Capacity Floors and Readout Limits for Bottlenecked Medical Image Classifiers

Medical image classifiers often route every input through a bounded store, a code- book, a concept layer, or a prototype bank, chosen for interpretability or for de- ployment cost. When such a model falls short, the cause may be a store too small to separate the classes, an encoder that underuses the store it has, or a decision rule that discards information the store holds, and each calls for a different fix. A vali- dation curve reports the total error alone and leaves the binding cause unidentified. This paper decomposes the error into one term for each of these three causes. The first is a capacity floor, the smallest error any sampled positive quadratic readout attains on a store of dimension m, equal to one minus the sum of the m largest prior masses and therefore fixed by the architecture and the prior alone. The sec- ond is the representation gap: the error of the best readout in that class on the learned store, obtained as a semidefinite program with a dual certificate, minus the floor. The third is the readout gap: the attained error minus that best error, the excess due to the decision rule. The floor is tight and holds for every decoder on a discrete store and for the positive-readout class on a continuous store, and a two- dimensional counterexample shows that this decoder boundary is sharp. Inverting the floor sizes a store before training: the prior and a target error fix the narrow- est bottleneck that could reach it, so a proposed architecture can be ruled out on paper. Across four benchmarks, five bottleneck families, and an unbottlenecked reference, the decomposition separates certified capacity limits from errors car- ried by the representation or the readout. On discrete stores and our own register, where it is exact, the representation gap dominates the readout gap in all 12 con- figurations, and in the 12 diagnostic ones continuous heads deployed with argmax beat the best positive readout of their store, as the decoder boundary predicts.