Theorem: Oai265-A separation

For a fixed finite $\mathsf{AL}$-record and logical dimension $d\ge 2$:

1. the record alone does not determine accessible recovery error or complete residual leakage;

2. selected accessible spectra, pure-input entropies, and the selected energy statistics of Proposition prop:thermal do not determine recovery;

3. a static area allowance, even accompanied by exact low-bond PEPS data and a uniformly gapped instantaneous ground-state family, does not determine a temporal entropy path;

4. a PEPS cut bound, a perfect-tensor network, or a decoder memory ledger does not by itself instantiate a gravitational entropy prescription;

5. no universal record-only inference can replace a missing channel, temporal, or geometric identification theorem.

These conclusions neither verify Family 265 nor exclude future bridges with genuinely additional hypotheses.

Proof. Append the two isometries in Proposition prop:goodbad to the same fixed record. Their dimensions agree, but recovery errors and complete leakages differ. This proves (1). Proposition prop:thermal proves the stronger spectral and energy non-identification in (2).

Propositions prop:stationary, prop:gappedpath, and prop:histories establish distinct static and temporal completions for (3).

For (4), the tensor-cut argument specifies a rank bound, while a decoder ledger specifies resources of an independently declared channel task. Neither supplies a spacetime, geometric surfaces, bulk state, area normalization, homology constraints, or entropy extremization data. A perfect tensor may add exact finite isometric identities, but those do not insert the missing gravitational data. Item (5) follows from the explicit unequal completions and the definition of determination.