A Mathematical Framework for Intra-Signal Phase Transitions in Neural Network Training

View a PDF of the paper titled The Spectral Edge Thesis: A Mathematical Framework for Intra-Signal Phase Transitions in Neural Network Training, by Yongzhong Xu

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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 position $k^* = \mathrm{argmax}\, \sigma_j/\sigma_{j+1}$.

From three axioms 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 $\alpha$-feedback loop requiring no assumption on the optimizer. The adiabatic parameter $\mathcal{A} = \|\Delta G\|_F / (\eta\, g^2)$ controls circuit stability: $\mathcal{A} \ll 1$ (plateau), $\mathcal{A} \sim 1$ (phase transition), $\mathcal{A} \gg 1$ (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.