Theorem: Mathematical release

For the explicitly regulated fixed-realization model:

1. The interacting source, scalar reservoir, coupling, and self-adjoint complete dynamics are specified.

2. A pure source preparation and message-independent parity-preserving diary insertion are constructed. Their energy and apparatus boundaries are retained.

3. Every logical matrix unit determines an accessible output and a complete complementary output, including the displaced carrier.

4. An explicit finite-mode, finite-occupation approximation controls the complete prepared channel uniformly on every finite observation interval.

5. Finite complementary Choi certificates and specified CPTP decoder certificates give the reference-preserving bounds in Section sec:certificates, with all code dimension factors and regulator allowances explicit.

6. The microscopic energy ledger, commuting-coupling recovery obstruction, disorder-averaging counterexample, coherent-cutoff check, and finite-proxy recurrence statement are proved.

7. A common-output endpoint construction has the continuity consequences of Theorem thm:endpoint-continuity. A unique gapped full-residual vacuum has the separate sufficient criterion of Proposition prop:residual-energy.

No item establishes a small dynamical complete-residual bound for the interacting model, a controlled gravitational correspondence, or a gravitational evaporation endpoint.

Proof. The items follow respectively from Theorem thm:selfadjoint, Propositions prop:preparation and prop:insertion, equations (eq:Vt)–(eq:matrix-units), Theorem thm:regulator, Theorem thm:certificate and Proposition prop:decoder, the results of Sections sec:complete, sec:influence, sec:debug, and sec:endpoint, and the two named endpoint results.

The excluded claims require additional dynamical and physical objects not constructed by these results.