Theorem: Mathematical release of this manuscript

Within the declared autonomous quadratic fermionic model:

1. The infinite-reservoir Hamiltonian is positive and self-adjoint, and its unitary evolution is constructed.

2. The loading operation is coherent and leaves its carrier in a fixed vacuum state.

3. The complete evolved channel and its full inaccessible output are specified without dropping uncollected modes.

4. The exact amplitudes, survival law, and factorial collection bound are derived.

5. A complete reference-preserving decoder has the explicit error bound (eq:main-bound), and Corollary cor:window proves an actual nonempty finite-time recovery regime.

6. For a one-particle code, both optimal recovery error and complete residual leakage are exactly (eq:exact-erasure).

7. Energy transfer, finite-chain channel approximation, checked perturbation allowances, and the fixed-regulator recurrence obstruction are proved.

8. The same Hamiltonian has a complete outgoing scattering endpoint with an exactly constant complete complement. No additional terminal release map is used.

These statements do not establish a fixed-realization interacting SYK recovery theorem, a physical black-hole code dictionary, gravitational backreaction, or a gravitational evaporation endpoint.

Proof. The items are proved respectively in Section sec:model, Proposition prop:loading, the access construction and Theorem thm:main, Proposition prop:spectral and Lemma lem:tails, Theorem thm:main and Corollary cor:window, Theorem thm:erasure, Section sec:energy and Section sec:regulators, and Theorem thm:endpoint. The final scope statement follows from the explicitly different Hamiltonian and the absence of the physical maps listed in Section sec:audit.