Theorem: Three-correlation subsystem certificate

For predeclared tasks ${\Phi }_{0,A}:\mathcal{B}(C\otimes S)\to \mathcal{B}(C)$, define the three correlations on $Q,C_{\mathrm{out}}$ in their normalized Choi states, ignoring $R$. Under the conditional validation and calibration contract of Assumption ass:validation, with the target comparison

the same estimates (eq:statistics) give simultaneous fidelity endpoints (eq:intervals). Replace the channel endpoints by

With probability at least $1-{\alpha }_{\mathrm{s}}-\beta$,

for every tested task. Acceptance by the upper endpoint has the same family-wise false-acceptance guarantee as (eq:falseaccept).

Proof. The effect $P_2^{Q{C}_{\mathrm{out}}}\otimes I_R$ has exactly the same three-correlation expansion as $P_2$, tensored with $I_R$. The statistical and calibration proof of Theorem thm:main is unchanged. Lemma lem:subsystemconversion supplies the new conversion. The transfer allowance is again added outside the square root.