Search over all positive sample sizes with explicit stopping certificates

Input the full planning specification Power and sample-size planning specifications and an exact evaluator $E$ on every positive integer $n$. Each call $E(n)$ must terminate and return a Boolean with the following meaning: $E(n)$ is true exactly when the specification has a feasible plan of total size $n$, including its sampling law/test, power or precision target, allowed allocation and resource conditions. With a fixed allocation rule this is the ordinary per-size power calculation. For several candidate plans at one size, this assumption requires exact resolution of whether any is feasible; it is not supplied by the procedure. The evaluator, and any stopping certificate below, are inputs, not a universal ability to compute probabilities of arbitrary models.

Choose one of these admissible modes.

1. Unbounded ascending search. Set $n=1$. Evaluate $E(n)$. If true, return $n$; otherwise replace $n$ by $n+1$ and repeat. This branch never returns $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ merely because a finite prefix failed.

2. Certified bounded search. Supply a nonnegative integer $B$ and a valid completeness certificate that, if any positive size is feasible, some feasible size is at most $B$. Evaluate $E(1),\dots ,E(B)$ in order, returning the first true index. If all are false, return $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$; for $B=0$, return $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ immediately. The certificate is a mathematical premise covering all positive sizes, not a claim inferred from those finite evaluations. A proved feasible witness of size $B$ is one sufficient certificate.

3. Certified infeasibility. Supply a valid proof that $E(n)$ is false for every positive integer $n$. Return $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ without an evaluation.

The output is a sample size, not a selected allocation. Selecting a plan at that size requires an explicit witness-producing evaluator or an independently specified selection rule. The extensional target remains the least feasible positive integer, or $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ if no such integer exists. Termination properties are separate proved results; no global infeasibility decision is asserted for unrestricted black-box input.