The forty-shot limitation, with its exact scope

The binomial coverage follows by testing the lower tail at each candidate $p_j$: the event that the upper bound is below the true parameter is contained in a lower-tail event of probability at most ${\alpha }_j$. A union bound completes the confidence accounting.

These propositions do not forbid a different certificate that uses independently justified structure, such as a trusted Pauli noise model. They do not constitute a universal laboratory sample complexity theorem.