The geometry of AI validation: Exact certification limits for iid best-of-N search

What the math gives to ML

This paper gives an exact, constructive limit on what reliability audits can certify for iid best-of-N selection when audits only measure smaller search budgets. The key transferable asset is an explicit ambiguity width, B_{m,N}, showing that repeated evaluation at n <= m cannot distinguish reliability worlds that differ substantially at deployment budget N; the relevant scale is m^2/N. This can become an audit-aware validation layer for best-of-N decoding, reward-model selection, or self-consistency, preventing false claims of reliability when the validation budget is structurally too small. It also provides a principled way to choose audit budgets rather than relying on ad hoc numbers of sampled candidates.