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

arXiv: https://arxiv.org/abs/2508.21186 · HTML: https://mlsanity.com/paper/2508.21186/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 17 references that are in the map.
- First posted on arXiv 28 Aug 2025.

## Builds on (most cited first, 1 of 1)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)

## Built on by (most cited first, 1 of 1)

- [Softmax as a Lagrangian-Legendrian Seam](https://mlsanity.com/paper/2511.11573/) (2025)

## Abstract (arXiv)

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.