Phantom Transitions in Language Model Fine-Tuning: A Density-Matrix Analysis

Fine-tuning a language model often fails silently when its correct completion must outrank a near-synonym competitor. Cross-entropy loss falls monotonically while the correct token never overtakes the competitor. We study five transformer architectures of two families over a fivefold parameter range, on ten contexts where the correct completion and nearest competitor overlap substantially. We construct an order parameter combining the predicted distribution with the geometric overlap between token embeddings. The density-matrix formulation is natural because the prediction distribution lives over a non-orthogonal embedding basis. It decomposes additively into a signal, tracking commitment to the correct token over its nearest competitor, and a background drag, set by how the embedding bulk leaks probability into the score. This isolates two failure modes. In kinematic failure the signal stays small and the model never commits. In structural failure the drag actively worsens as fine-tuning proceeds and the model degrades geometrically even as its loss decreases. Sharp jumps resemble phase transitions but are not. Tracking it after every gradient step rules out spontaneous symmetry breaking. Under LoRA fine-tuning the embedding matrix is frozen, so no geometric phase transition is possible and the discontinuity lies in the softmax readout. One dimensionless quantity is consistent across all five models under full fine-tuning, while a second sorts them into two classes by bulk embedding distribution and predicts whether LoRA alone can drive a sentence to commit. As a blind test, the framework predicts the critical learning rate of a held-out architecture, used to fit no parameter, to within 2.1% of a subsequent sweep. All findings concern the near-synonym geometric mechanism studied here and should not be extrapolated without re-calibration to instruction tuning or broader tasks.