Neuroscience
A geometric invariant computed from population neural activity that identifies causally important brain regions where existing similarity metrics fail. Standard metrics collapse when you control for the number of recorded neurons; bracket norm does not.
Bracket Norm Tracks Causally Important Brain Regions From Population Geometry
Zenodo 10.5281/zenodo.21246305
Existing metrics for identifying causally important brain regions — CKA, Procrustes distance, decoding accuracy, explained variance — all collapse when you control for the number of recorded neurons. A region with more electrodes looks more important, regardless of its actual causal role. All 19 geometric metrics from the neural population literature fail this test.
Bracket norm is a geometric invariant that does not have this problem. BN/sqrt(n) is stable across a 25x range of population sizes (CV = 2.0%). Top-3 and bottom-3 regions ranked by bracket norm match optogenetic silencing results (p = 0.0006). Directly perturbing a region (photoinhibition) increases BN/sqrt(n) while degrading behavior, suggesting the metric tracks per-neuron geometric expressiveness rather than computational quality. Validated on human ECoG data during speech production with zero electrode-count confound.