You Cannot Vote Your Way to the Truth: The Rank Law of Verification-Free Selection

Repeated sampling is a standard way to spend inference-time compute: coverage, the probability that at least one of $k$ samples is correct, can keep increasing with $k$. Yet a deployed system must return one answer without a gold label, and single-answer accuracy often saturates after only a few dozen samples. We characterize this selection ceiling by a single integer for each problem, $R^{\ast }$: the mass rank of the correct answer in the model's per-problem output distribution. The count-ranked shortlist frontier is then a functional of the distribution of $R^{\ast }$: self-consistency converges to $\mathrm{Pr}[R^{\ast }=1]$, a top-$L$ shortlist to $\mathrm{Pr}[R^{\ast }\le L]$, and coverage to $\mathrm{Pr}[R^{\ast }<\infty ]$, with exponential finite-$k$ convergence. This turns the gap into an information price: an ideal target-aware channel with $M$ messages is worth exactly the Bayes top-$M$ posterior mass, the count-ranked shortlist is a measurable realization of that capacity and coincides with it under rank calibration, so resolving a top-$L$ count-ranked ambiguity costs ${\log }_{2}L$ oracle bits, which a real verifier need not collect. Across 10 model–task pairs, three model families, and budgets to $k=10^4$, $R^{\ast }$ masses estimated from half of each problem's samples predict held-out selection accuracy across budgets and shortlist depths with median absolute error 0.002 and no fitted parameter. Across 12 bridgeable difficulty strata, a count-ranked four-answer shortlist plus two such ideal bits would recover 77–100% of the gap because, when plurality fails, the truth is usually the runner-up. Yet extracting those bits from the model is difficult: no count-based selector we test beats plurality, and only 1 of 26 implementable verifier conditions does; even there, spending the same tokens on a stronger generator performs better. Repeated sampling can therefore make the truth available without making it identifiable. The selection gap is not a decoding problem; it is missing information, and it can be priced.