The Depth Flow of Token Representations Is Nonlinear and Does Not Descend Its Own Density
Alexandre Quemy
Abstract
A token's representation is carried through the network layer by layer. The whole vocabulary carried together forms a flow. We fit this flow's equation of motion as a discrete Langevin model over corpus-mean trajectories of Pythia-160M and Pythia-410M, and score the predicted steps on held-out tokens. Linear maps are often used as cheap surrogates for a layer. The flow they summarize is not linear: a quadratic drift beats the linear linear map at every transition of both models, and the Kramers--Moyal estimator agrees wherever its neighborhoods stay local. We then characterize the flow further. First, we show that it does not descend its own log-density. The drift instead descends a potential that is not the density. Second, the rotational component is not negligible, $4$ to $45\%$ of the explainable drift, and the circulation shows in what the flow preserves: a token keeps its angular rank across all thirteen layers while its norm rank is shuffled and its concentration rank is reversed by the last block.
Topics
Classified with taxonomy v2 on Sat, 5 Sept 2026.