When Are Itemwise Rewards Enough? Finite-Sample Audits for Relational Preferences

Rubrics specify what to evaluate; representing pairwise evidence within each criterion remains a separate choice. A prompt–rubric comparison table may remain relational or be summarized by one score per response, imposing Bradley–Terry differences. We make this choice testable at prespecified approximation tolerances. A simultaneous finite-sample audit labels each row scalar-adequate, nonintegrable, or indeterminate, allowing itemwise and relational representations to coexist. For a fixed full-flow estimator, Hodge orthogonality, row-additive squared loss, and Cartesian actions reduce an explicit $2^R$-action source-risk problem to $R$ thresholds: projection pays squared cycle bias and retention pays cyclic estimation MSE. Independent audit and recovery blocks yield regret controlled by threshold-straddling intervals. Component-specific low-rank recovery and entropy-regularized Nash stability then produce policy certificates. In paired Bernoulli experiments, the selector lowers weighted source risk by 38.9% versus forced scalar and 23.4% versus forced relational representation, with no observed unsafe projection relative to the independently calibrated registry oracle. Separately, a continuous odd edgewise transform preserves every score-difference flow exactly when linear; expected-win margins can therefore make scalar source logits nonadditive. In 3,219 human votes across 183 exact Summarize-from-Feedback cells, all prompts remain indeterminate over the examined tolerances. In 3,115 UltraFeedback records, nonlinear scalar-score aggregation produces nonadditive payoffs with measurable policy effects. The resulting method projects only certified rows and retains relational structure when the audit is indeterminate.