Theorem: Simultaneous finite sector-label certificate

Fix ${\alpha }_{\mathrm{s}}\in (0,1)$. For all $K$ predeclared tested labels, set

Write $[u{]}_{[0,1]}=\max \{0,\min \{1,u\}\}$, and define

Under Assumption ass:validation, with probability at least $1-{\alpha }_{\mathrm{s}}-\beta$, simultaneously for every tested label,

If certification is declared only when $U_{\lambda }\le \epsilon$, the probability that any certified tested label has actual target error greater than $\epsilon$ is at most ${\alpha }_{\mathrm{s}}+\beta$. The same simultaneous event permits selection of one tested label after viewing all validation results.

Proof. Conditional Hoeffding inequalities give

A union bound over $3K$ pairs, followed by integration over $\mathcal{Z}$, proves statistical coverage with failure probability at most ${\alpha }_{\mathrm{s}}$.

On that event and $G$,

Clipping preserves coverage because the true fidelity lies in $[0,1]$. Lemma lem:bellbounds gives the channel-error bounds for ${\Lambda }_{0,\lambda }$. The reverse triangle inequality and the transfer allowance give (eq:errorinterval). Finally, every invalid acceptance lies outside the simultaneous coverage event.