Learning as Observable Matrix Dynamics: Diffusive Relaxations versus Phase Transitions

Observable Matrix Dynamics (OMD) is a diagnostic framework that probes the dynamics of high-dimensional internal representations of inputs by a neural network via a fixed-size N × N distance matrix M(t) on a held set of N inputs. OMD uses methods of random matrix theory and particle dynamics to explore spectral reorganisations that are missed by scalar loss functions, but are informative of the training process. We read M(t) against a perturbative ambient-versus-latent decomposition extending the Bogomolny--Bohigas--Schmit (BBS) theory of random distance matrices, with per-snapshot diagnostics for the top-of-spectrum band structure and ambient noise, trajectory-level observables linking snapshots, and a 3D MDS embedding (bottom-three eigenvectors) rendering training as a moving particle cloud. Across seven experiments, diffusive regimes lack stable top-of-spectrum band structure, while sharp endogenous or externally driven reorganisations produce stable fingerprints: consistent with smooth or product latent geometries in BBS-adjacent cases, and with finite-cluster or Fourier-soliton structures otherwise. OMD thus reads the geometric regime of a representation rather than reporting a single intrinsic dimension.