Spectral Edge Dynamics of Training Trajectories: Signal--Noise Geometry Across Scales

We present the first systematic study of weight matrix singular value spectra during transformer pretraining, tracking full SVD decompositions of every weight matrix at 25-step intervals across three model scales (30M--285M parameters). We discover three phenomena: (1)~Transient Compression Waves: stable rank compression propagates as a traveling wave from early to late layers, creating a dramatic gradient that peaks early then reverses -- late layers eventually over-compress past early layers. (2)~Persistent Spectral Gradients: the power-law exponent~$α$ develops a permanent depth gradient forming a non-monotonic inverted-U in deeper models, with peaks shifting toward earlier layers as depth increases. (3)~Q/K--V Functional Asymmetry: value/output projections compress uniformly while query/key projections carry the full depth-dependent dynamics. The dissociation between transient compression and persistent spectral shape reveals that rank and spectral shape encode fundamentally different information about training. We formalize this as a two-timescale dynamical model and derive scaling laws ($Δα\propto L^{0.26}$, $R^2{=}0.99$). We validate on nine models across three families (custom, GPT-2, Pythia; 30M--1B parameters; 8--36 layers), demonstrate that $α$ predicts layer importance ($ρ{=}0.69$--$0.84$, $p{<}0.02$), and show that spectral-guided pruning outperforms Last-N heuristics by $1.1{\times}$--$3.6{\times}$ across seven models in two families (GPT-2 124M--774M, Pythia 160M--1B), with worst-vs-best gaps up to $23.7{\times}$ confirming the causal role of spectral structure.

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 $P \sim 10^8$, 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 $k^* = \mathrm{argmax}\, σ_j/σ_{j+1}$. 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 $\mathcal{A} = \|ΔG\|_F / (η\, g^2)$ 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.

Training loss and throughput can hide distinct internal representation in language-model training. To examine these hidden mechanics, we use spectral measurements as practical and operational diagnostics. Using a controlled family of decoder-only models adapted from the modded NanoGPT codebase, we introduce an empirical protocol based on activation covariance and per-sample gradient SVD spectra. This dual-view reveals three empirical findings and one mechanistic explanation. First, batch size acts as a latent determinant of representation geometry: runs that reach equal loss settle into systematically distinct activation spectra. Second, the activation covariance tail measured early in training reliably forecasts downstream token efficiency. Third, movement of the activation spectrum head (leading modes), together with gradient spectra, characterizes underlying learning-dynamics changes, separating learning-side architectural improvements from primarily execution-side gains. These predictive and diagnostic signals persist across the 12-, 36-, and 48-layer model tiers. Finally, a mechanistic model proves the main observations and explains how activation covariance spectra correlate with task-aligned feature learning.