Theorem: Corrected Enter/Page operational certificate

Under the complete finite interface and specified destructive readout semantics:

1. Complete isometric evolution preserves the quantum code, including all external references.

2. Exact recovery from the declared accessible output is equivalent to replacement of its complete complement.

3. Every purely classical returned record has whole-code recovery error at least $1-1/d$, permanently.

4. The entropy of an unedited cumulative classical record is nondecreasing; a retained random record prevents purity of the raw hybrid output.

5. The conserving schedule of Section sec:schedule has

   $${\epsilon }_{\mathrm{rec}}({\mathcal{N}}_q)=q(1-d^{-2}),{\delta }_{\mathrm{full}}({\mathcal{C}}_q)=q(1-d^{-2}),$$

   while its pure-input quantum radiation entropy is $h_2(q)$. Its residual recovery error is instead $(1-q)(1-d^{-2})$.

6. Destructive readout of its flagged quantum output has exact recovery error

   $$q(1-d^{-2})+(1-q)(1-d^{-1}).$$

7. Matched pure-input entropy histories and matched selected thermal observations can have opposite endpoint recoverability.

8. Residual readout followed by classical discard is partial trace and leaves the already-returned accessible channel unchanged.

9. Approximate endpoint recovery follows from the four same-model obligations and the norm bounds of Theorem thm:endpoint, not from entropy, termination, or a channel-only thermal interpretation.

These statements establish a finite operational analogy. They do not construct a gravitational Page curve, a microscopic evaporation channel, or the inherited characteristic quantum coefficient.

Proof. Items (i)–(ii) are Theorems thm:global and thm:exact. Item (iii) is Theorem thm:classical. Item (iv) is Theorem thm:recordentropy. Item (v) is the corrected calibration (eq:correctflagleak), together with (eq:flagrec) and (eq:flagentropy). Item (vi) is Proposition prop:measuredflag. Item (vii) is Theorems thm:spectra and thm:histories. Item (viii) is Theorem thm:drop. Item (ix) is Theorem thm:endpoint and Proposition prop:cost. No gravitational identification or characteristic composition is an object constructed in these proofs.