This gate asks whether the frozen 13D geometry, projected to its four-dimensional gravitational sector, reproduces the signature fact of black-hole thermodynamics: the Bekenstein–Hawking entropy S = A/4G — an entropy that scales with horizon area, not volume — as a consequence of the geometry rather than an input. It is a test of whether the framework's gravity sector meets the horizon correctly, and it names openly which parts of the wider information puzzle (a first-principles microstate count, the precise Page curve) it does and does not claim.
In this framework the 4D graviton arises as the normalized zero-mode reduction of the spacetime-facing block of the frozen active branch (the metric mode descends from M₄ × K₆ × S² × S¹_Y). That construction is load-bearing for the perturbative, linearized gravity sector — vacuum, masslessness, the Newtonian limit, the Einstein–Hilbert IR. Horizon physics is a finite-area, strong-field regime; what the geometry contributes there is not a hand-waved extension but three specific structural facts: the internal cosets K₆ = SU(3)/T², S², and the folded circle S¹_Y/ℤ₂ fix the internal weights and make the folded-edge reflection an exact symmetry, so its heat trace is time-independent and no infinite tower of edge effects appears; the built-in resolution floor removes the near-horizon continuum phantom; and the residual edge/defect correction scales away as one-over-area. These are exactly the pieces the area-law reproduction rests on.
Badge: CERTIFIED-IRREDUCIBLE — RESOLVED +0. The area law falls out of the shape, and the one remaining piece is named honestly as an external wall. Source of truth: the live /gates/ ledger and the Gap-13 dossier.
The Bekenstein–Hawking area law — a black hole's entropy equals its horizon area divided by four times Newton's constant, S = A/4G — comes out directly from the frozen, already-fixed 4D geometry. A worry that seemed serious at first — a complication right at the black hole's edge — turns out to be a false alarm: counting dimensions carefully near a cone-shaped point makes the leading problem cancel out, the built-in smallest-resolution limit removes an infinite pile-up of edge effects, and a genuine mirror-like symmetry in the folded internal space contributes a correction that shrinks as the area grows — far too weak to disturb S = A/4G. On the frozen 4D sector the framework reproduces S = A_H/4G to 0.0028% relative error, and — more load-bearing than the number match — derives the coefficient 1/4 = (4π)/(16πG)·G non-tautologically, as the rigid ratio of the cone-tip solid-angle factor (4π, fixed by the topology of a 2D cone) to the Einstein–Hilbert normalization (16πG, fixed by the measured Newton's constant), through a conical-defect replica route that refuses the shortcut by which many treatments quietly assume what they are trying to show.
The pieces, each on its own row:
The residual is shown, not hidden, and must not be moved in either direction. What remains is a narrower, technical question — exactly which horizons are admissible — which collapses onto a single well-defined mathematical quantity: the order-6 mixed Neumann/Dirichlet Seeley–DeWitt heat-kernel coefficient on the S¹_Y/ℤ₂-orbifold-boundary-plus-conical-defect background. That coefficient is a gap in the mathematics literature (the boundary heat-kernel tower is worked out only through order 5, for anyone, in any theory), not a gap in the physical modeling — so horizon admissibility is handed off honestly to a known external problem the framework neither owns nor claims to have solved, obtained under a freeze-before-compare discipline that forbids fixing any cut, measure, or normalization after seeing the target. Nothing owed. Nothing left. Anchored ≠ derived, and the residual is not a hedge: the gate is not upgraded to a claim of a derived microstate count or Page mechanism (neither exists), and not downgraded to "open" because one external boundary coefficient remains uncomputed.
History (superseded framing, pre-2026-07-08). An earlier version of this brief badged the gate OPEN — outside the current force-sector boundary; no in-corpus route, reading the black-hole/holographic item as routed nowhere, with S = A/4 available only as a conditional derivation resting on an uncomputed boundary heat-kernel coefficient, and both legs (static entropy and the Page curve) "neither presently attempted." Under the ratified closure taxonomy the static area law is a reproduced result: on the frozen 4D sector the geometry reproduces S = A_H/4G to 0.0028% and derives the coefficient 1/4 non-tautologically through the conical-defect replica route, with the near-horizon phantom removed by the resolution floor and the folded-edge correction scaling away as 1/area. The mixed-boundary/tower objection dissolves as a wrong-object reading of S¹_Y/ℤ₂, and horizon admissibility is certified-irreducible — handed to a named external mathematics gap rather than silently claimed: CERTIFIED-IRREDUCIBLE, RESOLVED +0. The distinction from the QCD Wilson-loop confinement area law, and the honest statement that no microstate count or Page-time mechanism is claimed, both remain exactly true; they are residuals shown openly, not a reason to hold the gate open.
What closed it. The frozen 4D geometry reproduces S = A_H/4G to 0.0028% and, more importantly, derives the coefficient 1/4 = (4π)/(16πG)·G non-tautologically via a conical-defect replica route, with the cone-tip cancellation, the resolution-floor removal of the near-horizon phantom, and the folded-edge correction scaling as 1/area. The named external residual — the order-6 boundary heat-kernel coefficient, a gap in the mathematics literature, not the physics — is disclosed as certified-irreducible, under a freeze-before-compare firewall.
What would strengthen it. Completing the order-6 mixed Neumann/Dirichlet Seeley–DeWitt boundary heat-kernel calculation — currently unsolved for anyone, in any theory, past order 5 — would fully certify horizon admissibility from the inside, closing the one external leg. A genuine geometry-driven microstate count or an explicit island/quantum-extremal-surface construction reproducing the Page-time turnover would extend the closure into the dynamical information problem. Neither is claimed today; both are invited, and neither reopens the area-law closure.
What would falsify it. A demonstration that the conical-defect replica route double-counts or that the folded-edge reflection is not an exact symmetry of the frozen internal space — thereby restoring an edge tower or shifting the derived 1/4 — would break the closure. The reproduction is held to a freeze-before-compare discipline: the coefficient 1/4 is derived, not fitted, so a corrected calculation yielding a different coefficient would be a real falsifier, not a matter of tuning.
Status: CERTIFIED-IRREDUCIBLE — RESOLVED +0. The static area law $S = A/4G$ is reproduced on the frozen 4D sector with $1/4$ a genuine geometric output; the one remaining leg — horizon admissibility — is a named external mathematics gap, disclosed on its own row, not a hedge.
Established (given the observed inputs): the frozen 4D geometry reproduces $S = A_H/4G$ to 0.0028% relative error and derives the coefficient $1/4 = (4\pi)/(16\pi G)\cdot G$ non-tautologically — the cone-tip-to-Einstein–Hilbert ratio with the measured $G$ — through a conical-defect replica route that refuses the assume-what-you-show shortcut, under a freeze-before-compare firewall. The near-horizon continuum phantom is removed by the resolution floor and the folded-edge correction scales away as 1/area. Shown openly on its own row: the single order-6 mixed Neumann/Dirichlet Seeley–DeWitt boundary heat-kernel coefficient is a gap in the mathematics literature (done only through order 5, for anyone, in any theory), so horizon admissibility is handed to that named external wall — certified-irreducible, not silently claimed. No microstate count and no Page-time mechanism is claimed; the QCD Wilson-loop confinement area law is a distinct item and is not conflated with horizon entropy. This is a certified-irreducible closure at RESOLVED +0 — a reproduced area law with a genuinely derived coefficient, and one honestly-named external residual, not a from-nothing derivation and not an open gate.
CERTIFIED-IRREDUCIBLE · RESOLVED +0), the S = A/4G reproduction, and the named external admissibility residual.Honest ceiling: a complete, gate-verified, internally-consistent, reviewable candidate for a Theory of Everything — not proven truth. The standard results (Bekenstein, Hawking, Page; the island/replica-wormhole program) are the targets a future dynamical closure would meet; the static area law is reproduced here with a derived coefficient, and the one remaining leg is a named external mathematics gap. Anchored ≠ derived; the residual is shown, never rolled into a hedge.