When Does Sharing a Sampled Reasoning Prefix Pay Off?

Generating several reasoning candidates and keeping any correct one raises coverage, the fraction of problems that at least one candidate solves. Continuing one sampled prefix into several candidates makes this cheaper, but every candidate then inherits the prefix's early choices. We ask when sharing a sampled prefix pays off at an equal budget of generated tokens. When continuations keep the decoding settings and generation allowance of independent sampling, sharing the first $m$ tokens saves a fraction $s(m)$ of a second candidate's generation and removes a fraction $\rho (m)$ of its coverage gain, the share of success variance these tokens explain. On each problem, independent sampling at the same expected budget covers it at least as often as every preset allocation of candidates to sampled prefixes of length $m$ if and only if $\rho (m)\ge s(m)$. Measuring both fractions from continuations of stored prefixes, under plans fixed before sampling, on grade-school mathematics of intermediate difficulty with four base models and a thinking model, we find that early tokens resolve success variance faster than they consume generation: the shortest tested prefixes explain 1.8 to 3.5 times their share of generation. At the same expected budget, eight candidates sharing one prefix covered significantly fewer problems than independent sampling for every model and branch position tested. By the time candidates branch, they already share part of their chances of success.