Deciding which of two agents is stronger means playing games until skill outweighs luck, and every game costs money, model inference, or expert time. Since the number of games needed is unknown, fixed-budget evaluations either keep paying after the result is settled or stop before the agents can be told apart, while naive optional stopping with an ordinary confidence interval invalidates the stated level. We make such an evaluation stop as soon as its evidence suffices, with the guarantee intact. The Action-Informed Value Assessment Tool (AIVAT) reduces variance in imperfect-information games through conditional mean-zero corrections, by a median $54\times$ across 15 LLM agent configurations spanning 71,439 paired Heads-Up No-Limit Hold'em (HUNL) hands, but does not say when to stop. We combine AIVAT with continuously monitored Confidence Sequences (CSs) into anytime-valid AIVAT (AV-AIVAT), whose online value model learns only from past games so that no game scores its own correction. At the nominal 95\% level and a target precision of $\pm1$ Big Blind, raw outcomes need a median $74\times$ as many hands as AIVAT-corrected outcomes to stop under the Asymptotic CS (AsympCS). Exact finite-sample certification uses the Empirical-Bernstein CS (EB-CS), which needs an independently justified bound on corrected payoffs. We establish such a bound structurally for Leduc hold'em and characterize a width floor set by the CS's bet cap and that bound, which governs how much of a variance gain becomes earlier stopping; the descriptive HUNL EB-CS runs show a median $1.37\times$ stopping-time ratio. AV-AIVAT turns variance reduction into efficient, auditable early stopping while separating asymptotic screening from exact certification, so an evaluation can stop the moment its evidence suffices and hand a third party everything needed to recheck the verdict at that very stopping time.
Multi-agent large language model (LLM) loops, for example a Writer that drafts and a Critic that revises, are almost always terminated by a fixed iteration cap (max_iterations). This is a syntactic kill-switch: it is blind to whether the answer is still improving, so it over-spends tokens on easy inputs and truncates hard ones. We study semantic early-stopping: the loop halts when consecutive draft embeddings stop changing in meaning (cosine distance with a patience window) and the answer's measured quality stops improving. Our work makes three contributions. First, an honest theoretical footing: we prove deterministic termination and well-definedness and machine-check these claims, while treating the convergence of the distance sequence as an empirically tested conjecture rather than a (previously over-claimed) Banach contraction. Second, a judge-efficient evaluation protocol: we generate each question's full trajectory once, replay every stopping policy over the identical drafts, and cache every LLM-judge call, yielding a strictly paired efficiency-versus-quality comparison at low cost; we further separate operational tokens (charged to a policy) from evaluation tokens (a measurement instrument). Third, an empirical study on multi-hop retrieval-augmented question answering (HotpotQA). On the 60-question test split, a judge-free semantic stopper reduces operational tokens by 38% relative to max_iterations at parity quality (Delta-IS = -0.004, p = 0.81), whereas the full quality-gated variant is counter-productive because its per-round judging dominates cost. An oracle that selects the best round attains +0.115 Information Score over every practical policy (p ~ 4e-11), reframing the problem from "when to stop" (easy) to "which round is best" (open).