Independence over Accuracy: Generator–Verifier Error Correlation in Recursive Self-Improvement in Small Language Models

Verifier accuracy does not specify whether erroneous feedback repeats or corrects a policy’s mistaken judgments. We study this distinction using a binary verdict channel that fixes expected verifier accuracy while varying error correlation with a specified self-judge. Under an explicit conditional-independence assumption, the expected reward difference between correct and incorrect responses with the same self-judgment decreases linearly with correlation when the self-judge error rate is below one half. The resulting correlation tax defines an equivalent independent accuracy by matching conditional reward gaps. At a self-judge error rate of 0.30, a maximally correlated 90% verifier matches the gap of an independent 83.3% verifier, while a maximally correlated 97% verifier retains a 95% equivalent accuracy. Exact analysis identifies the feasible correlation range, the region where independence offsets lower accuracy, and the bounded tax as the self-judge error rate decreases. Reported Monte Carlo channel checks agree with these population calculations. A proposed five-round GRPO–LoRA protocol tests whether the conditional reward gap predicts learning, using controlled and natural verifiers and a frozen control. No language-model training outcome is claimed.