Two Calls, Two Moments, and the Vote-Accuracy Curve of Repeated LLM Inference

Repeated sampling is a standard way to spend test-time compute, but its benefit is controlled by the latent distribution of correctness across examples, not by one-call accuracy alone. We study the binary correctness layer of repeated LLM inference under conditional-i.i.d. calls. One labeled call identifies the mean latent success probability; two labeled calls identify its second moment and hence the same-example correctness correlation that separates stable errors from recoverable call-level randomness. From these two moments, every fixed majority-vote budget has a sharp distribution-free two-call interval. The key technical reduction is that the infinite-dimensional moment problem has three-atom extremizers and quadratic dual certificates for every finite budget, so the bounds are exact rather than discretized or parametric. The first useful budget, three votes, has a closed form, width at most $1/8$, and a certified-improvement criterion. The infinite-vote endpoint is the limit of majority voting as the number of calls tends to infinity; it is also sharply bounded, but remains threshold-sensitive because it depends on latent mass around $q=1/2$. We add maximum-entropy and Latent-difficulty Gaussian-probit (LDGP) point completions, and experiments on LLM calls over QNLI and QQP show that empirical three- and five-vote accuracies are contained in the projected two-call regions while temperature changes and randomized model mixtures can create voting gains not ordered by one-call accuracy.