Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training

We develop the spectral edge analysis: 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 ~ 108, window W ~ 10), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at position k^* = argmax, σj/σj+1. From three assumptions we derive: (i) gap dynamics governed by a Dyson-type ODE with curvature asymmetry, damping, and gradient driving; (ii) a spectral loss decomposition linking each mode's learning contribution to its Davis--Kahan stability coefficient; (iii) the Gap Maximality Principle, showing that k^* is the unique dynamically privileged position -- its collapse is the only one that disrupts learning, and it sustains itself through an α-feedback loop requiring no assumption on the optimizer. The adiabatic parameter A = |Delta G|F / (η, g2) controls circuit stability: A ll 1 (plateau), A ~ 1 (phase transition), A gg 1 (forgetting). Tested across six model families (150K--124M parameters): gap dynamics precede every grokking event (24/24 with weight decay, 1/24 without), the gap position is optimizer-dependent (Muon: k^*=1, AdamW: k^*=2 on the same model), and 19/20 quantitative predictions are confirmed. The framework is consistent with the edge of stability, Tensor Programs, Dyson Brownian motion, the Lottery Ticket Hypothesis, and neural scaling laws.