Checking Leakage Witnesses versus Certifying Bounded Non-Leakage

When a language-model audit finds no leak, what is needed to certify non-leakage? We study guarantees over a declared prompt domain under an executable leakage criterion and decoding rule. For general bounded polynomial-time evaluators, a supplied leaking execution is polynomial-time checkable, while leak existence is NP-complete and deterministic certification is coNP-complete. Exact stochastic certification is coNPPP-complete at every fixed rational cutoff in (0,1). Restricting the computation can change these bounds. For example, certification is in coNP when all randomness is a terminal draw from an efficiently computed finite probability table. Attention models admit polynomial-time certification when local dependency windows of logarithmic length precede one global head, given deterministic decoding, fixed vocabulary, exact rational weighted means, a direct binary affine readout and finite-automaton prompt domains. A construction with two global layers instead makes certification coNP-complete over template domains, with one head per layer, polynomial width, logarithmic precision and an inverse-polynomial logit margin. Planted-secret experiments measure what finite audits miss relative to complete references. Among 30 secret--model-state pairs that leak under greedy single-prompt execution on their secret's 4,096-prompt domain, uniformly selecting 256 recorded evaluations per pair misses every leak for an expected 41.06% of these pairs. Batched and single-prompt checks disagree on one complete-domain decision among all 48 fine-tuned pairs, while a same-order repeat reproduces every single-prompt output. These results distinguish computational conditions for certification from the coverage and execution conditions needed to interpret a negative audit.