Theorem: Complete observed-gap certificate

If the two attempted-run laws have the allowances $e_{+},e_{-}$ in (eq:obsledger), their score-expectation gap is at least

replaced by its positive part.

For $n_{+},n_{-}$ independent attempted trials with fixed parameters, let

With probability at least $1-\beta$,

The same statement holds for predetermined scores in $[0,1]$.

Proof. An effect or bounded-score expectation changes by at most its complete-law allowance. Apply this to both preparations.

For independent variables in $[0,1]$, the centered log moment-generating function is bounded by $t^2/8$: its tilted second derivative is a variance at most $1/4$. Exponential Markov inequality gives $\mathrm{Pr}(|\hat{p}-p|>u)\le 2e^{-2n{u}^2}$. Apply this to both ensembles and take a union bound. Then combine with the expectation-gap estimate.