Effective-rank collapse marks the semantic phase transition in neural networks

As a neural network learns, the singular-value spectrum of the matrix that reads out its representation undergoes a sharp noise-to-structure transition: its effective rank collapses from a Marchenko-Pastur-like bulk to a strongly concentrated spectrum. We show this collapse is a robust order parameter for the network's semantic phase transition, reproducing across five real substrates and a sixth controlled one, from frozen-backbone transformer heads up to the actual trained Qwen2.5-7B model (in its weights, its live activations, and over training), with a perfect null control. A grokking experiment decouples the collapse from the optimizer clock in time, establishing that it tracks the semantic transition specifically rather than co-occurring by a shared trigger. A label-corruption dose-response on the real 7B's features sharpens the claim: the collapse's coupling to generalization grades with the amount of learnable structure (|corr| 0.83 at 0% corruption to 0.41 at 100%), while the bare collapse magnitude is generic (a random-feature null collapses just as much) — the coupling, not the magnitude, is the semantic signature. We are deliberately honest about scope: the clean observable is effective-rank collapse, not strict heavy-tailed self-regularization (the power-law exponent barely moves), and a sibling random-matrix signature (level repulsion) demonstrably does not transfer to dense networks. The transferring signature is spectral support, a real but bounded bridge. Tier-2: a shared random-matrix description, not a physical identity.