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.

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.

Equations

$$\Delta^{V-1}=\Big{\{}p\in\mathbb{R}^{V}_{\geq 0}:\ \sum_{i=1}^{V}p_{i}=1\Big{\}},\qquad H(p)=-\sum_{i=1}^{V}p_{i}\log p_{i}\ \ \ (0\log 0:=0).$$

$$A(s)=T\log\!\Big{(}\sum_{i=1}^{V}e^{s_{i}/T}\Big{)},\qquad\pi_{i}(s)=\frac{e^{s_{i}/T}}{\sum_{j=1}^{V}e^{s_{j}/T}}.$$

$$p^{(t+1)}=\operatorname*{arg\,max}_{p\in\Delta^{V-1}}\Big{\{}\left\langle p,\,s\right\rangle+T\,H(p)-\tfrac{1}{\eta}D_{\mathrm{KL}}(p\|p^{(t)})\Big{\}}$$

$$\Delta^{V-1}\;=\;\big{\{}\,p\in\mathbb{R}^{V}_{\geq 0}\ :\ \sum_{i=1}^{V}p_{i}=1\,\big{\}},$$

$$\mathcal{F}(p)=⟨p,s⟩+TH(p)=\sum _{i=1}^V{p}_i{s}_i-T\sum _{i=1}^V{p}_i\log p_i,p\in {\Delta }^{V-1},$$

$$\underset{p\in {\Delta }^{V-1}}{\max }\mathcal{F}(p)=⟨p,s⟩+TH(p)$$

$${\pi }_i=\frac{e^{s_i/T}}{{\sum }_{j=1}^V{e}^{s_j/T}}(i=1,\dots ,V),$$

$$\mathcal{L}(p,\lambda)\;=\;\sum_{i}p_{i}s_{i}\;-\;T\sum_{i}p_{i}\log p_{i}\;+\;\lambda\Big{(}1-\sum_{i}p_{i}\Big{)}.$$

$$\frac{\partial\mathcal{L}}{\partial p_{i}}\;=\;s_{i}\;-\;T\big{(}1+\log p_{i}\big{)}\;-\;\lambda\;=\;0\quad\Rightarrow\quad\log p_{i}\;=\;\frac{s_{i}-\lambda-T}{T}.$$

$$A(s)\;=\;T\log\!\Big{(}\sum_{i=1}^{V}e^{s_{i}/T}\Big{)}.$$

$$A(s)\;=\;\max_{p\in\Delta^{V-1}}\ \Big{\{}\ \left\langle p,\,s\right\rangle\;+\;T\,H(p)\ \Big{\}},$$

$$\nabla A(s)=\pi(s)\quad\text{and}\quad\nabla^{2}A(s)\;=\;\frac{1}{T}\Big{(}\mathrm{diag}(\pi)\;-\;\pi\pi^{\top}\Big{)},$$

$$\mathcal{F}(p)=⟨p,s⟩+TH(p)\text{over }{\Delta }^{V-1}.$$

$$p^{(t+1)}\;=\;\operatorname*{arg\,max}_{p\in\Delta^{V-1}}\ \Big{\{}\,\left\langle p,\,s\right\rangle+T\,H(p)\;-\;\frac{1}{\eta}\,D_{\operatorname{KL}}\!\big{(}p\|p^{(t)}\big{)}\,\Big{\}}.$$

$$p^{(t+1)}_{i}\;=\;\frac{p^{(t)}_{i}\exp\!\big{(}\tfrac{\eta}{T}\,s_{i}\big{)}}{\sum_{j=1}^{V}p^{(t)}_{j}\exp\!\big{(}\tfrac{\eta}{T}\,s_{j}\big{)}}\,,\qquad i=1,\dots,V.$$

$$0=s_{i}-T(1+\log p_{i})-\tfrac{1}{\eta}\big{(}\log p_{i}-\log p^{(t)}_{i}\big{)}-\lambda\,,$$

$$\mathcal{F}\!\big{(}p^{(t+1)}\big{)}\;\geq\;\mathcal{F}\!\big{(}p^{(t)}\big{)}\;+\;\frac{1}{\eta}\,D_{\operatorname{KL}}\!\big{(}p^{(t+1)}\|p^{(t)}\big{)}\;\geq\;\mathcal{F}\!\big{(}p^{(t)}\big{)}.$$

$$\left\langle p^{(t+1)},\,s\right\rangle+T\,H(p^{(t+1)})-\tfrac{1}{\eta}D_{\operatorname{KL}}\!\big{(}p^{(t+1)}\|p^{(t)}\big{)}\;\geq\;\left\langle p^{(t)},\,s\right\rangle+T\,H(p^{(t)})\,.$$

$$\frac{p^{(t+1)}_{i}-p^{(t)}_{i}}{\eta}\ \Longrightarrow\ \dot{p}_{i}\;=\;\frac{1}{T}\,p_{i}\,\big{(}s_{i}-\bar{s}\big{)},\qquad\bar{s}=\sum_{j=1}^{V}p_{j}s_{j},$$

$$p^{(t+1)}_{i}=\frac{p^{(t)}_{i}\big{(}1+\tfrac{\eta}{T}s_{i}+o(\eta)\big{)}}{\sum_{j}p^{(t)}_{j}\big{(}1+\tfrac{\eta}{T}s_{j}+o(\eta)\big{)}}=p^{(t)}_{i}\Big{(}1+\tfrac{\eta}{T}(s_{i}-\bar{s})+o(\eta)\Big{)}.$$

$$\frac{d}{dt}\mathcal{F}(p(t))\ge 0,$$

$$\mathrm{int}\,\Delta^{V-1}\;=\;\Big{\{}p\in\mathbb{R}^{V}_{>0}:\sum_{i=1}^{V}p_{i}=1\Big{\}},$$

$$T_{p}\Delta^{V-1}\;=\;\Big{\{}u\in\mathbb{R}^{V}:\sum_{i=1}^{V}u_{i}=0\Big{\}}.$$

$$X_{i}(p)\;=\;\frac{1}{T}\,p_{i}\big{(}s_{i}-\bar{s}\big{)},\qquad\bar{s}=\sum_{j=1}^{V}p_{j}s_{j},$$

$$\dot{p}_{i}(t)\;=\;\frac{1}{T(t)}\,p_{i}(t)\,\big{(}s_{i}-\bar{s}(t)\big{)}.$$

$$\dot{p}_{i}\;=\;\frac{1}{T}\,p_{i}\big{(}s_{i}(p)-\bar{s}(p)\big{)}$$

Full text

1 Introduction

Large language models (LLMs) are typically described as choosing the next token by scoring a vocabulary and normalizing the scores with a softmax at some temperature. This account is operationally correct, but it leaves open a conceptual question often raised in the community: what does the decoding step look like as a dynamical process? In particular, when practitioners say that models “traverse a manifold” during decoding, is this merely a metaphor, or can one make it a theorem, precisely stated and proved from basic, verifiable ingredients?

This paper gives a minimal, self-contained answer at the level of the output distribution for a fixed context. We show that the next-token distribution is characterized by a classical free-energy principle and that the normalization-respecting ascent to that equilibrium admits both (i) a one-line discrete update—entropic mirror ascent, i.e., multiplicative weights—and (ii) a continuous-time limit—the replicator differential equation on the probability simplex. From these pieces we obtain a rigorous statement of the manifold picture: the decoding distribution follows a smooth trajectory in the open simplex and converges to the softmax equilibrium. The development uses only standard tools (entropy, convex duality, mirror methods, replicator dynamics); no geometric background is assumed in the body of the paper.

Contributions.

• •

We provide a derivation of next-token decoding as a constrained variational problem on the probability simplex. The discrete update is a KL-prox mirror step (multiplicative weights), and the continuous limit is the replicator ODE.

• •

We prove a manifold-traversal theorem (Thm. 4.1): for fixed scores and temperature, the decoding distribution traces a $C^{1}$ curve inside the open simplex and converges to softmax. This formalizes a common heuristic—“decoding moves along a manifold”— as a theorem in the next-token setting.

• •

We extract precise practitioner consequences: (a) temperature acts as an exact time-rescaling of the same trajectory; (b) top-$k$/nucleus truncation simply restricts the flow to a face of the simplex with identical guarantees; (c) mild path dependence in score adjustments can yield loops and brittle attractors, offering a controlled language for “hallucination”-like behavior.

Scope.

We focus exclusively on the output distribution over next tokens for a fixed context. We do not make claims about training dynamics, internal representations, or production decoders literally integrating an ODE; the ODE appears as a continuous-time limit of the standard multiplicative update. A broader geometric framework that treats hidden states as coordinates on a high-dimensional manifold and views the output distribution as a projection is prepared by the author and deferred to future work (see the pointer in §6).

1.1 Preliminaries (notation and classical facts)

Let $V\geq 2$ be the vocabulary size. We write

[TABLE]

For a fixed context, let $s\in\mathbb{R}^{V}$ denote the score/logit vector and $T>0$ the temperature. Define the temperature-scaled log-partition (log-sum-exp)

[TABLE]

Classically, $\pi(s)=\nabla A(s)$ and \nabla^{2}A(s)=\tfrac{1}{T}\big{(}\mathrm{diag}(\pi)-\pi\pi^{\top}\big{)}; see Jaynes [10], Cover and Thomas [7], Wainwright and Jordan [17], Murphy [12]. We use $\left\langle p,\,s\right\rangle=\sum_{i}p_{i}s_{i}$ and $\mathbbm{1}=(1,\dots,1)^{\top}$.

Two standard viewpoints will be used throughout. First, a variational view (Gibbs’ principle): $\pi$ uniquely maximizes the strictly concave free energy $\left\langle p,\,s\right\rangle+T\,H(p)$ over $\Delta^{V-1}$ [10, 7, 5]. Second, a constraint-respecting update view: the entropic mirror (KL-prox) step

[TABLE]

produces the multiplicative-weights update $p^{(t+1)}_{i}\propto p^{(t)}_{i}\exp((\eta/T)s_{i})$ [3, 2, 13]. As $\eta\to 0$, the continuous-time limit is the replicator ODE \dot{p}_{i}=\tfrac{1}{T}p_{i}\big{(}s_{i}-\sum_{j}p_{j}s_{j}\big{)} on $\Delta^{V-1}$ [16, 9]. These facts are classical; proofs are reproduced later for completeness and to keep the paper self-contained.

Adding a constant $c\mathbbm{1}$ to $s$ leaves $\pi$ unchanged; scaling $s\mapsto\alpha s$ is equivalent to $T\mapsto T/\alpha$. Temperature therefore sets the selection pressure and, in our dynamical view, the time scale of the trajectory we analyze in §3–4.

Positioning with prior work.

Our development relies only on well-established building blocks: Gibbs’ variational principle for softmax [10, 7], entropic mirror ascent and multiplicative weights [3, 2, 13], and the replicator flow with its Shahshahani geometry on the simplex [16, 9]. What is new here is the decoding-level formalization: we prove that the output distribution in next-token prediction follows a manifold trajectory and give exact, practice-facing corollaries for temperature, truncation, and mild path dependence. A broader contact geometry perspective is sketched in §6 with a pointer to the author’s preprint.

Organization.

Section 2 records the variational characterization and duality facts with proofs. Section 3 derives the mirror (multiplicative-weights) step, proves ascent, and passes to the replicator limit. Section 4 states the manifold-traversal theorem and corollaries. Section 5 discusses temperature, truncation, and controlled path dependence. Section 6 summarizes limitations and outlines a future extension to hidden-state manifolds and projection.

2 A Variational Principle for Next-Token Selection

Fix a decoding context and vocabulary size $V\geq 2$. Let $s\in\mathbb{R}^{V}$ denote the (fixed) score/logit vector produced by the model for the next token and $T>0$ the temperature. We work on the probability simplex

[TABLE]

and use Shannon entropy $H(p)=-\sum_{i}p_{i}\log p_{i}$ (convention $0\log 0:=0$). Consider the classical “free-energy” objective from statistical mechanics / information theory,

[TABLE]

which is a concave functional balancing score maximization against entropy. This setup is standard (see [10, 7]; convex-analytic background in 5, Ch. 3). We record the derivation for completeness and to fix notation.

2.1 Softmax as the unique maximizer (Gibbs variational principle)

Proposition 2.1** (Softmax variational characterization; standard).**

For fixed $s\in\mathbb{R}^{V}$ and $T>0$, the concave program

[TABLE]

has the unique solution

[TABLE]

i.e., the softmax distribution at temperature $T$.

Proof.

Introduce the Lagrangian for the affine constraint $\sum_{i}p_{i}=1$:

[TABLE]

On the relative interior of $\Delta^{V-1}$, stationarity gives, for each $i$,

[TABLE]

Hence $p_{i}=C\,e^{s_{i}/T}$ with $C=e^{-(\lambda+T)/T}$. Enforcing $\sum_{i}p_{i}=1$ yields $C=(\sum_{j}e^{s_{j}/T})^{-1}$ and thus $p=\pi$. Since $\mathcal{F}$ is strictly concave on the relative interior (Lemma 2.2), the maximizer is unique. [See also the classical Gibbs principle in 10, 7]. ∎

Lemma 2.2** (Strict concavity).**

The map $\mathcal{F}:\Delta^{V-1}\to\mathbb{R}$ in (2.1) is strictly concave on the relative interior of $\Delta^{V-1}$.

Proof.

The term $p\mapsto\left\langle p,\,s\right\rangle$ is linear. Since $-\,\sum_{i}p_{i}\log p_{i}$ is strictly concave on the relative interior of $\Delta^{V-1}$, $H$ is strictly concave there (and continuous on $\Delta^{V-1}$). Therefore $\mathcal{F}=\left\langle p,\,s\right\rangle+T\,H(p)$ is strictly concave on the relative interior, and for $T>0$ the unique maximizer lies in the interior. For background on entropy curvature, see Boyd and Vandenberghe [5, §3.1.4]. ∎

Remark 2.3* (Normalization and interiority).*

For $T>0$, the maximizer $\pi$ satisfies $\pi_{i}>0$ for all $i$; thus $\pi\in\mathrm{int}\,\Delta^{V-1}$. Intuitively, the entropic term at positive temperature rules out boundary optima unless some $s_{i}=-\infty$.

2.2 Dual (log-sum-exp) viewpoint

Define the log-partition function (log-sum-exp with temperature)

[TABLE]

Then $A$ is convex, smooth, and satisfies the Gibbs/Fenchel variational identity

[TABLE]

a special case of convex duality [cf. 14] and the classical Gibbs principle [10, 7]. In exponential-family form,

[TABLE]

the familiar Fisher information form; see Wainwright and Jordan [17, §3], Murphy [12, §8.3], Bishop [4, §4.3].

Remark 2.4* (Smoothness and sensitivity).*

The Jacobian (\partial\pi_{i}/\partial s_{j})_{ij}=\frac{1}{T}\big{(}\mathrm{diag}(\pi)-\pi\pi^{\top}\big{)} shows the map $s\mapsto\pi(s)$ is smooth on $\mathbb{R}^{V}$ and Lipschitz on compact sets, with sensitivity scaling like $1/T$; cf. Murphy [12, §8.3].

2.3 Temperature, limits, and basic properties

Proposition 2.5** (Temperature limits).**

Fix $s\in\mathbb{R}^{V}$ and let $\pi(T)$ denote the softmax at temperature $T>0$.

As $T\downarrow 0$, $\pi(T)$ concentrates on the set of indices attaining $\max_{i}s_{i}$. If the maximizer is unique, then $\pi(T)\to e_{i^{\star}}$ where $i^{\star}\in\operatorname*{arg\,max}_{i}s_{i}$. 2. 2.

As $T\uparrow\infty$, $\pi(T)\to\tfrac{1}{V}\,\mathbbm{1}$, the uniform distribution.

Proof.

Both statements follow directly from $\pi_{i}(T)=\exp(s_{i}/T)/\sum_{j}\exp(s_{j}/T)$ and standard Laplace-type asymptotics; see, e.g., Cover and Thomas [7, §12]. ∎

Remark 2.6* (Score shifts and invariances).*

Adding a constant to all scores $s\mapsto s+c\,\mathbbm{1}$ leaves $\pi$ unchanged and shifts $A(s)$ by $c$. Scaling $s\mapsto\alpha s$ is equivalent to rescaling the temperature $T\mapsto T/\alpha$. These invariances let us interpret $T$ as a selective pressure parameter.

We have established a minimal variational principle for decoding: the next-token distribution is the unique maximizer of the strictly concave free energy (2.1), equivalently the gradient of the convex potential $A$ in (2.2) via (2.3). In the next section we derive constraint-respecting updates that ascend $\mathcal{F}$ on the simplex—the discrete-time mirror (multiplicative-weights) step and its continuous-time limit, the replicator ODE—which inherit the normalization, interiority, and temperature behavior recorded here.

3 Constraint-Respecting Updates: Mirror $\rightarrow$ Replicator

In §2 we showed that for fixed scores $s\in\mathbb{R}^{V}$ and temperature $T>0$, the next-token distribution is the unique maximizer of the concave free energy

[TABLE]

We now derive a discrete-time update that ascends $\mathcal{F}$ while respecting the simplex constraint, and then pass to a continuous-time limit. Throughout this section, $s$ is fixed (context is frozen); extensions where $s$ depends on $p$ are deferred to §5.

3.1 Entropic mirror ascent equals multiplicative weights

We use the entropic mirror map (negative Shannon entropy), which yields the classical multiplicative-weights (MW) update; see Beck and Teboulle [3], Arora et al. [2], Raginsky and Bouvrie [13].

Definition 3.1** (KL divergence).**

For $p,q\in\mathrm{int}\,\Delta^{V-1}$, let $D_{\operatorname{KL}}(p\|q)=\sum_{i}p_{i}\log\!\frac{p_{i}}{q_{i}}$.

Proposition 3.2** (One-step mirror ascent).**

Given $p^{(t)}\in\mathrm{int}\,\Delta^{V-1}$ and step size $\eta>0$, the entropic mirror-ascent step for maximizing $\mathcal{F}$ is

[TABLE]

Its unique solution has the multiplicative-weights form

[TABLE]

Proof (for completeness).

First-order optimality for (3.1) with the affine constraint $\sum_{i}p_{i}=1$ gives

[TABLE]

hence $\log p_{i}=\log p^{(t)}_{i}+\tfrac{\eta}{T}s_{i}-\eta(1+\tfrac{\lambda}{T})$, which yields $p_{i}\propto p^{(t)}_{i}\exp((\eta/T)s_{i})$. Normalizing over $i$ gives (3.2). Strict concavity of the objective in (3.1) on $\mathrm{int}\,\Delta^{V-1}$ ensures uniqueness. ∎

Proposition 3.3** (Monotone ascent of $\mathcal{F}$).**

The update (3.1) satisfies

[TABLE]

In particular, t\mapsto\mathcal{F}\!\big{(}p^{(t)}\big{)} is nondecreasing, with equality iff $p^{(t+1)}=p^{(t)}$.

Proof.

By optimality of $p^{(t+1)}$ in (3.1),

[TABLE]

Rearrange to obtain the claim. ∎

Remark 3.4* (Interior invariance).*

If $p^{(t)}\in\mathrm{int}\,\Delta^{V-1}$ and $s_{i}\in\mathbb{R}$, then (3.2) yields $p^{(t+1)}\in\mathrm{int}\,\Delta^{V-1}$ and preserves normalization. Thus the update is constraint-respecting and stays strictly inside the simplex.

3.2 Continuous-time limit: the replicator ODE

We now pass to a continuous-time limit of (3.2) by letting $\eta\to 0$.

Proposition 3.5** (Replicator dynamics).**

Let $p^{(t+1)}$ be given by (3.2). Then

[TABLE]

as $\eta\to 0$. That is, the continuous-time limit is the replicator ODE on $\Delta^{V-1}$ with “fitness” $f_{i}=s_{i}/T$. [See, e.g., 9, 16, 15].

Proof (for completeness).

Using $\exp(\eta a)=1+\eta a+o(\eta)$,

[TABLE]

Subtract $p^{(t)}_{i}$, divide by $\eta$, and take $\eta\to 0$. ∎

The replicator ODE keeps trajectories on the simplex and preserves nonnegativity:

Lemma 3.6** (Forward invariance).**

For the ODE $\dot{p}_{i}=\tfrac{1}{T}p_{i}(s_{i}-\bar{s})$ with $p(0)\in\Delta^{V-1}$, we have $\sum_{i}p_{i}(t)=1$ for all $t$, and if $p_{i}(0)=0$ then $p_{i}(t)\equiv 0$; in particular, $\mathrm{int}\,\Delta^{V-1}$ and each face of $\Delta^{V-1}$ are forward-invariant.

Proof.

Summing gives $\sum_{i}\dot{p}_{i}=\tfrac{1}{T}\sum_{i}p_{i}(s_{i}-\bar{s})=0$. If $p_{i}(0)=0$, then $\dot{p}_{i}(0)=0$ and uniqueness of ODE solutions yields $p_{i}(t)\equiv 0$. ∎

3.3 Lyapunov ascent and convergence (fixed scores)

Replicator dynamics is a gradient flow of $\mathcal{F}$ with respect to the Shahshahani metric on the simplex [16, 9], hence $\mathcal{F}$ is a Lyapunov function.

Proposition 3.7** (Energy ascent).**

Along any solution of $\dot{p}_{i}=\tfrac{1}{T}p_{i}(s_{i}-\bar{s})$ with fixed $s$,

[TABLE]

with equality iff $p(t)$ is stationary (i.e., $p=\pi$ from Proposition 2.1).

Proof (sketch).

The replicator vector field equals the (natural) gradient of $\mathcal{F}$ under the Shahshahani inner product $\langle u,v\rangle_{p}=\sum_{i}\tfrac{u_{i}v_{i}}{p_{i}}$ on the tangent space to $\Delta^{V-1}$. Therefore $\tfrac{d}{dt}\mathcal{F}=\langle\nabla^{\text{nat}}\mathcal{F},\nabla^{\text{nat}}\mathcal{F}\rangle_{p}\geq 0$, with equality iff the field vanishes; see Shahshahani [16], Hofbauer and Sigmund [9]. A direct coordinate proof is also possible by differentiating $\mathcal{F}$ and substituting $\dot{p}$. ∎

Corollary 3.8** (Convergence to softmax).**

For fixed $s$ and $T>0$, any trajectory with $p(0)\in\mathrm{int}\,\Delta^{V-1}$ satisfies $p(t)\to\pi$ as $t\to\infty$, where $\pi$ is the softmax maximizer of §2.

Remark 3.9* (Top-$k$/nucleus as faces of the simplex).*

Practical decoders often restrict support before renormalization (top-$k$, nucleus). By Lemma 3.6, faces of $\Delta^{V-1}$ are invariant under replicator flow, and the mirror step (3.1) respects the constraint set. Thus the same analysis applies verbatim on any fixed face (subset of tokens), with the softmax equilibrium computed on that face.

In §4 we use these facts to state and prove the manifold-traversal theorem for decoding trajectories and to connect temperature and truncation to geometric features of the flow.

4 Manifold Traversal of Decoding Trajectories

We now formalize the common intuition that decoding “moves along a manifold.” In our setting the manifold is simply the open probability simplex

[TABLE]

a smooth $(V-1)$–dimensional submanifold of $\mathbb{R}^{V}$ with tangent space

[TABLE]

The replicator vector field from §3 is

[TABLE]

which is smooth on $\mathrm{int}\,\Delta^{V-1}$ and satisfies $\sum_{i}X_{i}(p)=0$, hence $X(p)\in T_{p}\Delta^{V-1}$ (tangency).

The following theorem concerns the output distribution for a fixed context; it does not address training dynamics or internal representations.

4.1 The manifold-traversal theorem

Theorem 4.1** (Manifold traversal of decoding trajectories).**

Fix $s\in\mathbb{R}^{V}$ and $T>0$. Consider the ODE $\dot{p}=X(p)$ on $\Delta^{V-1}$ with $X$ given by (4.1). For any initial condition $p(0)\in\mathrm{int}\,\Delta^{V-1}$:

(Well-posed flow on the manifold) There is a unique global solution $p:[0,\infty)\to\mathrm{int}\,\Delta^{V-1}$ with $p(t)\in\mathrm{int}\,\Delta^{V-1}$ for all $t\geq 0$, and $p(\cdot)$ is a $C^{1}$ curve on the manifold $\mathrm{int}\,\Delta^{V-1}$. 2. 2.

(Ascent of free energy) The free energy $\mathcal{F}(p)=\left\langle p,\,s\right\rangle+T\,H(p)$ from §2 is a Lyapunov function: $\frac{d}{dt}\,\mathcal{F}(p(t))\geq 0$, with equality iff $p(t)$ is stationary. 3. 3.

(Equilibrium and convergence) The unique stationary point in $\mathrm{int}\,\Delta^{V-1}$ is the softmax $\pi$ (Proposition 2.1), and $p(t)\to\pi$ as $t\to\infty$.

Proof.

(1) By Lemma 3.6, $\sum_{i}p_{i}(t)=1$ and faces of $\Delta^{V-1}$ are forward-invariant; since $p(0)\in\mathrm{int}\,\Delta^{V-1}$ and $X$ is smooth on $\mathrm{int}\,\Delta^{V-1}$, standard ODE theory yields a unique solution $p(t)\in\mathrm{int}\,\Delta^{V-1}$ for all $t\geq 0$. (2) Proposition 3.7 shows $\mathcal{F}$ is nondecreasing along trajectories, with strict increase unless stationary. (3) The stationary condition $\dot{p}=0$ with $p\in\mathrm{int}\,\Delta^{V-1}$ forces $s_{i}=\bar{s}$ for all $i$ in the support weighted by $p$, which coincides with the KKT conditions for the maximizer of $\mathcal{F}$; by Proposition 2.1 the unique interior maximizer is $\pi$. Since $\mathcal{F}$ is bounded above by $\mathcal{F}(\pi)$ and strictly increases unless at equilibrium, LaSalle’s invariance principle yields $p(t)\to\pi$. See also the replicator-as-gradient-flow viewpoint in Shahshahani [16], Hofbauer and Sigmund [9]. ∎

Remark 4.2* (Provenance).*

The dynamical ingredients—replicator flow on the simplex, its Shahshahani geometry, and free-energy ascent—are classical (e.g., 16, 9). The novelty here is to show that the decoding distribution in next-token prediction obeys this flow, thereby turning the common “manifold traversal” intuition into a theorem at the output-distribution level, and deriving exact corollaries for temperature (time rescaling) and truncation (face invariance).

Remark 4.3*.*

Under the Shahshahani inner product $\langle u,v\rangle_{p}=\sum_{i}\frac{u_{i}v_{i}}{p_{i}}$ on $T_{p}\Delta^{V-1}$, the replicator field is the natural gradient of $\mathcal{F}$ [16, 9]; cf. information-geometry perspectives [1, 13]. We do not rely on this fact in the main argument, but it clarifies why $\mathcal{F}$ is a Lyapunov function.

4.2 Basic geometric corollaries for decoding

Corollary 4.4** (Temperature is a time rescaling).**

For fixed $s$, replacing $T$ by $\alpha T$ in (4.1) scales the vector field by $1/\alpha$. Thus solutions are reparameterizations of time: lower $T$ yields faster motion along the same trajectory toward the same equilibrium $\pi$ (cf. Proposition 2.5).

Corollary 4.5** (Truncation acts on faces).**

If decoding restricts support to a fixed subset $S\subseteq\{1,\dots,V\}$ (e.g., top-$k$ or nucleus set) and renormalizes, the dynamics evolve on the face $\Delta^{V-1}_{S}=\{p\in\Delta^{V-1}:p_{i}=0\ \forall i\notin S\}$. By Lemma 3.6, $\Delta^{V-1}_{S}$ is forward-invariant for (4.1), and Theorem 4.1 holds verbatim with $\pi$ computed on $S$.

Remark 4.6* (Logit biases and affine invariances).*

Adding a constant $c$ to all scores leaves $X$ unchanged; scaling scores $s\mapsto\alpha s$ is equivalent to $T\mapsto T/\alpha$ (Remark 2.6). Thus, logit biases that uniformly shift preferences do not change the manifold trajectory, and temperature mediates the speed along that trajectory.

The next section (§5) discusses extensions where scores $s$ may weakly depend on $p$ (e.g., simple decoding heuristics) and records conditions ensuring forward invariance and ascent of a generalized free energy.

5 Phenomenology: Temperature, Truncation, and Path-Dependence

We keep the model in §2–4 and discuss three practical themes: (i) temperature as a time rescaling of the trajectory; (ii) truncation (top-$k$/nucleus) as an invariant-face restriction; (iii) path-dependent preferences as a mechanism for “hallucination”-like behavior. Throughout, we preserve the ML-facing language of scores and probabilities and avoid any geometric formalism beyond the simplex.

5.1 Temperature as selective pressure and effective time

With fixed scores $s$, the replicator field is $X_{i}(p)=\tfrac{1}{T}\,p_{i}(s_{i}-\bar{s})$ (§3). Replacing $T$ by a positive, time-varying schedule $T(t)$ gives

[TABLE]

Let $\tau(t)=\int_{0}^{t}\frac{1}{T(u)}\,du$. Then $p(t)=\tilde{p}(\tau(t))$, where $\tilde{p}$ solves the constant-temperature ODE $\frac{d}{d\tau}\tilde{p}_{i}=\tilde{p}_{i}(s_{i}-\bar{s})$. Thus temperature schedules reparameterize time along the same manifold trajectory (Corollary 4.4). Lower $T$ accelerates motion (higher selective pressure); higher $T$ slows motion and yields more uniform equilibria (Proposition 2.5).

Remark 5.1* (Practical reading).*

Annealing ($T\downarrow$) sharpens the distribution faster along the same path; cooling too abruptly can cause early lock-in on spurious modes, while warming ($T\uparrow$) slows convergence and increases diversity.

5.2 Truncation as dynamics on a face of the simplex

Let $S\subseteq\{1,\dots,V\}$ be a fixed support set (top-$k$ or nucleus). Define the face $\Delta^{V-1}_{S}=\{p\in\Delta^{V-1}:p_{i}=0\ \forall i\notin S\}$. By Lemma 3.6, $\Delta^{V-1}_{S}$ is forward-invariant for the replicator ODE, and the mirror step (3.1) respects $\Delta^{V-1}_{S}$ by construction. All results from §3–4 hold verbatim on $\Delta^{V-1}_{S}$, with softmax and free energy computed using indices in $S$ only. Truncation therefore acts as an ecological bottleneck: the same ascent principle operates on a lower-dimensional manifold.

5.3 Path-dependent preferences and “hallucination”-like behavior

Sections 3–4 assumed $s\in\mathbb{R}^{V}$ fixed (context frozen). In practice, the scores used during decoding can acquire path dependence, e.g., via token-dependent biases, heuristics modifying logits after each step, or implicit feedback from the partial output. To discuss this briefly, consider a mild dependence $s=s(p)$ that is Lipschitz on $\mathrm{int}\,\Delta^{V-1}$. Then the vector field

[TABLE]

is well-defined and tangent to the simplex. When $s$ derives from a potential, the flow remains a gradient ascent of a generalized free energy; but in general $s(p)$ can include non-potential (curl) components, producing nonconservative dynamics on $\Delta^{V-1}$.

Nonconservative preference fields can yield loops and lock-ins: along a closed path in $p$, the net change in alignment need not cancel, permitting cycles or self-reinforcing excursions away from the original context. Phenomenologically, such path-dependent drifts can present as “hallucination”-like behavior: local selection is strong (the replicator field is large in norm), yet global coherence is weak because successive local preferences are not globally integrable. We deliberately avoid stronger claims: formal analysis of path-dependent $s(p)$ lies beyond our scope here, see Figure 1 for a schematic of this perspective.

Our results justify the following practical guidance without additional assumptions. (i) Temperature directly modulates the speed along a fixed manifold path (§5.1); (ii) truncation restricts dynamics to a face with identical guarantees (§5.2); (iii) when auxiliary heuristics introduce strong path dependence, the system can develop cycles or brittle attractors. Mitigations include moderating selective pressure (raising $T$), delaying truncation, or dampening path-dependent logit shifts.

6 Discussion and Outlook

What is new.

The mathematical ingredients we use—Gibbs’ variational principle for softmax, entropic mirror ascent as multiplicative weights, the replicator flow and its Shahshahani geometry—are classical (§2–4). Our contribution is to show that next-token decoding in LLMs admits a rigorous dynamical description on the probability simplex, thereby turning the common “manifold traversal” intuition into a theorem at the output-distribution level (Thm. 4.1), and to extract exact, practice-facing corollaries: temperature as a time rescaling (Cor. 4.4) and truncation as restriction to a face with identical guarantees (Cor. 4.5).

Limitations and scope.

Our results are restricted to a fixed decoding context and address only the output distribution over next tokens. We deliberately avoid claims about (i) training dynamics, (ii) internal representation geometry, or (iii) production decoders literally integrating the ODE; the ODE is a continuous-time limit of the standard multiplicative update. Section 5 sketches safe phenomenology when scores acquire mild path dependence; a full analysis of nonconservative preference fields is beyond our scope here.

Hidden states, constraints, and future work.

While the present work treats only the output distribution, a parallel variational/dynamical viewpoint can, in principle, be extended to internal states. One may regard hidden activations for a fixed prompt as coordinates on a high-dimensional manifold shaped jointly by (a) the user input and (b) the learned parameters, which impose complementary constraints. The observable token distribution then arises as a projection from this internal state manifold to the probability simplex.

There exists a geometric formalism in which entropy-regularized, normalization-preserving flows arise as contact-type dynamics; in that language, the entropic mirror step (3.1) discretizes a contact descent and the replicator field (4.1) is its continuous-time limit (see 6, 8 for background). A folded-symplectic bulk inducing a contact structure on a fold and projecting to screen-level, normalized dynamics provides a compact global mechanism that recovers softmax/replicator; see the author’s preprint [11] for a complete treatment of this bulk–fold–screen construction. In that framework, the contact Reeb direction implements an effective time reparameterization, consistent with our temperature-as-time result in §5.1. We keep these details out of the main text; a full development is deferred to future work. The goal of such an extension would be to relate internal state flows to the output-level replicator dynamics established here, providing a principled bridge between representation dynamics and decoding behavior.