Theorem: Diagnostic separation

For the declared extraction and output-noise experiments:

1. The signed extraction map and its complete complement are physical for every $\eta \in [0,1]$.

2. Occupation, coherence, purity, Bell-input spectra, reference mutual information, and reference negativity obey the proved identities.

3. The specified decoder has error $1-\eta$, Bell fidelity $(1+3\eta )/4$, worst-case target overlap $\eta$, and decoded concurrence $\eta$.

4. Exact recovery is impossible at $\eta <1$, but the specified decoder is not presumed optimal there. Optimal recovery errors at $\eta =0,1$ are $3/4,0$.

5. Population-preserving dephasing retains an additional complete environment. At full transfer its optimal recovery error is $(1-\lambda )/2$, although the Bell-input accessible marginal is independent of $\lambda$.

6. Accessible marginal purity, occupation delivery, and even knowledge of the complete accessible marginal for a single Bell-input experiment do not replace a referenced channel-recovery criterion.

These are finite-channel information-flow results. They do not establish black-hole evaporation, a gravitational Page curve, or a solution to the black-hole information paradox.

Proof. Theorem thm:channels proves (i). Theorem thm:marginals, Theorem thm:information, and Proposition prop:rawneg prove (ii). Theorem thm:specified proves (iii). Proposition prop:optimal proves (iv). Proposition prop:noisy and Theorem thm:fullNoise prove (v). The purity endpoints and Corollary cor:separation prove (vi). None of the hypotheses specifies a gravitational identification.