Anytime-Valid Stopping for Self-Consistency on Open Answer Sets

Self-consistency samples reasoning chains and returns the most frequent answer, and adaptive variants stop sampling early to save compute. Deployments act on the stopped answer, yet these rules recheck a fixed-sample criterion as draws arrive, so the chance of stopping on an answer that is not the population mode has no time-uniform bound. In simulation, Adaptive Consistency (AC) at its published setting significantly exceeds the tolerance implied by its credible level in 27 of 70 cells. Existing anytime-valid mode certificates need the number of answers in advance, control nulls that must hold at every round rather than the error of the answer returned at stopping, certify a target fixed before sampling, or restart fixed-sample tests on fresh draws; a tuple-indexed repair for a data-selected leader is stated under a unique mode. We give a stopping rule for i.i.d. draws over an unknown, possibly countably infinite answer set that certifies at level δ that an answer it selects from the data is a population mode. Each discovered answer must beat every rival, seen or unseen, against a threshold set by the order in which it was first seen rather than by the size of the answer set. Under a unique mode and runner-up, the expected stopping time of this complete rule attains the optimal leading constant as δ → 0 without knowledge of the answer set. The rule and its four variants never exceed δ in 140 i.i.d. simulation cells; where the rule and AC both certify at least 90% of replicates, the rule costs a median 3.2 times AC's draws. On GSM8K with Qwen2.5-1.5B at δ = 0.05, the rule, set by δ alone, lands within one wrong certificate and four certified questions in 200 of the coverage–error frontier that AC reaches with a threshold tuned in-sample; on Qwen3.5-2B and 9B it makes at most one more wrong certificate in 200 than AC at matched coverage.