Theorem: Finite-channel diagnostic separation

For the declared extraction and noise experiments:

1. The signed isometry, accessible channel, and complete complements are CPTP constructions with the stated diagnostics.

2. Exact recovery is equivalent to a constant complete complement. For the noiseless extractor it occurs exactly at $\eta =1$.

3. The complementary pair-distance is $1-{\eta }^2$, and the associated lower bound is proved by a Bell-projection witness.

4. The specified complete decoder has error $1-\eta$, Bell fidelity $(1+3\eta )/4$, worst-case target overlap $\eta$, and decoded concurrence $\eta$. Its instrument makes off-code events visible without identifying a flag with successful recovery.

5. The exact Bell-fidelity optimum is $(1+4\eta -{\eta }^2)/4$. A balanced decoder proves (eq:improvedSandwich), including the sharp endpoint recovery errors $3/4$ and $0$.

6. Full-transfer dephasing has exact optimal recovery error $(1-\lambda )/2$. Complete dephasing attains the entanglement-breaking floor $1/2$, despite unchanged Bell-input accessible occupation and purity.

7. Normalized-Choi, matrix-unit, and instrument estimates provide typed referenced certification interfaces.

A single input's accessible marginal does not replace an all-input referenced recovery criterion.

Proof. Theorem thm:channels and Proposition prop:noisyChannel prove (i). Theorem thm:constantComplement and Corollary cor:exactEndpoint prove (ii). Lemma lem:leakageBell and Proposition prop:pairDistances prove (iii). Theorem thm:specified and Proposition prop:instrument prove (iv). Theorems thm:optimalBell and thm:balanced prove (v). Lemma lem:witnesses, Proposition prop:noiseBoundary, and Theorem thm:fullNoise prove (vi). Section sec:certification proves (vii). Corollary cor:separation supplies the marginal counterexample.