# Gap-13 — Black-Hole Entropy / Page Curve

## What this gate must establish

This gate asks whether the frozen 13D geometry, projected to its four-dimensional gravitational sector, can reproduce the two signature facts of black-hole thermodynamics: the Bekenstein–Hawking entropy (S = A/4 in Planck units, an entropy that scales with horizon *area*, not volume) and the *Page curve* — the unitary rise-and-fall of an evaporating black hole's radiation entropy that any quantum-consistent theory of gravity must produce instead of Hawking's monotonic increase. Closing it would mean these are *consequences* of the geometry, not inputs. It is a test of whether the framework's gravity sector is quantum-unitary at the horizon, not merely classically correct.

## The geometry's role (load-bearing, not validation)

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 genuinely load-bearing for the *perturbative, linearized* gravity sector — vacuum, masslessness, the Newtonian limit, and the Einstein–Hilbert IR are recovered as a route statement. But horizon physics is a *nonperturbative*, strong-field, finite-area regime that this zero-mode projection does not reach. The geometry supplies the carrier of gravity; it does not, as written, supply a horizon microstate count or an entanglement-entropy island prescription.

## Current honest status

**Badge: OPEN — outside the current force-sector boundary; no in-corpus route.**

The corpus is explicit and consistent on this: "Black-hole entropy, gravitational anomalies on horizons, holographic completions" are listed as **outside-scope** items, marked *"not in UQFC; outside the force-sector boundary."* Unlike most outside-scope graviton items (UV fixed point, BRST positivity, compactification stability), which are *routed* to named internal modules (UQF-9, UQF-10), the black-hole/holographic item is routed *nowhere* — there is no internal module that claims it, and no Page-curve computation, island construction, replica-wormhole calculation, or microstate count exists anywhere in the corpus.

Honest caveats against over-reading:
- The framework does contain an **area law** — but it is the *Wilson-loop confinement* area law of the inherited SU(3)_c sector (an AUDIT-tier item, numerically verified on the lattice, not proven). This is QCD confinement, **not** horizon entropy. The two must not be conflated, and the corpus does not link them.
- There is **no DERIVED, DERIVED-GIVEN-E, AXIOM-CLOSED, or even SCHEME-ANCHORED** content for S = A/4 or the Page curve. There is no named axiom that, if granted, would deliver them — which distinguishes this gate from gates that are reduced to one clean posit.
- **No status was ever upgraded.** Nothing here is derived; no number is produced.

## What would close it

A genuine closure has two distinct legs, neither presently attempted:
1. **Static entropy (S = A/4).** A microstate count — or a horizon entanglement-entropy calculation — performed *within* the frozen geometry that yields the area coefficient 1/4 with **no fitted prefactor**. Reproducing 1/4 by adjusting a normalization would be SCHEME-ANCHORED at best, not a derivation.
2. **Dynamics (the Page curve).** A demonstration that the radiation entropy turns over at the Page time — the modern route requires an island/quantum-extremal-surface prescription or a replica-wormhole saddle, which in turn presupposes a controlled nonperturbative or holographic completion of the gravity sector. That completion is precisely what is declared outside the present boundary.

The honest near-term verdict is **a no-go for closure within the current scope**: the gate cannot move until the gravity sector is extended past its perturbative zero-mode projection into a horizon-capable (nonperturbative or holographic) formulation, and that extension is itself unbuilt. Until then this remains genuinely open, with the gravity sector standing as a *serious candidate / partial unification — NOT validated*.

## Sources

- *Gaps & Walls Register* (Fable programme, four-paper set), status-vocabulary table and gravity-sector scope rows — programme status "serious candidate / partial unification — NOT validated," No status was ever upgraded.
- *Quantum* paper (Fable corpus), §8 gravitational-sector scope table: black-hole entropy / horizon gravitational anomalies / holographic completions listed as outside the force-sector boundary ("not in UQFC"); §8.1 graviton zero-mode reduction; §12.2–12.3 Wilson-loop area-law AUDIT row (distinct QCD confinement item).

No external derivation of S = A/4 or the Page curve is claimed by this framework; standard results (Bekenstein, Hawking, Page; island/replica-wormhole program) are referenced as the targets a future closure would have to meet, not as results obtained here.