Proximal Policy Optimization (PPO) for large language models typically trains its critic by mean-squared-error (MSE) regression on scalar value targets. Although scalar MSE is statistically valid for estimating the conditional expected return, sparse binary rewards in reinforcement learning with verifiable rewards (RLVR) make critic optimization and calibration especially consequential: small value errors directly distort the scalar advantages used by PPO. We study whether a classification-based training objective can improve this critic signal. HL-Gauss PPO replaces the scalar MSE head with a categorical predictor over a discretized value support, trained by cross-entropy against smoothed HL-Gauss targets. Its output is decoded to a scalar expectation for standard GAE and PPO; the actor update is therefore unchanged and is not distributional. Across mathematical reasoning, tool-augmented math, and Search-R1, and on both Qwen2.5 and Qwen3 backbones, HL-Gauss PPO consistently improves over strong PPO and DAPO baselines. Controls with one-hot, two-hot, and Bernoulli two-bin critics show that neither a larger output head nor binary classification alone explains the gains. On a common collection of reasoning prefixes, HL-Gauss improves Brier score and calibration error and yields more symmetric, lower-variance advantages. These results position categorical value learning as an effective optimization surrogate for PPO critics in RLVR.
Good action rankings do not make a contrastive critic safe to maximize. These critics increasingly act as value-like objectives for best-of-$K$ selection, planning, and critic-guided generation. Unbounded bilinear scores can let large embedding norms inflate off-support values, but cosine bounding does not remove the failure. A controlled support decomposition attributes most raw bilinear regret to norm drift. Cosine and hybrid critics nevertheless select off-support actions from most pools and incur comparable regret. Contrastive scores are weakly calibrated or inverted in the top score decile across four OGBench navigation tasks, and they fail to order fixed-query actions by value. Bellman-trained TD-Q succeeds, including in a parameter-matched function-class control. Realized costs depend on the task: simulator rollouts reveal single-step selection costs on PointMaze and the exact-$Q^*$ toy but well-powered nulls on AntMaze and HumanoidMaze, where the controller can self-correct. A training/readout decomposition traces the lost ordering to the cosine training objective; raw-trained embeddings retain weak ordering after inference-time normalization. Candidate maximization can therefore exploit false positives caused by norm drift, score saturation, or in-support misranking. Contrastive critics remain useful compatibility rankers on navigation and manipulation tasks, but action selection requires a value-calibrated scalar.