When no positive bounded gap is supplied

A valid alternative is a simultaneous confidence region for the complete finite histogram. For $m$ bins and $N$ independent attempted runs,

holds for all bins with probability at least $1-\alpha$. The bins include setting labels, preparation flags, split outcomes, and terminal failures.

A model family is rejected only if no admissible joint nuisance value, with its declared complete-law error, intersects this region. This is a calibrated procedure even when no common linear statistic has been found. It is not an asserted apparatus-specific rejection.