Theorem: Simultaneous connected boundary certificate

Under Assumption ass:validation, with probability at least $1-{\alpha }_{\mathrm{s}}-\beta$, simultaneously for all $A\in \mathcal{A}$,

If an acceptance requires both the declared access audit and $U_A\le \epsilon$, then

The simultaneous event allows selection of a tested task after viewing all validation results.

Proof. Conditional Hoeffding inequalities Hoeffding give

A union bound over $3K$ pairs and integration over $\mathcal{Z}$ give simultaneous statistical coverage with failure probability at most ${\alpha }_{\mathrm{s}}$.

On that event, $\left|{\hat{F}}_A-F_{\mathrm{obs},A}\right|\le R_A$. Intersecting with $G$ adds $b_{\mathrm{cal},A}$. Clipping preserves coverage because the true fidelity lies in $[0,1]$. Lemma lem:bellbounds converts the fidelity interval to bounds on ${\Lambda }_{0,A}$. The triangle and reverse triangle inequalities add the transfer allowance in channel-distance units, giving (eq:intervals).

Every invalid acceptance lies outside the simultaneous coverage event. An additional audit gate cannot enlarge that false-acceptance event.