The Geometry of Language Model Logits: Generic Structure and Learned Departures

Autoregressive language models produce, at every prediction step, a vocabulary-sized logit vector whose structure determines the conditional next-token distribution. We empirically identify that the ordered vocabulary-head logits follow a remarkably consistent log-linear relation $z_{(r)} \approx \mu - \alpha \log r.$ across prediction contexts, architectures, and scales, emerging progressively during pretraining. To our knowledge, this is the first systematic analysis of the logit geometry underlying the distributional assumptions used in the prior decoding work such as \cite{basu2021mirostat}. While controlled order-statistic baselines spanning bounded, light-tailed, and heavier-tailed distributions can largely explain the observed log-linear geometry, they do not explain the systematic shift in the slope $\alpha$: trained language models exhibit $\alpha \approx 1.4$--$1.7$, sitting well above the initialization baseline ($\alpha \approx 0.2$) . Moreover, $\alpha$ correlates with predictive performance, suggesting that it captures learned structure beyond generic extreme-value effects.