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

• URL: http://arxiv.org/abs/2508.21186v1
• Date: Thu, 28 Aug 2025 20:00:22 GMT
• Title: Manifold Trajectories in Next-Token Prediction: From Replicator Dynamics to Softmax Equilibrium
• Authors: Christopher R. Lee-Jenkins,
• Abstract summary: Decoding in large language models is often described as scoring tokens and normalizing with softmax.<n>We give a self-contained hallucination of this step as a constrained variational principle on the probability simplex.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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