Accuracy, Confidence, and Sample Budgets: Exact Certificates for Mathematical Benchmark Claims

When does a benchmark's restatement of a theorem request an unattainable guarantee? We examine a mean-field control terminal in MemoryArena whose reference substitutes confidence parameter $\delta$ for accuracy parameter $\epsilon$. A proved admissibility lemma connects a two-step, three-model construction to the source assumptions and observation model. For transition parameter $q$, the exact minimax failure risk under a deterministic trajectory cap $N$ is $(1-q)^N/2$ below accuracy $q/4$, and zero at or above that threshold. Below the threshold, the exact zero-error minimax expected cost is $2/q-1/2$. A reusable testing certificate also yields matching $\Theta(\delta^{-2}\log(1/\delta))$ sample-complexity order for a shared-support noisy extension. Both constructions show why fixed-accuracy, polylogarithmic confidence budgets cannot uniformly deliver shrinking accuracy $\delta$. Source-preserving statement repairs and offline pipeline regressions provide checkable remedies at the specification and serialization levels. Exploratory judge and solver summaries previously reported 120 candidates and 120 outputs. Twelve-record archives delimit reproducibility and support no causal performance claim.