Decision-sufficient predictive representations for symmetric fixed-cardinality losses

What information about uncertainty must a predictor expose to support optimal decisions? We answer this question for symmetric fixed-cardinality overlap losses, where decisions and outcomes are fixed-size subsets and the loss depends only on their overlap. Using the Johnson decomposition, we derive coefficients directly from the scalar loss profile to identify the active interaction layers, without constructing the full loss matrix. Projection onto these layers preserves every expected-loss comparison. Conversely, every nonzero direction in the active subspace yields two feasible outcome distributions with no common optimal decision. These results establish the minimum affine-linear interface dimension required to support optimal decisions for every outcome distribution. Applying existing convex-calibration bounds gives the same exact dimension, attained by a standard squared embedding with an explicit regret transfer bound. Required interaction orders need not form a prefix: a threshold loss can require third-order information while ignoring the second-order layer. Synthetic experiments and frozen-predictor ablations on a candidate-conditioned WikiSeeAlso task show that retaining active higher-order information can improve decisions, with its value shaped by sample size, estimation accuracy, and decision margins. Matched training-objective comparisons distinguish the information required for decision-making from the choice of estimator used to learn it.