Theorem: One-model finite compatibility

For the same $J,V,U,\beta$, and for every reference-message state ${\rho }_{QM}$:

1. the emission dynamics are linear and arise from the specified finite unitaries;

2. recovery from $E$ is exact, with complete residual decoupling;

3. the partner algebra is the state-independent algebra of Theorem thm:partner;

4. the pair-proxy instrument passes with probability one and leaves the emitted state unchanged;

5. every declared logical $AB$ instrument, lifted by $J$, has exactly the same complete output as its logical counterpart, including references and adaptive histories.

These statements modify the independent-factor benchmark by identifying the partner algebra with a subalgebra of radiation. They do not establish physical horizon smoothness.

Proof. Items 1–3 follow from the specified evolution and Theorems thm:modelchannels and thm:partner. The range of $V$ lies in the range of ${\Pi }_{\beta }\otimes I_K$, so ${\Pi }_{\beta }V=V$. This proves item 4, including the selective residual state. Item 5 is Lemma lem:instrument, with the logical instrument acting as the identity on $M$. Because $A$ is represented inside $E$, no joint state on independent factors $A,B,E$ is asserted. The final sentence records the absence of a spacetime and field-state construction.