[2603.28964] The Spectral Edge Thesis: A Mathematical Framework for Intra-Signal Phase Transitions in Neural Network Training

Computer Science > Machine Learning arXiv:2603.28964 (cs) [Submitted on 30 Mar 2026] Title:The Spectral Edge Thesis: A Mathematical Framework for Intra-Signal Phase Transitions in Neural Network Training Authors:Yongzhong Xu 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 View PDF HTML (experimental) 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 / (\e...