Theorem: Page derivation with typed access

For the explicitly prepared, Haar-scrambled, conserving finite emission model:

1. coherent quantum radiation has the exact finite Page mean (eq:page) for every fixed pure input independent of $U$;

2. one sampled unitary supplies an all-prefix normalized leading Page history as in (eq:leadingpage);

3. destructive Enter has the exact cumulative-record mean (eq:recordmean), not a Page turnover, and record-only whole-code recovery error is at least $1-1/d$;

4. the same coherent Page ensemble can describe a full code whose exact radiation recovery error is $1-{b_k}^{-2}$, or a smaller code with the separate decoupling certificate (eq:prob-decouple);

5. the corrected single-excitation calibration is (eq:flaggederrors), with its complete classical history given by (eq:flagrecordentropy);

6. entropy history, selected thermal data, complete-complement constancy, and executable quantum recovery remain separate certification targets;

7. a physical Page-history or endpoint conclusion follows only after its same-model comparison hypotheses are supplied.

The finite random-state Page curve is constructed. A microscopic gravitational evaporation channel and the inherited characteristic quantum coefficient are not constructed.

Proof. Items (i)–(ii) are Theorems thm:page and thm:uniform. Item (iii) is Theorem thm:classicalpage and Proposition prop:classical. Item (iv) is Theorems thm:fullcode and thm:decoupling. Item (v) is Theorem thm:flagged. Item (vi) follows from the explicit separations and the recovery criterion. Item (vii) is Theorems thm:endpoint and thm:transfer. No gravitational comparison or characteristic product is supplied by the objects in these proofs.