Theorem: Oai265-A mathematical release

Without assuming either external Family 265 claim, this manuscript establishes:

1. exact boundary accounting and disconnected-component handling;

2. a finite entropy-cancellation lemma and an explicit conditional terminal area-law constant;

3. finite PEPS, entropy-continuity, Schmidt-tail, and energy-accuracy accounting;

4. explicit failures of unrestricted degeneracy, subsystem-gap inference, strict-equality inference, and spectrum-only $\alpha <1$ control;

5. exact complete-output restoration, the complete-complement recovery criterion, and the sharp replacement-channel benchmark;

6. record-only non-identification of recovery, complete leakage, selected thermal spectra, temporal paths, and gravitational entropy prescriptions;

7. a separately declared finite qubit-memory certificate and calibration counterexample;

8. a complete stage ledger with one unresolved quantitative successor.

No uniform two-dimensional area-law theorem, polynomial-PEPS existence theorem, efficient tensor-production algorithm, or gravitational entropy history is newly proved by these results.

Proof. Items (1)–(3) are Lemmas lem:boundary–lem:tail, Propositions prop:Cstar–prop:energy, and Theorem thm:terminal, on their explicit hypotheses. Item (4) is Section sec:hypotheses. Item (5) is Lemmas lem:global, lem:constant, and Theorem thm:recovery. Items (6)–(7) are Theorem thm:separation, Propositions prop:stationary–prop:fake, and Theorem thm:statistics. Section sec:successor supplies (8).

The external records are never used as mathematical premises. Their universal conclusions therefore are not outputs of the finite local arguments. The missing quantitative and physical interfaces remain explicitly identified.