Clean Sheets and Checkpoints: Evidence Complexity for Certifying Sequential Agent Risk

Dangerous-capability evaluations often test an AI agent from prepared states at ordered milestones. A clean sheet records no successes. What it certifies about a campaign of B attempts depends on what a failed attempt erases. We model the task as a chain of k milestones divided into checkpoint segments and ask how many zero audit responses certify, at level $α$, a completion probability of at most $ε$. For segments chosen before the audit, counting possible completion windows gives a certificate for any specified budget. The bound also covers adaptive deployment processes that follow the chain's transitions, provided their success probabilities after every complete history are capped by the corresponding audit success probabilities. The audit does not test this transport assumption. In rare-risk phase interiors, each segment is either rarely completed within the budget (active) or nearly always completed (free). The minimum expected number of zero responses is asymptotic to log(1/α)\,(Bᵃ/(a!\,ε))1/KA, where the a active segments contain KA milestones and are those at least as long as a budget-dependent threshold. The certificate attains this constant. Every asymptotically optimal evaluator must divide its expected response count equally among active milestones in the limit, with a vanishing share elsewhere. Where a single segment passes from active to free, each of its milestones receives a smaller limiting share than an active milestone; a crossover-aware certificate attains the optimal constant. For selected segments with known or unqueried prefix steps, counting failed cycles can sharpen the certificate.