Theorem: Simultaneous coverage and acceptance

Consider finitely many frozen tasks. For every validation setting allocate a failure probability, with total at most ${\alpha }_{\mathrm{s}}$. Suppose all required calibration and transfer statements hold jointly except with probability $\beta$. Then all intervals from Theorem thm:tomography or Corollary cor:bell cover their respective errors simultaneously except with probability at most ${\alpha }_{\mathrm{s}}+\beta$. In particular,

The same event controls false specified-decoder exclusions.

Proof. Condition on the frozen information. Each setting's concentration bound remains valid by Contract con:validation. Union over every setting in every task and then over the calibration/transfer failure event. On the complementary event every true error lies in its interval. The decision implications follow immediately. No independence between different tasks is required.