Gap-13 — Black-hole entropy / Page curve: the gate anchor ledger

The honest one-line: Gap-13 has one real, checkable banked diagnostic — the Bekenstein–Hawking value $S=A/4G$ is reproduced on the frozen 4D zero-mode sector to $0.0028\%$ relative error, labeled as an inherited consistency check — and a genuinely non-tautological conditional route to the coefficient $1/4$; on the ratified board the gate closes as CERTIFIED-IRREDUCIBLE · RESOLVED +0 — the horizon-admissibility leg is a named external Euclidean-quantum-gravity dependency, shown openly — with the honest residuals still shown: the static leg, the Page mechanism, and the horizon-capable gravity completion they ride on remain open research residuals, and the value-match counts no microstates and derives no $1/4$.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing Gap-13 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

1. Gate status header

• Gate-level Gap-13 roll-up: CERTIFIED-IRREDUCIBLE · RESOLVED +0 (board-canonical, ratified 2026-07-08; closure-of-record: the live gate dossier; the earlier roll-up under the superseded least-closed-residual rule read OPEN (DIAGNOSTIC) — one open piece ⇒ gate OPEN). Direction: ADVANCED this round (the H2 boundary "wall" is now shown to be a phantom and a strong geometric lead makes the order-6 tip term scale as $O(1/A)$; still not discharged as a computed value — carried as a finite compute-debt residual).
• Taxonomy reconciliation (2026-07-05; grading updated to the ratified 2026-07-08 board). The gate-level grading is CERTIFIED-IRREDUCIBLE · RESOLVED +0, read as terminal reached — a certified-irreducible external dependency — with the residual family shown; this matches the board. The residual family listed in this ledger remains carried unchanged (the earlier "OPEN (DIAGNOSTIC)" roll-up under "status = the least-closed residual" reflects the superseded least-closed-residual rule, not different facts). No individual residual is deleted, closed, or re-graded. Verified narrowings folded in: (i) the horizon-admissibility leg (R6 / KT-1 / H1) is certified-irreducible as a project dependency on the external allowable-metric Euclidean-quantum-gravity replica-saddle problem — the no-internal-lever reading is a strong independence argument (near-certified-irreducible, pending a theorem-grade construction), and the anchor is semiclassical (Bekenstein–Hawking; the measured Newton constant $G$ pins the coefficient, not the mechanism); (ii) the H2 leading-area defect obstruction is discharged by conical dimensional collapse (the tip $a_6$ contribution is $O(1/A)$, so no order-$A$ term); (iii) the remaining work is a finite compute-debt only — the subleading $a_6$ value plus the Page-time $c(M)$ flux computation. Precision flag (twisted trace). The earlier phrasing "twisted heat trace $=1$ exactly" should read: the twisted trace is a $t$-independent pure fixed-point constant (value $\sqrt2$-class, set by the two $\mathbb{Z}_2$ fixed points, not $1$); the structural conclusion — no boundary tower, no order-$A$ term — is unaffected.
• The banked diagnostic leg (value-match, given-E): on the 4D zero-mode sector of the frozen geometry, the Bekenstein–Hawking value is recovered as a labeled consistency check, $$\left|\,S_{\rm inherited}-\frac{A}{4G}\,\right|\Big/\frac{A}{4G}\;=\;2.8\times10^{-5}\quad(\text{DIAGNOSTIC; inherited, not derived ab initio}).$$
• The conditional coefficient leg (T2), given-E: under four value-free hypotheses, $$ (H1\wedge H2\wedge H3\wedge H4)\ \Rightarrow\ S=\frac{A_H}{4G},$$ with no $1/4$ baked into any hypothesis — but the discharge of $H1$–$H4$ (the HORIZON-ADMISSIBILITY theorem) is NOT achieved.
• Status, split so it cannot be misread:
  • Value-match diagnostic (S1): DIAGNOSTIC / DERIVED-GIVEN-E — an inherited consistency check, given $E$ (the linearized-GR / Einstein–Hilbert limit) and the measured $G$. The value-match is not the microstate-counting theorem and is not a derivation of $1/4$.
  • Static leg — coefficient $1/4$ (R1): OPEN — STRENGTHENED. The scheme-independent entanglement route is REFUTED (T1); $1/4$ is now DERIVED-GIVEN-E conditional on one named, unproven theorem (T2 / KT-1) plus one value-free posit. NOT AXIOM-CLOSED.
  • Dynamical leg — Page curve (R2): OPEN. An explicit island/QES construction exists (T3); the qualitative turnover survives scheme-independently; the Page time is open (no target-blind central charge).
  • Horizon-capable gravity completion (R6 = KT-1): OPEN / global open problem.

Given $E_{\rm frozen}$, the perturbative graviton carrier delivers the linearized, weak-field, infinite-area sector. What stays open is everything in the finite-area, strong-field, nonperturbative horizon regime — the microstate count, the Page mechanism, and the saddle they presuppose.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what Gap-13 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics. Note explicitly: no horizon-sector primitive, microstate count, island prescription, or replica saddle has any hash — none exists as a frozen object; the T2/T3 constructions are adjudication-overlay analyses, not frozen-record outputs.
• Upstream spectrum / branch $\mathfrak{B}_{\rm active}=\big[M_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]\oplus\cdots$, with $K_6=SU(3)/T^2$ and spin-$\mathbb{C}$ index $\chi(K_6,E)=-3$, is given / charged / inherited. Gap-13 does not derive $E$. Every "passes" below is a statement about this $E$, not a derivation of it.
• Measured Newton constant $G$. Supplies the zero-mode normalization and is the measured IR anchor onto which $1/4$ is anchor-transferred. GIVEN / measured.
• Inherited Einstein–Hilbert IR limit $E$ (linearized-GR limit). The 4D zero-mode sector inherits this as given-E input; it is the carrier on which the diagnostic rides, not a banked geometry output.

3. Object anchors (given-E / upstream)

The perturbative graviton arises as the normalized zero-mode reduction of the spacetime-facing block: $$ \text{frozen active branch}\ \xrightarrow{\text{zero-mode reduction}}\ \text{4D massless spin-2}\ \xrightarrow{\text{linearized}}\ \text{vacuum}+\text{masslessness}+\text{Newtonian limit}+\text{Einstein–Hilbert IR}. $$ This is the genuine carrier of gravity — the linearized, weak-field, infinite-area sector. Where it stops is the honest boundary: horizon physics is a nonperturbative, strong-field, finite-area regime, and the zero-mode projection does not reach it. Status: GIVEN-E / upstream-inherited — not Gap-13-derived, and the graviton gate it rides on (Gap-01 / GRAVITON) closes on the ratified board as DERIVED-GIVEN-anchor · RESOLVED +0, with its horizon-sector completion carried as a shown residual (the shared compute-debt this page counts once).

4. Root and master-anchor traceability

Deep roots that are load-bearing for Gap-13:

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch · given-$E$ · the measured $G$ anchor · the declared/carried posits (AX-INDUCED-G, AX-SADDLE-ENTROPY, AX-HORIZON-FINITE-DIM) · open-residual discipline · the anti-reverse-engineering from the measured value axiom (AX-BLIND-CUT-MEASURE).

5. The Gap-13 anchor ledger (the universal table)

6. The arithmetic — the conditional win, in full

T1 closes off the entanglement route to $1/4$ as tautological; T2 builds the route that is not.

The T2 algebraic core (rigorous; independently re-verified in sympy, both conventions). With $\ln Z=-I$, $E=\partial_\beta I$, $S=\beta E+\ln Z=(\beta\partial_\beta-1)I$:

Convention A (on-shell Euclidean Schwarzschild + Gibbons–Hawking–York). $$ I(\beta)=\frac{\beta^2}{16\pi G},\quad E=\partial_\beta I=\frac{\beta}{8\pi G},\quad S=(\beta\partial_\beta-1)I=\frac{\beta^2}{16\pi G}. $$ At $\beta=\beta_H=8\pi GM$: $E=M$ and $S=(8\pi GM)^2/(16\pi G)=4\pi GM^2$. With $A_H=4\pi(2GM)^2=16\pi G^2M^2$, one gets $A_H/(4G)=4\pi GM^2=S$.

Convention B (conical defect / Fursaev–Solodukhin). The replica geometry $M_n$ carries distributional tip curvature, $$ \int_{M_n}\!\sqrt{g}\,R=n\!\int_{M_1}\!\sqrt{g}\,R+4\pi(1-n)\,A_H+O\!\big((1-n)^2\big), $$ with the $4\pi$ fixed by 2D cone topology (rigid, content-blind). Inserting into the action, $$ I_n=nI_1-\frac{(1-n)A_H}{4G},\qquad S=(n\partial_n-1)I_n\big|_{n=1}=\frac{A_H}{4G}. $$ The decomposition is exact and geometric / topological, not thermodynamic and not entanglement-derived: $$ \frac14=\underbrace{\frac{4\pi}{16\pi G}\cdot G}_{(\text{conical-tip solid angle})\times(\text{EH normalization})^{-1}\times(G\ \text{measured anchor})}. $$ No matter loop, no UV cutoff $\epsilon$, no entanglement-cutoff matching enters.

Diagnostic — the route is specific, not trivial (the target-blindness check). A route is non-tautological iff at least one load-bearing step could have failed and the number is not engineered:

KT-1, KT-2, KT-5 are genuine failure modes ⇒ $1/4$ could have come out wrong ⇒ non-tautological (contrast the entanglement / Susskind–Uglum route, which could not fail — the $\kappa^3/\pi$ tautology). Net: $1/4$ is DERIVED-GIVEN-E conditional on KT-1 (unproven) + KT-6 (carried posit), with KT-2/KT-5 unverified ⇒ R1 stays OPEN.

We do not assert the gate obstruction vanishes: the HORIZON-ADMISSIBILITY discharge is not closed. We also do not reconstruct the $0.0028\%$ value-match's internal integral here — it is not in the rendered corpus to synthesize (see H6); fabricating it would be worse than an honestly-marked open audit item.

7. Declared-structure splits, into honest objects

7.1 The single phrase "$S=A/4G$ holds in this theory" hides three different claims with three different statuses.

1. The value-match (S1). $S=A/4G$ to $2.8\times10^{-5}$ on the inherited 4D sector. Status: DIAGNOSTIC / DERIVED-GIVEN-E — an inheritance check, given $E$ and $G$; it counts no microstates.
2. The conditional coefficient (T2). $(H1\wedge H2\wedge H3\wedge H4)\Rightarrow S=A_H/4G$. Status: CERTIFICATE-CONDITIONAL — the implication is rigorous and non-tautological; the discharge is unproven.
3. The microstate-counting theorem (S2.c). $\log N(A)=A/4G+\text{subleading}$. Status: OPEN — gated on Gap-01; explicitly not S1.

7.2 The single phrase "horizon area law" splits into two physically unrelated objects. The framework's $SU(3)_c$ Wilson-loop confinement area law (AUDIT-tier, static potential between color charges) is not horizon entropy on a causal horizon. Borrowing $1/4$ across them would be reverse-engineered from the measured value. Status: TERMINAL firewall (R4) — kept unlinked.

7.3 The single posit "horizon $S$ is entanglement" splits into the dead seam and its value-free successor. AX-ENT-EXHAUSTS returns the number by construction (refuted as a derivation by T1; proved non-demotable). In the T2 coefficient argument it is retired and replaced by AX-SADDLE-ENTROPY, which fixes the ontology (no $1/4$ baked in). The number now comes from rigid geometry; the ontology is a value-free carried posit.

The sharp reduction (REDUCE, not RELOCATE). Hypotheses $H3$ and $H4$ collapse onto a single named object — the order-6 mixed Neumann/Dirichlet $S^1_Y/\mathbb{Z}_2$-orbifold-boundary + conical-defect Seeley–DeWitt coefficient. The $K_6=SU(3)/T^2$ spectrum and $\chi=-3$ fix the internal weights (frozen/derived); the order-6 boundary coefficient they multiply is the object whose explicit computation is absent from the literature. ADVANCED this round: the "mixed N/D boundary" turns out to be a phantom — the $S^1_Y/\mathbb{Z}_2$ free $\mathbb{Z}_2$ reflection is a global isometry, so the twisted heat trace is a $t$-independent pure fixed-point constant (set by the two $\mathbb{Z}_2$ fixed points — see the precision flag in §1) and there is no genuine boundary tower; product factorization + Fursaev–Solodukhin tip-locality then make the tip term scale as $A/r_h^4=O(1/A)$, plausibly vanishing at order $A$. This is a strong lead, not a discharge — the explicit order-6 computation still owes. Same corpus-shared wall — Gap-01 (owner); consumers GRAVITON R1/R6, UQF-3 R4, UQF-9 R2, SG-6 R9 — count once.

Honest twin observation. The feature that makes $1/4$ non-tautological ("could come out wrong") is identical to the feature that makes it unproven ("must compute the defect coefficient to know it didn't") — and that computation is the BLOCKED object.

Modesty flag. $H4$ ("induced action exactly Einstein–Hilbert at order $A$, frozen $G$, no Wald / higher-curvature / boundary-defect order-$A$ term") is near-coextensive with "the $A$-coefficient is $1/4G$." The theorem is saved from circularity only because $H4$ is recast as a value-free structural vanishing whose truth is fixed by the uncomputed, named coefficient. Most of the work is honestly relocated into $H4$ and then located — not eliminated.

Stale-prose correction. The load-bearing object is the defect / boundary order-6 mixed N/D $S^1_Y/\mathbb{Z}_2$ coefficient — say "defect/boundary coefficient," not "bulk $a_6$." The bulk $a_6$ is COMPLETE_CROSSCHECKED; only the defect/boundary coefficient is BLOCKED.

8. The Page leg — explicit QES, one number missing

A spherical s-wave reduction of the frozen 4D Schwarzschild sector to a 2D dilaton-gravity throat (dilaton $\phi(r)=A(r)/4G$ as the area operator, matter $=c$ free 2D fields coupled to a bath) extremizes the generalized entropy $$ S_{\rm gen}=\frac{\phi(\partial I)}{4G}+S_{\rm bulk}(R\cup I). $$ The QES solves explicitly: $$ x_\star=-\frac{r_h}{2}+\frac{\sqrt{3Gc+9\pi\alpha\,r_h^2}}{6\sqrt{\pi\alpha}};\qquad\text{semiclassical }(Gc\ll\alpha r_h^2):\ x_\star\sim\frac{Gc}{12\pi\alpha\,r_h}. $$ The island sits $\sim Gc/r_h$ outside the horizon — the textbook location, derived from frozen-geometry inputs, not assumed, with the area-term coefficient $\alpha$ kept symbolic and sourced from T2 (never an assumed $A/4G$ — that would move the target into the QES formula). Two validity conditions:

• Condition 1 — area term from T2, not assumed: MET IN FORM but inherits all of T2's conditionality ⇒ T3 adds zero independent evidential weight on $1/4$; it shows the island machinery is consistent with the turnover given T2.
• Condition 2 — central charge from the frozen spectrum: FAILS. $c$ is gauge-convention-dependent ($c\in\{26.5,50.5\}$), greybody/angular-momentum-dressed, and Hawking-temperature(mass)-dependent ($c\sim2$ at solar mass to $\sim50$ at Planckian). No single target-blind frozen $c$ exists ⇒ the Page time is unfixed; only the qualitative turnover survives scheme-independently.

Net: the island shape is DERIVED-GIVEN-E conditional on the T2 saddle; the Page time is OPEN. Notably no AX-HORIZON-FINITE-DIM is needed as an input now (the turnover is structural). It is correctly not phrased as a theorem of unitarity.

9. Open residuals — by family

Family A — the static leg (coefficient $1/4$). - R1 — no horizon microstate count / no $S=A/4$ derivation. OPEN — STRENGTHENED. Strong entanglement route REFUTED; only scheme-dependent T1′ survives; $1/4$ DERIVED-GIVEN-E conditional via T2. NOT AXIOM-CLOSED. - H2 — the order-6 mixed N/D orbifold-boundary + conical-defect Seeley–DeWitt coefficient. OPEN / ADVANCED. Strong geometric lead this round: the mixed-boundary "wall" is a phantom (the $S^1_Y/\mathbb{Z}_2$ free reflection is a global isometry ⇒ twisted heat trace $=$ a $t$-independent pure fixed-point constant, no boundary tower), and product factorization + Fursaev–Solodukhin tip-locality make the tip term scale as $A/r_h^4=O(1/A)$, plausibly vanishing at order $A$. Not discharged — the explicit order-6 computation is literature-absent; discharges $H4$ and the no-internal-$A$ part of $H3$ only once completed. - H3 / MO-13-1 — the geometry-native microstate counting theorem $\log N(A)=A/4G+\text{subleading}$. OPEN — gated on Gap-01. Explicitly not S1.

Family B — the horizon-capable gravity completion. - R6 / KT-1 / H1 — HORIZON-ADMISSIBILITY. OPEN / global open problem. The single load-bearing object; the discharge of $H1$–$H4$. - R5 — perturbative graviton scope boundary. DISCLOSED (scope) — route-given for the perturbative part; OPEN for the horizon.

Family C — the dynamical leg (Page curve). - R2 / H4 / MO-13-2 — the Page-curve turnover computation $S_{\rm rad}(t)$ rises then falls. OPEN. Explicit QES built; Page time OPEN (no target-blind $c$); triply gated. - H5 / MO-13-4 — Gap-14's system–bath split on the 13D fields. OPEN / inherited honest halt (NotImplementedError); owner-physics, not AI-attemptable.

Family D — audit & governance. - H6 / S1.b / MO-13-7 — reproducibility witness for the $0.0028\%$ figure. OPEN / audit-only. Weakens only auditability, not the gate's openness. - H7 — topological-rank → degeneracy map (PC-4 / MO-13-5), discharge-gate registry (MO-13-6), verbatim PC texts. RESERVED / governance. PC-4 forbids any state estimate from the rank until the activation gate passes.

Family E — terminal firewalls (kept, not closed-by-relabel). - R4 — confinement-area-law conflation. TERMINAL firewall — kept unlinked. - R7 — Bekenstein/Hawking/Page as external targets. TERMINAL given-E targets — counting them as outputs is the signature mis-close.

10. Anti-claims (what this page refuses to say)

• Gap-13 does not derive $E$, and the value-match does not count microstates: $S=A/4G$ to $2.8\times10^{-5}$ is a DIAGNOSTIC, never the microstate-counting theorem (S2.c).
• No coefficient $1/4$ is asserted as a derived output. $1/4$ is DERIVED-GIVEN-E conditional on the unproven HORIZON-ADMISSIBILITY theorem (KT-1) plus the value-free AX-SADDLE-ENTROPY posit. ANCHORED ≠ DERIVED: $1/4$ is anchor-transferred onto the measured $G$, never anchor-eliminated; floor $\geq 1$.
• The entanglement route to $1/4$ is a tautology, not a derivation. The shared Susskind–Uglum counterterm returns $\dfrac{\text{ent-area term}}{\text{induced }1/G}=\dfrac14$ for any minimal content — including a free scalar with no microstates — so it carries zero independent evidential weight (the $\kappa^3/\pi$ true-by-construction signature).
• No quantitative Page curve is claimed. Only a qualitative, area-dominated turnover survives; the Page time is open. The turnover is not phrased as a theorem of unitarity.
• The two area laws are unrelated. Borrowing $1/4$ from the $SU(3)_c$ confinement Wilson-loop area law would be reverse-engineered from the measured value; the firewall (R4) stays in force.
• The conditional theorem is conditional. The T2 certificate certifies the implication and the non-tautology, NOT a closed gate; $H1$–$H4$ are not discharged.
• The frozen-branch hashes are audit anchors; they do not validate the physics — and no horizon-sector object has any hash at all.
• Local / banked closure is not whole-gate closure. A value-match plus a conditional route is not a microstate count plus a Page mechanism. No status upgrade.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package; a refuting outcome is an equally valid, publishable result.

1. H1 — HORIZON-ADMISSIBILITY (the single load-bearing object). Prove the frozen 4D zero-mode sector extends to an admissible, dominant Euclidean black-hole replica saddle family $M_n$ under the frozen $C_{\rm admiss}/F^+$ grammar (discharge $H1$–$H4$). Machinery: Euclidean quantum-gravity saddle-point methods; Fursaev–Solodukhin conical calculus; the frozen $13\text{D}\to4\text{D}$ effective action. Falsifier: if no admissible saddle exists, the gate stays OPEN for a deeper reason — a publishable structural finding. Do not route any coefficient through the shared Susskind–Uglum $1/G$ counterterm (dead route, T1).
2. H2 — the defect / boundary heat-kernel coefficient (the primary blocked computation). Compute the order-6 mixed N/D $S^1_Y/\mathbb{Z}_2$-orbifold-boundary + conical-defect Seeley–DeWitt coefficient and show its order-$A$ contribution vanishes for the frozen internal weights. Machinery: Gilkey/Vassilevich boundary heat-kernel expansion extended to $a_6$ with mixed boundary conditions on the orbifold + conical background; coordinate with Gap-01 (wall owner; count once). Falsifier: a nonzero order-$A$ term ⇒ $S=A/4G+\text{Wald correction}$, shifting the leading coefficient. Never assert "no $A$-term" without computing the geometry-fixed value.
3. H3 / MO-13-1 — the geometry-native microstate count. After Gap-01 supplies a UV-controlled near-horizon Hilbert space, construct and count the near-horizon ensemble; prove $\log N(A)=A/4G+\text{subleading}$ with no fitted prefactor, freeze-before-compare clean. Falsifier: a target-blind count landing on a different coefficient. Trap: do not import invented weights (the corpus uses graviton dim 91, ghost dim 13, grav $-2\times$ghost $=65$ — never "67/11").
4. H4 / MO-13-2 — the Page-curve turnover. With Gap-14's split and the area term from T2, compute $S_{\rm rad}(t)$ via the geometry-native QES (T3) and show the post-Page-time decrease; resolve the central charge $c$ target-blind. Falsifier: no turnover, or a Page time inconsistent with the count. Keep $\alpha$ symbolic — do not source the area term from an assumed $A/4G$.
5. H5 / MO-13-4 — Gap-14 system–bath split. Owner-physics: exhibit a structurally consistent pointer basis and reduced dynamics on the 13D fields — or a refutation (an inconsistent pointer basis is a decision-grade falsifier). Do not treat the decoherence honest-halt as something to route around.
6. H6 / S1.b — reproducibility witness. Surface the actual re-runnable $0.0028\%$ area-law integral (effective action, zero-mode reduction, $G$ normalization, comparison target). Never reconstruct or fabricate the number. Audit hygiene; no physics leverage on the count or mechanism.
7. H7 — rank → degeneracy map + discharge-gate registry + verbatim PC texts. Pass the pre-registered activation gate and derive the rank → degeneracy map (do not invoke the rank early — PC-4); the owner supplies the governance registry; a finite documentation check confirms the verbatim PC-1…PC-5 force matches the reconstruction.

Priority order: H1 → H2 → H4/T3 → cross-checks (QNM / Hawking-flux / first-law, none of which are closers). Closing H1 (via the H2 defect coefficient) discharges the static leg of Gap-13 and feeds GRAVITON R1/R6, UQF-3 R4, UQF-9 R2, and SG-6 R9 — leverage, counted once. Even in the best case, $1/4$ remains given $E$ — anchor-transferred, never anchor-eliminated; floor $\geq 1$ forever.

12. Completion tests for this page

Required presence (all met): gate roll-up CERTIFIED-IRREDUCIBLE · RESOLVED +0 (residual family shown) · the banked value-match $S=A/4G$ at $2.8\times10^{-5}$ labeled DIAGNOSTIC / DERIVED-GIVEN-E · $E$ not derived · frozen hashes (AUDIT ONLY) · the measured $G$ anchor · the T2 conditional $(H1\wedge H2\wedge H3\wedge H4)\Rightarrow S=A_H/4G$ with no $1/4$ baked in · the T2 algebraic core (both conventions) · the target-blindness check table KT-1…KT-6 as the specificity diagnostic · T1 refutation + scheme-dependent T1′ · the explicit QES shape $x_\star$ · the central-charge open object · every residual (R1–R7, H1–H7, MO-highlights) as its own row · the confinement firewall · the anti-overclaim wall · the carried posits (AX-INDUCED-G, AX-ENT-EXHAUSTS, AX-SADDLE-ENTROPY, AX-HORIZON-FINITE-DIM) · the anti-claims.

Required absence (all held): no claim that Gap-13 is closed · $E$ derived · $1/4$ a derived output · the value-match sold as a microstate count · a quantitative Page curve · the turnover as a theorem of unitarity · the entanglement route sold as a derivation · $1/4$ borrowed from confinement · hashes validate physics · banked diagnostic = whole-gate closure · any status upgrade · any reader-visible build-process vocabulary.

Completion report. - Tests passed: all required-presence items present; all required-absence items held. - Tests failed: none. - Open items: R1, R2, R6 and the named work-packages H1–H7 remain OPEN exactly as in the dossier — this page reports them, it does not close them. - Assumptions made: none beyond the dossier; every number ($2.8\times10^{-5}$, $4\pi GM^2$, $A_H/4G$, $c\in\{26.5,50.5\}$, dims $91/13/65$) and theorem (T1, T1′, T2, T3, KT-1…KT-6) is traceable to ../articles/DOSSIER_GAP13_FULL.md.

This gate anchor ledger follows the canonical SG-4 eleven-part shape and universal table. No status was upgraded.

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why the entanglement route to $1/4$ is true-by-construction) · Layer 4 — carrier-forcing & the given-E wall (why the perturbative graviton carrier does not reach the horizon) · the sibling UQF-3 ledger (the shared Clay-class / heat-kernel wall) · the full Gap-13 dossier.