Coverage Is Not Selection: Exactly Attributing the Gains of Test-Time Scaling

Test-time scaling methods (self-consistency, best-of-$n$ under a trained verifier, adaptive sampling) are compared by one number: accuracy at budget $n$. Per question that number is an exact product: coverage $C_q(n)$, the probability a correct answer exists among $n$ samples, times selection, the probability of returning it given it exists; the first depends on the sampler, the second on sampler and rule. We turn it into an audit instrument: a remainder-free identity splitting any accuracy difference into base-rate, headroom and selection channels; an efficiency ${\sigma }_R(n)$ normalized between a content-blind floor and the oracle ceiling; and unbiased estimators pricing any rule at any budget $n\le N$ from one pool of $N$ cached samples. On 149,000 generations from six models it does three things accuracy alone cannot. It attributes: a token cap and 63 hours at a fixed configuration leave accuracy unresolved at $Q=60$ yet move the selection channel (cap: $\sigma$ $0.60\to 0.33$; 63 hours: $-0.36$ at $n=32$, $p=0.005$, and $-0.17$, $p=0.03$, at $n=16$; one unreplicated pair). It prices verifiers: a 7B process reward model is the best selector we measure, yet selectors lose efficiency as the budget grows on all 9 (cell, selector) pairs (6 resolved, uncorrected), and how fast a vote loses it tracks how concentrated its per-sample weights are (Spearman $-0.76$ with effective sample size, 17 pairs). It audits adaptive schedules at matched expected budget: adaptive-consistency gains on the three cells with ample coverage ($+0.016$ to $+0.022$), though which channel pays is unresolved. All code, per-cell results and cached pools behind every number here are released at https://github.com/xxxxx/xxxxx (URL anonymized for review; public upon acceptance); code and results also accompany this submission, the pools being too large for it.