Friction as a Measurement of the Model: What Per-Token Entropy Density Measures, and When It Is Diagnostic

Per-token entropy — the spread of probability mass over the routes a language model did not take, here called density — is a free, always-available signal. But what does it actually measure about a model, how does it relate to the quantities a practitioner already has, and when is it diagnostic of error? This paper characterises it as a measurement instrument. Four results: (1) on near-greedy short answer spans density’s 0.90–0.98 correlation with perplexity is mechanical geometry, not an empirical equivalence — a two-model sampling sweep shows it degrade as the realised token departs from the argmax; (2) density predicts errors strongly inside a model’s competent regime (per-substrate AUROC 0.79–0.91, cluster-robust CIs) but drops toward chance once the model is crashed or overloaded (0.57–0.60) — its diagnosticity is itself capability-gated, and an effect-size check separates this from class imbalance; (3) the absolute friction level carries a proximity-to-capacity signal that per-substrate normalisation deletes, so a single pooled error-threshold on raw density out-performs the normalised version; (4) surface readability (LIX) is not LLM friction, and the length-weighted sum is a length artefact — density is the clean, length-free estimator. Item-level data and a script-to-dataset manifest are released for full reproducibility. Series position. Paper 27 in the Friction Theory series. It characterises the measurement of the model; it does not claim the unifying law (Paper 0), the critical-slowing-down early-warning (Paper 2E), the measurement of the mind (Paper 21), or the human-readability side (Paper 16), which it cites as companions.