Measuring Transformation-Coverage Savings from Geometric Priors: A Fixed-Data Evaluation of Equivariance, Augmentation, and Capacity

Geometric priors can reduce the range of transformations needed during training, but measured savings depend on the baseline's training and inference budgets. We evaluate this dependence by fixing image count and test transformations, varying training rotation coverage, and recording the smallest sampled training half-width that reaches a shared accuracy target. A 60-run ImageNet-100 grid and a separate 45-run CIFAR-10 replication both show that exact $C_4$ saves an average of 156 degrees of training-band half-width on five test angles at a fixed epoch budget. Stronger baselines change the interpretation. On ImageNet-100, a plain model trained for the measured $C_4$ time and evaluated with four-view averaging reaches 10 of 11 fixed-epoch targets from the augmentation cohort at a half-width of 30 degrees, while $C_4$ retains accuracy advantages of 4.03 percentage points at that width and 2.97 percentage points at full coverage. On CIFAR-10, the corresponding advantage falls from 14.83 percentage points at 30 degrees to 0.07 percentage points at 90 degrees. Against the stronger full-band target, the primary operator gives 42 degrees of mean per-angle saving but zero saving for the macro-average target; crossings under the alternative operator and at the seed level are partly censored. These results expose sensitivity to the target and sampled grid. Separate learned-prior controls find smaller gains on hierarchical backbones than on flat networks and matched tasks. Coverage savings at fixed epochs therefore need not imply persistent accuracy gains under stronger baselines; transformation support, computation, uncertainty, and implementation fidelity must be reported together.