When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation

Increasing a voting budget can improve or reduce population accuracy, and an initial gain or loss need not persist. We give a complete structural classification of these curves for binary unweighted majority under infinitely exchangeable correctness indicators, with an arbitrary latent success-probability law $\Pi$. The classification is encoded by the signed voting signature ${\omega }_{\Pi }(B)=\int 1\{q(1-q)\in B\}(2q-1)\Pi (\mathrm{d}q).$ The complete odd-budget curve and this signature determine each other: the initial accuracy gives its total signed mass, and successive budget increments give its higher moments. Reflection-symmetric differences between latent laws are exactly the changes invisible to voting. A weighted total-variation constraint characterizes every attainable signature, and a minimum-mass allocation to the two reflected branches, completed by symmetric probability mass, gives every population producing its curve. Consequently, the full curve identifies the latent population precisely when no symmetric mass remains to redistribute.