Theorem: Hawk07-B operational extrapolation and function theorem

Within the finite preparation, complete-output, and access declarations of Section sec:scaling, the following conclusions hold.

1. For any declared qubit payload dimension $d$, any $0<\epsilon <1$, and any $0<p<1$, the design in Theorem thm:design supplies a positive-residual retained window and a family of Haar devices of measure at least $1-p$ with one fixed receiver core and full-diamond error at most $\epsilon$.

2. Such a receiver supplies referenced quantum return, entanglement transfer, classical basis-message transmission, and post-return logical tasks with the errors in Theorem thm:functions. The residual privacy allowance is the separate statement of Theorem thm:privacy.

3. The sufficient released-dimension law is $d^2{\epsilon }^{-2}\sqrt{f/p}$, up to the displayed constant, not the proposed $d^2{\epsilon }^{-1}\sqrt{f}$. The complete receiver dimension and batch workspace must additionally price $h$.

4. For the square-power family $d=4^s$, $s\ge 2$, the newly proved turnover family has measure at least $1/(4d)$, accurate return through every retained cut, strict coherent rise-and-fall, and a final residual consisting of one physical qubit.

5. Finite convoys and serial links are certified under the independent-resource or complete memory-reset contracts of Section sec:convoy, including entanglement among payloads.

6. Actual channel and receiver approximations preserve the stated return target only within the reserved error budget of Section sec:noise.

7. Purely classical return obeys the entanglement-breaking floor; hybrid return can avoid it only by retaining its quantum resource. Append-only records do not turn over. Terminal residual discard neither retracts a completed return nor deletes the complete physical complement.

8. No second commuting code-preserving action of the same logical matrix algebra is produced.

These are operational predictions, not predictions of a spacetime wormhole, gravitational travel, a quantum extremal surface, a replica wormhole, a gravitational Page curve, or spacetime emergence.

Proof. The arbitrary-dimension recovery statement follows from the rederived moment, the threshold theorem, the explicit dimension design, and the selected receiver program. The operational functions and residual privacy are proved in Section sec:functions. Equation (eq:fullthreshold) and the distinction $\dim R_{\ast }=r_{\ast }h$ establish the corrected resource statement.

Theorem thm:turnover proves the new scaled entropy claim. The tensor-product, memory-reset, and serial telescoping arguments prove the composition statements. The noise theorem and its budgeted corollary prove transfer to a compatible actual receiver. The classical, teleportation, record-chain-rule, and residual-discard arguments prove the return-type and endpoint assertions. Proposition prop:noduplicate proves the algebraic exclusion.

No metric, gravitational stress equation, geometric throat, or signal-crossing experiment is among the model's declarations. The excluded physical conclusions are therefore not established by this theorem.