Can a committee of weak reasoning-model calls reach the performance of much stronger models? We study verifier-backed committee search as inference-time boosting for reasoning language models. The mechanism is not simply that ``more agents help'': samples expose latent correct solutions, while critics and comparators must recover them without access to the hidden verifier. We formalize this view by separating proposal coverage, local identifiability, progress, and diversity. We prove that coverage can be amplified by repeated sampling, but cannot by itself create useful critics or comparators; reliable amplification requires an additional local soundness signal, such as execution, proof checking, type checking, tests, or constraint solving. We give rank-based bounds showing when local selection errors compose into reliable trajectories, and characterize the proposer-side ceiling: oracle best-of-\(k\) converges only to the mass of task slices on which the proposal system assigns nonzero useful probability. Empirically, on SWE-bench Verified, a single \texttt{GPT-5.4 nano} proposal solves \(67.0\%\) of tasks. Using the same nano model, our critic--comparator orchestration reaches \(76.4\%\) with \(k=8\) proposals, matching the standalone performance of \texttt{Gemini 3 Pro} and \texttt{Claude Opus 4.5} Thinking and approaching the \(79.0\%\) oracle best-of-\(8\) upper bound. Thus, many correct patches are already present in weak-model proposal pools; the main challenge is selecting them. The remaining failures are mostly proposal-coverage failures, indicating shared blind spots that stronger selection alone cannot close.