# Manifold Trajectories in Next-Token Prediction: From Replicator Dynamics to Softmax Equilibrium

## TL;DR

This paper models the decoding process in large language models as a smooth trajectory within a probability manifold, formalizing the intuition of manifold traversal and analyzing the effects of temperature and sampling methods.

## Contribution

It provides a formal, mathematical framework connecting softmax decoding to replicator dynamics and manifold traversal, with practical implications for decoding strategies.

## Key findings

- Next-token distribution follows a smooth trajectory inside the simplex.
- Temperature rescales the decoding trajectory in time.
- Top-k and nucleus sampling restrict the flow to specific faces of the simplex.

## Abstract

Decoding in large language models is often described as scoring tokens and normalizing with softmax. We give a minimal, self-contained account of this step as a constrained variational principle on the probability simplex. The discrete, normalization-respecting ascent is the classical multiplicative-weights (entropic mirror) update; its continuous-time limit is the replicator flow. From these ingredients we prove that, for a fixed context and temperature, the next-token distribution follows a smooth trajectory inside the simplex and converges to the softmax equilibrium. This formalizes the common ``manifold traversal'' intuition at the output-distribution level. The analysis yields precise, practice-facing consequences: temperature acts as an exact rescaling of time along the same trajectory, while top-k and nucleus sampling restrict the flow to a face with identical guarantees. We also outline a controlled account of path-dependent score adjustments and their connection to loop-like, hallucination-style behavior. We make no claims about training dynamics or internal representations; those are deferred to future work.