标题： 流形轨迹在下一个标记预测中的应用：从复制者动力学到Softmax平衡 显示英文标题

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

摘要： 在大型语言模型中的解码通常被描述为对标记进行评分并通过softmax进行归一化。 我们对此步骤提供一个最小且自包含的描述，将其视为概率单纯形上的约束变分原理。 离散的、尊重归一化的上升过程是经典的乘法权重（熵镜像）更新；其连续时间极限是复制者流。 从这些要素出发，我们证明了对于固定的上下文和温度，下一个标记分布沿着单纯形内的平滑轨迹移动并收敛到softmax平衡状态。 这在输出分布层面上形式化了常见的“流形遍历”直觉。 该分析得出精确且面向实践的后果：温度沿同一轨迹精确地对时间进行缩放，而top-k和核采样将流动限制在一个具有相同保证的面内。 我们还概述了路径依赖的分数调整的受控解释及其与类似循环、幻觉风格行为的联系。 我们不对训练动态或内部表示做出任何声明；这些将推迟到未来的工作中。 显示英文摘要

摘要： 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.