Theorem: An explicit anytime-valid certificate

Suppose the conditional null bounds $c_i$ hold indefinitely. For $N\ge 1$, define

Then

Thus stopping at the first such crossing is licensed. Repeatedly inspecting the fixed-$N$ radius $r_N(\delta )$ is not licensed by Theorem thm:fixedtest.

Proof. At each $N$, allocate

The fixed-sample conditional bound gives failure probability at most ${\delta }_N$. Since ${\sum }_{N\ge 1}1/(N(N+1))=1$, a union bound proves the result.