Wrong Votes, Right Gradients: Why Majority-Vote Rewards Track Verifiable RL

Reinforcement learning with verifiable rewards (RLVR) enhances the reasoning of large language models (LLMs), but standard RLVR often depends on human-annotated answers or carefully curated reward specifications. In machine-checkable domains, label-free alternatives such as majority voting or LLM-as-a-judge remove annotation cost but can introduce false positives that destabilize training. We introduce JURY-RL, a label-free RLVR framework that decouples answer proposal from reward disposal: votes from model rollouts propose a candidate answer, and a formal verifier determines whether that candidate can receive positive reward. Concretely, only rollouts matching the plurality-voted answer are rewarded when that answer is successfully verified in Lean. When verification is inconclusive, we invoke ResZero (Residual-Zero), a fallback reward that discards the unverified plurality proposal and redistributes a zero-mean, variance-preserving signal over the residual answers. This design maintains a stable optimization gradient without reinforcing unverifiable consensus. Across three backbone models trained on mathematical data, JURY-RL consistently outperforms other label-free baselines on mathematical reasoning benchmarks and transfers competitively to code generation and general benchmarks. It attains pass@1 performance comparable to supervised ground-truth training, with superior generalization demonstrated by higher pass@k and response diversity.

On unlabeled test data, reinforcement learning lacks a ground-truth reward; test-time RL methods derive one from the model's own roll-outs, rewarding those that match the majority vote over $N$ sampled answers. That vote discards a correct answer whenever it is a minority and scores every majority-matching roll-out identically. We replace it with CoRE (Consensus Rewards via Equilibrium): the $N$ roll-outs form a graph whose edges combine answer agreement, reasoning similarity, and generation confidence, and replicator dynamics extract its dominant set, yielding a refined pseudo-label, a graded per-roll-out reward, and a per-question cohesiveness gate. CoRE strictly generalizes voting: majority voting is recovered as a special case; a block-value analysis gives a sharp threshold for when consensus recovers a correct minority against a larger wrong plurality; and confidence calibration provably lowers that threshold multiplicatively. Across seven backbones and five benchmarks (42 model--benchmark cells, three seeds each), CoRE improves the untrained base by $+21.7$ points on average versus $+20.4$ for majority-vote TTRL, wins wherever agreement is contestable with margins over the vote of up to $+7.5$ points, and reaches the voting baseline's plateau accuracy in $54$--$70$% fewer steps. Consensus, not counting: treating the roll-out group as a graph rather than a ballot box turns a brittle vote into a calibrated, graded, self-supervised reward at no extra roll-out cost.

Reinforcement Learning from Verifiable Rewards (RLVR) has improved language models in domains such as mathematics and code, where correctness can be checked automatically. However, many important tasks are only partially verifiable: prompts contain multiple requirements, responses may satisfy some but not all of them, or no single reference answer might exist. We introduce Soft-RLVR, a framework for reinforcement learning from decomposed, learned verification signals. Soft-RLVR converts each prompt into a checklist of atomic requirements, scores candidate responses item by item with an LLM verifier, and trains on the resulting soft reward. Checklist-based rewards turn sparse pass/fail supervision into a denser partial-credit signal, but they also introduce a tradeoff: averaging item-level judgments can reduce verifier noise, while partial credit can reward incomplete responses. We formalize this tradeoff and identify conditions under which checklist-based verification gives a more reliable RL training signal than holistic verification. We further introduce Soft-SVeRL, a self-verifying variant of Soft-RLVR in which the policy also acts as the verifier. We show that self-verification is prone to reward inflation from overly permissive self-judgments, and that explicit stabilization is needed to prevent this collapse. In a controlled instruction-following setting with rule-based ground-truth evaluation, checklist-based Soft-RLVR improves IFEval by up to 11.1 points using only learned verifier rewards. Our experiments further show that verifier quality and checklist quality both affect downstream RL outcomes, and that explicit stabilization is essential for effective self-verification.