Theorem: Conditional extrapolation certificate

Within the stated finite interface discipline:

1. nonuniform nested tensor release has the exact coherent and record means (eq:twomeans);

2. its fixed-dimensional capacities obey (eq:sumconstraint); a coarse entropy-production ratio $\xi \ne 1$ cannot be inserted as a mere clock change;

3. the compatible inverse-square example has the explicit rounded two-curve sketch (eq:sketchQ)–(eq:sketchL), while the general supplied capacity envelope has the crossover and cutoff conditions of Theorem thm:crossover;

4. additive-charge-sector Haar scrambling has the exact direct-sum means (eq:sectorpage)–(eq:sectorrecord);

5. physical two-curve transfer is conditional on the separate capacity, ensemble, channel, and detector allowances in (eq:Qbudget)–(eq:Ybudget);

6. whole-code recovery is a separate complete-complement or decoded-channel certificate, with bounds (eq:recoverybudget) and (eq:specifiedbudget);

7. every record-only endpoint has the permanent error lower bound $1-1/d$, regardless of the clock or capacity plot.

These results establish finite extrapolation possibilities and their compatibility tests. They do not construct a gravitational Page curve, an island, spacetime emergence, or the inherited characteristic quantum coefficient.

Proof. Items (i)–(iv) are Theorems thm:two, thm:compatibility, thm:crossover, thm:sketch, and thm:sector, with Proposition prop:xi. Item (v) is Theorem thm:transfer. Item (vi) is Theorem thm:recovery and the exact criterion of Theorem thm:exact. Item (vii) is Proposition prop:classical. No physical comparison theorem supplying the excluded identifications is among the constructed objects.