Reliable Local Credit from Stratified Continuation Evidence

Sample-based local credit uses auxiliary continuations to evaluate intermediate reasoning decisions. With a finite evidence budget, accurate value estimates are only part of the problem: their errors interact as local credits are converted into a policy update. We characterize this evidence-to-update relationship and show how reliability depends on both evidence covariance and update geometry. We develop Stratified Segment Policy Optimization (S-SPO), which changes the joint design of auxiliary continuations while preserving the operational value target, local-credit rule, and continuation count per query. Under an explicit independence and reuse contract, equal-mass stratification guarantees non-increase of every fixed linear quadratic update risk; a first-order guarantee with a controlled remainder extends the analysis through normalization. At nine continuations per query, S-SPO improves full-parameter frozen-gradient reliability. Across three matched RhoMath-1.1B training seeds, it raises fixed-endpoint validation accuracy by 2.75 percentage points on average and official GSM8K test accuracy by 2.26 points, with every seed pair positive on both evaluations. A second-model comparison on Llama-3.2-1B-Instruct yields a 2.85-point validation gain. These results connect the design of finite continuation evidence to reliable local credit and improved learning.