When Are Classes Cones? Scale Invariance, Linear Sufficiency, and Conic Decision Geometry

Representation-learning methods implicitly choose a geometry for each class. We derive conic decision geometry from positive-scaling invariance and affine Bayes realizability: invariance alone gives cones, while an invariant affine readout admits a bias-free representation and polyhedral convex decision regions. Support containment additionally requires separability, and positive angular margin rules out class hulls containing a line. Under a shared-concentration von Mises–Fisher model with equal priors, the readout is Bayes-optimal and pairwise Bhattacharyya overlap yields a closed-form error-bound objective. On CIFAR-100 with 15% labels, a five-seed factorial study separates the architecture and objective effects: the full method reaches 71.2% accuracy and exceeds homogeneous cosine cross-entropy by 4.2 percentage points (paired 95% CI [4.05, 4.35]). Across 30 conditions, a finite-sample geometric quantity predicts error with cross-validated $R^2=0.71$. The experiments also show improved shifted calibration, density-based OOD detection, and gain agreement of 100% over the tested range. These findings support the mechanism without establishing exact vMF fit, universal OOD optimality, or a positive interaction between architecture and objective.