The Spectral Edge Thesis: A Mathematical Framework for Intra-Signal Phase Transitions in Neural Network Training

arXiv:2603.28964v1 Announce Type: cross Abstract: We develop the spectral edge thesis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme aspect ratio regime (parameters $P \sim 10^8$, window $W \sim 10$), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at po