Certified Topological Interaction in Neural Representations: Class Disentanglement Is Mostly Pairwise

Abstract

Class disentanglement (the separation of a representation's class-conditional point clouds along depth and over training) is usually read off descriptive curves. We measure it as certified topological interaction between labeled point clouds, using the recently introduced Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, computed by one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test: exact permutation tests in both directions, a guarded separation certificate, and a paired test for the comparative claims applications make. Across 111 trained networks and 52,650 certified measurements, disentanglement is depth-graded and concentrated in the first epochs, and interaction quotients rank class pairs by confusability (Spearman rho=0.83), on par with cheap separability statistics. In a 96-model factorial population, augmentation is the one training choice that separates classes relative to chance; weight decay compresses the overlap without separating, and depth and width do nothing. The structural finding is one only a k-fold statistic can pose: the joint entanglement of a class triple sits below that of its strongest pair in 97% of triple-layer cells and 99.5% of deep cells, far below a measured null floor, in vision encoders and frozen language models alike. This pairwise dominance is a regularity, not a law: expected from the nesting of overlaps but not forced by geometry, present at initialization and in raw pixels, and manufactured in the last stage alone when a network memorizes random labels. The unnormalized profile mass predicts test accuracy (R^2=0.94), the quotient does not, and neither beats a linear probe. One lesson is reported in full: the paired test must use a scale-free statistic, or it certifies feature-norm dynamics as disentanglement.