Self-consistency samples many reasoning trajectories and aggregates their final answers, treating the LLM as a black box that returns one answer per trajectory.
Yet the final answer of each trajectory is sampled from a softmax vector that is available from the model's log-probabilities. We refer to this as the grey-box setting in which each trajectory reveals this answer distribution rather than a single draw from it.
We formulate efficient inference in this setting as sequential mode identification with distribution-valued observations: sample trajectories one at a time and stop as soon as the LLM's modal answer is identified at a prescribed confidence level.
We characterize the asymptotic stopping rate of mode identification with distribution-valued observations exactly and show that it is never worse than the black-box rate.
We then propose the ASC-D algorithm, a betting stopping rule that attains this asymptotic stopping rate.
On MMLU-Redux, ASC-D uses $46.4$--$95.6\%$ fewer trajectories than answer-only adaptive self-consistency baselines and achieves the highest fixed-budget correct-certification rate across three open-source models.