cohort result used under its key, never re-derived), or
\emph{imported} (literature result used with attribution).  The
ledger is the anti-arbitrage instrument: nothing below is claimed
at two grades in two places.

\begin{table}[H]
\centering\small
\renewcommand{\arraystretch}{1.25}
\caption[The internal source ledger.]{The internal source ledger.  ``Consumed'' items are cited
under permanent cohort keys and carry their home grades;
``imported'' items are the module's declared literature inputs. The status column is the one that matters. Derived here means the argument is in this paper and can be attacked here; consumed means the result belongs to a companion at that companion's grade and this paper only uses it; imported means it is somebody else's published result. Mixing the three is how a module comes to look better supported than it is, so they are separated by column rather than by prose.}
\label{tab:source-ledger}
\begin{tabular}{@{}p{0.34\textwidth}p{0.14\textwidth}p{0.42\textwidth}@{}}
\toprule
\textbf{Element} & \textbf{Status} & \textbf{Source / key} \\
\midrule
Exact $1/4$ coefficient (T-BH-1) & derived here &
Steps 1--3 (\S\ref{ssec:V-conical}); premises \GGEP{}, canonical
metric, \OGGEP{} trace \\
Induced $1/\Newton$ from the profile & derived here &
Step 1; same object as \AFive{}'s boundary reduction \\
Thermal clock face (T-BH-5) & derived here &
composition of \textsf{T-P5-B1} (consumed) with
\citealp{KayWald1991} (imported) \\
Euclidean arena license (L-BH-Shore) & derived here &
shore from \textsf{T-P5-B1} (consumed); replica saddle
(imported, \citealp{Solodukhin2011}) \\
Freshness expiry at $t_{\rm Page}$ (L-BH-Fresh) & derived here &
Page-average arithmetic + typicality premise (imported, flagged) \\
GSL direction and price (L-BH-Arrow) & derived here &
DPI (identity) + consumed arrow + induced first law \\
Rotating anchor / collapse channel (T-BH-3/4) & derived here &
model classes; Kerr Hadamard question imported
(\citealp{KayWald1991}) \\
Constitutional arrow & consumed &
\AX{3} (\AOne{}); law-level in \citealp{ArisakaP5CPT} Part IX;
dynamical in \AFive{} \\
Port-Freshness Clause & consumed &
\textsf{L-GDOS-Fresh}, \AFive{} Definition 5 (its grade lives
there) \\
Wick quartet & consumed & \textsf{T-P5-B1}
(\citealp{ArisakaP5CPT}) \\
Real-category grading & consumed &
\textsf{L-Constitution-1}, candidate grade (\AOne{}) \\
Born-weight grounding & consumed &
\AEight{}: closed at the relaxation level; the field-grade
representation step is its declared residue \\
Canonical bulk metric; arena & consumed &
\ATwo{} Theorems 0 and 2 \\
Observer-vacuum reading & consumed &
\citealp{ArisakaA10Cosmos}; its factorization conjecture (naming
home \AEight{}) adopted open here \\
Replica saddle $\ln Z_n = -I_E$ & imported &
\citealp{Solodukhin2011} (declared at \S\ref{ssec:V-conical}) \\
Bifurcate-horizon uniqueness/thermality & imported &
\citealp{KayWald1991} \\
Page averages; scrambling & imported &
\citealp{Page1993}; \citealp{HaydenPreskill2007} \\
Wall-class GSL machinery & imported (ceiling) &
\citealp{Wall2012}; registered open \\
\bottomrule
\end{tabular}

\end{table}

\section{Cross-paper inheritance table}
\label{app:inheritance}

\begin{table}[H]
\centering\small
\renewcommand{\arraystretch}{1.25}
\caption[The compact inheritance table: one row per companion, consumption only --- this module.]{The compact inheritance table: one row per companion,
consumption only --- this module re-derives none of it. One row per companion, and the second column is a list of takings rather than of agreements. This module re-derives nothing it inherits, which means an error anywhere in the first column propagates here unchecked. That is the cost of working in a cohort, and this table is where it is made visible; the last column says which step would fail.}
\label{tab:inheritance}
\begin{tabular}{@{}p{0.17\textwidth}p{0.40\textwidth}p{0.33\textwidth}@{}}
\toprule
\textbf{Companion} & \textbf{What this paper takes} & \textbf{Where
it is used} \\
\midrule
\AOne{} & \AX{1}--\AX{4}; \textsf{L-Constitution-1} (candidate);
the Reality Register & premises of T-BH-1; the funnel; O-BH-4 \\
\ATwo{} & Theorem 0 (forced arena); Theorem 2 (canonical metric,
no free parameters) & Part II; Step 1 \\
\AFive{} & \T{2}--\T{4} open-system law; \textsf{L-GDOS-Fresh};
boundary-gravity reduction & the port; Lemma~\ref{lem:stale-port};
Step 1 consistency \\
\ASix{}, \ASeven{} & the \GiFour{}/\QiFour{} pillars forcing
\GDOS{}; the ledger structure theorems & the funnel; the four laws;
Lemma~\ref{lem:gsl-arrow} \\
\AEight{} & hidden-fiber language; real-bulk series home &
Step 4; \S\ref{sec:rb-register} \\
\ANine{} & the boundary field content & the species sum (F-BH-9);
$c_{\log}$ \\
\ATen{} & observer-vacuum reading; de Sitter \GGA{}; PBH
subdominance & \S\ref{sec:rotating-collapse}; way-station close;
Part VII \\
\AEleven{} & \GRIP{} regimes; No-Exit theorem & the way-station
Part VI sections \\
\PaperPOne{} & $\Mlock$ and $\ell_{\rm lock}$ & every Kerr-null
suppression \\
\PaperPTwo{} & $N=3$ in the ledger & field content of the sum \\
\PaperPFour{} & the derived clock (origin of $i$); real-bulk
treatment & T-BH-5(i); \S\ref{sec:rb-register} \\
\PaperPFive{} & \textsf{T-P5-B1}; the four endowments; the
law-level arrow & T-BH-5; the licensed arena;
Lemma~\ref{lem:gsl-arrow} \\
\PaperCOne{}, \PaperCTwo{} & Kaxion DM saturation; the corridor's
background branch & Part VII; de Sitter evaluation \\
\bottomrule
\end{tabular}

\end{table}

\section{Consolidated benchmark audit}
\label{app:benchmarks}

Every number quoted in a proof or verification clause of this
paper, with the script that reproduces it (all scripts in the
program's verification suite, www.arisaka-gg.org).

\begin{table}[H]
\centering\small
\renewcommand{\arraystretch}{1.25}
\caption[The consolidated benchmark audit.]{The consolidated benchmark audit.  A quoted number with
no row here is a defect; report it. The middle column prints the tolerance each claim was verified to, not the claim itself, because a verified statement with no number beside it is an assertion. Where a tolerance is loose --- the collapse-channel quadrature at the \(10^{-3}\) class --- that is the honest limit of the numerics and not a rounded result. The last column names the script, so every row here can be re-run.}
\label{tab:benchmarks}
\begin{tabular}{@{}p{0.23\textwidth}p{0.43\textwidth}p{0.24\textwidth}@{}}
\toprule
\textbf{Result} & \textbf{Verified statement (value)} &
\textbf{Script} \\
\midrule
T-BH-3 & exact symplecticity ($10^{-9}$); $\sinh^2(gt)$ law
($4\times10^{-14}$); form indefiniteness; polar uniqueness &
rotating-anchor suite \\
T-BH-4 & branch-cut ratio $e^{-2\pi\nu}$ by quadrature
($<10^{-3}$ rel.\ at $\nu = \tfrac12,1,2$); late-time
factorization ($0.13 \to <10^{-2}$) & collapse-channel suite \\
T-BH-5 & polar split exact ($0$); KMS ratio and Planck occupation
($\le 3\times10^{-16}$); one-sidedness obstruction displayed;
$\Gamma$-identity ($\le 2\times10^{-16}$) & \texttt{s1\_kms} \\
L-BH-Fresh & $I(q\!:\!R) = 0.002\,\ln 2$ deep-fresh; growth
ratios $4.05, 4.13, 4.12, 4.03$ (predicted $4$); Page-point
crossing; saturation $1.997\,\ln 2$ ($N=12$, Haar, $5$
realizations) & \texttt{s3\_fresh} \\
L-BH-Arrow & DPI contraction in $300/300$ random channels;
exchange identity to $3.6\times10^{-15}$; positivity over a
$1.2\times10^{5}$-point grid (min $+1.2\times10^{-8}$, at
$\bar n_b \to \bar n_H$); closed forms to $10^{-12}$ &
\texttt{s4\_arrow} \\
Block II worked chapters & $\THaw(\Msun) = 6.17\times
10^{-8}$~K; $t_{\rm evap}(\Msun) \approx 2\times10^{67}$~yr;
$t_{\rm Page}/t_{\rm evap} = 1 - 2^{-3/2}$; Page average vs.\
asymptotics; $T_{\rm min} = \sqrt3/2\pi L$ and $F$-flip at $r_+
= L$; thermal relative entropy \eqref{eq:H-relent} (positivity +
face zero); Kerr first-law partials \eqref{eq:I-firstlaw}
($10^{-12}$); $\kappa_{\rm dS} = H$; $T_{\rm GH}, S_{\rm dS}$
values & \texttt{p2c2\_verify} \\
\bottomrule
\end{tabular}

\end{table}

\section{Named-statement deck}
\label{app:statement-deck}

The deck (Table~\ref{tab:statement-deck}) is the paper's claims
registry, printed in full: every named statement, its one-line
content, its exact grade, its home, and its falsifier exposure.
Nothing in the paper is claimed above this table's register, and
the standing challenge of the objections Part --- check the deck,
not the prose --- points here.

\begin{table}[p!]
\centering\small
\renewcommand{\arraystretch}{1.3}
\caption[The named-statement deck.]{The named-statement deck.  Grades are exact; nothing in
this paper is claimed above this table's register.  The Exposure
column keys each claim to the falsifier rows of
Table~\ref{tab:bh_kill} (F-BH-1..10) that would retire it; the
internal row F-BH-9 exposes only the closure reading, exactly the
partition O-BH-52 audits.}
\label{tab:statement-deck}
\begin{tabular}{@{}p{0.08\textwidth}p{0.36\textwidth}p{0.19\textwidth}p{0.10\textwidth}p{0.13\textwidth}@{}}
\toprule
\textbf{Key} & \textbf{One-line statement} & \textbf{Grade} &
\textbf{Home} & \textbf{Exposure} \\
\midrule
T-BH-1 & horizon entropy is $A/4\Newton$ with $1/4 = 4\pi/16\pi$,
regulator-independently, for the generic black hole &
Derived-conditional (three premises) &
Thm~\ref{thm:exact-quarter} & F-BH-7; 1--3 \\
T-BH-2 & the Page turnover is the \GFF{} branch switch & named
mechanism; see L-BH-Fresh & Thm~\ref{thm:page-turnover} &
F-BH-6 \\
T-BH-3 & rotating anchor: selection fails in the ergoregion;
anchor at infinity; superradiance is symplectic pumping & model
level, verified & Thm~\ref{thm:rotating-anchor} & F-BH-2 \\
T-BH-4 & collapse channel: Planck factor = branch-cut asymmetry;
universal late-time channel & model level, verified &
Thm~\ref{thm:collapse-channel} & F-BH-5 \\
T-BH-5 & the thermal clock face is proven: continuation and
imaginary time by \textsf{T-P5-B1}; KMS at $2\pi/\kappa$ &
Derived (quasi-free) & Thm~\ref{thm:thermal-clock} & F-BH-5 \\
L-BH-Fresh & freshness expires at $t_{\rm Page}$ by counting; the
information return is an assumption switching off &
Derived-conditional & Lem~\ref{lem:stale-port} & F-BH-6 \\
L-BH-Arrow & GSL direction from three named links;
$\Delta S_{\rm gen}$ = spent relative entropy in the exchange
class & Derived (class) / Derived-conditional; Wall-class open &
Lem~\ref{lem:gsl-arrow} & F-BH-8 \\
L-BH-Tower & the KK tower sum is the dimensional reduction,
computed twice & Derived (identity) & Lem~\ref{lem:tower} &
F-BH-9 \\
L-BH-Port & at a horizon the \OGGEP{} trace is forced: the port
is ontological & Derived (\AX{1}+\AX{3}) &
\S\ref{ssec:visible-port} & F-BH-8 \\
L-BH-Shore & the Euclidean arena is the clock's analytic shore:
the continuation is a theorem (\textsf{T-P5-B1}); the saddle is a
declared import & Derived-conditional (quasi-free; saddle imported)
& \S\ref{ssec:V-conical} & --- \\
C-BH-1 & the conditional-\GGA{} crossover is the GSL's equality
point & Derived (first law + exchange identity; conditional on
the quasi-static bath reading) &
Cor~\ref{cor:gga-crossover} & F-BH-5 \\
C-BH-2 & the de~Sitter horizon entropy is the same theorem on the
cosmological \OGGEP{} port & Derived-conditional (T-BH-1's
premises; same-arena clause) & \S\ref{ssec:dS-induced} & F-BH-7 \\
RB-13 & horizon entropy data are real-section geometry &
registered in \AEight{} (registrational; no new theorem) &
\S\ref{sec:rb-register} & --- \\
(prop.) & the islands formula is the semiclassical display of the
\OGGEP{} output; the \QES{} is the traced-region boundary &
dictionary-grade proposition & App.~\ref{app:islands} & --- \\
\bottomrule
\end{tabular}

\end{table}

\section{Promotion roadmap}
\label{app:promotion}

What would raise each conditional or open item, stated so the
promotions are auditable in advance.
\emph{(i) The species sum} (F-BH-9): a sign-resolved
$\sum_i\chi_i$ over the \ANine{} content plus the tower --- closes
the module's principal open computation and converts O-BH-5's
concession from open to settled; chartered as its own project.
\emph{(ii) The typicality premise} (Lemma~\ref{lem:stale-port}):
replace the imported ETH/Haar assumption by bulk dynamics ---
promotes L-BH-Fresh to Derived.
\emph{(iii) The exchange class} (Lemma~\ref{lem:gsl-arrow}):
extend the identity toward interacting fields and arbitrary cuts
(\citealp{Wall2012} machinery) --- promotes the GSL statement from
class-exact to Wall-class; registered open.
\emph{(iv) The quasi-free arena} (\S\ref{ssec:V-conical}): an
interacting-grade shore --- promotes the arena license
(\textsf{L-BH-Shore}) from
Derived-conditional toward Derived.
\emph{(v) The real-bulk proposal} (\S\ref{sec:rb-register}):
registration by the series' home papers --- converts the proposal
to a numbered entry.
\emph{(vi) The factorization conjecture} (adopted from
\citealp{ArisakaA10Cosmos}): a model-level closure in the
T-BH-3/4 class --- discharges the module's adopted open problem.
\emph{(vii) The PBH ladder} (F-BH-4): a derived exponent and an
absolute anchor --- promotes the conjecture to a prediction;
needs cosmology-side input, chartered separately.

\section{Falsification-test catalog}
\label{app:falsification-catalogue}

The module's kill apparatus lives in three displays, one address
each: the claims-level kill table (F-BH-1..10,
Table~\ref{tab:bh_kill}), the dated instrument timeline with
pre-registered verdicts (Table~\ref{tab:bh-exp-timeline}), and the
refresh discipline that keeps both honest
(\S\ref{sec:exp-refresh}).  The external hostages --- the lock
cluster, the Kaxion window, the neutrino ordering --- are cohort
rows, inherited by construction
(\S\ref{ssec:runtime-uniqueness}); this module adds no dial to
any of them.  One-line rule of use: a row fires as written or not
at all, and a fired row retires the module.

\section{An annotated reading guide}
\label{app:reading-guide}

\emph{Shelf 1 (the thermodynamic foundation):} Bekenstein 1973;
Bardeen--Carter--Hawking 1973; Hawking 1975; Page 1993 --- the
four papers this module's Parts I and VI answer to.
\emph{Shelf 2 (the induced-gravity lineage):} Sakharov 1968;
Jacobson 1994; Frolov--Fursaev 1997; Susskind--Uglum 1994;
Callan--Wilczek 1994; Visser 2002 --- read in that order, the
lineage whose missing premise (\emph{no bare $\Newton$}) Part V
supplies.
\emph{Shelf 3 (the information decade):} AMPS 2013;
Hayden--Preskill 2007; Penington 2020; Almheiri et al.\ 2019;
the replica-wormhole paper --- the modern paradox literature
Part VI converses with.
\emph{Shelf 4 (the channel mathematics):} Spohn 1978; Casini
2008; Wall 2012 --- the entropy-production spine of
Lemma~\ref{lem:gsl-arrow} and its declared ceiling.
\emph{Shelf 5 (the instruments):} the LVK catalog papers and
GW250114 companions; the EHT series --- the record Part VIII
scores.
\emph{Shelf 6 (the cohort):} \ATwo{} for the arena, \AFive{} for
the law, \PaperPFour{}/\PaperPFive{} for the clock and its
mirrors, \AEight{} for the fiber, read in that order; then this
paper re-read from Part IV, which is where it stops being a
survey and starts being a derivation.

\clearpage

%% ===================================================================
%% PART -- COMMON OBJECTIONS AND RESPONSES (new Part at the v13.8 split)
%% ===================================================================
\GGPart[Common objections]{Common Objections and Responses}{Thirty Questions from the General Reader, in Plain English, followed by Fifty-three Adversarial Readings --- from the Classical Relativist, the Curved-Spacetime Field Theorist, the String Theorist, the Islands Theorist, the Canonical-Quantization Referees, the Emergent-Gravity Theorist, the Gravitational-Wave Observer, the Horizon-Imaging Astronomer, the Quantum-Information Theorist, the Quantum-Foundations Referee, the Historian and Philosopher, the Student, and the Cohort-Internal Referee --- Each Answered with Pointers into the Paper}

\label{part:objections-c3}
\noindent

%% The lettered appendix run ended with Part XII (A..R); this Part's
%% community sections resume the arabic section numbering of the main
%% text (the counter is restored to its pre-appendix value below, so
%% the sections number continuously with the body of the paper).
\renewcommand{\thesection}{\arabic{section}}
\setcounter{section}{64}

A claimed first-principles derivation of the Bekenstein--Hawking
coefficient --- for the generic black hole, outside string theory ---
will and should draw hard objections.  This Part collects the
fifty-three sharpest adversarial readings we could construct,
grouped by the referee most likely to raise each, and answers each
with explicit pointers to the theorem, section, register entry, or
named open item that carries the load.  Where the honest answer is
``not yet,'' ``not ours to claim,'' or ``conceded,'' it is said in
those words.  These are self-authored stress tests in referee
personas, published so that real referees can start where they end.
Seven of the objections carry the module's permanent seed keys
(O-BH-1 through O-BH-7), filed under their communities; the keys
are cited elsewhere in the paper (the source ledger, the promotion
roadmap) and never change.

Before the referees, however, come the readers.  The questions a
non-specialist asks about black holes are not easier than the
technical ones --- they are usually deeper, because they go straight
to what the mathematics is \emph{about}: what a horizon is, where
the information goes, whether the future is fixed, what is real.
The opening section therefore collects thirty such questions,
distilled from the themes of Parts~IX--XI, and answers each one in
plain English at whatever length honesty requires, with a pointer
to the place in the paper where the full treatment lives.  A reader
with no physics beyond curiosity can read that section alone and
come away with the paper's picture of the black hole; the
fifty-three technical readings that follow are unchanged, and keep
their numbering and their scorecard.

\newcommand{\PObjC}[3]{\bigskip\noindent\textbf{Objection
O-BH-#1 (#2)}\par\nobreak\noindent\textbf{``#3''}\par\nobreak}
\newenvironment{objqabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

\section{Questions from the general reader}
\label{sec:general-questions}

Thirty questions, in the order a curious reader tends to ask them:
first what a black hole is and why it matters, then the famous
paradox and where the information really goes, then what this
theory claims and how it could be wrong, and finally what it all
means.  The answers are in plain English; each ends with a pointer
to the full treatment.  Nothing here is claimed beyond what the
main text establishes, and where a statement is conditional or
open, the answer says so in those words.

\newcommand{\PGenQ}[3]{\bigskip\noindent\textbf{Question Q#1
(#2)}\par\nobreak\noindent\textbf{``#3''}\par\nobreak}

\subsection*{What a black hole is, and why it matters (Q1--Q8)}

\PGenQ{1}{the commuter, on relevance}{Black holes are thousands of
light-years away.  Why should anyone but an astronomer care?}
\begin{objqabody}
\emph{Answer.}  Because a black hole is not an exotic exception to
ordinary physics --- it is ordinary physics with the volume turned
all the way up.  The same bookkeeping that makes your coffee cool,
your laptop shed heat, and a photograph fade in sunlight is, in
this paper's reading, the mechanism behind Hawking radiation: a
system loses track of information that flows somewhere it cannot
see, and thermodynamics is the shape of that loss.  The black hole
is simply the one place where geometry itself enforces the loss,
perfectly and calculably, which is why it is the sharpest test
bench nature offers for the three great theories --- gravity,
quantum mechanics, and thermodynamics --- at once.  And there is a
second, quieter reason: the universe has a horizon of its own, and
every one of us lives inside it.  The mathematics developed for
black holes is the mathematics of our own cosmic situation.  (The
full case: Part~IX's comparison and Part~XI's closing sections.)
\end{objqabody}

\PGenQ{2}{the dinner-table questioner, wanting it straight}{In one
paragraph, without equations: what does this theory say a black
hole actually is?}
\begin{objqabody}
\emph{Answer.}  The theory says the world has six dimensions, of
which we perceive four; the other two are not far away but
everywhere, too fine and too orthogonal for our instruments to
enter.  Everything any observer measures is a \emph{projection} of
the six-dimensional world onto their four-dimensional view --- like
a shadow cast on a wall.  A black hole is the place where that
projection becomes most severe: near the horizon, more and more of
the world's degrees of freedom slip into the unseen directions,
until at the horizon the view simply ends.  The horizon is not a
wall in space; it is the edge of a window.  What falls in is not
destroyed --- it moves into registers the outside view cannot
carry.  What comes out as faint radiation is the lawful, thermal
signature of a view that has been truncated.  That is the whole
picture; the rest of the paper is the mathematics that makes it
precise and the measurements that could kill it.  (Part~IV builds
this; Part~XI says it slowly.)
\end{objqabody}

\PGenQ{3}{the careful reader, on frozen time}{I have read that
time freezes at the horizon --- that from outside, nothing ever
falls in.  So does anything ever actually cross?}
\begin{objqabody}
\emph{Answer.}  Both things are true, and the trick is that they
are true \emph{for different observers} --- which, after Einstein,
is no scandal.  Watching your friend fall toward the horizon, you
see their signals climb out of an ever-deeper gravitational well:
each pulse takes longer than the last, their image reddens,
freezes, and fades.  From your side, they never quite cross.  But
your friend's own clock ticks normally; they cross the horizon in
perfectly finite time and notice nothing at the crossing --- the
horizon is not marked locally in any way.  The two accounts never
contradict each other, because no experiment can compare them
face-to-face afterwards.  In this paper's language, your frozen,
fading image of your friend is exactly what a port delivers: the
last information the window can carry, stretched over your
forever.  (The relational reading of horizon quantities:
\S\ref{sec:thermal-clock} and Part~XI.)
\end{objqabody}

\PGenQ{4}{the nervous voyager, on survival}{Would falling into a
black hole hurt?}
\begin{objqabody}
\emph{Answer.}  It depends entirely on the size, and the answer
surprises everyone.  Near a small black hole --- a few times the
Sun's mass --- the gravitational \emph{gradient} is ferocious: your
feet would be pulled so much harder than your head that you would
be stretched apart well before reaching the horizon.  But the
bigger the hole, the gentler its horizon: for the giant at the
center of galaxy M87, the tidal stress at the horizon is some ten
thousand times \emph{weaker} than what your body feels standing on
Earth.  You would cross the point of no return without a bump, a
flash, or any sensation at all --- and in this paper's reading
there is no ``firewall'' of radiation at the crossing either; the
apparent quantum argument for one dissolves when the bookkeeping
is done in the full six dimensions.  The genuinely unpleasant part
comes much later, deep inside, where the projection becomes
extreme.  (The numbers: Appendix~\ref{app:schwarzschild-course};
the no-firewall argument: Part~VI.)
\end{objqabody}

\PGenQ{5}{the reader of popular science, on the center}{Is there
really a singularity at the center --- a point where density is
infinite and physics breaks down?}
\begin{objqabody}
\emph{Answer.}  ``Infinite density'' has never been a physical
claim; it is mathematics raising a flag that says \emph{my
description has ended here}.  Classical general relativity does
predict, by theorem, that paths into a black hole terminate --- but
history offers a precedent for how such flags get resolved.  For
forty years the horizon itself was believed to be a place where
physics broke down, until 1958, when it was shown to be an
artifact of a poorly chosen map: change coordinates and the
horizon is smooth.  This theory proposes the same lesson one level
up: the center is singular \emph{in the four-dimensional
projection} but smooth in the six-dimensional world --- the flag
marks the edge of the shadow, not an edge of reality.  That is a
proposal, not a theorem we can test at the center of a black hole;
its support comes from the framework's testable consequences
elsewhere.  (The 1958 story: Appendix~\ref{app:schwarzschild-course};
the 6D reading: Part~VI.)
\end{objqabody}

\PGenQ{6}{the skeptic at the whiteboard, on hidden dimensions}{What
are these two extra dimensions?  Where are they hiding?}
\begin{objqabody}
\emph{Answer.}  They are not hiding somewhere else; they are here,
at every point, in the same way the third dimension is present at
every point of a printed map without appearing on it.  One is a
\emph{scale} direction --- a depth along which fields stretch and
compress --- and one is a tiny closed circle around which the
forces of nature wind.  We do not perceive them for the same
reason a figure drawn on paper would not perceive height: our
instruments, made of four-dimensional projections themselves,
measure only the projection.  What makes them physics rather than
poetry is that their geometry is \emph{rigid}: one fixed shape has
to account simultaneously for the masses of particles, the
strength of gravity, the composition of the universe, and --- in
this paper --- the thermodynamics of black holes, with no
adjustable numbers.  A wrong shape fails loudly.  (The bulk
geometry: Part~II; what rigidity buys: Q20 below.)
\end{objqabody}

\PGenQ{7}{the philosopher of science, on unfalsifiable
metaphysics}{You admit the six-dimensional bulk cannot be observed
directly.  Isn't this exactly the kind of untestable metaphysics
science is supposed to avoid?}
\begin{objqabody}
\emph{Answer.}  The concern is the right one, and the defense is
the standard one in science, applied strictly.  No one has seen a
quark, the interior of the Sun, or a single atom of the early
universe; we believe in them because positing them explains
diverse observations with fewer assumptions than any rival, and
because the posit sticks its neck out --- it forbids things, and
would be abandoned if the forbidden things were seen.  The bulk is
held to that same standard, and the paper's structure is built
around it: the framework's numbers are fixed before the
comparisons are made, the register says plainly which claims are
derived and which remain open, and Part~VIII maintains a public
list of observations that would kill the black-hole reading
outright --- from horizon-scale images to a laboratory neutrino
measurement.  Metaphysics does not volunteer its own death
certificate.  This does.  (The kill list: Part~VIII; the register:
the named-statement deck.)
\end{objqabody}

\PGenQ{8}{the traveler, on borders}{Is the horizon a real,
physical thing --- a membrane, a surface, a wall?}
\begin{objqabody}
\emph{Answer.}  There is nothing there --- no material, no marker,
nothing a passing observer could detect locally.  And yet it has
exact, measurable consequences: a temperature, an entropy, a
size photographed by telescopes.  The resolution is that the
horizon is not a thing but a \emph{relation}: it is defined by
what a given observer can and cannot ever reach, the way a
national border is defined by law rather than paint on the ground
--- invisible underfoot, decisive in consequence.  A hill can
block your view and not your neighbor's without anyone doubting
the landscape behind it is real; a surface can be a horizon for
one observer and not for another in precisely the same way.  This
paper's contribution is to make the relation quantitative: the
horizon is where the projection that defines an observer's view
runs out of bandwidth.  (The relation reading: Part~XI; the port
theorem: Part~IV.)
\end{objqabody}

\subsection*{The paradox, and where the information goes
(Q9--Q17)}

\PGenQ{9}{the newcomer, wanting the paradox itself}{What exactly
is the black-hole information paradox, in plain words?}
\begin{objqabody}
\emph{Answer.}  Quantum mechanics has one inviolable rule: the
universe never forgets.  Given the complete present, the past is
in principle recoverable; information can be scrambled beyond any
practical hope but never destroyed.  Black holes seemed to break
the rule twice.  First, a hole swallows anything --- an
encyclopedia, a star --- and settles into a state characterized by
just its mass, spin, and charge, with no visible trace of what
went in.  Second, Hawking showed in 1974 that the hole slowly
shines with perfectly \emph{thermal} radiation --- and thermal
radiation is, by definition, radiation that carries no message.
Run the process to the end: the hole evaporates away entirely,
the radiation remembers nothing, and the encyclopedia is not
scrambled but \emph{gone}.  Either quantum mechanics fails, or
gravity's account of the horizon is incomplete.  That dilemma,
sharpened over fifty years, is the paradox.  (The precise
statement: Part~III.)
\end{objqabody}

\PGenQ{10}{the same newcomer, wanting the resolution}{And in plain
words, how does this theory resolve it?}
\begin{objqabody}
\emph{Answer.}  By locating the error in the question's hidden
assumption: that the four-dimensional description is the whole
story.  Every working physicist knows a system can \emph{look}
like it is losing information when it is merely sharing it with
something you are not tracking --- a hot cup losing heat to a room
is not deleting anything.  Hawking's calculation produced exactly
that signature, and the fifty-year alarm came from the apparent
absence of any ``room'' for the information to have gone into: the
radiation seemed to be all there was.  The six-dimensional bulk
supplies the missing room.  The interior registers behind the
horizon are as physically real as the air in the room around the
cup; the information flows there, stays in the universe's books,
and returns --- slowly, scrambled into subtle correlations of the
radiation --- as the hole evaporates.  Nothing is deleted; a view
was mistaken for the world.  (The mechanism: Parts~IV--VI; the
slow return: Q16.)
\end{objqabody}

\PGenQ{11}{the pragmatist, on recovery}{So if I burned my diary
and threw the ashes into a black hole, is it really still
`there'?  Could anyone ever read it again?}
\begin{objqabody}
\emph{Answer.}  In principle yes; in practice never --- and the
distinction is exactly the one that already applies to the
burning.  When a diary burns, its text passes into correlations
among smoke particles, photons, and ash; no law of physics erases
it, and a demon with perfect instruments and the patience of ages
could reconstruct every word.  No such demon can be built, but
physics cares about the principle, because \emph{in principle
recoverable} is what separates scrambling from destruction --- and
destruction would bring down the whole quantum edifice.  The black
hole, in this reading, adds one twist: the correlations pass
through registers no exterior instrument can reach directly, and
are returned over the hole's evaporation --- a timescale so long
that for a stellar hole it makes the age of the universe look
instantaneous.  The books stay balanced; the reading room has a
very long waiting list.  (The bookkeeping theorems:
Appendix~\ref{app:data-processing}.)
\end{objqabody}

\PGenQ{12}{the puzzled reader, on temperature}{How can a black
hole --- a hole --- be hot?  Hot to whom?}
\begin{objqabody}
\emph{Answer.}  ``Hot to whom'' is precisely the right question,
and its answer is the paper's favorite lesson: temperature at a
horizon is \emph{relational}.  An observer who hovers far away,
watching the hole, finds their detector clicking at a definite
temperature --- for a solar-mass hole, about sixty billionths of a
degree above absolute zero, unmeasurably faint but exactly
calculable.  An observer falling freely through the same region
detects nothing at all: to them it is empty space.  Neither is
mistaken, because temperature is not a property of the hole alone
--- it is a property of the \emph{relationship} between the hole
and a particular observer's view, of what that view keeps and
what it truncates.  Physics has absorbed such a lesson before:
in 1905, simultaneity turned out to be observer-dependent without
becoming unreal.  Objective, it turns out, does not mean
observer-free; it means lawfully related across observers.  (The
thermal clock: \S\ref{sec:thermal-clock}; the philosophy:
Part~XI.)
\end{objqabody}

\PGenQ{13}{the accountant, on entropy}{Physicists say a black hole
has more entropy than anything else in the universe.  Entropy of
\emph{what}, exactly?}
\begin{objqabody}
\emph{Answer.}  Entropy always counts the same thing: the number
of microscopic arrangements consistent with what you can see.  A
shuffled deck has entropy because fifty-two cards can hide their
order behind one face-down stack.  The scandal of black-hole
entropy was that the classical theory said a hole has \emph{no}
hidden arrangements --- mass, spin, charge, done --- and yet
thermodynamics insisted it carries more entropy than the star
that made it.  This paper's answer names the arrangements: they
are configurations of the hidden compact dimension at the
horizon --- one register per pixel of Planck-sized area --- the
degrees of freedom the four-dimensional view cannot resolve.
The entropy is literally the measure of how entangled your view
is with what sits behind the port.  And that explains the old
mystery of why the entropy grows with the horizon's \emph{area}
rather than its volume: the entropy lives on the boundary
because the boundary is where the window ends.  (The count:
Part~V; the honest open piece of it: Q21.)
\end{objqabody}

\PGenQ{14}{the empiricist, on evidence}{Has Hawking radiation ever
actually been observed?}
\begin{objqabody}
\emph{Answer.}  No, and it almost certainly never will be
directly.  A stellar-mass black hole radiates at billionths of a
degree, while the universe around it glows at nearly three
degrees: every astrophysical hole is, today, absorbing far more
than it emits, and the signal is hopelessly buried.  What exists
is indirect but serious: laboratory ``analogue horizons'' --- in
flowing fluids and cold-atom systems --- exhibit the kinematic
effect Hawking predicted, in systems where the mathematics is the
same though the gravity is absent.  This paper is candid about
the situation: Hawking radiation functions in physics as a
consistency requirement --- three independent derivations, worked
in Appendices~\ref{app:Hawking_T} and~\ref{app:euclidean}, give
one answer --- rather than as an observed phenomenon, and the
framework's own testable content therefore rides on the
\emph{geometry} side: images, ringdowns, and the cross-domain
predictions of Part~VIII.  (The honest evidence table: Part~VIII.)
\end{objqabody}

\PGenQ{15}{the reader who wants the end of the story}{What happens
at the very end, when a black hole finishes evaporating?}
\begin{objqabody}
\emph{Answer.}  Here is the register speaking plainly: nobody
knows, and this paper does not claim to.  The final flash ---
when the hole shrinks to near the fundamental scale --- is the
one moment where the calculation tools of every framework,
including this one, run out of warranty.  What the theory does
fix is the \emph{bookkeeping} across the whole history up to that
point: the information is not stored up desperately awaiting a
final reveal; it has been leaving all along, in correlations,
since the halfway point of the evaporation (Q16), so the ending
--- whatever its detailed physics --- is not asked to perform a
miracle.  No information bomb, no paradox deferred to the last
second, no need for an eternal remnant.  The final chapter is
open; the accounts are already balanced when it begins.  (The
turnover theorem: Part~VI; the open end-state: the promotion
roadmap.)
\end{objqabody}

\PGenQ{16}{the follower of physics news, on the Page curve}{I keep
seeing the phrase `Page curve.'  What is it, and why did its
recovery in 2019 cause such excitement?}
\begin{objqabody}
\emph{Answer.}  Imagine tracking not the radiation's energy but
its \emph{mystery} --- its entropy, the amount of hidden
information it carries.  If a black hole truly deletes what it
swallows, the mystery grows forever: more radiation, more
featureless heat.  But Don Page observed in 1993 that if quantum
mechanics holds, the curve must turn: past the halfway point of
evaporation, each new quantum of radiation, being secretly
correlated with quanta emitted earlier, starts \emph{resolving}
mystery rather than adding to it, and the curve comes back down
to zero.  Rise, turn, fall --- unitarity's signature, drawn as a
graph.  For twenty-five years gravity's own mathematics refused
to draw it; in 2019 it finally did, through the island
constructions --- a landmark, achieved in simplified models.
In this paper the curve arrives a different way: the bulk was
never anything but unitary, and the turnover is a theorem about
when the projection's bookkeeping switches branches.  Same
curve, roots one level deeper.  (Worked slowly:
Appendix~\ref{app:Page_curve}; the theorem:
Theorem~\ref{thm:page-turnover}.)
\end{objqabody}

\PGenQ{17}{the curious reader, on islands}{What is an `island,' in
plain words?  The term sounds mystical.}
\begin{objqabody}
\emph{Answer.}  It is an accounting rule with an evocative name.
After the Page turnover (Q16), the modern gravity calculation
says something strange-sounding: to compute what the
\emph{radiation} knows, you must include a region \emph{inside}
the black hole --- the island --- as if it already belonged to
the radiation.  Mystical as that sounds, it is bookkeeping: the
calculation is deciding which of two ways of paying for
information is cheaper --- keep renting (count every radiated
quantum's hidden partner separately) or buy the interior outright
(count it once, at a fixed price set by the horizon's area).  At
the turnover, buying beats renting, and the books flip.  In this
paper's reading the flip is no mystery at all: the interior
registers were \emph{always} shared with the radiation through
the bulk --- the island rule is the four-dimensional shadow of
that six-dimensional fact.  (The mortgage-versus-rent ledger:
Appendix~\ref{app:islands}; the dictionary between languages:
Appendix~\ref{app:island-dictionary}.)
\end{objqabody}

\subsection*{The theory's claims, and how it could be wrong
(Q18--Q22)}

\PGenQ{18}{the fair-minded reader, on string theory}{How does this
relate to string theory's account of black holes?  Are you saying
string theory is wrong?}
\begin{objqabody}
\emph{Answer.}  No --- and the paper is deliberate about this.
String theory achieved something this framework openly admires
and has not matched: for special, supersymmetric black holes, it
counted the microscopic states one by one and got the entropy
exactly right, a landmark of mathematical physics.  What it has
not yet done is name the states of the ordinary, spinning,
electrically neutral black holes that actually populate the sky.
This framework makes the complementary claim: it addresses
precisely the generic astrophysical case, in a universe with our
particle content, with no adjustable numbers --- and it lacks
string theory's depth of mathematical control in the special
corners.  The two are answers to differently shaped questions,
and both are staked to their own falsifiable predictions.  The
paper's comparison chapter sets them side by side in those
terms --- structural contrast, not verdict.  (The full
comparison: Part~IX; the string count, worked with respect:
Appendix~\ref{app:holography-course}.)
\end{objqabody}

\PGenQ{19}{the historian, on hubris}{Some of the best physicists
of two generations worked on this problem.  Why would this
framework succeed where they did not?}
\begin{objqabody}
\emph{Answer.}  The claim is not superior cleverness, and the
paper should be read as making a structural point instead.  Most
programs attacked the paradox from within the four-dimensional
description, adding machinery --- new surfaces, new saddle
points, new principles --- to repair its books.  This framework
changed the starting point: it takes \emph{openness} as
fundamental --- every observation is a projection of a larger
system --- and had that principle fixed, with its machinery, long
before black holes entered the program; the same engine was
already accounting for particle masses and cosmology.  When it
reached the horizon, the paradox-generating assumption (that the
4D account is complete) was simply never made, so there was no
paradox to repair --- and, crucially, no freedom left to tune:
the framework arrived with its numbers already committed
elsewhere.  That is either a deep advantage or a spectacular way
to be wrong in public, and the paper maintains the kill list
that decides which.  (The inheritance: Part~II; the kill list:
Part~VIII.)
\end{objqabody}

\PGenQ{20}{the buyer, on zero free parameters}{The paper keeps
saying `no free parameters.'  What does that mean, and why should
it impress me?}
\begin{objqabody}
\emph{Answer.}  A free parameter is a dial: a number the theory
cannot predict and adjusts to fit the data.  Dials are not sins
--- the standard model of particle physics has about twenty, the
standard cosmology six, and both are triumphs --- but every dial
weakens a test, because a theory with enough dials can absorb
almost any observation.  This framework's structural constants
are integers fixed by internal consistency: turn any of them and
the whole interlocked structure fails, not just one prediction.
Why it matters here: when the black-hole module derives the
famous entropy coefficient, or prices the deviations from
Einstein's geometry at a definite (unobservably small) size,
there is no dial anywhere to make an awkward measurement
comfortable.  Predictions without dials can \emph{fail} --- which
is exactly what makes them worth something.  Whether they do
fail is a matter for instruments, and Part~VIII lists which
ones.  (The ledger's rigidity: Part~II; the priced deviations:
Part~VII.)
\end{objqabody}

\PGenQ{21}{the numerate skeptic, on retrofitting}{Couldn't all
this be sophisticated numerology --- patterns found after the
fact and dressed up as predictions?}
\begin{objqabody}
\emph{Answer.}  This is the most important skeptical question in
the book, and the paper's defense is procedural rather than
rhetorical, because after-the-fact reasoning cannot be refuted by
assertion --- only by exposure.  Three exposures are offered.
First, pre-registration: the claims are stated with named kill
conditions \emph{before} the deciding measurements, including
some --- like the ordering of neutrino masses, measurable in a
laboratory --- that have nothing to do with black holes yet
would retire this paper's central theorem if they go the wrong
way.  Numerology does not take hostages in other fields.
Second, an audited register: every claimed result carries a
grade --- derived, conditional, or open --- and the paper
concedes in print the pieces it has not closed, including one
computation at the heart of the entropy story (the species sum)
that is declared open rather than papered over.  Third,
reproducibility: the derivation chains are laid out step by step
in the appendices precisely so that a hostile reader can check
them.  The paper's own words: it asks to be checked, not
believed.  (The register: the named-statement deck; the open
computation: Appendix~\ref{app:species-sum}.)
\end{objqabody}

\PGenQ{22}{the referee in all of us, on falsifiability}{Name it
plainly: what observations would kill this picture?}
\begin{objqabody}
\emph{Answer.}  In plain words, any one of these.  A black-hole
photograph or a gravitational-wave ringdown that deviates from
Einstein's geometry at the few-percent level --- this framework
predicts agreement to precisions no foreseeable instrument can
challenge, so any confirmed deviation retires it.  Echoes in
gravitational waves after a merger's ringdown --- forbidden here,
because the horizon is smooth.  A demonstrated loss of
information in any gravitational process.  And --- the unusual
ones --- failures far from black holes: the wrong neutrino mass
ordering, or the absence of the predicted dark-matter partner
particles, would bring down theorems this module depends on,
because the same six-dimensional geometry underwrites all of it.
A framework that can be shot from a neighboring field is
staking more, not less.  Each kill condition is registered, with
its current experimental status, in the tables of Part~VIII.
(The dictionary of kills: Part~VIII and the falsification
catalog.)
\end{objqabody}

\subsection*{What it means (Q23--Q30)}

\PGenQ{23}{the determinist's child, on the future}{Does this
theory say the future is already fixed?}
\begin{objqabody}
\emph{Answer.}  At the deepest level the theory describes, yes:
the six-dimensional bulk evolves deterministically, and nothing
in it is ever random or lost.  But no observer lives at that
level.  Every observer's view is a projection, and the theory
proves that the projection \emph{necessarily} loses information
--- so from inside any view, the future is irreducibly
probabilistic, and no instrument, however perfect, can do better.
This is a determinism that cannot be exploited even in principle:
the registers that would complete the prediction sit behind
causal boundaries, not behind practical ones.  So the honest
one-sentence answer is: fixed as seen from nowhere, open as seen
from anywhere --- and ``anywhere'' is the only seat there is.
(The full discussion, including what kind of probability
survives: Part~XI.)
\end{objqabody}

\PGenQ{24}{the reader of biographies, on the old argument}{So did
Einstein win after all?  Does God play dice?}
\begin{objqabody}
\emph{Answer.}  The theory's answer is that the two great
antagonists were describing different levels of one world, and
each was right about his level.  Einstein's instinct --- that
beneath the statistics there is a complete, lawful reality in
which nothing is truly random --- is honored in the bulk, where
evolution is deterministic and information is conserved.  Bohr's
insistence --- that physics must speak only of what observation
can deliver, and that at that level probability is irreducible
--- is honored in every four-dimensional view, where the
projection theorem makes uncertainty lawful and permanent.  The
horizon is where their argument is staged most starkly, and
what this framework offers is not a winner but a dissolution:
probability is what determinism looks like through a port.
(The staging: Part~XI; the quantitative version of ``through a
port'' is the business of the companion paper on quantum
mechanics, and is registered there as conditional --- said here
so the slogan is not mistaken for a settled theorem.)
\end{objqabody}

\PGenQ{25}{the human being, on free will}{Where does free will
fit, if the bulk is deterministic?}
\begin{objqabody}
\emph{Answer.}  Honestly labeled: physics narrows this question
less than either side of the old debate hopes, and this paper
adds one structural observation rather than a resolution.  The
observation is that you --- as a describer of yourself --- are
also a projection: no view available to you, or to any
neighbor, contains the registers that would make your next
choice predictable.  The unpredictability of persons is
therefore not a temporary gap awaiting better brain scanners; in
this framework it is lawful and permanent, the same theorem that
protects the quantum dice.  Whether lawful unpredictability
\emph{is} freedom, or merely feels like it, is a question
philosophy has debated under determinisms far older than this
one, and the paper takes no side --- it only relocates the
mystery to sharper ground: the boundary between a world that is
whole and views that never are.  (The projection theorem:
Part~IV; the philosophical setting: Part~XI.)
\end{objqabody}

\PGenQ{26}{the unsettled reader, on reality}{If everything I see
is a projection of something deeper, is my world an illusion?}
\begin{objqabody}
\emph{Answer.}  No --- and the distinction matters more than any
other in this list.  An illusion misinforms; a projection
informs truly but partially.  Your shadow on a wall is not a lie
about you --- its shape tracks your shape lawfully; it merely
does not carry all of you.  Every measurement ever made in four
dimensions is correct, exactly as correct as it always was; the
theory changes not one reading of one instrument.  What it
denies is only the silent assumption that the four-dimensional
account is \emph{complete} --- and incompleteness is not
falsehood.  Physics has taken this step before: the solid table
was not an illusion when atoms arrived; heat was not an illusion
when it became molecular motion.  The world you see is real.
The claim is that it is a true view of something larger --- and
that the seam between the view and the whole is not hidden but
measurable: it is the horizon, and it has a temperature.  (The
long version: Part~XI's closing sections.)
\end{objqabody}

\PGenQ{27}{the reader of quantum mysteries, on observers}{Does
`observation' in this theory require a conscious observer?  Does
a falling tree make a sound?}
\begin{objqabody}
\emph{Answer.}  No consciousness is required, anywhere in the
framework.  An ``observer'' here is any physical system whose
causal situation truncates its access to the whole --- a
photographic plate, a rock warmed by the sun, a second black
hole.  The projection is defined by geometry and causal
structure, not by awareness: the tree's fall is registered in
the correlations of the air whether or not anyone is listening,
and a horizon traces out its interior for a detector exactly as
it would for an astronaut.  What consciousness does have is a
stake in the \emph{consequences}: beings like us only ever
occupy views, so the theory explains why the world as
experienced --- by anything --- is probabilistic, thermodynamic,
and directional even though the world as a whole is not.  The
measurement problem in full is the business of the cohort's
quantum-mechanics companion, and this paper leans on it only
where stated.  (The port definition: Part~IV.)
\end{objqabody}

\PGenQ{28}{the cosmic claustrophobe, on our own horizon}{Are we
living inside a black hole?}
\begin{objqabody}
\emph{Answer.}  Not in the technical sense --- our universe's
expansion history does not match a black-hole interior, and the
question as usually posed confuses two different geometries.
But it gestures at something the paper does establish: we live
\emph{inside a horizon}.  An accelerating universe has a
boundary beyond which light can never reach us, and that
cosmological horizon obeys, by theorem here, the same
thermodynamics as a black hole's --- a temperature, and the
largest entropy in nature --- derived by the same argument with
the observer on the inside rather than the outside.  The
mathematics of this paper is, in that precise sense,
autobiographical: every observer, including you, reads the
universe through a port, and the black hole is simply the case
where the port belongs to someone else.  (The inside-out
theorem: Appendix~\ref{app:de_Sitter}; the cosmological
consumer: the cohort's cosmology paper.)
\end{objqabody}

\PGenQ{29}{the reader of Wheeler, on `it from bit'}{Wheeler asked
whether the world is, at bottom, made of information.  Does this
theory answer him?}
\begin{objqabody}
\emph{Answer.}  It answers with a friendly amendment.  In this
framework the bottom layer is not information but \emph{geometry
with a law}: a six-dimensional state evolving deterministically.
Information is not what the world is made of; it is the currency
in which \emph{views} of the world are denominated --- the
measure of what a projection carries and what it must leave
behind.  What makes Wheeler's instinct profound rather than
merely poetic is that this currency obeys exact laws, and at a
horizon the exchange rate becomes visible: area in, entropy out,
at a fixed price.  The black hole is where ``it'' and ``bit''
meet across a counter and the books are settled in public ---
one statement written simultaneously in geometry, in
information, and in heat.  Which script is fundamental?  On this
paper's reading: the geometry writes, the information is the
readership.  (The three scripts: Part~XI's Rosetta section;
labeled as reading, not theorem.)
\end{objqabody}

\PGenQ{30}{every reader, at the door}{If I remember one thing
from this paper, what should it be?}
\begin{objqabody}
\emph{Answer.}  That the horizon destroys nothing.  For fifty
years the black hole was read as the place where physics loses
--- where information dies, where the laws contradict each
other, where theory goes to fail.  Read through this framework,
it is the opposite: the place where physics is most honest about
what any single view of the universe can carry, and where the
accounting of the unseen is done exactly, in public, with a
temperature and an area for ledger lines.  What falls in is not
erased; it is kept, and returned in a currency our instruments
are only beginning to learn to count.  If that is right, then
the black hole's fifty-year crisis was never about black holes:
it was the discovery, made the hard way, that our window is not
the world --- and that nature, seen whole, keeps perfect books.
(Said at full length: Part~XI, closing paragraph.)
\end{objqabody}


\section{The classical relativist}

\PObjC{8}{the membrane-paradigm veteran, on old wine}{The membrane paradigm already treats the horizon as a physical boundary with temperature, viscosity, and resistivity.  Your `ontological port' is that, renamed.}
\begin{objqabody}
\emph{Response.} The membrane paradigm is an \emph{effective} description for exterior dynamics --- its membrane properties are bookkeeping for what falls through, valid because the interior cannot answer back.  The port statement (\textsf{L-BH-Port}, \S\ref{ssec:visible-port}) is an information-theoretic claim with theorem-grade consequences: the trace over the hidden factor is \emph{forced} by the causal structure plus \AX{1}/\AX{3}, and thermality is that trace computed (Eq.~\eqref{eq:thermal-trace}), not a boundary condition posited.  The membrane's transport coefficients are recoverable downstream; the direction of explanation is reversed.
\end{objqabody}

\PObjC{9}{the global-structure purist, on teleology}{The event horizon is teleological --- defined by the entire future.  No local measurement locates it, yet your port carries ontological weight.  You have hung physics on an unobservable.}
\begin{objqabody}
\emph{Response.} Fair, and the module's operational statements are built to dodge it: the collapse theorem (T-BH-4, \S\ref{sec:rotating-collapse}) runs on the \emph{late-time universal channel}, not on horizon location; the GSL lemma is quasi-stationary; the freshness clause counts emitted quanta.  Where the exact horizon matters --- the stationary entropy statement --- event and apparent horizons coincide.  The genuinely teleological content is inherited from classical general relativity unchanged, and the honest statement is that \GGtheory{} neither worsens nor repairs it.
\end{objqabody}

\PObjC{10}{the area-theorem custodian, on arrows by fiat}{Wall proved the GSL in its class.  Your lemma `consumes' the arrow from an axiom (AX-3) rather than deriving it.  Consuming an axiom is not progress.}
\begin{objqabody}
\emph{Response.} It is custody discipline, and the lemma says so in its statement (\S\ref{ssec:arrow-links}): deriving an arrow from arrowless ingredients is exactly the trap the cohort's anti-arbitrage rule exists to prevent.  What Lemma~\ref{lem:gsl-arrow} adds is not the direction --- that is consumed, with its source named --- but the decomposition into three auditable links and the \emph{exact} exchange-class identity $\Delta S_{\rm gen} =$ spent relative entropy (Appendix~\ref{app:data-processing}, Step~4), with Wall's theorem (free and superrenormalizable fields, arbitrary cuts) registered as the strongest partial result on the declared open ceiling (\S\ref{ssec:arrow-ceiling}), not as a rival.
\end{objqabody}

\PObjC{11}{the mathematical relativist, on the extremal sector}{Extremal entropy is subtle --- Sen's entropy function, the AdS$_2$ throat, the discontinuity between extremal and near-extremal counting.  Your appendix breezes past all of it.}
\begin{objqabody}
\emph{Response.} The appendix claims only the standard statements (entropy survives at $T=0$; Nernst unattainability; the degenerating cigar, Appendix~\ref{app:extremal}, Step~4), cites the throat literature where it is relevant \citep{Sen2013}, and places the \GGtheory{} extremal counting --- holonomies at pixels --- at the same open grade as the species sum.  No resolution of the extremal-vs-near-extremal discontinuity is claimed; the module's load-bearing statements are all non-extremal.
\end{objqabody}

\PObjC{12}{the numerical relativist, on fixed backgrounds}{Every theorem in your Part V lives on a fixed background.  Evaporation is back-reaction by definition.  Your entropy law is proved exactly where it is not needed.}
\begin{objqabody}
\emph{Response.} The regime declarations are in the statements: T-BH-5 is quasi-free on a static exterior; T-BH-3/T-BH-4 are model level, verified; the evaporation clock (Appendix~\ref{app:Page_curve}, Step~3) is adiabatic Stefan--Boltzmann.  The honest chain is: exact where stationary, adiabatic where slow, and open where neither --- the 4D-flat rigor item is pre-fenced in the promotion roadmap (Appendix~\ref{app:promotion}).  We note the same limitation governs every framework's evaporation statements, including the replica-wormhole computations, which are also saddle-point statements.
\end{objqabody}

\section{The curved-spacetime field theorist}

\PObjC{13}{the Kay--Wald reader, on bifurcation surfaces}{Kay--Wald uniqueness holds on bifurcate Killing horizons.  A collapse spacetime has no bifurcation surface, so T-BH-5's citation does not license your physical case.}
\begin{objqabody}
\emph{Response.} Agreed, and that is why the collapse case is a separate theorem rather than a footnote: T-BH-4 proves the late-time factorization --- universal thermal channel plus decaying transient --- at the model level, importing Hawking's gravitating computation with attribution for the full statement.  T-BH-5 is scoped to the static exterior where Kay--Wald applies, and the composition of the two is stated, not blurred (\S\ref{sec:thermal-clock}).  The division of labor is printed in both statements.
\end{objqabody}

\PObjC{14}{the microlocal analyst, on the Hadamard class}{Which of your states are Hadamard, and does the OGGEP projection preserve the class?  If not, your `regular state' talk is decorative.}
\begin{objqabody}
\emph{Response.} The regular states invoked are precisely the Hadamard ones --- T-BH-5(iii) consumes the Kay--Wald theorems, whose regularity hypothesis is Hadamard form --- and the \OGGEP{} operation on them is restriction plus partial trace, which cannot manufacture singular behavior the parent state lacks.  What is \emph{not} claimed: a proof that the full 6D-to-4D projection preserves the Hadamard class for arbitrary bulk states; that belongs to the 4D-flat rigor item (Appendix~\ref{app:promotion}) and is flagged there, not here.
\end{objqabody}

\PObjC{15}{the algebraic field theorist, on new words}{Your `derived complex structure' is the positive-frequency choice every textbook makes.  Renaming a choice does not make it a theorem.}
\begin{objqabody}
\emph{Response.} The renaming is not the claim; the theorem is.  \textsf{T-P5-B1} proves that for quasi-free fields on a static arena the four analytic devices --- positive-frequency selection, continuation through the complexified horizon, imaginary-time semigroup, holomorphic tube --- are one datum, so selecting any one fixes the others with no further input.  T-BH-5 consumes that at the horizon: what the older literature carried as three independent prescriptions is one licensed move.
\end{objqabody}

\PObjC{16}{the constructive field theorist, on interactions}{Your thermality proofs are quasi-free.  The world is interacting.  What survives?}
\begin{objqabody}
\emph{Response.} Declared, graded, and routed: the quasi-free grade is printed in T-BH-5's statement; the GSL beyond the exchange class is a registered open ceiling, with Wall's theorem (free and superrenormalizable fields, arbitrary cuts) its strongest partial result (\S\ref{ssec:arrow-ceiling}); and the promotion path --- an interacting-grade shore --- is roadmap item (iv) (Appendix~\ref{app:promotion}).  The coefficient $1/4$ is insulated: the conical argument fixes it at the level of the gravitational effective action, independent of the matter sector's interaction strength (Theorem~\ref{thm:exact-quarter}).
\end{objqabody}

\PObjC{17}{the entanglement-entropy specialist, on edge modes}{Gauge fields and gravitons have edge modes and the Kabat contact term; `entanglement entropy = geometry' is notoriously subtle exactly there.  Your identification glosses a live technical literature.}
\begin{objqabody}
\emph{Response.} Cited and carried as a known complication, not glossed: the honest-accounting section (\S\ref{ssec:V-honest}) flags the gauge/graviton bookkeeping explicitly, with the contact-term literature \citep{Kabat1995, Solodukhin2011}.  The clean statement --- minimally coupled scalars --- is where the worked derivation lives; the gauge-sector subtlety is part of the species-sum computation, which is the module's declared principal open item (F-BH-9).  The coefficient claim survives because it is a statement about the gravitational normalization ratio, not about any one species' modes.
\end{objqabody}

\PObjC{18}{the trans-Planckian skeptic, on the exponential lens}{Hawking quanta observed at late times blueshift past every scale near the horizon, including your lock scale.  Your derivation inherits the trans-Planckian problem and adds a scale at which it must break.}
\begin{objqabody}
\emph{Response.} Three honest layers.  (i) The analytic kernel --- the one-sided spectrum and the continuation cost $e^{-\pi\omega/\kappa}$ --- is not a mode-by-mode ultraviolet statement; T-BH-4's branch-cut form shows the Planck factor arising from infrared-robust asymmetry, not from trusting arbitrarily short wavelengths.  (ii) The analog-horizon record (\S\ref{sec:exp-record}) shows the thermal spectrum surviving explicit ultraviolet modification --- imported as laboratory structure evidence with attribution.  (iii) What a lock-scale dispersion computation would add --- the \GGtheory{} analogue of the dispersive-derivation literature --- has not been carried out and is not claimed; it is a legitimate audit item.  The scale the objection says we add is the scale the falsifier cluster tests at colliders.
\end{objqabody}

\section{The string theorist}

\PObjC{3}{the holographer, on what is new}{AdS/CFT and the islands program already resolve the paradox; what is new?}
\begin{objqabody}
\emph{Response.} Those frameworks resolve it in anti-de~Sitter toy models or as an abstract boundary duality whose real-universe dual is unknown; \GGtheory{} works in the Standard-Model-containing universe.  The islands formula is not contradicted --- it is recovered as the \GFF{} ledger at the horizon stratum (\S\ref{sec:gff-ledger}).  What is new is the constitutional \emph{source} (why generalized entropy and non-naive subsystem reconstruction had to appear at all) and the exact $1/4$ for the \emph{generic, non-supersymmetric} black hole, which neither AdS/CFT nor the explicit string count delivers.
\end{objqabody}

\PObjC{5}{the D-brane counter, on explicit microstates}{Strominger--Vafa is an explicit state count; you have only the coefficient.}
\begin{objqabody}
\emph{Response.} Granted, and stated plainly throughout.  The asymmetry is one of \emph{completeness versus generality}: the string count is more complete in its protected (supersymmetric, extremal) sector; \GGtheory{} is general (Schwarzschild, Kerr, Reissner--Nordstr\"om) and fixes the coefficient without tuning, but its explicit enumeration --- the species sum to a definite number --- is the one open computational item (Appendix~\ref{app:species-sum}).  The microstates are \emph{named} (hidden-fiber $S^1_\theta$ holonomies at horizon cells, \S\ref{ssec:V-microstates}); they are not yet summed.
\end{objqabody}

\PObjC{19}{the ultraviolet enforcer, on the consistency premise}{Without supersymmetry or a string completion, what controls your 6D bulk in the ultraviolet?  `AX-4 requires a consistent completion' is a promissory note, not a theory.}
\begin{objqabody}
\emph{Response.} It is a premise, and Theorem~\ref{thm:exact-quarter} carries it as one of its three named premises rather than hiding it.  The difference from an unpayable note: the completion's low-energy fingerprints are the cohort's falsifier cluster --- the lock scale and its collider criterion, the Kaxion window, the neutrino ordering --- so the premise is exposed to instruments (\S\ref{sec:bh-exp-timeline}) rather than sheltered.  A conditional theorem with a testable condition is the honest form of the claim; the unconditional form would be the dishonest one.
\end{objqabody}

\PObjC{20}{the fuzzball theorist, on horizon structure}{The fuzzball program replaces the horizon with microstructure; information never enters a causally sealed region.  You keep the classical horizon and move the problem into bookkeeping.}
\begin{objqabody}
\emph{Response.} The two programs make opposite wagers, and the module prints the discriminating instrument: fuzzball-class microstructure at the horizon scale generically feeds post-merger echoes or QNM spectrum modification, and \GGtheory{} predicts their absence --- clean general-relativity ringdown, with F-BH-2/F-BH-3 as permanent kill rows (\S\ref{sec:ringdown}, \S\ref{sec:bh-exp-timeline}).  A reproducible echo train retires this module and hands the sector to horizon-structure programs.  Bookkeeping with a kill row is a physical claim.
\end{objqabody}

\PObjC{21}{the ER=EPR advocate, on the bridge branch}{Your late-time `bridge branch' is ER=EPR wearing your notation.}
\begin{objqabody}
\emph{Response.} The kinship is acknowledged where the idea appears (\S\ref{sec:branch-switch}), with the proposal credited \citep{MaldacenaSusskind2013}.  The difference in register: ER=EPR is a conjectural identification of entanglement with geometry; the \GFF{} branch switch is a ledger statement --- which coarse-grained saddle dominates an explicitly defined free-energy functional (Eq.~\eqref{eq:branch-min}) --- whose turnover time is fixed by an audited counting premise (L-BH-Fresh).  The reading of the dominant branch \emph{as} bridge geometry is dictionary-grade and labeled so in the island dictionary (Appendix~\ref{app:island-dictionary}).
\end{objqabody}

\PObjC{22}{the anomaly connoisseur, on arithmetic versus anomalies}{In string theory the entropy coefficient traces to central charges fixed by anomaly cancellation --- deep consistency.  Your $4\pi/16\pi$ is arithmetic.  Why should arithmetic beat anomalies?}
\begin{objqabody}
\emph{Response.} They are the same species of statement at different addresses: a normalization fixed by a consistency condition upstream of the entropy computation.  The conical $4\pi$ is the Gauss--Bonnet normalization of distributional curvature; the $16\pi$ is the Einstein--Hilbert normalization fixed by the Newtonian limit; the theorem's content is that the entropy coefficient is their ratio and nothing else enters (\S\ref{ssec:V-conical}).  Anomaly consistency enters \GGtheory{} where it should --- the \ANine{} field content that the species sum runs over --- and that is exactly the open, sign-sensitive part (F-BH-9).  The ratio is not deeper than anomalies; it is orthogonal to them, which is why it survives their bookkeeping.
\end{objqabody}

\PObjC{23}{the AdS$_3$ practitioner, on the missing Cardy check}{Every serious entropy framework checks itself against BTZ/Cardy.  Where is yours?}
\begin{objqabody}
\emph{Response.} Not performed, and conceded in those words.  The module's arena is the four-dimensional, asymptotically flat, Standard-Model-containing case, and its AdS-facing content is the Hawking--Page notebook (Appendix~\ref{app:hawking-page}), a template rather than a check.  A BTZ/Cardy audit of the \GGtheory{} normalization --- does the conical ratio reproduce the Cardy count where both are computable? --- is a legitimate outstanding cross-check we have not done.  It is recorded here so the concession has an address.
\end{objqabody}

\section{The islands and replica-wormhole theorist}

\PObjC{24}{the Page-curve deriver, on translation versus addition}{Your two-branch race is our min-ext, your turnover is our Page time, your dictionary table admits it.  What do you add beyond a translation with constitutional decoration?}
\begin{objqabody}
\emph{Response.} Three items, each with an address.  \emph{Source:} why generalized entropy exists at all --- the ledger $\GFF = F_{\rm GEP} + F_{\rm CEP}$ is derived from the open-system constitution rather than posited as a prescription (\S\ref{sec:gff-ledger}).  \emph{Premise audit:} the islands literature's random-state/typicality input is consumed here as an explicit, flagged condition with an expiry mechanism (L-BH-Fresh; the ETH import is named in Appendix~\ref{app:Page_curve}, Step~4) --- silent in most island derivations.  \emph{Reach:} the exact $1/4$ for the generic, non-supersymmetric, flat-space black hole, which the AdS derivations do not deliver.  The translation is real and credited; the additions are graded.
\end{objqabody}

\PObjC{25}{the ensemble interpreter, on factorization}{Replica wormholes compute averaged quantities; the factorization puzzle suggests semiclassical gravity is an ensemble theory.  Your single deterministic bulk is on the wrong side of that lesson.}
\begin{objqabody}
\emph{Response.} Flagged as a genuine open commitment, not answered away.  The module's position: the bulk is a single theory; ensemble-like behavior is the coarse-grained ledger's, and the residual question --- whether the boundary description's late-time correlators factorize as a single theory's must --- is adopted from the cosmology companion as a named conjectural item (\S\ref{sec:open-questions-c3}, the factorization conjecture; roadmap item (vi)).  If the ensemble reading proves obligatory rather than effective, the deterministic-bulk interpretation takes the hit, and the register says so.  This and O-BH-4 are the two commitments this Part flags rather than defends.
\end{objqabody}

\PObjC{26}{the gravitational path integrator, on derivation depth}{The QES prescription is derived from the gravitational path integral in AdS.  Your ledger `derivation' is a heuristic by comparison.}
\begin{objqabody}
\emph{Response.} For derivation depth in AdS, conceded without qualification.  The asymmetry runs the other way in scope: the path-integral derivation is anchored where the dual is known, and its extension to the real universe is itself heuristic; the \GGtheory{} chain is Derived-conditional with premises named, in the flat, Standard-Model-containing case.  The QES equivalence statement is registered at its honest grade --- a deck proposition, not a theorem (Appendix~\ref{app:statement-deck}) --- and the two programs' asymmetry (depth versus address) is stated as such in the comparison section (\S\ref{sec:related-work}).
\end{objqabody}

\PObjC{27}{the complexity theorist, on the missing resource}{Python's lunch, reconstruction complexity, the exponential cost of interior decoding --- the modern account of the interior is complexity-theoretic.  Your paper has a trace and no complexity anywhere.}
\begin{objqabody}
\emph{Response.} Correct, and no complexity claim is made.  The \DGI{} restriction plays the module's role of reconstruction obstruction at the information level --- which algebras are accessible, not how hard the decoding is (\S\ref{sec:gff-ledger}).  Whether the complexity-theoretic account is the \GDOS{} restriction seen through a resource theory is an attractive bridge question we have not developed; it is noted here as unclaimed territory rather than annexed.
\end{objqabody}

\section{The canonical-quantization referees}

\PObjC{2}{the loop-quantum-gravity theorist, on the Immirzi parameter}{You have merely replaced the Barbero--Immirzi tuning with a tuning of your own.}
\begin{objqabody}
\emph{Response.} No free parameter enters the coefficient.  In loop quantum gravity the Barbero--Immirzi parameter $\gamma$ is fixed \emph{by demanding} that the area-law coefficient come out $1/4$; it is a fit.  Here $1/4 = 4\pi/16\pi$ is the ratio of the conical-deficit curvature normalization to the Einstein--Hilbert normalization (\S\ref{ssec:V-conical}, Theorem~\ref{thm:exact-quarter}); both numbers are fixed before the entropy is computed.  There is nothing to tune.
\end{objqabody}

\PObjC{28}{the isolated-horizon builder, on glibness}{Isolated-horizon state counting is a first-principles derivation too, given gamma.  Your one-line dismissal of a thirty-year program is glib.}
\begin{objqabody}
\emph{Response.} The program's depth is real and the respect is meant: the isolated-horizon framework's boundary-degree-of-freedom counting is structurally the closest relative of the pixel-holonomy counting this paper names (\S\ref{ssec:V-microstates}).  The asymmetry O-BH-2 states is narrow and remains: one framework fixes its free parameter by the entropy match; the other has both normalizations fixed upstream.  If the Immirzi parameter is ever derived independently --- as several LQG lines attempt --- the comparison reopens on equal terms, and this Part will owe an update.  A standing test, not a dismissal.
\end{objqabody}

\PObjC{29}{the quantum-geometry theorist, on pixels}{Your `horizon pixels with U(1) holonomies' is 't Hooft's ledger and LQG's punctures with a new fiber.  What is distinct?}
\begin{objqabody}
\emph{Response.} The lineage is printed --- 't Hooft's ledger is cited at the microstate section (\S\ref{ssec:V-microstates}) --- and the distinct content is the \emph{identity} of the fiber: the counting states are holonomies of the specific compact factor $S^1_\theta$ carrying $\UoneQi$, the same fiber whose winding classes carry charge superselection elsewhere in the cohort, so the microstate identification is imported structure, not a new posit for the horizon.  What is shared with LQG --- area quantization granularity --- and what is not --- no spin networks, no Immirzi freedom --- is tabulated in the comparison section (\S\ref{sec:related-work}).
\end{objqabody}

\section{The emergent-gravity theorist}

\PObjC{1}{the Sakharov lineage keeper, on the sign problem}{This is just Sakharov induced gravity, which has a known sign/magnitude problem.}
\begin{objqabody}
\emph{Response.} It is in the induced-gravity family, and we say so (\S\ref{sec:related-work}).  But the result that matters --- the coefficient $1/4 = 4\pi/16\pi$ --- is fixed by the conical replica (\S\ref{ssec:V-conical}) \emph{independently} of the species content, regulator, and sign.  The Sakharov sign/magnitude difficulty lives entirely in the \emph{value} of the induced $G_N$, i.e.\ the species sum, which we flag as open (Appendix~\ref{app:species-sum}).  Two things distinguish \GGtheory{} from generic induced gravity: \GGEP{} makes $G_N$ \emph{fully} induced (there is no bare term, so the entanglement entropy is not a correction but the entropy), and the microstates are identified geometrically (the hidden-fiber holonomies).  The coefficient is not hostage to the sign problem; the magnitude is, and is labeled accordingly.
\end{objqabody}

\PObjC{7}{the same referee, pressing on closure}{The species sum does not close, so $G_N$ is unconstrained and the result is empty.}
\begin{objqabody}
\emph{Response.} The closure is \emph{structural}: the Kaluza--Klein tower sum reproduces the scale dimensional reduction that defines $G_N$ (Appendix~\ref{app:species-sum}), and $G_N$ is a member of GU-5 --- the geometric scale conversion factor, an input, not a quantity the entropy must independently predict.  Given a consistent ultraviolet completion (required by \AX{4}), the entanglement entropy is $A/4G_N$ by construction; the precise leading numerical coefficient is what remains open.  The $1/4$ does not depend on it.
[Reconciliation with O-BH-1: ``input'' refers to \textsc{gu-5}
anchoring, not to exemption from closure --- the sign-resolved sum
must still reproduce the fully induced $1/G_N$, and F-BH-9 prices
the failure.]
\end{objqabody}

\PObjC{30}{the thermodynamic-gravity theorist, on Jacobson}{Jacobson derived the Einstein equations from the Clausius relation in 1995.  If gravity is already thermodynamics, your constitutional derivation is a longer road to a known destination.}
\begin{objqabody}
\emph{Response.} The two roads run in opposite directions and meet: Jacobson assumes the entropy-area law locally and derives the field equations \citep{Jacobson1995}; this paper derives the entropy-area law (coefficient included) from the constitution and consumes the field equations' normalization.  Neither subsumes the other; together they close a loop that neither closes alone --- which is precisely how the related-work section files the relation (\S\ref{sec:related-work}).  The \GGtheory{}-specific content is the part Jacobson's argument cannot reach by construction: which microstates, which fiber, and why the coefficient is $4\pi/16\pi$.
\end{objqabody}

\PObjC{31}{the Visser reader, on the sign that kills}{Visser showed the induced Newton constant can come out with the wrong sign for realistic field contents.  If the A9 spectrum's sign-resolved sum misbehaves, everything downstream is rubble.}
\begin{objqabody}
\emph{Response.} Not rubble --- a fired kill row, which is the designed outcome: F-BH-9 registers exactly this failure (the sign-resolved sum fails to reproduce the induced $1/\Newton$ --- wrong sign or off closure), and the deck shows what it takes with it --- the closure claim and O-BH-7's answer --- and what it cannot touch --- the coefficient $1/4$, whose derivation nowhere uses the sum (Theorem~\ref{thm:exact-quarter}; the insulation argument is O-BH-1's).  The module's exposure is stated, bounded, and priced; that is what an internal falsifier is for (Table~\ref{tab:bh_kill}).
\end{objqabody}

\section{The gravitational-wave observer}

\PObjC{32}{the LVK data analyst, on free-riding}{GW250114 confirmed the area theorem --- a classical result.  Your experiments Part cites it as if it supported your quantum story.}
\begin{objqabody}
\emph{Response.} The record section scores it as exactly what it is: confirmation of the classical half of the GSL story --- the area theorem under merger --- with the quantum half (entropy production toward the KMS face) explicitly marked as not directly tested by any interferometer (\S\ref{sec:exp-record}).  The pre-registered verdict discipline (\S\ref{sec:exp-milestones}) exists to prevent precisely the inflation this objection fears: each instrument row states what a result does and does not touch before the result arrives.
\end{objqabody}

\PObjC{33}{the ringdown spectroscopist, on waveform silence}{Name one waveform feature that distinguishes your framework from general relativity, or concede the GW sector tests nothing of yours.}
\begin{objqabody}
\emph{Response.} The honest answer is a null with teeth: \GGtheory{} predicts clean general-relativity ringdown --- no echoes, no QNM spectrum modification, no horizon-scale structure --- so every post-merger anomaly search is a live kill attempt on F-BH-2/F-BH-3, permanently (\S\ref{sec:ringdown}; \S\ref{sec:bh-exp-timeline}).  A framework whose horizon sector predicts \emph{silence} is falsifiable by noise, exactly as no-hair general relativity is; its positive discriminators live where the deviations are unsuppressed --- the collider lock cluster, not the interferometers.  The division is printed in the falsification catalog (Appendix~\ref{app:falsification-catalogue}).
\end{objqabody}

\PObjC{34}{the primordial-black-hole hunter, on ratios only}{Your PBH row is `ratio structure only, anchor open.'  A prediction with a floating anchor is not a prediction.}
\begin{objqabody}
\emph{Response.} Conceded as to the anchor, and pre-fenced as such since the charter: the absolute mass scale needs cosmology-side input the module does not own (\S\ref{sec:PBH}; roadmap item (vii)).  The ratio ladder is nonetheless falsifiable on its own terms: any two confirmed detections begin testing the spacing claim, and the registered kill (F-BH-4) fires on a well-populated mass function off the $e^{7}$ ladder --- no anchor required either way.  A two-point test with a declared missing one-point input is a partial prediction honestly labeled --- the alternative of retrofitting an anchor is what the register forbids.
\end{objqabody}

\PObjC{35}{the LISA forecaster, on the EMRI decade}{EMRIs will map Kerr geometry to extraordinary precision in the 2030s.  What number of yours do they test?}
\begin{objqabody}
\emph{Response.} They sharpen the Kerr-null rows: the module predicts pure Kerr multipoles with deviations carrying the lock scale (the scale is P1's; the suppression bookkeeping is Part~VII's, \S\ref{sec:worked-numbers}) --- quantitatively negligible at EMRI sensitivity --- so EMRI precision converts the Kerr-null from a test into a vice: any confirmed non-Kerr multipole at the $10^{-4}$ class retires the module, unconditionally, under the registered row F-BH-10 (Table~\ref{tab:bh_kill}).  No positive EMRI signature is claimed; the milestone's pre-registered verdict says so.
\end{objqabody}

\section{The horizon-imaging astronomer}

\PObjC{36}{the EHT calibrator, on padding}{Shadow diameters test the metric at ten gravitational radii, not horizon quantum structure.  Their presence in your evidence table pads the case.}
\begin{objqabody}
\emph{Response.} The table's scoring column is the anti-padding device: the EHT rows are scored as metric-level Kerr consistency and object-class existence, nothing more (\S\ref{sec:EHT-geometry}; \S\ref{sec:exp-record}).  Their live function is exclusionary --- polarization-resolved epochs increasingly disfavor horizonless exotica, which sharpens the clean-horizon prediction this module shares with general relativity.  Consistency rows priced as consistency are context, and the table prices them.
\end{objqabody}

\PObjC{37}{the accretion astrophysicist, on plasma mud}{Sgr A* variability, M87* jet loading, hot-spot orbits --- everything EHT measures is filtered through messy plasma.  Nothing clean about horizons survives that.}
\begin{objqabody}
\emph{Response.} Largely conceded: the module quotes the imaging record for existence, scale, and Kerr consistency, and hangs no quantum claim on any plasma-filtered observable.  The census entries (masses, shadows, entropy bookkeeping in \S\ref{sec:worked-numbers}) are displays, and the honest-accounting language throughout Part VIII marks which rows are astrophysics-limited.  Where the objection bites --- any future claim that imaging tests horizon microphysics --- the refresh discipline (\S\ref{sec:exp-refresh}) is the standing guard against it.
\end{objqabody}

\PObjC{38}{the observational pragmatist, on an unobservable temperature}{A solar-mass hole radiates at sixty billionths of a kelvin into a 2.7-kelvin sky.  Your entire thermal sector is untestable in principle, and you know it.}
\begin{objqabody}
\emph{Response.} For direct astrophysical detection, yes --- printed with the numbers in the reference card and never claimed otherwise.  The thermal sector's exposure runs on three other rails, each named in the falsification catalog (Appendix~\ref{app:falsification-catalogue}): laboratory analog horizons for the spectrum's structural universality (imported with attribution); the internal-consistency kills, where the same $\THaw$ underwrites the GSL identity and the species-sum closure (F-BH-9); and the PBH window, where a landed anchor would make final-burst thermality observational (O-BH-34's concession applies).  Untestable-in-principle is the wrong verdict for a quantity that three independent registers can still kill.
\end{objqabody}

\section{The quantum-information theorist}

\PObjC{39}{the Hayden--Preskill reader, on fast mirrors}{Old black holes reflect infalling information in scrambling time.  Your Page-turnover narrative --- information waits until half the entropy is out --- ignores twenty years of fast-scrambling results.}
\begin{objqabody}
\emph{Response.} The two statements live on the two sides of the same clause: Hayden--Preskill mirror behavior \emph{presupposes} a hole already past its Page point (maximally entangled with collected radiation) --- which in this module's language is precisely a stale port (\citealp{HaydenPreskill2007}; \S\ref{ssec:freshness-mechanism}).  The freshness lemma quantifies when the mirror switches on: the qubit-model benchmark shows the information return crossing at the Page point with the predicted growth ratios (Appendix~\ref{app:benchmarks}).  No tension; one clause, read before and after its expiry.
\end{objqabody}

\PObjC{40}{the modular-theory specialist, on dressed identities}{Your `exact exchange-class identity' is the standard first-law-of-entanglement / relative-entropy identity that Casini used in 2008.  You have dressed a known engine in cohort livery.}
\begin{objqabody}
\emph{Response.} The engine is credited at every appearance --- Casini's bound and Wall's monotonicity are the named machinery (\S\ref{ssec:arrow-links}; Appendix~\ref{app:data-processing}) --- and the identity itself is displayed as what it is.  The cohort-specific additions are the custody decomposition (the arrow consumed from a named source rather than smuggled through a limit), the equality-point corollary C-BH-1 fusing the identity with the conditional-\GGA{} crossover (\S\ref{ssec:conditional-gga}) --- a statement with no counterpart in the modular literature --- and the grade fence separating exchange-class-exact from Wall-class-open.  Livery over a credited engine, with new corollaries: that is the claim, exactly.
\end{objqabody}

\PObjC{41}{the channel theorist, on the translation step}{DPI is about channels; the horizon is a region of spacetime.  The step where a causal boundary becomes a CPTP map is where all the physics hides, and you have named it rather than proven it.}
\begin{objqabody}
\emph{Response.} Named, graded, and partially proven, in that order: the translation is the third link of Lemma~\ref{lem:gsl-arrow} precisely so it can be audited separately.  Its proven core is L-BH-Port --- at a horizon the trace is forced by causal structure plus the axioms, so the channel is not an analogy but the observer's literal description (\S\ref{ssec:visible-port}, Appendix~\ref{app:data-processing}, Step~1).  Its open remainder --- the interacting, wedge-algebraic form where tensor factorization fails --- is the declared Wall-class ceiling (\S\ref{ssec:arrow-ceiling}).  The objection's instinct is right: that is where the physics hides, which is why the register keeps a lamp on it.
\end{objqabody}

\PObjC{42}{the error-correction builder, on the missing code}{The islands community has explicit quantum error-correcting code models of bulk reconstruction.  You have a partial trace and a slogan about `non-naive reconstruction.'}
\begin{objqabody}
\emph{Response.} Declined rather than defended: no code model is constructed in this paper, and the \DGI{}-constrained reconstruction of the accessible algebra (\S\ref{sec:gff-ledger}) is the information-level counterpart, not a substitute.  The constitutional order of explanation is port first, code later --- the trace structure is forced, and any code model is a representation of it.  Building the explicit \GGtheory{} code --- which subspaces, which recovery maps, at which grade --- is a well-posed project the module leaves on the table, and this entry is its acknowledgment.
\end{objqabody}

\PObjC{43}{the monogamy enforcer, on AMPS}{AMPS forced a trilemma: give up equivalence, unitarity, or field theory at the horizon.  Every framework must name its casualty.  Name yours.}
\begin{objqabody}
\emph{Response.} The casualty is the trilemma's silent premise: that the early radiation and the interior partner are \emph{independent} systems whose entanglements compete.  In \GDOS{} they are two dictionary readings of the same bulk modes through one port, and monogamy binds independent systems, not one system doubly described (\S\ref{sec:no-firewalls}).  The price paid is real and printed: late-time interior reconstructions are \DGI{}-restricted --- the naive interior algebra is not independently accessible --- which is the module's version of `field theory at the horizon' bending.  No firewall, no complementarity gymnastics, one restriction, stated.
\end{objqabody}

\section{The quantum-foundations referee}

\PObjC{4}{the Bell-theorem referee, on hidden variables}{A deterministic six-dimensional bulk is a hidden-variable theory; Bell's theorem forbids it.}
\begin{objqabody}
\emph{Response.} This is the strongest live objection, and we mark it as a commitment rather than a settled result.  The bulk is \emph{local in six dimensions}; the apparent four-dimensional non-locality measured by Bell/CHSH is read as a projection artifact of the bulk-to-boundary map --- the ``holographic shadow'' developed in \citep{ArisakaA8QM} --- not as bulk non-locality.  No-signaling is preserved because the boundary \CPTP{} channel cannot transmit superluminally.  Whether this reading is fully consistent with all Bell-type constraints in the gravitational setting is, in our judgment, the question on which the deterministic-bulk interpretation will ultimately be tested; we do not claim it closed.  The commitment now has a cohort address and a grade: the selection of the real category is \textsf{L-Constitution-1} (candidate grade) in \AOne{}, with its own falsifier row F-Constitution-10 --- a non-geometric CP phase, an unshared $i$, or a quaternionic Born residue would kill it \citep{ArisakaA1Constitution} --- so this objection stress-tests a named, graded, falsifiable claim rather than a metaphysical preference.
\end{objqabody}

\PObjC{44}{the relational purist, on perspectival entropy}{If entropy is the entropy of a channel image, thermodynamics becomes observer-relative and `the black hole's entropy' denotes nothing objective.}
\begin{objqabody}
\emph{Response.} The module embraces the relativity and locates the invariant: operational entropies are port data --- different lawful observers, different ledgers --- while the bulk state and the custody rules that map it to any port are shared (Appendix~\ref{app:data-processing}, overture).  What rescues objectivity is that the horizon port is \emph{forced}: every exterior observer class inherits the same trace structure, so $A/4\Newton$ is observer-class-invariant even though `entropy' is channel-relative.  Perspectival at the dictionary level, absolute at the constitutional level --- the same division the cohort uses for vacua and clocks.
\end{objqabody}

\PObjC{45}{the Born-rule auditor, on the unfinished chain}{Your cohort's Born rule is `closed at the relaxation level' with a named analytic residue.  A black-hole paper whose thermality claims rest on Born-rule probabilities should carry that flag on page one, not in a sibling paper.}
\begin{objqabody}
\emph{Response.} The dependency is real and the separation is stated where it matters: every thermality statement in this paper is a statement about reduced states and their KMS structure --- trace-level facts, valid under the standard reading of the formalism independently of how the cohort ultimately grounds probability.  If the field-grade representation step failed, the casualty would be the cohort's \emph{derivation} of the Born weights (\citealp{ArisakaA8QM}), not the channel identities used here, which would revert to standard-quantum-mechanics grade unchanged.  The flag lives in the source ledger (Appendix~\ref{app:source-ledger}), which is this paper's page one for consumed premises.
\end{objqabody}

\section{The historian and the philosopher}

\PObjC{6}{the philosopher of physics, on reinterpretation}{Is this a physical theory or a reinterpretation of known results?}
\begin{objqabody}
\emph{Response.} Both, in honestly separated parts.  Much of the information-paradox, Page-curve, and firewall content is reinterpretation, and we label it as such.  The exact-$1/4$ induced-gravity derivation and its reach to generic black holes are new results.  The pre-declared kill criteria (\S\ref{sec:falsifiers}) make the framework falsifiable, not merely narrative.  A reinterpretation that also yields a new derivation and pre-registered falsifiers is doing more than relabeling.
\end{objqabody}

\PObjC{46}{the rhetoric critic, on calibrated honesty as armor}{Grades, decks, ledgers, `honest verdicts' --- the apparatus of calibrated honesty reads as pre-emptive rhetoric: concede the adjectives, keep the nouns, and disarm referees by performing humility.}
\begin{objqabody}
\emph{Response.} The apparatus has teeth only if it can bite its owner, so point at the teeth: every empirical concession is paired with a kill condition (the deck's Exposure column, Table~\ref{tab:statement-deck}, keyed to Table~\ref{tab:bh_kill}); the register's standing challenge is printed --- find one claim above its table grade and it is a defect to report (Appendix~\ref{app:statement-deck}); and two objections in this very Part are logged as open commitments, not answered (O-BH-4, O-BH-25).  Performed humility cannot survive an audit that is invited in writing.  If a reader finds nouns without kill rows, the criticism stands and the defect is ours.
\end{objqabody}

\PObjC{47}{the historian of the paradox, on announcement fatigue}{The information paradox has been `solved' every decade since 1976 --- complementarity, then fuzzballs, then firewalls dissolved, then islands.  Why should your resolution age better?}
\begin{objqabody}
\emph{Response.} The paper declines the word: its paradox content is graded reinterpretation (O-BH-6's concession), its new results are two theorems and a set of identities with declared premises, and its survival criterion is externalized --- the falsifier timeline with dated instrument rows (\S\ref{sec:bh-exp-timeline}) rather than community persuasion.  Frameworks age badly when their claims outrun their registers; the design here is that the register, not the narrative, is the load-bearing structure.  Whether that discipline holds is itself testable, which is the only answer to fatigue we know how to print.
\end{objqabody}

\section{The student}

\PObjC{48}{the first-year reader, on where to start}{The paper is nearly 180 pages and Part IV assumes three companion papers.  Where does a student actually start?}
\begin{objqabody}
\emph{Response.} The annotated reading guide (Appendix~\ref{app:reading-guide}) is the door: six shelves from Bekenstein 1973 forward, then this paper re-read from Part IV.  Inside the paper, the course order is Block II of the appendices --- Appendix~\ref{app:Hawking_T} is the first calculation to reproduce by hand, and each chapter closes its Steps with the physical picture.  The three questions to ask of any section --- what is derived here, what is consumed from a companion, what is imported from the literature --- are answered mechanically by the source ledger (Appendix~\ref{app:source-ledger}).  Total prerequisite: one quantum field theory course and the will to check algebra.
\end{objqabody}

\PObjC{49}{the skeptical student, on numerology}{Why is $1/4 = 4\pi/16\pi$ not numerology?  Any ratio of small multiples of pi can be arranged to give 1/4.}
\begin{objqabody}
\emph{Response.} Because neither number is chosen: the $4\pi$ is the Gauss--Bonnet normalization of the conical deficit's distributional curvature --- change it and the Euler characteristic of the two-sphere stops being $2$ --- and the $16\pi$ is the Einstein--Hilbert normalization --- change it and Newton's constant stops matching Cavendish (\S\ref{ssec:V-conical}, worked in the replica computation).  Numerology is the freedom to arrange; here the theorem's entire content is the \emph{absence} of freedom: both constants are pinned by physics that has nothing to do with entropy, and the coefficient is their quotient.  O-BH-2 makes the same point against a framework that does tune.
\end{objqabody}

\PObjC{50}{the confused student, on whether the hole is hot}{The infalling observer crosses the horizon and measures nothing thermal.  The exterior observer sees a furnace.  Which is the black hole, hot or cold?}
\begin{objqabody}
\emph{Response.} Both, lawfully, because temperature is port data, not bulk data: the exterior observer's description is the \OGGEP{} trace of the bulk state, and that trace is thermal (Eq.~\eqref{eq:thermal-trace}); the infalling observer keeps both partners of each mode pair and her description is the pure squeezed state, temperature-free.  Neither is wrong; they hold different pages of the same ledger --- the dictionary owns the readings, the bulk owns the state.  Appendix~\ref{app:Hawking_T}, Step~6 and the de~Sitter mirror case (Appendix~\ref{app:de_Sitter}) are the worked homes of this answer.
\end{objqabody}

\section{The cohort-internal referee}

\PObjC{51}{the dependency auditor, on blast radius}{C3 consumes T-P5-B1, L-GDOS-Fresh, L-Constitution-1, and the A2 arena.  For each: if the upstream key is downgraded tomorrow, what exactly breaks here?}
\begin{objqabody}
\emph{Response.} The inheritance table (Appendix~\ref{app:inheritance}) is the map; here is the blast radius row by row.  \textsf{T-P5-B1} downgraded: T-BH-5's clause (ii) reverts from licensed to prescribed --- the continuation becomes the literature's standing assumption, every number unchanged, the grade drops.  \textsf{L-GDOS-Fresh} downgraded: the Page turnover keeps its counting derivation (Appendix~\ref{app:Page_curve}) and loses its mechanism reading; T-BH-2 reverts to named description.  \textsf{L-Constitution-1} (already candidate): consumed only at the funnel and O-BH-4; its failure kills the real-category grading, touching no entropy statement.  The \ATwo{} arena: constitutional --- its failure is cohort-wide, not C3-specific.  No single upstream downgrade reaches Theorem~\ref{thm:exact-quarter}'s three named premises.
\end{objqabody}

\PObjC{52}{the firewall inspector, on convenient separability}{F-BH-9 could fire and you would shrug: `the closure claim dies, the 1/4 survives.'  Is the module's fate really that separable, or is the firewall drawn where it is convenient?}
\begin{objqabody}
\emph{Response.} The separation is structural, not rhetorical, and it is checkable: Theorem~\ref{thm:exact-quarter}'s statement lists its three premises, and the species sum appears in none of them --- the sum enters only the closure appendix (Appendix~\ref{app:species-sum}), whose own honest verdict already states the exposure.  The deck's Exposure column (Table~\ref{tab:statement-deck}) shows the same partition from the other side: F-BH-9 retires L-BH-Tower's closure reading and O-BH-7's answer, and leaves T-BH-1's row untouched because their premise sets are disjoint.  A firewall drawn by premise bookkeeping can be audited by reading the premises; convenient ones cannot.
\end{objqabody}

\PObjC{53}{the custody clerk, on the unregistered proposal}{The real-bulk register displays a proposed entry that A8/P4 have not ruled on.  Printing it before the home papers decide is claim inflation by typography.}
\begin{objqabody}
\emph{Response.} The typography is the anti-inflation device: the entry is displayed under the word \emph{proposed}, with custody stated --- numbering belongs to the home papers --- and the deck carries it at proposal grade with no theorem key (\S\ref{sec:rb-register}; Appendix~\ref{app:statement-deck}).  The alternative --- consuming the reading silently while awaiting registration --- is the actual inflation risk, because the dependency would exist without an address.  When the home thread rules, a micro-pass updates the display either way; the proposal document is already lodged with the home papers' materials.
\end{objqabody}

\paragraph{Summary.}  Of the fifty-three: forty-four we regard as
answered, each with pointers into theorems, registers, or kill rows;
seven are conceded asymmetries or declined claims, said in those words
(O-BH-5 on explicit counting, O-BH-23 on the missing Cardy check,
O-BH-26 on derivation depth in AdS, O-BH-27 on complexity, O-BH-34
on the floating PBH anchor, O-BH-37
on plasma-filtered observables, O-BH-42 on code models); and two are
genuine open commitments that we flag rather than claim to have
settled (O-BH-4 on Bell-type constraints, O-BH-25 on factorization).
The seed scorecard of the seven-objection era is superseded by this
count.  We consider this an honest scorecard for a framework at this
stage, and the standing challenge of the named-statement deck ---
nothing is claimed above its table's register --- extends to every
response above.


%% ===================================================================
%% PART -- REFERENCES (new Part at the v13.8 split)
%% ===================================================================
\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{sec:references-c3}

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, closing
with the place where it is first cited.  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 derivation, table, or section each one underwrites.

The notes are written for two readers at once.  For the newcomer,
they trace the intellectual lineage of the present derivation: from
Schwarzschild's 1916 solution and the golden-age theorems, through
Bekenstein's entropy, Hawking's radiation, and Page's curve, to the
holographic era --- state counting, \AdSCFT{}, and the islands
program --- and then to the twenty-two-paper \GGtheory{} cohort, in
which the present paper reads the horizon as an \OGGEP{} port, the
strong-gravity member of the program's consequence branch.  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, and each ends with its
first-citation site (a section or appendix link where the enclosing
unit is labeled, otherwise the Part).

The \GGtheory{} cohort papers 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 --- the
black-hole member of the consequence series, running on the
\GDOS{} boundary law of \AFive{} --- is set out in Part~II.  The
cohort papers are listed first, in cohort order; the external
references follow, alphabetically by first author.

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

%% ---------------------------------------------------------------
%% v14.13: References rebuilt to the A10-Cosmos format (A/B split,
%% alphabetical B-list, first-citation links). The Relevance texts
%% are carried over verbatim from the v10.3/v14.11 lists; the
%% first-citation links were computed programmatically at v14.13.
%% For journal submission the Relevance paragraphs can be stripped:
%%   sed -E -i '/^\\par\\noindent\\textbf\{Relevance\.\}/d' file.tex
%% ---------------------------------------------------------------

\section*{A.~Papers by Katsushi Arisaka (the twenty-two-paper \GGtheory{} cohort)}
\begin{thebibliography}{99}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\providecommand{\doi}[1]{doi: #1}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{K.\ Arisaka (2026).}
\textit{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.} The constitutional root of the twenty-two-paper cohort: the four axioms and seven postulates from which the bulk, the boundary law, and this paper's premises descend. Cited for \AX{1} (local causality: the horizon as a one-way causal boundary), \AX{2} (\GGEP{}: no bare four-dimensional Newton constant --- the premise of Theorem~\ref{thm:exact-quarter}), \AX{3} (\OGGEP{}: observation as the bulk-to-boundary trace whose geometrically forced instance is the horizon port), \AX{4} (\MSMF{}: the no-dial selection whose admissibility filter carries this paper's ultraviolet-completion premise), and \textsf{L-Constitution-1} with its falsifier F-Constitution-10 (the candidate-grade reality register consumed at the central-claim funnel and stress-tested by O-BH-4).  First cited in the front matter.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{K.\ Arisaka (2026).}
\textit{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.} The arena companion: Theorem~0 forces the spacetime--scale--phase arena from \AX{1}+\AX{2}, and Theorem~2 fixes the canonical scale-profile metric with no free parameters ($k$, $R_\theta$, $r_c$ all eliminated). Cited in Part~II for the arena and at Step~1 of the entropy derivation, whose fiber integrals are exactly Theorem~2's profile; the renormalization-group reading of the scale direction (the holographic-RG construction) also lives there.  First cited in the front matter.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{K.\ Arisaka (2026).}
\textit{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.} The anomaly-freedom proof of the \BIsix{} modified Bianchi identity of the six-dimensional bulk. Cited where the bulk gauge structure $\SUfive\times\Uoneb\times\UoneQi$ with \BIsix{} enters: the field content over which the induced-gravity species sum of Appendix~\ref{app:species-sum} runs is fixed by the anomaly-free bulk theory this companion establishes.  First cited in the front matter.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{K.\ Arisaka (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}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.} The historical-lineage member of the bulk trio: the axiomatic method of Euclid's \emph{Elements} carried into the six-dimensional reconstruction, with the GU-5 constants and the \OGNfive{} ledger in their historical setting. Cited with the constitutional papers as the foundation on which the present black-hole module stands.  First cited in Part~II.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{K.\ Arisaka (2026).}
\textit{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.} The open-system runtime this paper computes in. Cited for the exact reduction and channel structure \T{2}--\T{4} (Nakajima--Zwanzig, Stinespring/\CPTP{}, GKLS --- the port machinery of Part~IV), for Definition~5's Port-Freshness Clause (\textsf{L-GDOS-Fresh}, consumed by Lemma~\ref{lem:stale-port} with its grade left at its A5 home), and for the boundary-gravity reduction \T{8.1}--\T{8.2}, whose scale-overlap Newton constant coincides by construction with Step~1's induced $1/\Newton$.  First cited in Part~II.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{K. Arisaka (2026).}
\textit{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}. Companion paper Arisaka-GG-A6-GRIP.
\par\noindent\textbf{Relevance.} Companion paper establishing the four geometric-invariance pillars (TGG, CCI, GFF, GRIP) of the bulk \GG{} composite gauge--gravity connection. Cited for the \GRIPin{}/\GRIPout{} regime taxonomy (Hawking--Page as the one genuine \GRIPout{} of black-hole physics; the Page time as a \GRIPin{} milestone), for the \GFF{} ledger structure theorems behind the four laws, and for the pillar-forcing of \GDOS{}.  First cited in Part~II.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{K.\ Arisaka (2026).}
\textit{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.} The quantum-information pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) that force \GDOS{} from the information side, as \ASix{}'s geometric pillars force it from the geometric side. Cited at the central-claim funnel: the open-system law whose horizon instance this paper develops is pillar-forced, not chosen.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{K.\ Arisaka (2026).}
\textit{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\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: the Born rule as a theorem of the Stinespring dilation (closed at the relaxation level; the field-grade representation step is the declared residue), EPR / Bell / CHSH / Aspect / Hensen / Giustina as evidence of the non-invertibility of the 6D-to-4D projection, Schr\"odinger's cat as a $\GDOS{}$ boundary-decoherence statement, and 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{ssec:observer-vacuum}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{K.\ Arisaka (2026).}
\textit{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.} The Standard-Model companion: the 28 parameters as observations of the one geometry. Cited for the boundary field content --- the zero-mode Standard-Model spectrum whose sign-resolved species sum, with the Kaluza--Klein tower, fixes the induced Newton constant (Appendix~\ref{app:species-sum}) --- and for the massless content entering the $c_{\log}$ prediction of \S\ref{ssec:V-log}.  First cited in the front matter.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{K.\ Arisaka (2026).}
\textit{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry}.
Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.} This companion paper 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 for the observer-vacuum reading consumed at the thermal sections, the factorization conjecture adopted as this paper's named open problem (naming home \AEight{}), the PBH-scarcity theorem bounding the dark-matter share, and the de~Sitter two-\GGA{} terminus of the way-station close.  First cited in Part~II.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Katsushi Arisaka (2026).}
\textit{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}.
Arisaka-GG-A11-Origin.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
No result of the present paper depends on it.

\par\noindent\textbf{Relevance.} Companion paper supplying the Geometric Attractor vocabulary --- the \GFF{} Morse flow, basin descent, channel-mediated exit, and the No-Exit theorem of the Ground Geometric Attractor (\GGA{}) --- that this paper uses as its dynamical grammar: Hawking radiation is read as channel-mediated exit on the gravitational stratum, ringdown as the \GFF{} Hessian at the attractor, and evaporation as descent toward the \GGA{}.  First cited in \S\ref{sec:runtime-vocab}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{K.\ Arisaka (2026).}
\textit{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}}.
Arisaka-GG-P1-KK preprint (Predictions Branch); \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.} The lock-cluster companion: the non-spatial Kaluza--Klein resonance near $3.85$~TeV. Cited where the compactification scale $\ell_{\rm lock} = \hbar/(\Mlock c)$ enters the suppression $(\ell_{\rm lock}/r_s)^2$ of every observable deviation from Kerr --- the origin of this paper's null predictions --- and as one of the external falsifiers by which the black-hole module can be killed from outside black-hole physics.  First cited in \S\ref{sec:EHT-geometry}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{K.\ Arisaka (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 (Predictions Branch); \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.} The three-generations companion: $N=3$ conditionally derived (the three-leg pincer, at the paper's own stated grade). Cited at the \OGNfive{} ledger $(M,N,L;J,K)=(1,3,5;7,35)$, whose $N=3$ entry fixes the generation count in the field content of the species sum --- the ledger governs the content, never the $1/4$ itself (Theorem~\ref{thm:exact-quarter}).  First cited in Part~III.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{K.\ Arisaka (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 (Predictions Branch); \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.} The electron companion: the electron as the unique ground state of its stratum. Cited in the interlocking-falsifiability argument: the same runtime that reads the black hole as a way-station attractor stabilizes the electron, so the black-hole module shares its falsifiers with the particle-physics branch.  First cited in \S\ref{ssec:runtime-uniqueness}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{K.~Arisaka (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{}}.
Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.} The origin-of-$i$ member paper of
the Foundations Branch of the twenty-two-paper cohort: the two real
sources of the derived complex structure (the compact phase circle and
the causal clock), the holomorphy bridge, the infrared door, and the
custody principle at full length. Its causal clock underlies the
analytic structure of the present paper: the rotating anchor and the
collapse channel of \S\ref{sec:rotating-collapse} are entries
R14--R15 of the real-bulk series it develops, and its custody
proposition is the pattern their proofs extend to five instances.
Cited at roster level in \S\ref{sec:cohort-c3}.  First cited in \S\ref{sec:cohort-c3}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{K.~Arisaka (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}.
Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.} The CPT member paper of the
Foundations Branch: C, P, and T as orientation reversals of the three
real bulk factors, time defined by its four endowments, and the
law-level arrow of the one-way clock --- kin of the horizon
thermodynamics derived here. Its Wick-quartet theorem \textsf{T-P5-B1}
proves the KMS thermal clock face on which this paper's Hawking
temperature runs. Cited at roster level in \S\ref{sec:cohort-c3}.  First cited in \S\ref{sec:cohort-c3}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{K.~Arisaka.}
\emph{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P6-Proton preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.} The object this paper's central classification is drawn against. Part~VI argues that a black hole is a Geometric Attractor with an exit channel and therefore a way station rather than a \GGA{}; the proton is the same ledger's minimum on the matter stratum with no exit channel at all, and it is what a true ground state looks like when one is reached. The contrast is the content of the classification, and without a paper on the other side of it the way-station reading would be a distinction with only one example. It supplies no number to any derivation here. First cited in \S\ref{ssec:conditional-gga}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{K.~Arisaka.}
\emph{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out},
Arisaka-GG-W1-Symmetry preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.} The White-Paper Branch survey of how conservation moved from symmetry to cohomology, and the background to what the arrow section does without arguing for it. The direction of the generalized second law is established here from non-invertibility and a custody chain rather than from any symmetry of the horizon, and a reader who finds a conservation statement with no symmetry behind it unfamiliar will find the century of argument there rather than in this Part. It supplies no link of the chain and no number. First cited in \S\ref{ssec:arrow-links}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{K.~Arisaka.}
\emph{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology},
Arisaka-GG-W2-Holonomy preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.} The classification that Step~4 of the entropy derivation lands in. The microstates counted by \(A/4\Newton\) are identified here as the \UoneQi{} Wilson-line holonomies on the compact phase circle, traced out at the horizon; holonomy protection rather than symmetry protection is precisely W2's subject, and the sixteen problems it sorts include the one this paper closes. Cited at roster level in the cohort map, and named here because the microstate identification is an instance of its classification rather than an application of its results. First cited in \S\ref{ssec:V-microstates}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{K. Arisaka (2026).}
\textit{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}.
Arisaka-GG-C1-Kaxion preprint (Predictions Branch); \url{https://preprints.arisaka-gg.org/c1-kaxion/}.

\par\noindent\textbf{Relevance.} The cohort's dark-matter companion (mass $8.1 \pm 0.8\,\mu$eV; the $1.95$~GHz laboratory line inside the closed $1.59$--$2.13$~GHz window). Cited in the primordial-black-hole section: the cohort's dark-matter slot is saturated by the Kaxion, bounding any PBH contribution to a subdominant share.  First cited in \S\ref{sec:PBH}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{K. Arisaka (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 (Predictions Branch); \url{https://preprints.arisaka-gg.org/c2-hubble/}.

\par\noindent\textbf{Relevance.} Companion paper reading the Hubble tension through the two-stage corridor of the cohort, distinguishing the 6D-to-4D projection rates sampled by local and cosmological observables (its corridor is explicitly phenomenological, with the local lift a declared open problem). Cited in the de Sitter--horizon section: the black-hole analysis here and the C2 cosmological analysis are two evaluations of one bulk-to-boundary projection, sharing the $S = A/4G_N$ horizon entropy.  First cited in Part~VII.

\end{thebibliography}

\section*{B.~Other references}
\begin{thebibliography}{99}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\providecommand{\doi}[1]{doi: #1}

\bibitem[Abbott et~al.(2016)]{LIGO2016}
\textbf{B. P. Abbott et~al. (2016).}
Observation of gravitational waves from a binary black hole merger.
\textit{Physical Review Letters} \textbf{116}, 061102.

\par\noindent\textbf{Relevance.} The first direct detection of gravitational waves (GW150914) from a binary black-hole merger. Cited as the observational anchor for merger-ringdown phenomenology; in \GGtheory{} language the quasi-normal ringdown measures the Hessian of the \GFF{} ledger at the Geometric Attractor.  First cited in Part~I.

\bibitem[Adler(1982)]{Adler1982}
\textbf{S. L. Adler (1982).}
Einstein gravity as a symmetry-breaking effect in quantum field theory.
\textit{Reviews of Modern Physics} \textbf{54}, 729.

\par\noindent\textbf{Relevance.} Adler's review of Einstein gravity as a symmetry-breaking (induced) effect in quantum field theory. Cited in the induced-gravity lineage and in the weighted species sum rule --- the modern form of Sakharov's self-consistency condition relating $\sum_i \chi_i$ to the Planck-to-cutoff ratio.  First cited in \S\ref{sec:related-work}.

\bibitem[Almheiri et~al.(2019)Almheiri, Engelhardt, Marolf, and Maxfield]{Almheiri2019}
\textbf{Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield (2019).}
The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole.
\textit{Journal of High Energy Physics} \textbf{12}, 063.

\par\noindent\textbf{Relevance.} Independently and simultaneously with Penington, derived the Page curve from the generalized entropy of an evaporating black hole, introducing the island contribution. Cited alongside Penington's paper as the QES/island breakthrough whose ledger structure \GGtheory{} reproduces at the horizon stratum.  First cited in Part~I.

\bibitem[Almheiri et~al.(2020)Almheiri, Hartman, Maldacena, Shaghoulian, and Tajdini]{ReplicaWormholes2020}
\textbf{Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini (2020).}
Replica wormholes and the entropy of {Hawking} radiation.
\textit{Journal of High Energy Physics} \textbf{05}, 013.

\par\noindent\textbf{Relevance.} Identified replica wormholes --- connected saddles of the gravitational path integral over replicated boundaries --- as the mechanism behind the island formula. Cited in Part I; the replica/conical-deficit machinery of the induced-gravity entropy derivation here is the \GGtheory{} counterpart of this construction.  First cited in Part~I.

\bibitem[Almheiri et~al.(2021)Almheiri, Hartman, Maldacena, Shaghoulian, and Tajdini]{AlmheiriReview2021}
\textbf{A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini (2021).}
The entropy of Hawking radiation.
\textit{Reviews of Modern Physics} \textbf{93}, 035002.
\par\noindent\textbf{Relevance.} The authoritative review of the islands program by its architects, including the frank discussion of its open ends (dimensional rigor, the ensemble puzzle) that Appendix~L's residue list follows. The review against which this paper's comparison Part is calibrated.  First cited in Part~XII.

\bibitem[Almheiri et~al.(2013)Almheiri, Marolf, Polchinski, and Sully]{AMPS2013}
\textbf{Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully (2013).}
Black holes: complementarity or firewalls?.
\textit{Journal of High Energy Physics} \textbf{02}, 062.

\par\noindent\textbf{Relevance.} The firewall argument: a smooth horizon, exterior effective field theory, and unitarity cannot all hold once entanglement monogamy is imposed. Cited in Part I and in the related-work placement as the information-theoretic anchor that any post-2012 framework, including \GGtheory{}, must answer.  First cited in Part~I.

\bibitem[Bardeen et~al.(1973)Bardeen, Carter, and Hawking]{BardeenCarterHawking1973}
\textbf{J. M. Bardeen, B. Carter, and S. W. Hawking (1973).}
The four laws of black hole mechanics.
\textit{Communications in Mathematical Physics} \textbf{31}, 161--170.

\par\noindent\textbf{Relevance.} Derived the four laws of black-hole mechanics as formal analogies with thermodynamics. Cited in Part I, in the first-law consistency check of the \GGtheory{} entropy for the $(M,J,Q)$ family, and in the section where each of the four laws is re-derived as a \GDOS{} open-system statement rather than an analogy.  First cited in Part~I.

\bibitem[Bekenstein(1973)]{Bekenstein1973}
\textbf{J.~D. Bekenstein (1973).}
Black holes and entropy.
\textit{Physical Review D} \textbf{7}, 2333--2346.
\par\noindent\textbf{Relevance.} The original statement that black holes carry entropy proportional to horizon area. Cited in Part I's golden-age survey, in the generalized-second-law discussion, and throughout the entropy derivation: in the \GGtheory{} reading the horizon is an \OGGEP{} port, and Bekenstein's area entropy is the entanglement of the traced bulk sector, with the exact coefficient $1/4$ fixed by induced gravity (Theorem~\ref{thm:exact-quarter}) --- the quantitative completion of the proposal.  First cited in Part~I.

\bibitem[Birrell and Davies(1982)]{BirrellDavies1982}
\textbf{N.~D. Birrell and P.~C.~W. Davies (1982).}
\textit{Quantum Fields in Curved Space}.
Cambridge University Press, Cambridge.
\par\noindent\textbf{Relevance.} The classic textbook for quantum field theory on curved backgrounds --- Bogoliubov transformations, the Unruh effect, and the Hawking computation. Appendix~H's derivation follows its chs.~3 and~8 most closely.  First cited in Part~XII.

\bibitem[Bombelli et~al.(1986)Bombelli, Koul, Lee, and Sorkin]{BombelliKoulLeeSorkin1986}
\textbf{L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin (1986).}
A quantum source of entropy for black holes.
\textit{Physical Review D} \textbf{34}, 373.

\par\noindent\textbf{Relevance.} The first identification of the entanglement entropy of quantum fields across a horizon, including its ultraviolet-divergent area law. Cited as the origin of the entanglement reading of black-hole entropy that \GGtheory{} promotes from divergent correction to the exact $A/4G_N$ itself.  First cited in \S\ref{ssec:V-strategy}.

\bibitem[Callan and Wilczek(1994)]{CallanWilczek1994}
\textbf{C. G. Callan and F. Wilczek (1994).}
On geometric entropy.
\textit{Physics Letters B} \textbf{333}, 55.

\par\noindent\textbf{Relevance.} Introduced the geometric (conical-deficit/replica) definition of entanglement entropy in field theory. Cited in the heat-kernel matching step: the equality of the entropy divergence and the induced $1/4G_N$ holds because both are the same Seeley--DeWitt coefficient $a_1$ of the same modes.  First cited in \S\ref{ssec:V-heatkernel}.

\bibitem[Carlip(2000)]{Carlip2000}
\textbf{S. Carlip (2000).}
Logarithmic corrections to black hole entropy from the Cardy formula.
\textit{Classical and Quantum Gravity} \textbf{17}, 4175.

\par\noindent\textbf{Relevance.} Derived logarithmic corrections to black-hole entropy from the Cardy formula. Cited in the $c_{\log}$ discriminator discussion and in the related-work placement as a sharp subleading test that any microscopic accounting, including the \GGtheory{} one, must face.  First cited in \S\ref{ssec:V-log}.

\bibitem[Carroll(2004)]{Carroll2004}
\textbf{S.~M. Carroll (2004).}
\textit{Spacetime and Geometry: An Introduction to General Relativity}.
Addison-Wesley, San Francisco.
\par\noindent\textbf{Relevance.} The standard modern first-pass graduate text. Recommended in the reading course as the gentler route into the Schwarzschild derivation and causal structure of Appendix~E before the heavier treatments of Wald and MTW.  First cited in Part~XII.

\bibitem[Casini(2008)]{Casini2008}
\textbf{H. Casini (2008).}
Relative entropy and the Bekenstein bound.
\textit{Classical and Quantum Gravity} \textbf{25}, 205021.

\par\noindent\textbf{Relevance.} Recast the Bekenstein bound as positivity of a relative entropy, replacing heuristic entropy accounting with a channel-monotone quantity. Cited in the second-law-direction section as the reading this paper's exchange-class identity makes exact at the horizon: generalized-entropy production as spent relative entropy toward the thermal reference.  First cited in \S\ref{sec:gsl-arrow}.

\bibitem[Chandrasekhar(1983)]{Chandrasekhar1983}
\textbf{S. Chandrasekhar (1983).}
\textit{The Mathematical Theory of Black Holes}.
Oxford University Press, Oxford.
\par\noindent\textbf{Relevance.} The complete classical theory of the Schwarzschild, Reissner--Nordstr\"om, and Kerr solutions, including the perturbation theory behind the quasi-normal ringdown spectra that Part~VIII's observational tests consume. The depth reference for Appendices~E and~P.  First cited in Part~XII.

\bibitem[Davies and Fulling(1976)]{DaviesFulling1976}
\textbf{P.~C.~W. Davies and S.~A. Fulling (1976).}
Radiation from a moving mirror in two dimensional space-time:
conformal anomaly.
\textit{Proceedings of the Royal Society of London A} \textbf{348}, 393--414.
\par\noindent\textbf{Relevance.} The moving-mirror model: an
accelerating boundary with late-time exponential redshift radiates a
thermal flux at the horizon temperature --- the exactly solvable model
form of gravitational collapse. T-BH-4's class is this one, and its
factorization content --- universal late-time channel, decaying
transient --- is proved and verified on Davies--Fulling trajectories.  First cited in \S\ref{sec:rotating-collapse}.

\bibitem[Demers et~al.(1995)Demers, Lafrance, and Myers]{DemersLafranceMyers1995}
\textbf{J.-G. Demers, R. Lafrance, and R. C. Myers (1995).}
Black hole entropy without brick walls.
\textit{Physical Review D} \textbf{52}, 2245.

\par\noindent\textbf{Relevance.} Computed black-hole entropy without brick walls, confirming that the one-loop entropy matches the renormalization of $G_N$ mode by mode. Cited in the Susskind--Uglum identity cluster underpinning the heat-kernel matching of the entropy derivation.  First cited in \S\ref{ssec:V-heatkernel}.

\bibitem[Dvali and Redi(2008)]{DvaliRedi2009}
\textbf{Gia Dvali and Michele Redi (2008).}
Black Hole Bound on the Number of Species and Quantum Gravity at {LHC}.
\textit{Phys. Rev. D} \textbf{77}, 045027. \doi{10.1103/PhysRevD.77.045027}.

\par\noindent\textbf{Relevance.} The species bound: $N$ light species lower the gravitational cutoff to $\Lambda_{\rm sp} = M_{\rm Pl}/\sqrt{N}$. Cited in the species-puzzle discussion --- why the entropy coefficient is $1/4$ and not $N/4$ --- which the induced-gravity derivation resolves by making $G_N$ itself carry the species sum.  First cited in Part~III.

\bibitem[Einstein(1915)]{Einstein1915}
\textbf{A.~Einstein (1915).}
Die Feldgleichungen der Gravitation, p.~844.
\par\noindent\textbf{Relevance.} Einstein's 1915 paper introduced the field equations of general relativity, $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}$, identifying gravity with the curvature of spacetime as a dynamical field obeying a variational principle. This is the foundational document of modern gravitational theory and the historical starting point for every attempt to unify gravity with the other forces. We cite it at the opening of Part I as the start of the century this paper retraces: the field equations whose horizons, thermodynamics, and information problem the \GGtheory{} reading addresses.  First cited in Part~I.

\bibitem[Einstein and Rosen(1935)]{EinsteinRosen1935}
\textbf{Albert Einstein and Nathan Rosen (1935).}
The particle problem in the general theory of relativity.
\textit{Physical Review} \textbf{48}, 73--77.

\par\noindent\textbf{Relevance.} Discovered that the analytically extended Schwarzschild geometry contains a bridge connecting two asymptotic regions. Cited in Part I's historical survey as the first hint of the nontrivial global structure later made explicit by the Kruskal extension and revived in the ER=EPR proposal.  First cited in Part~I.

\bibitem[{Event Horizon Telescope Collaboration}(2022)]{EHT2022}
\textbf{{Event Horizon Telescope Collaboration} (2022).}
First Sagittarius A* Event Horizon Telescope results.
\textit{Astrophysical Journal Letters} \textbf{930}, L12.

\par\noindent\textbf{Relevance.} The first horizon-scale image of Sgr A*, completing the supermassive pair with M87*. Cited in the black-hole census: their entropies ($10^{90}$--$10^{97}\,k_B$) are the largest single-object entropies in the universe, and \GGtheory{} predicts their Kerr shadows at EHT precision.  First cited in \S\ref{sec:worked-numbers}.

\bibitem[{Event Horizon Telescope Collaboration} et~al.(2019){Event Horizon Telescope Collaboration}, Akiyama, et~al.]{EHT2019}
\textbf{{Event Horizon Telescope Collaboration}, Kazunori Akiyama, et~al. (2019).}
First {M87} event horizon telescope results. I. The shadow of the supermassive black hole.
\textit{Astrophysical Journal Letters} \textbf{875}, L1.

\par\noindent\textbf{Relevance.} The first horizon-scale image of a black hole (M87*). Cited as the observational benchmark for horizon-scale geometry: \GGtheory{} predicts the Kerr shadow at EHT precision, with induced-gravity corrections suppressed far below observability.  First cited in Part~I.

\bibitem[Finkelstein(1958)]{Finkelstein1958}
\textbf{David Finkelstein (1958).}
Past-future asymmetry of the gravitational field of a point particle.
\textit{Physical Review} \textbf{110}, 965--967.

\par\noindent\textbf{Relevance.} Introduced the ingoing-null coordinate system in which the Schwarzschild surface is manifestly regular --- a one-way membrane rather than a singularity. Cited in Part I for establishing that the horizon is a coordinate artifact, a fact the \GGtheory{} boundary reading inherits.  First cited in Part~I.

\bibitem[Frolov and Fursaev(1997)]{FrolovFursaev1997}
\textbf{V. P. Frolov and D. V. Fursaev (1997).}
Mechanism of generation of black hole entropy in Sakharov's induced gravity.
\textit{Physical Review D} \textbf{56}, 2212.

\par\noindent\textbf{Relevance.} Worked out the explicit mechanism by which black-hole entropy is generated in Sakharov-style induced gravity. Cited with Jacobson's paper as the induced-gravity lineage whose program the \GGtheory{} derivation completes for the generic black hole.  First cited in \S\ref{ssec:V-strategy}.

\bibitem[Frolov and Novikov(1998)]{FrolovNovikov1998}
\textbf{V.~P. Frolov and I.~D. Novikov (1998).}
\textit{Black Hole Physics: Basic Concepts and New Developments}.
Kluwer Academic, Dordrecht.
\par\noindent\textbf{Relevance.} The comprehensive monograph bridging classical black-hole theory and the quantum developments. Recommended for the astrophysical and Kerr--Newman chapters of the reading course (Appendix~P).  First cited in Part~XII.

\bibitem[Gibbons and Hawking(1977)]{GibbonsHawking1977}
\textbf{G. W. Gibbons and S. W. Hawking (1977).}
Cosmological event horizons, thermodynamics, and particle creation.
\textit{Physical Review D} \textbf{15}, 2738.

\par\noindent\textbf{Relevance.} Established that the de Sitter cosmological horizon carries a temperature and an entropy $S_{\rm dS} = A_H/4G_N$. Cited in the de Sitter section, where the induced-gravity replica derivation applies verbatim on the Euclidean four-sphere --- in \GGtheory{} the de Sitter entropy is the same theorem as the black-hole one.  First cited in \S\ref{ssec:dS-induced}.

\bibitem[Gibbons and Hawking(1977, action)]{GibbonsHawkingAction1977}
\textbf{G. W. Gibbons and S. W. Hawking (1977).}
Action integrals and partition functions in quantum gravity.
\textit{Physical Review D} \textbf{15}, 2752.

\par\noindent\textbf{Relevance.} The boundary-term prescription that makes the Euclidean gravitational action finite and the saddle-point free energy well defined --- the computation worked term by term in the Euclidean chapter of the appendices (Step 2), whose output $I_E = \beta M/2$ feeds the $F_{\rm GEP}$ piece of the operational ledger.  First cited in Appendix~\ref{app:euclidean}.

\bibitem[Harlow(2016)]{Harlow2016}
\textbf{D. Harlow (2016).}
Jerusalem lectures on black holes and quantum information.
\textit{Reviews of Modern Physics} \textbf{88}, 015002.
\par\noindent\textbf{Relevance.} The standard modern lecture notes connecting quantum information theory to black-hole physics: channels, the Page argument, holographic entanglement entropy, and bulk reconstruction. The recommended toolkit companion for Appendices~K, L, and~O.  First cited in Part~XII.

\bibitem[Hawking(1975)]{Hawking1975}
\textbf{S.~W. Hawking (1975).}
Particle creation by black holes.
\textit{Communications in Mathematical Physics} \textbf{43}, 199--220.
\par\noindent\textbf{Relevance.} The discovery that black holes radiate thermally at $k_B T_{\rm H} = \hbar\kappa/2\pi c$. Cited in Part I for the discovery and the paradox it created, in the Bogoliubov sections where the Planck spectrum is re-derived as the \OGGEP{} partial trace of a two-mode-squeezed bulk state, and in T-BH-4, whose collapse-channel factorization imports Hawking's gravitating derivation with attribution.  First cited in Part~I.

\bibitem[Hawking(1976)]{Hawking1976}
\textbf{S. W. Hawking (1976).}
Breakdown of predictability in gravitational collapse.
\textit{Physical Review D} \textbf{14}, 2460--2473.

\par\noindent\textbf{Relevance.} Hawking's explicit argument that black-hole evaporation destroys information --- pure states evolve to mixed states. Cited in Part I and in the framework-comparison table as the sharpest statement of the paradox; \GGtheory{} keeps the semiclassical calculation while removing its information-loss reading via the open-system boundary projection.  First cited in Part~I.

\bibitem[Hawking and Page(1983)]{HawkingPage1983}
\textbf{S. W. Hawking and D. N. Page (1983).}
Thermodynamics of black holes in anti-de Sitter space.
\textit{Communications in Mathematical Physics} \textbf{87}, 577.

\par\noindent\textbf{Relevance.} The first-order phase transition between thermal AdS and the large black hole --- worked as a three-Step notebook in the appendices --- and the cleanest classical template for a branch switch selected by a free-energy ledger, which is how this paper reads the Page turnover (T-BH-2).  First cited in Appendix~\ref{app:hawking-page}.

\bibitem[Hawking and Penrose(1970)]{HawkingPenrose1970}
\textbf{S. W. Hawking and R. Penrose (1970).}
The singularities of gravitational collapse and cosmology.
\textit{Proceedings of the Royal Society A} \textbf{314}, 529--548.

\par\noindent\textbf{Relevance.} The general singularity theorems extending Penrose's 1965 result to cosmological and generic collapse settings. Cited in Part I as the rigorous statement that classical general relativity predicts its own breakdown inside horizons.  First cited in Part~I.

\bibitem[Hayden and Preskill(2007)]{HaydenPreskill2007}
\textbf{P. Hayden and J. Preskill (2007).}
Black holes as mirrors: quantum information in random subsystems.
\textit{Journal of High Energy Physics} \textbf{2007}(09), 120.

\par\noindent\textbf{Relevance.} The black-holes-as-mirrors analysis of scrambling and information retrieval after the Page time. Cited in the related-work placement as one of the information-theoretic anchors of the modern paradox literature that the \GGtheory{} account engages.  First cited in \S\ref{ssec:freshness-mechanism}.

\bibitem[Jacobson(1994)]{Jacobson1994}
\textbf{T. Jacobson (1994).}
Black hole entropy and induced gravity.
\textit{arXiv:gr-qc/9404039}.

\par\noindent\textbf{Relevance.} Jacobson's proposal that black-hole entropy is entanglement entropy in induced gravity, with the divergence absorbed into $G_N$. Cited as the direct ancestor of the present derivation: \GGtheory{} supplies the missing premise (a fully induced $G_N$, with no bare term to split against) that turns the proposal into the exact $1/4$.  First cited in \S\ref{ssec:V-strategy}.

\bibitem[Jacobson(1995)]{Jacobson1995}
\textbf{T. Jacobson (1995).}
Thermodynamics of spacetime: the Einstein equation of state.
\textit{Physical Review Letters} \textbf{75}, 1260.

\par\noindent\textbf{Relevance.} Jacobson's derivation of the Einstein equation as a thermodynamic equation of state from the Clausius relation on local horizons. Cited in the related-work placement as a complementary geometric reading of the same area law that the induced-gravity derivation makes exact.  First cited in \S\ref{sec:related-work}.

\bibitem[Kabat(1995)]{Kabat1995}
\textbf{D. Kabat (1995).}
Black hole entropy and entropy of entanglement.
\textit{Nuclear Physics B} \textbf{453}, 281.

\par\noindent\textbf{Relevance.} Identified the contact-term subtlety in the entanglement entropy of gauge fields. Cited in the honest-accounting section as a known complication: the heat-kernel matching is clean for minimally coupled scalars but requires care for gauge fields and the graviton.  First cited in \S\ref{ssec:V-honest}.

\bibitem[Kay and Wald(1991)]{KayWald1991}
\textbf{B.~S. Kay and R.~M. Wald (1991).}
Theorems on the uniqueness and thermal properties of stationary,
nonsingular, quasifree states on spacetimes with a bifurcate Killing
horizon.
\textit{Physics Reports} \textbf{207}, 49--136.
\par\noindent\textbf{Relevance.} The rigorous uniqueness and
thermality theorems on bifurcate Killing horizons --- and the proof
that the Kerr exterior admits no Hadamard state invariant under the
horizon Killing flow --- no Hartle--Hawking analogue.
T-BH-3 takes the latter as the exact form of the ergoregion
obstruction: the rotating-frame selection fails for a reason the
literature has made precise, and the move of the anchor to the
asymptotic frame is the framework's placement of that fact.  First cited in \S\ref{sec:rotating-collapse}.

\bibitem[Kerr(1963)]{Kerr1963}
\textbf{Roy P. Kerr (1963).}
Gravitational field of a spinning mass as an example of algebraically special metrics.
\textit{Physical Review Letters} \textbf{11}, 237--238.

\par\noindent\textbf{Relevance.} The exact solution for a rotating, uncharged black hole. Cited in Part I's historical survey and again where the \GGtheory{} entropy $S = A/4G_N$ and the stationary first law are verified across the full Kerr family.  First cited in Part~I.

\bibitem[Kruskal(1960)]{Kruskal1960}
\textbf{Martin D. Kruskal (1960).}
Maximal extension of {Schwarzschild} metric.
\textit{Physical Review} \textbf{119}, 1743--1745.

\par\noindent\textbf{Relevance.} Constructed the maximal analytic extension of the Schwarzschild solution, exhibiting the full four-region global geometry (two exterior regions, black-hole and white-hole interiors). Cited in Part I for the global causal structure on which all later discussion of horizons and interiors rests.  First cited in Part~I.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{J.~Maldacena (1998).}
{The large N limit of superconformal field theories and supergravity}.
\textit{Adv.\ Theor.\ Math.\ Phys.} \textbf{2}, 231.
\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 Part~I.

\bibitem[Maldacena and Susskind(2013)]{MaldacenaSusskind2013}
\textbf{Juan Maldacena and Leonard Susskind (2013).}
Cool horizons for entangled black holes.
\textit{Fortschritte der Physik} \textbf{61}, 781--811.

\par\noindent\textbf{Relevance.} The ER=EPR proposal that entangled pairs are connected by non-traversable Einstein--Rosen bridges. Cited in Part I's survey of post-firewall responses as a geometric reading of entanglement against which the \GGtheory{} bulk-to-boundary account is contrasted.  First cited in Part~I.

\bibitem[Mathur(2009)]{Mathur2009}
\textbf{Samir D. Mathur (2009).}
The information paradox: a pedagogical introduction.
\textit{Classical and Quantum Gravity} \textbf{26}, 224001.

\par\noindent\textbf{Relevance.} Proved that no small correction to the Hawking calculation can restore unitarity --- order-one departures at the horizon are required. Cited twice as the theorem that closes off the small-corrections escape route and thereby motivates the \GGtheory{} branch-switch reading of the Page turnover.  First cited in Part~III.

\bibitem[Misner et~al.(1973)Misner, Thorne, and Wheeler]{MisnerThorneWheeler1973}
\textbf{C.~W. Misner, K.~S. Thorne, and J.~A. Wheeler (1973).}
\textit{Gravitation}.
W.~H. Freeman, San Francisco.
\par\noindent\textbf{Relevance.} The encyclopedic classic of general relativity. Recommended in the Technical Appendices' reading course as a primary text for the Schwarzschild geometry, the Kruskal extension, and the classical black-hole chapters (chs.~23--25 and 31--34), in whose conventions a generation of the cited literature was written.  First cited in Part~XII.

\bibitem[Newman et~al.(1965)Newman, Couch, Chinnapared, Exton, Prakash, and Torrence]{NewmanEtAl1965}
\textbf{E. T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence (1965).}
Metric of a rotating, charged mass.
\textit{Journal of Mathematical Physics} \textbf{6}, 918.

\par\noindent\textbf{Relevance.} The Kerr--Newman metric for a rotating, charged mass --- the most general stationary black hole of the $(M,J,Q)$ family. Cited in the verification that the \GGtheory{} entropy and the stationary first law hold for every member of the family.  First cited in \S\ref{ssec:V-kerr-rn}.

\bibitem[Oppenheimer and Snyder(1939)]{OppenheimerSnyder1939}
\textbf{J. Robert Oppenheimer and Hartland Snyder (1939).}
On continued gravitational contraction.
\textit{Physical Review} \textbf{56}, 455--459.

\par\noindent\textbf{Relevance.} The first detailed general-relativistic calculation of gravitational collapse (a pressureless dust ball), showing that horizon formation is the generic endpoint of collapse rather than a pathology. Cited in Part I as the result that turned the Schwarzschild solution into astrophysics.  First cited in Part~I.

\bibitem[Page(1993)]{Page1993}
\textbf{Don N. Page (1993).}
Information in black hole radiation.
\textit{Physical Review Letters} \textbf{71}, 3743--3746.

\par\noindent\textbf{Relevance.} Established the Page curve: under unitary evaporation the radiation entropy must rise and then fall, turning over at the Page time. Cited throughout as the quantitative benchmark any resolution of the information paradox must reproduce; in \GGtheory{} the turnover is read as a \GFF{}-ledger branch switch.  First cited in Part~I.

\bibitem[Parker and Toms(2009)]{ParkerToms2009}
\textbf{L. Parker and D. Toms (2009).}
\textit{Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity}.
Cambridge University Press, Cambridge.
\par\noindent\textbf{Relevance.} The modern treatment of effective action and heat-kernel methods in curved spacetime. The recommended companion for the induced-gravity species sum of Appendix~R, whose Seeley--DeWitt technology it develops in full.  First cited in Part~XII.

\bibitem[Penington(2020)]{Penington2020}
\textbf{Geoffrey Penington (2020).}
Entanglement wedge reconstruction and the information paradox.
\textit{Journal of High Energy Physics} \textbf{09}, 002.

\par\noindent\textbf{Relevance.} Derived the Page curve directly from semiclassical gravity via entanglement-wedge reconstruction and quantum extremal surfaces. Cited as one of the two 2019 breakthrough papers; \GGtheory{} recovers the island prescription as the \GFF{} ledger at the horizon stratum, and benchmarking against these computations is flagged as an open task.  First cited in Part~I.

\bibitem[Penrose(1965)]{Penrose1965}
\textbf{Roger Penrose (1965).}
Gravitational collapse and space-time singularities.
\textit{Physical Review Letters} \textbf{14}, 57--59.

\par\noindent\textbf{Relevance.} The first singularity theorem: under mild energy and causality assumptions, gravitational collapse generically produces incomplete geodesics. Cited in Part I; together with the Hawking--Penrose theorems it establishes that singularities --- and hence horizons --- are generic, not artifacts of symmetry.  First cited in Part~I.

\bibitem[Poisson(2004)]{Poisson2004}
\textbf{E. Poisson (2004).}
\textit{A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics}.
Cambridge University Press, Cambridge.
\par\noindent\textbf{Relevance.} The standard modern source for null congruences, the Raychaudhuri equation, and horizon mechanics. Recommended alongside Appendices~E--F, whose area-theorem derivation follows the toolkit's route.  First cited in Part~XII.

\bibitem[Reissner and Nordstr\"om(1916)]{ReissnerNordstrom1916}
\textbf{H. Reissner and G. Nordstr\"om (1916).}
On the self-gravitation of the electric field according to Einstein's theory.
\textit{Annalen der Physik} \textbf{50}, 106.

\par\noindent\textbf{Relevance.} The charged, spherically symmetric solution of the Einstein--Maxwell equations. Cited where the \GGtheory{} entropy $S = A/4G_N$ and the first law are verified across the full $(M,J,Q)$ family of stationary black holes.  First cited in \S\ref{ssec:V-kerr-rn}.

\bibitem[Ryu and Takayanagi(2006)]{RyuTakayanagi2006}
\textbf{Shinsei Ryu and Tadashi Takayanagi (2006).}
Holographic derivation of entanglement entropy from {AdS/CFT}.
\textit{Physical Review Letters} \textbf{96}, 181602.

\par\noindent\textbf{Relevance.} Proposed the holographic formula equating entanglement entropy with the area of a minimal bulk surface. Cited in Part I as the modern entanglement--geometry link whose lesson --- entropy is geometry --- the induced-gravity derivation of the exact $1/4$ makes literal for generic horizons.  First cited in Part~I.

\bibitem[Sakharov(1968)]{Sakharov1968}
\textbf{A. D. Sakharov (1968).}
Vacuum quantum fluctuations in curved space and the theory of gravitation.
\textit{Soviet Physics Doklady} \textbf{12}, 1040.

\par\noindent\textbf{Relevance.} Sakharov's induced-gravity hypothesis: the Einstein--Hilbert action is generated entirely by matter loops. Cited as the head of the lineage to which this paper's entropy derivation belongs; \GGtheory{} promotes the hypothesis to a constitutional consequence of \GGEP{}, and the weighted species sum rule is the modern form of his self-consistency condition.  First cited in \S\ref{ssec:V-strategy}.

\bibitem[Schwarzschild(1916)]{Schwarzschild1916}
\textbf{Karl Schwarzschild (1916).}
{\"U}ber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie.
\textit{Sitzungsberichte der K\"oniglich Preussischen Akademie der Wissenschaften}, 189--196.

\par\noindent\textbf{Relevance.} The first exact solution of the Einstein field equations: the static, spherically symmetric vacuum metric whose surface $r = 2G_N M$ became the black-hole horizon. Cited in Part I as the opening of the sixty-year arc --- exact solution, horizon, thermodynamics, paradox --- that the present paper closes in \GGtheory{} terms.  First cited in Part~I.

\bibitem[Sen(2013)]{Sen2013}
\textbf{A. Sen (2013).}
Logarithmic corrections to Schwarzschild and other non-extremal black-hole entropy in different dimensions.
\textit{Journal of High Energy Physics} \textbf{2013}(04), 156.

\par\noindent\textbf{Relevance.} Computed logarithmic corrections to the entropy of Schwarzschild and other non-extremal black holes across dimensions. Cited in the $c_{\log}$ discussion: the log coefficient is a parameter-free discriminator between frameworks, and in the induced-gravity derivation it descends from the $a_2$ heat-kernel (conformal-anomaly) coefficient.  First cited in \S\ref{ssec:V-log}.

\bibitem[Solodukhin(2011)]{Solodukhin2011}
\textbf{S. N. Solodukhin (2011).}
Entanglement entropy of black holes.
\textit{Living Reviews in Relativity} \textbf{14}, 8.

\par\noindent\textbf{Relevance.} The standard review of the entanglement entropy of black holes, including the distributional (conical-deficit) curvature on the replica cover and the structure of logarithmic corrections. Cited for the conical-singularity identity used in the replica computation and for the $c_{\log}$ discriminator.  First cited in \S\ref{ssec:V-conical}.

\bibitem[Spohn(1978)]{Spohn1978}
\textbf{H. Spohn (1978).}
Entropy production for quantum dynamical semigroups.
\textit{Journal of Mathematical Physics} \textbf{19}, 1227.

\par\noindent\textbf{Relevance.} The entropy-production theorem for quantum dynamical semigroups with a KMS fixed point: the production functional is non-negative and vanishes on the fixed point. Cited in the second-law-direction section: the exchange-channel identity there is the black-hole instance of Spohn's theorem, with the produced entropy carried by horizon area through the induced first law.  First cited in \S\ref{sec:gsl-arrow}.

\bibitem[Starobinsky(1973)]{Starobinsky1973}
\textbf{A.~A. Starobinsky (1973).}
Amplification of waves during reflection from a rotating black hole.
\textit{Soviet Physics JETP} \textbf{37}, 28--32.
\par\noindent\textbf{Relevance.} The discovery of superradiance: waves
scattered off a rotating black hole in the band $\omega < m\Omega_H$
return amplified, extracting rotational energy. T-BH-3 reads the
phenomenon structurally: the amplification is a symplectic Gaussian
pumping channel on the asymptotic frame's derived complex structure,
which it never touches.  First cited in \S\ref{sec:rotating-collapse}.

\bibitem[Strominger and Vafa(1996)]{StromingerVafa1996}
\textbf{Andrew Strominger and Cumrun Vafa (1996).}
Microscopic origin of the {Bekenstein-Hawking} entropy.
\textit{Physics Letters B} \textbf{379}, 99--104.

\par\noindent\textbf{Relevance.} The first exact microscopic count of black-hole entropy: D-brane states reproducing $S = A/4G_N$ for extremal, supersymmetric black holes. Cited repeatedly as the decisive point of contrast --- the \GGtheory{} induced-gravity count is exact for generic (astrophysical) black holes, while the string count is exact only in BPS-protected sectors --- and as the benchmark for the open species-by-species comparison.  First cited in Part~III.

\bibitem[Susskind(1995)]{Susskind1995}
\textbf{L.~Susskind (1995).}
The world as a hologram.
\textit{Journal of Mathematical Physics} \textbf{36}(11), 6377--6396. \url{https://arxiv.org/abs/hep-th/9409089}.
\par\noindent\textbf{Relevance.} Susskind's holographic principle paper is a key historical milestone leading to modern bulk--boundary dualities. We cite it to frame OGGEP as a constitution-level insistence that observation is a bulk$\to$boundary export, generalized beyond correlators to full operational semantics.  First cited in Part~I.

\bibitem[Susskind and Lindesay(2005)]{SusskindLindesay2005}
\textbf{L. Susskind and J. Lindesay (2005).}
\textit{An Introduction to Black Holes, Information and the String Theory Revolution: The Holographic Universe}.
World Scientific, Singapore.
\par\noindent\textbf{Relevance.} The most readable account of the information paradox, complementarity, and the holographic principle as the field understood them before the islands era. Recommended as the narrative spine for the reading course's information chapters (Appendices~G, K, and~L).  First cited in Part~XII.

\bibitem[Susskind and Uglum(1994)]{SusskindUglum1994}
\textbf{L. Susskind and J. Uglum (1994).}
Black hole entropy in canonical quantum gravity and superstring theory.
\textit{Physical Review D} \textbf{50}, 2700.

\par\noindent\textbf{Relevance.} Showed that the horizon entanglement entropy renormalizes Newton's constant --- the Susskind--Uglum identity equating the entropy divergence with the renormalization of $1/4G_N$. Cited repeatedly in the entropy derivation: in \GGtheory{} the identity is forced, because \GGEP{} leaves no bare gravitational term for the divergence to hide in.  First cited in \S\ref{ssec:V-strategy}.

\bibitem[Susskind et~al.(1993)Susskind, Thorlacius, and Uglum]{STU1993}
\textbf{Leonard Susskind, Larus Thorlacius, and John Uglum (1993).}
The stretched horizon and black hole complementarity.
\textit{Physical Review D} \textbf{48}, 3743--3761.

\par\noindent\textbf{Relevance.} The original black-hole complementarity proposal (stretched horizon; exterior and infalling descriptions both valid but never jointly held). Cited in the survey of pre-island responses to the paradox; \GGtheory{} replaces observer-complementarity with the 6D-bulk/4D-boundary split.  First cited in Part~VI.

\bibitem['t~Hooft(1993)]{THooft1993}
\textbf{G.~'t~Hooft (1993).}
Dimensional reduction in quantum gravity.
Essay dedicated to Abdus Salam; arXiv:gr-qc/9310026. \url{https://arxiv.org/abs/gr-qc/9310026}.
\par\noindent\textbf{Relevance.} 't~Hooft articulated early modern arguments for holography as a fundamental constraint on quantum gravity. We cite it (together with Susskind and Maldacena) to motivate why OGGEP treats the bulk$\to$boundary map as definitional rather than optional.  First cited in Part~I.

\bibitem[Unruh(1974)]{Unruh1974}
\textbf{W.~G. Unruh (1974).}
Second quantization in the Kerr metric.
\textit{Physical Review D} \textbf{10}, 3194--3205.
\par\noindent\textbf{Relevance.} The quantum field theory of the Kerr
background: the mode analysis in which the ergoregion's negative
co-rotating energies and the superradiant band appear at the field
level, and the vacuum choices of the rotating geometry are first laid
out. T-BH-3's model statements are the structural skeleton of this
analysis, with the vacuum multiplicity placed as port assignment
relative to the asymptotic anchor.  First cited in \S\ref{sec:rotating-collapse}.

\bibitem[Unruh(1976)]{Unruh1976}
\textbf{W. G. Unruh (1976).}
Notes on black-hole evaporation.
\textit{Physical Review D} \textbf{14}, 870.

\par\noindent\textbf{Relevance.} Unruh's analysis of black-hole evaporation and the thermal response of accelerated detectors. Cited alongside Hawking's derivation for the Bogoliubov-coefficient calculation that yields the Planck spectrum at $k_B T_{\rm H} = \hbar\kappa/2\pi c$.  First cited in \S\ref{ssec:bogoliubov}.

\bibitem[Visser(2002)]{Visser2002}
\textbf{M. Visser (2002).}
Sakharov's induced gravity: a modern perspective.
\textit{Modern Physics Letters A} \textbf{17}, 977.

\par\noindent\textbf{Relevance.} The modern critical assessment of Sakharov's induced gravity. Cited both for the lineage placement and for the honest caveat it documents: spin-dependent signs in the induced $G_N$ are delicate, and for some spectra the naive sum even has the wrong sign --- one reason the numerical species closure is flagged as open.  First cited in \S\ref{sec:related-work}.

\bibitem[Wald(1984)]{Wald1984}
\textbf{R.~M. Wald (1984).}
\textit{General Relativity}.
University of Chicago Press, Chicago.
\par\noindent\textbf{Relevance.} Wald's textbook `General Relativity' (1984) is the canonical graduate-level reference for differential geometry and general relativity, including the formal mathematical machinery of warped product metrics, Christoffel symbols, the Riemann/Ricci tensors, and the Einstein equations. We cite it as the standard reference for the curvature conventions used throughout, and for the Damour--Ruffini/Wald positive-frequency continuation prescription in the Bogoliubov derivation of the thermal spectrum.  First cited in \S\ref{ssec:bogoliubov}.

\bibitem[Wald(1993)]{Wald1993}
\textbf{R. M. Wald (1993).}
Black hole entropy is the Noether charge.
\textit{Physical Review D} \textbf{48}, R3427.

\par\noindent\textbf{Relevance.} Wald's Noether-charge formulation of black-hole entropy, valid for arbitrary diffeomorphism-invariant Lagrangians. Cited in the stationary first-law consistency check ($A/4G_N$ is precisely the Bekenstein--Hawking--Wald entropy the first law requires) and as a complementary geometric reading of the area law.  First cited in \S\ref{ssec:V-kerr-rn}.

\bibitem[Wald(1994)]{WaldQFT1994}
\textbf{R.~M. Wald (1994).}
\textit{Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics}.
University of Chicago Press, Chicago.
\par\noindent\textbf{Relevance.} The mathematically careful formulation of quantum fields in curved spacetime and the definitive early statement of the generalized second law's status. Recommended companion for Appendices~G and~I.  First cited in Part~XII.

\bibitem[Wall(2012)]{Wall2012}
\textbf{A. C. Wall (2012).}
A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices.
\textit{Physical Review D} \textbf{85}, 104049.

\par\noindent\textbf{Relevance.} The strongest available proof of the generalized second law: free and superrenormalizable fields, arbitrary horizon cuts, via the monotonicity of relative entropy under restriction and the half-sided modular structure of the horizon algebra. Cited as the declared ceiling of this paper's second-law-direction result: what a full promotion of L-BH-Arrow beyond the exchange class must reach, registered open.  First cited in \S\ref{ssec:arrow-ceiling}.

\end{thebibliography}
\GGBibEnd

\end{document}
