Human-Centered Learning Mechanics: A Dynamical Framework for Entropy-Regulated Representation Learning

# Human-Centered Learning Mechanics: A Dynamical Framework for Entropy-Regulated Representation Learning
Source: [https://arxiv.org/html/2605.22940](https://arxiv.org/html/2605.22940)
22footnotetext:Corresponding author: Kim Phuc Tran, Email: kim\-phuc\.tran@ensait\.frKim Phuc TranUniv\. Lille, ENSAIT, ULR 2461 – GEMTEX – Génie et Matériaux Textiles, F\-59000 Lille, FranceInternational Chair in DS & XAI, International Research Institute for Artificial Intelligence and Data Science, Dong A University, Danang, Vietnam

###### Abstract

While deep learning is increasingly understood as a dynamical process in parameter space, many existing perspectives primarily model training as a closed optimization system\. This closed\-system view is insufficient for real\-world artificial intelligence, where models operate under uncertainty, resource constraints, distribution shift, downstream decision risk, and continuous human feedback\. To address this gap, we propose*Human\-Centered Learning Mechanics*\(HCLM\), a dynamical and information\-theoretic framework for studying open and controlled learning systems\. The central premise of HCLM is that entropy regularization is not beneficial by mere inclusion in the objective\. It becomes meaningful only when the chosen entropy surrogate induces a non\-degenerate*information force*along the optimization trajectory\. Naively defined entropy penalties may produce negligible, unstable, or poorly aligned gradients, in which case entropy\-regularized dynamics effectively collapse to ordinary loss minimization\. We therefore introduce the notion of*effective entropy*and study tractable geometric entropy surrogates, including variance\-based and log\-determinant covariance proxies\. The paper makes three concrete contributions\. First, we formalize entropy regularization through effective information force and characterize degenerate entropy regimes\. Second, we derive convergence, entropy\-flow, Wasserstein\-gradient\-flow, and noisy\-representation generalization results under explicit assumptions\. Third, we provide a conditional dynamical interpretation of scaling\-law\-like behavior through the balance between information injection, entropy dissipation, and residual risk\. This interpretation is not intended as an unconditional derivation of empirical neural scaling laws\. Empirical analyses on controlled representation\-learning tasks support the hypothesis that geometric entropy surrogates, especially log\-determinant covariance entropy, induce stronger and more stable information forces than softmax\-normalized entropy\.

Keywords:Human\-Centered Learning Mechanics, entropy\-regulated learning, representation learning, information force, noisy representation compression, Wasserstein gradient flow, controlled learning dynamics

## 1Introduction

Deep learning has significantly advanced science, engineering, and industry\. Despite this progress, the theoretical understanding of how these systems learn remains incomplete\. In practice, neural networks are predominantly trained using empirical heuristics, large\-scale hyperparameter tuning, and extensive trial\-and\-error engineering\. A growing body of theoretical work argues that deep learning is a physical phenomenon that admits a rigorous scientific theory, moving beyond empirical heuristics\. This emerging paradigm is increasingly referred to as the*mechanics of learning*\[Meiet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib10), Chizat and Bach,[2018](https://arxiv.org/html/2605.22940#bib.bib9)\]\. Within this mechanics viewpoint, the training of a neural network is modeled as a continuous dynamical system in a high\-dimensional parameter space\. A model evolves along a trajectory driven by gradient vectors, much like a physical particle moving through an energy landscape under the influence of conservative forces\. The network’s architecture, the statistical structure of the data distribution, the task objective, and the specific learning rule collectively determine the effective forces acting on the model parameters\. This closed\-system perspective has successfully explained several universal phenomena, such as the edge\-of\-stability in gradient descentCohenet al\.\[[2021](https://arxiv.org/html/2605.22940#bib.bib3)\]and the infinite\-width mean\-field limits\[Jacotet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib2)\]\. This closed\-system mechanics, while useful, is incomplete when applied to modern real\-world Artificial Intelligence \(AI\) systems\. Real\-world AI does not operate in an isolated vacuum\. These systems are deployed in open environments characterized by deep uncertainty, partial observability, and constant distributional shifts\. They are subject to strict latency constraints, privacy restrictions, energy budgets, safety guardrails, and continuous human decision processes\. In safety\-critical, industrial, and human\-centered settings—such as autonomous driving, medical diagnostics, or Large Language Models \(LLMs\) aligned via human feedback—learning evolves from minimizing a static prediction error to a controlled information process embedded within a broader decision loop\.

To bridge this gap, this paper develops*Human\-Centered Learning Mechanics*\(HCLM\), a unified theoretical framework wherein learning is explicitly treated as entropy\-constrained dynamics under practical constraints and active human control\. By reformulating deep learning as an open, entropy\-regulated thermodynamic system, we establish the central principle of HCLM:*learning is controlled entropy shaping under uncertainty and constraints*\. A critical departure of HCLM from classical information\-theoretic learning\[Tishbyet al\.,[2000](https://arxiv.org/html/2605.22940#bib.bib1)\]is the operationalization of entropy\. In traditional literature, exact representation entropy is often treated as an abstract, intractable quantity used primarily for post\-hoc bounding\. In contrast, HCLM introduces tractable entropy surrogates that actively participate in the optimization process\. The goal of this work is not to replace stochastic gradient descent \(SGD\) or adaptive optimizers like Adam, but to provide a cohesive framework that explains, quantifies, and dynamically regulates the flow of information during learning\. We find that simply appending an entropy regularization term to the objective function is insufficient\. Standard entropy regularizers are often dynamically inactive, leading to degenerate optimization\. To resolve this, we introduce the concept of*effective entropy*, demonstrating that an entropy surrogate is dynamically useful only if it induces a non\-degenerate, measurable*information force*that actively sculpts the geometry of the representation space\. Additionally, HCLM provides a dynamical interpretation of human or reward\-based feedback—including signals used in Reinforcement Learning from Human Feedback \(RLHF\)\[Ouyanget al\.,[2022](https://arxiv.org/html/2605.22940#bib.bib14)\]—as a thermodynamic control mechanism that can regulate the rate of entropy dissipation\. By synthesizing task\-driven information injection with controlled entropy dissipation, this framework provides a mechanistic and mathematically explicit perspective on generalization, representation compression, and human\-aligned adaptation\. Rather than claiming that scaling laws follow unconditionally from entropy dynamics, we show that power\-law behavior can be recovered under explicit balance assumptions linking information injection, dissipation, and excess risk\. In this sense, HCLM should be understood as a controlled\-dynamics framework that clarifies when entropy regulation can support stable learning, rather than as a replacement for existing optimization or statistical learning theory\. This paper should be read as a mechanistic study rather than as a claim of universal superiority over existing learning algorithms\. Our goal is to isolate a basic dynamical phenomenon: entropy affects learning only when it generates an effective force\. Controlled synthetic experiments are therefore used deliberately, because they make information\-forced degeneracy, collapse, and stabilization observable without the confounding effects of large\-scale architectures, data augmentation, or optimizer engineering\. In particular, HCLM does not claim that entropy regularization alone explains generalization, alignment, or neural scaling laws\. Rather, it isolates a specific dynamical mechanism: learning is affected by entropy only when entropy induces a non\-degenerate force along the optimization trajectory\.

The paper is organized as follows\. Section 2 reviews related literature\. Section 3 introduces the HCLM conceptual architecture\. Section 4 defines the practical energy formulation and stochastic representation surrogates\. Section 5 defines effective entropy and the information force\. Section 6 establishes the theoretical foundations of HCLM, including convergence guarantees, entropy\-flow identities, representation\-compression generalization, and Wasserstein gradient\-flow formulations\. Section 7 provides a conditional mechanistic interpretation of neural scaling laws\. Section 8 presents the empirical analysis\. Detailed mathematical proofs are deferred to the Appendix\.

## 2Related Work

The HCLM framework lies at the intersection of learning mechanics, information\-theoretic learning, PAC\-Bayes generalization, mean\-field dynamics, sharpness\-based analysis, and human\-in\-the\-loop learning\. The mechanics view studies neural network training through dynamical systems, limiting regimes, and statistical\-physics\-inspired principles\. Neural Tangent Kernel theory explains the training dynamics of infinitely wide networks by reducing nonlinear optimization to an approximately linear evolution in function space\[Jacotet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib2)\], while studies of optimization geometry reveal phenomena such as the edge of stability, where gradient\-based learning operates near the boundary of stable curvature\[Cohenet al\.,[2021](https://arxiv.org/html/2605.22940#bib.bib3)\]\. These approaches provide powerful tools for understanding training as a dynamical process, but they primarily describe closed learning systems governed by fixed losses, fixed data distributions, and fixed optimization rules\. HCLM builds on this trajectory\-based perspective while extending it to open and controlled systems by introducing entropy dissipation, representation geometry, decision constraints, and human or reward\-based feedback as explicit components of the learning dynamics\.

A closely related line of work studies generalization through robustness to perturbations, sharpness, and flatness of the loss landscape\. PAC\-Bayes analyses have been used to compute nonvacuous generalization bounds for deep stochastic neural networks and to derive spectrally normalized margin bounds for neural networks\[Dziugaite and Roy,[2017](https://arxiv.org/html/2605.22940#bib.bib8), Neyshaburet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib20)\]\. Sharpness\-Aware Minimization explicitly optimizes for solutions that remain robust within local neighborhoods of parameter space\[Foretet al\.,[2021](https://arxiv.org/html/2605.22940#bib.bib21)\]\. HCLM is complementary to these approaches: whereas sharpness and PAC\-Bayes flatness mainly characterize stability in parameter space, HCLM focuses on entropy\-induced information forces acting on representation geometry\. In this sense, HCLM can be viewed as a representation\-geometric counterpart to sharpness\-based generalization analysis, with the important distinction that it studies whether a chosen entropy surrogate actively shapes the learning trajectory rather than merely measuring complexity after training\.

Information\-theoretic learning has classically been studied through the Information Bottleneck principle, which formulates learning as a trade\-off between compressing the input and preserving information relevant to the prediction target\[Tishbyet al\.,[2000](https://arxiv.org/html/2605.22940#bib.bib1)\]\. Related approaches include variational information bottlenecks\[Alemiet al\.,[2016](https://arxiv.org/html/2605.22940#bib.bib5)\]and empirical analyses of compression in deep networks\[Shwartz\-Ziv and Tishby,[2017](https://arxiv.org/html/2605.22940#bib.bib4)\]\. However, exact mutual information is difficult to estimate reliably in high\-dimensional deterministic neural networks, and an information quantity may be mathematically meaningful while remaining dynamically inactive if it does not generate a useful gradient during training\. HCLM therefore departs from standard information\-bottleneck formulations by embedding tractable geometric entropy surrogates directly into the learning dynamics\. Instead of treating information only as a static complexity measure, HCLM analyzes its temporal injection, dissipation, and induced information force along the optimization trajectory\.

PAC\-Bayes theory relates generalization to the divergence between a learned posterior distribution over predictors and a prior distribution\[McAllester,[1999](https://arxiv.org/html/2605.22940#bib.bib6), Catoni,[2007](https://arxiv.org/html/2605.22940#bib.bib7)\]\. Recent work has shown that PAC\-Bayes bounds can become nonvacuous for deep networks when the posterior and prior are carefully constructed\[Dziugaite and Roy,[2017](https://arxiv.org/html/2605.22940#bib.bib8)\]\. A complementary line of work derives generalization bounds using mutual information between the training data and the learned hypothesis\[Russo and Zou,[2016](https://arxiv.org/html/2605.22940#bib.bib17), Xu and Raginsky,[2017](https://arxiv.org/html/2605.22940#bib.bib16), Buet al\.,[2020](https://arxiv.org/html/2605.22940#bib.bib18)\]\. HCLM is aligned with this second perspective but avoids assuming a direct equivalence between parameter\-space PAC\-Bayes complexity and representation entropy\. Instead, it uses noisy representation compression as a more explicit route from geometric entropy control to generalization\. This distinction is central: HCLM does not claim that lower entropy automatically improves generalization\. Rather, it asks whether the entropy surrogate induces a measurable information force and whether this force regulates representation geometry in a way that supports stable prediction\.

Mean\-field theories model wide neural networks as probability measures evolving over parameter space\[Meiet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib10)\]\. This connects neural network training to optimal transport and Wasserstein gradient flows, where learning can be viewed as the evolution of a probability distribution minimizing a free\-energy functional\[Chizat and Bach,[2018](https://arxiv.org/html/2605.22940#bib.bib9)\]\. The variational formulation of Fokker–Planck equations as Wasserstein gradient flows was established by Jordan, Kinderlehrer, and Otto\[Jordanet al\.,[1998](https://arxiv.org/html/2605.22940#bib.bib19)\]\. Stochastic optimization methods such as Stochastic Gradient Langevin Dynamics further connect learning dynamics with Fokker–Planck equations and non\-equilibrium thermodynamics\[Raginskyet al\.,[2017](https://arxiv.org/html/2605.22940#bib.bib11)\]\. HCLM builds on this line of work by introducing entropy\-controlled and human\-controlled terms into the free\-energy formulation, allowing learning to be interpreted as a dissipative stochastic process in which prediction loss, entropy production, representation compression, and external control jointly shape the evolution of the system\.

Finally, HCLM is related to decision\-aware and human\-in\-the\-loop learning\. Standard empirical risk minimization assumes that prediction accuracy is the primary goal, whereas many real\-world systems use predictions only as intermediate quantities for downstream decisions\. Decision\-focused learning addresses this issue by integrating decision objectives, such as cost, safety, or operational constraints, directly into training\[Dontiet al\.,[2017](https://arxiv.org/html/2605.22940#bib.bib12)\]\. Human feedback has also become central to the alignment of large\-scale models\. Reinforcement Learning from Human Feedback trains reward models from human preferences and uses them to guide policy optimization\[Christianoet al\.,[2017](https://arxiv.org/html/2605.22940#bib.bib13), Ouyanget al\.,[2022](https://arxiv.org/html/2605.22940#bib.bib14)\]\. While RLHF is practically successful, its interpretation as a dynamical process remains less developed\. HCLM provides a complementary view: human or reward feedback can be interpreted as a control signal that regulates the information\-force equilibrium of learning dynamics\. Importantly, the thermostat mechanism proposed in HCLM should not be understood as a replacement for RLHF\. Classical RLHF remains necessary for learning preference or reward models\. HCLM instead clarifies how such reward signals may regulate the rate of entropy dissipation through an adaptive coefficientβt\\beta\_\{t\}\. In this view, human feedback does not need to directly perturb high\-dimensional model parameters; it can act indirectly by modulating the thermodynamic balance between information injection and entropy dissipation\. Overall, HCLM differs from prior information\-theoretic, PAC\-Bayes, sharpness\-based, and human\-feedback approaches by treating entropy not as an automatically useful regularizer, but as a dynamical object whose usefulness depends on the force it induces during optimization\. This makes HCLM a diagnostic and mechanistic framework for controlled learning dynamics, rather than a universal complexity bound or a replacement for existing optimization and alignment methods\.

## 3The Human\-Centered Learning Mechanics \(HCLM\) Framework

We begin by outlining the conceptual architecture of Human\-Centered Learning Mechanics \(HCLM\)\. Classical deep learning relies predominantly on empirical risk minimization \(ERM\), where the objective is to find a set of parameters that minimizes a scalar prediction error over a training dataset\. This view is consistent with standard optimization\-based interpretations of neural\-network training, including kernel, mean\-field, and gradient\-flow perspectives\[Jacotet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib2), Meiet al\.,[2018](https://arxiv.org/html/2605.22940#bib.bib10), Chizat and Bach,[2018](https://arxiv.org/html/2605.22940#bib.bib9)\]\. In contrast, HCLM reformulates learning as the search for a dynamic equilibrium within an open, entropy\-regulated learning system\. This perspective is related to information\-theoretic learning and stochastic dynamical views of optimization, but differs by treating entropy as a controlled trajectory\-level force rather than only as a static complexity or compression measure\[Tishbyet al\.,[2000](https://arxiv.org/html/2605.22940#bib.bib1), Raginskyet al\.,[2017](https://arxiv.org/html/2605.22940#bib.bib11), Jordanet al\.,[1998](https://arxiv.org/html/2605.22940#bib.bib19)\]\.

In our framework, the continuous learning process is governed by four interacting conceptual forces:

1. 1\.Task\-driven information injection\.The standard predictive loss acts as an information\-injection mechanism\. It forces the model to absorb task\-relevant structure from the training data and drives the formation of hidden representations\. Without additional regulation, this injection may also promote excessive representation expansion, memorization, and sensitivity to spurious sample\-specific patterns\.
2. 2\.Entropy\-induced dissipation\.HCLM introduces an explicit entropy\-based dissipative mechanism to regulate the geometry of the learned representation\. The objective is not to add entropy for its own sake, but to identify entropy surrogates whose gradients generate a non\-degenerate information force along the optimization trajectory\. In this sense, entropy is useful only when it actively shapes representation dynamics\.
3. 3\.Structural constraints\.Priors regarding geometry, sparsity, modularity, invariance, smoothness, or interpretability act as structural forces\. These constraints shape the admissible learning trajectory and connect HCLM to explainable and constrained learning systems, where prediction accuracy alone is not sufficient for reliable deployment\.
4. 4\.Human\-in\-the\-loop control\.Human feedback—whether through explicit preference labels, safety guardrails, expert corrections, or reinforcement\-learning rewards—is modeled as an external control input\. This view is consistent with recent human\-feedback paradigms in reinforcement learning and language\-model alignment\[Christianoet al\.,[2017](https://arxiv.org/html/2605.22940#bib.bib13), Ouyanget al\.,[2022](https://arxiv.org/html/2605.22940#bib.bib14)\]\. In HCLM, such feedback does not need to perturb high\-dimensional model parameters directly; it may instead regulate the balance between information injection and entropy dissipation\.

By viewing learning through this mechanistic lens, HCLM bridges the gap between microscopic parameter updates and the macroscopic behavior of representation geometry\. The framework does not replace empirical risk minimization, PAC\-Bayes theory, reinforcement learning, or RLHF\[McAllester,[1999](https://arxiv.org/html/2605.22940#bib.bib6), Catoni,[2007](https://arxiv.org/html/2605.22940#bib.bib7), Christianoet al\.,[2017](https://arxiv.org/html/2605.22940#bib.bib13), Ouyanget al\.,[2022](https://arxiv.org/html/2605.22940#bib.bib14)\]\. Rather, it provides a dynamical layer describing how prediction, entropy dissipation, structural constraints, and feedback interact during optimization\.

RepresentationManifold \(ZθZ\_\{\\theta\}\)Dynamic EquilibriumTask & DataEnvironmentInformationSinkHuman Feedback\(Alignment\)Structural ConstraintsΩ​\(θ\)\\Omega\(\\theta\)\(Priors, Architecture, Sparsity\)Information Injectionℒpred\\mathcal\{L\}\_\{\\mathrm\{pred\}\}Entropy Dissipationβ​ℋ~​\(Zθ\)\\beta\\widetilde\{\\mathcal\{H\}\}\(Z\_\{\\theta\}\)Thermodynamic ControlSteering & Calibrationℛhuman\\mathcal\{R\}\_\{\\mathrm\{human\}\}Figure 1:The Human\-Centered Learning Mechanics \(HCLM\) conceptual framework\. Learning is modeled as an open controlled system governed by task\-driven information injection, entropy\-induced dissipation, structural constraints, and human\-in\-the\-loop feedback\.The central premise of HCLM is that learning is not merely the minimization of a fixed prediction loss, but a controlled process of entropy shaping under uncertainty and constraints\. As illustrated in Figure[1](https://arxiv.org/html/2605.22940#S3.F1), the predictive objective injects information into the representation space, while entropy dissipation regulates the expansion or compression of this space\. Structural constraints guide the trajectory toward admissible and interpretable solutions, and human or reward\-based feedback provides an external control signal that can calibrate the strength of entropy regulation\.

This formulation is intentionally mechanism\-oriented\. Its objective is not to derive universal laws of deep learning, nor to claim that entropy regularization automatically improves generalization or alignment\. Instead, HCLM identifies observable dynamical regimes: entropy\-force degeneracy, unstable representation expansion, controlled information\-force collapse, and feedback\-regulated stabilization\. These regimes motivate the practical energy formulation and entropy surrogates introduced in the next section\.

## 4Practical HCLM Energy and Stochastic Representation Surrogates

Letf​\(x;θ\)f\(x;\\theta\)be a neural network parameterized byθ∈ℝd\\theta\\in\\mathbb\{R\}^\{d\}, and let

Zθ=hθ​\(X\)Z\_\{\\theta\}=h\_\{\\theta\}\(X\)denote a hidden representation induced by an input random variableX∼PXX\\sim P\_\{X\}\. Since deterministic high\-dimensional representations may have singular or ill\-defined differential entropy, we introduce a noisy representation

Z~θ=hθ​\(X\)\+ξ,ξ∼𝒩​\(0,σξ2​I\),\\widetilde\{Z\}\_\{\\theta\}=h\_\{\\theta\}\(X\)\+\\xi,\\qquad\\xi\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\xi\}^\{2\}I\),\(1\)whereξ\\xiis independent ofXX\. This stochastic representation makes entropy and mutual\-information quantities well\-defined and prevents degenerate differential\-entropy pathologies\.

The practical HCLM energy is defined as

ℱ​\(θ\)=ℒpred​\(θ\)\+β​ℋ~​\(Z~θ\)\+γ​Ω​\(θ\)\+λ​ℛdec​\(θ\),\\mathcal\{F\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\)\+\\beta\\widetilde\{\\mathcal\{H\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)\+\\gamma\\Omega\(\\theta\)\+\\lambda\\mathcal\{R\}\_\{\\mathrm\{dec\}\}\(\\theta\),\(2\)whereℒpred\\mathcal\{L\}\_\{\\mathrm\{pred\}\}is the predictive loss,ℋ~​\(Z~θ\)\\widetilde\{\\mathcal\{H\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)is a tractable entropy surrogate,Ω​\(θ\)\\Omega\(\\theta\)encodes structural constraints, andℛdec\\mathcal\{R\}\_\{\\mathrm\{dec\}\}captures decision\-aware risk\.

A central design requirement is that the entropy surrogate should reflect the geometry of the representation rather than merely the marginal dispersion of normalized activations\. For a mini\-batch representation matrixZ∈ℝB×pZ\\in\\mathbb\{R\}^\{B\\times p\}, with empirical covariance

Σ^Z=1B−1​\(Z−Z¯\)⊤​\(Z−Z¯\),\\widehat\{\\Sigma\}\_\{Z\}=\\frac\{1\}\{B\-1\}\(Z\-\\bar\{Z\}\)^\{\\top\}\(Z\-\\bar\{Z\}\),we consider the regularized log\-determinant surrogate

ℋ~logdet​\(Z\)=12​log​det\(Σ^Z\+ϵ​I\),\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(Z\)=\\frac\{1\}\{2\}\\log\\det\\left\(\\widehat\{\\Sigma\}\_\{Z\}\+\\epsilon I\\right\),\(3\)whereϵ\>0\\epsilon\>0ensures numerical stability\. This quantity is motivated by the Gaussian maximum\-entropy identity

H​\(U\)≤12​log​det\(2​π​e​ΣU\),H\(U\)\\leq\\frac\{1\}\{2\}\\log\\det\(2\\pi e\\,\\Sigma\_\{U\}\),with equality whenUUis Gaussian\. Hence, the log\-determinant surrogate should be interpreted not as exact entropy, but as a geometry\-sensitive upper proxy for representation volume\. The log\-determinant surrogate is particularly useful because it captures multivariate representation volume while remaining differentiable and computationally tractable for moderate representation dimensions\. Unlike marginal variance penalties, it is sensitive to correlation structure; unlike softmax\-normalized activation entropy, it does not erase geometric scale through normalization\. For this reason, it is a natural candidate for studying whether entropy can generate a non\-degenerate information force in representation space\.

We also consider a simpler variance surrogate

ℋ~var​\(Z\)=1p​tr​\(Σ^Z\),\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{var\}\}\(Z\)=\\frac\{1\}\{p\}\\mathrm\{tr\}\(\\widehat\{\\Sigma\}\_\{Z\}\),\(4\)which controls marginal dispersion but does not capture correlation structure\. By contrast, softmax entropy applied to activations may be dynamically weak because normalization can erase geometric scale and induce gradients that are poorly aligned with representation\-volume control\.

To make the HCLM framework operational, we now introduce a generic entropy\-regulated learning procedure\. The objective of this algorithm is not to define a new optimizer replacing SGD or Adam, but to formalize how entropy dissipation, information force, and adaptive feedback can be integrated into the training dynamics\. In particular, the adaptive thermostat coefficient allows external feedback or reward signals to regulate the balance between information injection and entropy dissipation during optimization\.

Algorithm 1Entropy\-Regulated Human\-Centered Learning Dynamics \(ER\-HCLM\)1:Initialize parameters

θ0\\theta\_\{0\}
2:Initialize thermostat coefficient

β0\>0\\beta\_\{0\}\>0
3:Choose entropy surrogate

ℋ~\\widetilde\{\\mathcal\{H\}\}
4:Choose learning rate

η\\eta
5:for

t=0,…,T−1t=0,\\dots,T\-1do

6:Sample mini\-batch

ℬt\\mathcal\{B\}\_\{t\}
7:Compute hidden representations

Zt=hθt​\(Xt\)Z\_\{t\}=h\_\{\\theta\_\{t\}\}\(X\_\{t\}\)
8:Construct noisy representations

Z~t=Zt\+ξt,ξt∼𝒩​\(0,σξ2​I\)\\widetilde\{Z\}\_\{t\}=Z\_\{t\}\+\\xi\_\{t\},\\qquad\\xi\_\{t\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\xi\}^\{2\}I\)
9:Compute predictive loss

ℒpred​\(θt\)\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)
10:Compute entropy surrogate

Ht=ℋ~​\(Z~t\)H\_\{t\}=\\widetilde\{\\mathcal\{H\}\}\(\\widetilde\{Z\}\_\{t\}\)
11:Compute information force

Gt=‖∇θHt‖G\_\{t\}=\\left\\\|\\nabla\_\{\\theta\}H\_\{t\}\\right\\\|
12:Compute structural and decision\-aware penalties

Ω​\(θt\),ℛdec​\(θt\)\\Omega\(\\theta\_\{t\}\),\\qquad\\mathcal\{R\}\_\{\\mathrm\{dec\}\}\(\\theta\_\{t\}\)
13:Form HCLM energy

ℱt=ℒpred\+βt​Ht\+γ​Ω​\(θt\)\+λ​ℛdec​\(θt\)\\mathcal\{F\}\_\{t\}=\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\+\\beta\_\{t\}H\_\{t\}\+\\gamma\\Omega\(\\theta\_\{t\}\)\+\\lambda\\mathcal\{R\}\_\{\\mathrm\{dec\}\}\(\\theta\_\{t\}\)
14:Update parameters

θt\+1=θt−η​∇θℱt\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\nabla\_\{\\theta\}\\mathcal\{F\}\_\{t\}
15:Observe reward or feedback signal

rtr\_\{t\}
16:Update thermostat coefficient

βt\+1=Π\[βmin,βmax\]​\(βt\+αr​\(rt−r⋆\)−αg​\(Gt−G⋆\)\)\\beta\_\{t\+1\}=\\Pi\_\{\[\\beta\_\{\\min\},\\beta\_\{\\max\}\]\}\\left\(\\beta\_\{t\}\+\\alpha\_\{r\}\(r\_\{t\}\-r^\{\\star\}\)\-\\alpha\_\{g\}\(G\_\{t\}\-G^\{\\star\}\)\\right\)
17:endfor

18:Return trained parameters

θT\\theta\_\{T\}

Algorithm[1](https://arxiv.org/html/2605.22940#alg1)summarizes the operational interpretation of HCLM\. The predictive objective injects task\-relevant information into the representation space, while the entropy surrogate generates a dissipative information force regulating representation expansion\. The adaptive thermostat coefficientβt\\beta\_\{t\}dynamically balances these competing effects using external reward or feedback signals\. Importantly, this thermostat mechanism should not be interpreted as a replacement for RLHF or reinforcement learning itself\. Rather, it provides a dynamical interpretation of how reward\-based feedback may regulate entropy dissipation during learning\.

## 5Effective Entropy and Information Force

The utility of Eq\. \([2](https://arxiv.org/html/2605.22940#S4.E2)\) depends on whether the entropy surrogate produces a meaningful force along the training trajectory\. LetH​\(θ\)=ℋ~​\(Z~θ\)H\(\\theta\)=\\widetilde\{\\mathcal\{H\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)\. The Euclidean information force is

𝒢H​\(θ\)=∇θH​\(θ\)\.\\mathcal\{G\}\_\{H\}\(\\theta\)=\\nabla\_\{\\theta\}H\(\\theta\)\.However, this quantity is parameterization\-dependent\. For this reason, whenever a local metricM​\(θ\)≻0M\(\\theta\)\\succ 0is available, for example a damped Fisher or Gauss–Newton metric, we define the metric\-adjusted information\-force magnitude as

‖𝒢H​\(θ\)‖M−1=∇θH​\(θ\)⊤​M​\(θ\)−1​∇θH​\(θ\)\.\\\|\\mathcal\{G\}\_\{H\}\(\\theta\)\\\|\_\{M^\{\-1\}\}=\\sqrt\{\\nabla\_\{\\theta\}H\(\\theta\)^\{\\top\}M\(\\theta\)^\{\-1\}\\nabla\_\{\\theta\}H\(\\theta\)\}\.\(5\)In empirical sections, we use the Euclidean norm for computational tractability, while Eq\. \([5](https://arxiv.org/html/2605.22940#S5.E5)\) gives the geometrically preferred form\.

###### Definition 1\(Effective entropy surrogate\)\.

Let\{θt\}t=0T\\\{\\theta\_\{t\}\\\}\_\{t=0\}^\{T\}be a training trajectory\. An entropy surrogateHHis called*dynamically effective*on this trajectory if there exist constantsc\>0c\>0and0<τ≤10<\\tau\\leq 1such that

1T\+1​∑t=0T𝟏​\{‖𝒢H​\(θt\)‖M−1≥c\}≥τ\.\\frac\{1\}\{T\+1\}\\sum\_\{t=0\}^\{T\}\\mathbf\{1\}\\left\\\{\\\|\\mathcal\{G\}\_\{H\}\(\\theta\_\{t\}\)\\\|\_\{M^\{\-1\}\}\\geq c\\right\\\}\\geq\\tau\.Equivalently, the surrogate is effective if it produces a non\-negligible information force over a nontrivial portion of the optimization path\.

This trajectory\-based definition is intentionally operational\. It avoids treating entropy as a purely post\-hoc complexity measure and instead requires the entropy surrogate to participate actively in the dynamics\.

###### Proposition 1\(Degenerate collapse\)\.

Consider the deterministic HCLM update

θt\+1=θt−η​\(∇ℒpred​\(θt\)\+β​∇H​\(θt\)\+γ​∇Ω​\(θt\)\+λ​∇ℛdec​\(θt\)\)\.\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\left\(\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\+\\beta\\nabla H\(\\theta\_\{t\}\)\+\\gamma\\nabla\\Omega\(\\theta\_\{t\}\)\+\\lambda\\nabla\\mathcal\{R\}\_\{\\mathrm\{dec\}\}\(\\theta\_\{t\}\)\\right\)\.If‖∇H​\(θt\)‖=o​\(‖∇ℒpred​\(θt\)‖\)\\\|\\nabla H\(\\theta\_\{t\}\)\\\|=o\(\\\|\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\\\|\)along the trajectory andβ<∞\\beta<\\infty, then the entropy contribution is asymptotically negligible\. In the simplified caseγ=λ=0\\gamma=\\lambda=0, the update reduces to standard gradient descent on the predictive loss up to a vanishing perturbation\.

Proof provided in Appendix[A\.1](https://arxiv.org/html/2605.22940#A1.SS1)\.A high entropy value is not sufficient for effectiveness\. Effectiveness is a property of the trajectory\-level gradient field induced by the surrogate, not of the scalar entropy magnitude alone\.

## 6Theoretical Foundations of HCLM

### 6\.1Convergence of Entropy\-Constrained Dynamics

###### Assumption 1\(Trajectory\-local smoothness\)\.

There existsL\>0L\>0such that, for all iteratesθt\\theta\_\{t\}andθt\+1\\theta\_\{t\+1\},

ℱ​\(θt\+1\)≤ℱ​\(θt\)\+∇ℱ​\(θt\)⊤​\(θt\+1−θt\)\+L2​‖θt\+1−θt‖2\.\\mathcal\{F\}\(\\theta\_\{t\+1\}\)\\leq\\mathcal\{F\}\(\\theta\_\{t\}\)\+\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)^\{\\top\}\(\\theta\_\{t\+1\}\-\\theta\_\{t\}\)\+\\frac\{L\}\{2\}\\\|\\theta\_\{t\+1\}\-\\theta\_\{t\}\\\|^\{2\}\.

###### Assumption 2\(Lower bounded energy\)\.

There existsℱ⋆\>−∞\\mathcal\{F\}^\{\\star\}\>\-\\inftysuch thatℱ​\(θ\)≥ℱ⋆\\mathcal\{F\}\(\\theta\)\\geq\\mathcal\{F\}^\{\\star\}\.

###### Theorem 1\(Convergence to first\-order stationarity\)\.

Under Assumptions[1](https://arxiv.org/html/2605.22940#Thmassumption1)and[2](https://arxiv.org/html/2605.22940#Thmassumption2), let\{θt\}\\\{\\theta\_\{t\}\\\}be generated byθt\+1=θt−η​∇ℱ​\(θt\)\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)with0<η<1/L0<\\eta<1/L\. Then

lim inft→∞‖∇ℱ​\(θt\)‖=0\.\\liminf\_\{t\\to\\infty\}\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|=0\.

Proof provided in Appendix[A\.2](https://arxiv.org/html/2605.22940#A1.SS2)\.

### 6\.2Entropy Flow and Critical Balance

Under the continuous\-time flow

d​θd​t=−∇ℒ​\(θ\)−β​∇H​\(θ\),\\frac\{d\\theta\}\{dt\}=\-\\nabla\\mathcal\{L\}\(\\theta\)\-\\beta\\nabla H\(\\theta\),entropy evolves through injection and dissipation\.

###### Theorem 2\(Entropy\-flow identity\)\.

Along the above flow,

dd​t​H​\(θt\)=−∇H​\(θt\)⊤​∇ℒ​\(θt\)⏟injection​Iθ​\(t\)−β​‖∇H​\(θt\)‖2⏟dissipation​Dθ​\(t\)\.\\frac\{d\}\{dt\}H\(\\theta\_\{t\}\)=\\underbrace\{\-\\nabla H\(\\theta\_\{t\}\)^\{\\top\}\\nabla\\mathcal\{L\}\(\\theta\_\{t\}\)\}\_\{\\text\{injection \}I\_\{\\theta\}\(t\)\}\-\\underbrace\{\\beta\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}\}\_\{\\text\{dissipation \}D\_\{\\theta\}\(t\)\}\.

Proof provided in Appendix[A\.3](https://arxiv.org/html/2605.22940#A1.SS3)\.

###### Proposition 2\(Instantaneous critical entropy coefficient\)\.

If∇H​\(θt\)≠0\\nabla H\(\\theta\_\{t\}\)\\neq 0, setting

βc​\(t\)=Iθ​\(t\)‖∇H​\(θt\)‖2\\beta\_\{c\}\(t\)=\\frac\{I\_\{\\theta\}\(t\)\}\{\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}\}yields locally stationary entropy, i\.e\.,dd​t​H​\(θt\)=0\\frac\{d\}\{dt\}H\(\\theta\_\{t\}\)=0\.

Proof provided in Appendix[A\.4](https://arxiv.org/html/2605.22940#A1.SS4)\.

### 6\.3Representation\-Compression Generalization

The previous energy formulation regulates representation geometry\. To connect this regulation to generalization, we avoid assuming a direct and generally unjustified inequality between parameter\-space complexityK​L​\(Q∥P\)KL\(Q\\\|P\)and representation entropy\. Instead, we use a noisy representation\-compression argument\. LetS=\{\(Xi,Yi\)\}i=1nS=\\\{\(X\_\{i\},Y\_\{i\}\)\\\}\_\{i=1\}^\{n\}be the training sample and letZ~θ=hθ​\(X\)\+ξ\\widetilde\{Z\}\_\{\\theta\}=h\_\{\\theta\}\(X\)\+\\xibe the noisy representation defined in Eq\. \([1](https://arxiv.org/html/2605.22940#S4.E1)\)\. Assume that the lossℓ​\(Y^,Y\)\\ell\(\\hat\{Y\},Y\)isσ\\sigma\-sub\-Gaussian and bounded or sub\-Gaussian under the data distribution\.

###### Assumption 3\(Representation\-compression control\)\.

There exist constantsA\>0A\>0andB≥0B\\geq 0such that the information carried by the noisy representation about the input satisfies

I​\(X;Z~θ\)≤A​ℋ~logdet​\(Z~θ\)\+B\.I\(X;\\widetilde\{Z\}\_\{\\theta\}\)\\leq A\\,\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)\+B\.For additive Gaussian noise, this type of control is natural because

I​\(X;Z~θ\)=H​\(Z~θ\)−H​\(ξ\),I\(X;\\widetilde\{Z\}\_\{\\theta\}\)=H\(\\widetilde\{Z\}\_\{\\theta\}\)\-H\(\\xi\),and the Gaussian maximum\-entropy inequality gives

H​\(Z~θ\)≤12​log​det\(2​π​e​ΣZ~θ\)\.H\(\\widetilde\{Z\}\_\{\\theta\}\)\\leq\\frac\{1\}\{2\}\\log\\det\(2\\pi e\\,\\Sigma\_\{\\widetilde\{Z\}\_\{\\theta\}\}\)\.Thus, the log\-determinant covariance surrogate controls an upper bound on representation information up to constants induced by the injected noise\.

###### Theorem 3\(Information\-theoretic generalization through noisy representations\)\.

Assume that the loss isσ\\sigma\-sub\-Gaussian and that Assumption[3](https://arxiv.org/html/2605.22940#Thmassumption3)holds\. Then the expected generalization gap of predictors that depend on the data only throughZ~θ\\widetilde\{Z\}\_\{\\theta\}satisfies

\|𝔼​\[ℒ​\(θ\)−ℒ^S​\(θ\)\]\|≤2​σ2n​\(A​ℋ~logdet​\(Z~θ\)\+B\)\.\\left\|\\mathbb\{E\}\\left\[\\mathcal\{L\}\(\\theta\)\-\\widehat\{\\mathcal\{L\}\}\_\{S\}\(\\theta\)\\right\]\\right\|\\leq\\sqrt\{\\frac\{2\\sigma^\{2\}\}\{n\}\\left\(A\\,\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)\+B\\right\)\}\.

Proof provided in Appendix[A\.5](https://arxiv.org/html/2605.22940#A1.SS5)\.This result should be interpreted as an information\-flow diagnostic rather than as a universal characterization of deep\-network generalization\. It depends critically on the noisy\-representation assumption and on the existence of a stable entropy surrogate whose geometry is meaningfully related to representation information\.

###### Corollary 1\(Entropy\-scaling diagnostic\)\.

If the controlled representation entropy satisfies

ℋ~logdet​\(Z~θ\)=O​\(nα\),\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)=O\(n^\{\\alpha\}\),withα<1\\alpha<1, then the expected generalization gap obeys

ℒgen=O​\(n\(α−1\)/2\)\.\\mathcal\{L\}\_\{\\mathrm\{gen\}\}=O\\left\(n^\{\(\\alpha\-1\)/2\}\\right\)\.Thus, the HCLM diagnostic is meaningful only when representation entropy grows sublinearly with sample size\.

### 6\.4Wasserstein Gradient\-Flow Formulation

To describe stochastic learning dynamics, letρt​\(θ\)\\rho\_\{t\}\(\\theta\)be a probability density over parameters\. Consider the free\-energy functional

ℰ​\(ρ\)=∫𝒰​\(θ\)​ρ​\(θ\)​𝑑θ\+β​∫ρ​\(θ\)​log⁡ρ​\(θ\)​𝑑θ,\\mathcal\{E\}\(\\rho\)=\\int\\mathcal\{U\}\(\\theta\)\\rho\(\\theta\)\\,d\\theta\+\\beta\\int\\rho\(\\theta\)\\log\\rho\(\\theta\)\\,d\\theta,\(6\)where

𝒰​\(θ\)=ℒpred​\(θ\)\+γ​Ω​\(θ\)\+λ​ℛdec​\(θ\)\.\\mathcal\{U\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\)\+\\gamma\\Omega\(\\theta\)\+\\lambda\\mathcal\{R\}\_\{\\mathrm\{dec\}\}\(\\theta\)\.The corresponding Wasserstein gradient flow is

∂tρt=∇θ⋅\(ρt​∇θδ​ℰδ​ρ\)=∇θ⋅\(ρt​∇θ𝒰\)\+β​Δ​ρt\.\\partial\_\{t\}\\rho\_\{t\}=\\nabla\_\{\\theta\}\\cdot\\left\(\\rho\_\{t\}\\nabla\_\{\\theta\}\\frac\{\\delta\\mathcal\{E\}\}\{\\delta\\rho\}\\right\)=\\nabla\_\{\\theta\}\\cdot\(\\rho\_\{t\}\\nabla\_\{\\theta\}\\mathcal\{U\}\)\+\\beta\\Delta\\rho\_\{t\}\.\(7\)This is the Fokker–Planck equation associated with the Langevin dynamics

d​θt=−∇θ𝒰​\(θt\)​d​t\+2​β​d​Wt\.d\\theta\_\{t\}=\-\\nabla\_\{\\theta\}\\mathcal\{U\}\(\\theta\_\{t\}\)\\,dt\+\\sqrt\{2\\beta\}\\,dW\_\{t\}\.
###### Theorem 4\(Free\-energy dissipation\)\.

Assume thatρt\\rho\_\{t\}is smooth, decays sufficiently fast at infinity, and evolves according to Eq\. \([7](https://arxiv.org/html/2605.22940#S6.E7)\)\. Then

dd​t​ℰ​\(ρt\)=−∫ρt​‖∇θδ​ℰδ​ρ‖2​𝑑θ≤0\.\\frac\{d\}\{dt\}\\mathcal\{E\}\(\\rho\_\{t\}\)=\-\\int\\rho\_\{t\}\\left\\\|\\nabla\_\{\\theta\}\\frac\{\\delta\\mathcal\{E\}\}\{\\delta\\rho\}\\right\\\|^\{2\}d\\theta\\leq 0\.

Proof provided in Appendix[A\.6](https://arxiv.org/html/2605.22940#A1.SS6)\.

###### Theorem 5\(Distributional entropy production\)\.

Let

𝒮​\(ρt\)=−∫ρt​\(θ\)​log⁡ρt​\(θ\)​𝑑θ\\mathcal\{S\}\(\\rho\_\{t\}\)=\-\\int\\rho\_\{t\}\(\\theta\)\\log\\rho\_\{t\}\(\\theta\)d\\thetabe the Shannon entropy of the parameter distribution\. Under Eq\. \([7](https://arxiv.org/html/2605.22940#S6.E7)\),

dd​t​𝒮​\(ρt\)=−∫ρt​\(θ\)​Δ​𝒰​\(θ\)​𝑑θ⏟drift\-induced entropy change\+β​∫ρt​\(θ\)​‖∇log⁡ρt​\(θ\)‖2​𝑑θ⏟diffusive entropy production\.\\frac\{d\}\{dt\}\\mathcal\{S\}\(\\rho\_\{t\}\)=\\underbrace\{\-\\int\\rho\_\{t\}\(\\theta\)\\Delta\\mathcal\{U\}\(\\theta\)d\\theta\}\_\{\\text\{drift\-induced entropy change\}\}\+\\underbrace\{\\beta\\int\\rho\_\{t\}\(\\theta\)\\\|\\nabla\\log\\rho\_\{t\}\(\\theta\)\\\|^\{2\}d\\theta\}\_\{\\text\{diffusive entropy production\}\}\.

Proof provided in Appendix[A\.7](https://arxiv.org/html/2605.22940#A1.SS7)\.

### 6\.5Information Force Collapse

###### Proposition 3\(Information\-force stabilization\)\.

Under continuous HCLM dynamics withβ\>0\\beta\>0, suppose that there exist constantsmH\>0m\_\{H\}\>0,CL≥0C\_\{L\}\\geq 0, andR\>0R\>0such that, outside a ball of radiusRR,

∇H⊤​∇2H​∇H≥mH​‖∇H‖2,\|∇H⊤​∇2H​∇ℒ\|≤CL​‖∇H‖\.\\nabla H^\{\\top\}\\nabla^\{2\}H\\nabla H\\geq m\_\{H\}\\\|\\nabla H\\\|^\{2\},\\qquad\\left\|\\nabla H^\{\\top\}\\nabla^\{2\}H\\nabla\\mathcal\{L\}\\right\|\\leq C\_\{L\}\\\|\\nabla H\\\|\.ThenG​\(t\)=‖∇H​\(θt\)‖2G\(t\)=\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}enters and remains in a bounded attracting region\.

Proof provided in Appendix[A\.8](https://arxiv.org/html/2605.22940#A1.SS8)\.

## 7Conditional Interpretation of Scaling Laws

HCLM does not aim to derive empirical neural scaling laws as universal consequences of entropy regularization\. Rather, it provides a conditional mechanistic interpretation of why power\-law\-like improvements may emerge when scale\-dependent information injection, entropy dissipation, and residual risk are coupled in compatible ways\. LetSSdenote a scale variable, such as dataset size, model size, or compute budget\. We model the scale\-dependent information injection and entropy dissipation rates as

I​\(S\)=a​Sα,D​\(S\)=b​Sγ,I\(S\)=aS^\{\\alpha\},\\qquad D\(S\)=bS^\{\\gamma\},wherea,b\>0a,b\>0, andα,γ≥0\\alpha,\\gamma\\geq 0\. Their ratio is

R​\(S\)=I​\(S\)D​\(S\)=ab​Sα−γ\.R\(S\)=\\frac\{I\(S\)\}\{D\(S\)\}=\\frac\{a\}\{b\}S^\{\\alpha\-\\gamma\}\.This quantity measures the amount of effective information injection that remains after entropy\-controlled dissipation\.

###### Assumption 4\(Risk\-response relation\)\.

There exists a monotone response functionΨ\\Psisuch that the excess loss satisfies

ℒ​\(S\)−ℒ∞=Ψ​\(R​\(S\)\)\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}=\\Psi\(R\(S\)\)\.In a locally balanced regime, we assume the response has the asymptotic form

Ψ​\(r\)≍r−q,q\>0\.\\Psi\(r\)\\asymp r^\{\-q\},\\qquad q\>0\.

###### Proposition 4\(Power\-law behavior under entropy\-balance assumptions\)\.

Under the scale lawsI​\(S\)=a​SαI\(S\)=aS^\{\\alpha\},D​\(S\)=b​SγD\(S\)=bS^\{\\gamma\}, and Assumption[4](https://arxiv.org/html/2605.22940#Thmassumption4), ifα\>γ\\alpha\>\\gamma, the excess loss satisfies

ℒ​\(S\)−ℒ∞≍S−q​\(α−γ\)\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}\\asymp S^\{\-q\(\\alpha\-\\gamma\)\}\.In particular, whenq=1/2q=1/2,

ℒ​\(S\)−ℒ∞≍S−\(α−γ\)/2\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}\\asymp S^\{\-\(\\alpha\-\\gamma\)/2\}\.

Proof provided in Appendix[A\.9](https://arxiv.org/html/2605.22940#A1.SS9)\.

The novelty of Proposition[4](https://arxiv.org/html/2605.22940#Thmproposition4)is not the recovery of a power law itself, since a regular risk\-response relation is explicitly assumed\. Rather, the contribution is to identify the effective information ratioR​\(S\)=I​\(S\)/D​\(S\)R\(S\)=I\(S\)/D\(S\)as the scale\-dependent dynamical variable governing the balance between information injection and entropy dissipation\. This makes the scaling interpretation conditional, falsifiable, and tied to measurable quantities in the HCLM framework\. This proposition should be read as a mechanistic scaling model, not as a universal theorem for all architectures and datasets\. It states that power\-law behavior can arise when three conditions hold: information injection grows with scale, entropy dissipation grows in a compatible but slower way, and the residual risk responds regularly to the effective information ratioR​\(S\)R\(S\)\.

The interpretation is therefore qualitative as well as mathematical\. If dissipation is too weak,R​\(S\)R\(S\)may grow too rapidly, producing uncontrolled representation expansion, instability, or overfitting\. If dissipation is too strong, useful information is suppressed and the loss may plateau\. Stable scaling corresponds to a controlled growth ofI​\(S\)/D​\(S\)I\(S\)/D\(S\), rather than to pure information accumulation\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/hclm_scaling_loss.png)Figure 2:HCLM interpretation of scaling laws\. The balanced regime produces stable power\-law\-like improvement, whereas weak or excessive dissipation leads respectively to instability or slower improvement\.Figure[3](https://arxiv.org/html/2605.22940#S7.F3)illustrates the two competing scale\-dependent processes\. Increasing scale injects more usable information into the model, but entropy dissipation regulates how much of this information is retained in the representation geometry\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/hclm_injection_dissipation.png)Figure 3:Information injection and entropy dissipation as competing scale\-dependent processes\. Scaling improves performance when information growth is not overwhelmed by dissipation and not left completely uncontrolled\.The effective information ratioR​\(S\)=I​\(S\)/D​\(S\)R\(S\)=I\(S\)/D\(S\)summarizes this balance\. As shown in Figure[4](https://arxiv.org/html/2605.22940#S7.F4), weak dissipation yields an excessively increasing ratio, while excessive dissipation keeps the ratio too small\. The balanced HCLM regime maintains controlled growth ofR​\(S\)R\(S\), which supports stable scaling behavior under Assumption[4](https://arxiv.org/html/2605.22940#Thmassumption4)\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/hclm_ratio.png)Figure 4:Effective information ratio across scale\. Stable scaling corresponds to controlled growth ofI​\(S\)/D​\(S\)I\(S\)/D\(S\), rather than maximal information accumulation\.
## 8Empirical Analysis: Evaluating Surrogate Effectiveness and Dynamics

All empirical experiments reported in Section 8 follow the generic learning structure summarized in Algorithm[1](https://arxiv.org/html/2605.22940#alg1)\. The experiments are designed as mechanism\-level probes\. Their purpose is not to establish state\-of\-the\-art benchmark performance, but to test whether entropy surrogates generate measurable information forces, whether such forces collapse under regularization, and whether this collapse correlates with generalization and reward behavior\. Large\-scale systems introduce confounding factors such as optimizer engineering, data augmentation, curriculum learning, batch normalization, and architecture\-specific inductive biases\. Controlled experiments are therefore used deliberately to isolate the dynamical mechanism predicted by HCLM\. To validate the theoretical framework empirically, we isolate the effect of the entropy coefficientβ\\betaon learning dynamics\. We evaluate three families of entropy surrogates: softmax entropy, variance entropy, and log\-determinant covariance entropy\. Their behaviors are compared across Fixed Hybrid, Thermostat, and RL\-Thermostat regimes\.

### 8\.1Softmax Entropy as a Weak Surrogate

Our empirical findings indicate that softmax entropy behaves as a weak information surrogate\. As depicted in Figure[5](https://arxiv.org/html/2605.22940#S8.F5), increasingβ\\betaunder the softmax formulation does not produce a stable improvement in test loss\. Furthermore, the generalization gap in Figure[6](https://arxiv.org/html/2605.22940#S8.F6)fails to contract meaningfully and deteriorates at higherβ\\betavalues, suggesting that the entropy suppression is either excessive or poorly targeted\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_softmax_test_loss_vs_beta.png)Figure 5:Transformer RL\-HCLM: test loss versusβ\\betausing softmax entropy\. Softmax entropy produces weak and unstable improvements, confirming its limited effectiveness as an information surrogate\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_softmax_gen_gap_vs_beta.png)Figure 6:Transformer RL\-HCLM: generalization gap versusβ\\betausing softmax entropy\. The gap remains high and becomes worse for largeβ\\beta, suggesting excessive or poorly targeted entropy suppression\.The underlying cause of this failure is illuminated in Figure[7](https://arxiv.org/html/2605.22940#S8.F7), which tracks the information force\. The force induced by softmax entropy is relatively small and noisy, rendering it dynamically ineffective at steering the optimizer\. Even when representation entropy quantitatively decreases, as shown in Figure[8](https://arxiv.org/html/2605.22940#S8.F8), this reduction mainly squashes activation distributions without capturing geometric structure, and therefore fails to translate into robust generalization\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_softmax_info_force_vs_beta.png)Figure 7:Transformer RL\-HCLM: information force versusβ\\betausing softmax entropy\. The force is relatively small and unstable, indicating weak dynamic effectiveness\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_softmax_entropy_vs_beta.png)Figure 8:Transformer RL\-HCLM: representation entropy versusβ\\betausing softmax entropy\. Entropy decreases for largeβ\\beta, but this reduction does not translate into robust generalization\.Although the adaptive thermostat controller attempts to regulate learning \(Figure[9](https://arxiv.org/html/2605.22940#S8.F9)\), the fundamental weakness of the surrogate limits its utility\. Consequently, while the RL\-thermostat modestly improves human/RL reward relative to fixed hybrid control \(Figure[10](https://arxiv.org/html/2605.22940#S8.F10)\), the system remains unstable in high\-β\\betaregimes\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_softmax_beta_t_vs_beta.png)Figure 9:Transformer RL\-HCLM: thermostat coefficientβt\\beta\_\{t\}versus baseβ\\betausing softmax entropy\. The adaptive controller activates, but the weak surrogate limits its usefulness\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_softmax_reward_test_vs_beta.png)Figure 10:Transformer RL\-HCLM: human/RL reward versusβ\\betausing softmax entropy\. RL\-thermostat improves reward relative to fixed hybrid control, but the softmax surrogate remains unstable for highβ\\beta\.
### 8\.2Variance Entropy and Marginal Dispersion

Moving beyond simple activations, variance entropy provides a stronger surrogate by directly measuring marginal representation spread\. As observed in Figure[11](https://arxiv.org/html/2605.22940#S8.F11), both Thermostat and RL\-thermostat methods stabilize and lower the test loss compared to fixed hybrid control\. More importantly, Figure[12](https://arxiv.org/html/2605.22940#S8.F12)shows a contraction of the generalization gap across a broad range ofβ\\beta, supporting the usefulness of marginal dispersion control\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_variance_test_loss_vs_beta.png)Figure 11:Transformer RL\-HCLM: test loss versusβ\\betausing variance entropy\. Thermostat and RL\-thermostat regimes improve stability compared with fixed hybrid HCLM\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_variance_gen_gap_vs_beta.png)Figure 12:Transformer RL\-HCLM: generalization gap versusβ\\betausing variance entropy\. Adaptive thermostat control significantly reduces the gap compared with fixed hybrid HCLM\.The mechanism of this stability is visible in Figure[13](https://arxiv.org/html/2605.22940#S8.F13), where the RL\-thermostat maintains a lower, controlled information force\. The thermostat computes a nonzero adaptive dissipation coefficientβt\\beta\_\{t\}dynamically \(Figure[14](https://arxiv.org/html/2605.22940#S8.F14)\)\. This targeted control directly translates to higher and more stable human/RL reward \(Figure[15](https://arxiv.org/html/2605.22940#S8.F15)\), supporting the hypothesis that human feedback can be interpreted as adaptive entropy control\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_variance_info_force_vs_beta.png)Figure 13:Transformer RL\-HCLM: information force versusβ\\betausing variance entropy\. RL\-thermostat maintains a lower information force than fixed hybrid control for small and moderateβ\\beta, supporting the thermostat interpretation\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_variance_beta_t_vs_beta.png)Figure 14:Transformer RL\-HCLM: thermostat coefficientβt\\beta\_\{t\}versus baseβ\\betausing variance entropy\. The controller produces a nonzero adaptive dissipation coefficient even when the baseβ\\betais small\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_variance_reward_test_vs_beta.png)Figure 15:Transformer RL\-HCLM: human/RL reward versusβ\\betausing variance entropy\. RL\-thermostat achieves the highest reward, showing that human/RL feedback can be interpreted as adaptive entropy control\.
### 8\.3Log\-Determinant Entropy and the Three Regimes of Geometry

Log\-determinant covariance entropy provides the clearest empirical support for the HCLM theory because it captures multivariate representation volume\. Figure[16](https://arxiv.org/html/2605.22940#S8.F16)shows a contrast between control regimes: fixed hybrid control suffers at extremes, whereas adaptive thermostats stabilize the test loss\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_logdet_test_loss_vs_beta.png)Figure 16:Transformer RL\-HCLM: test loss versusβ\\betausing log\-determinant entropy\. Thermostat and RL\-thermostat regimes stabilize the test loss compared with fixed hybrid control\.The superiority of this geometric surrogate is highlighted in Figure[17](https://arxiv.org/html/2605.22940#S8.F17), where the generalization gap is sharply reduced\. Furthermore, Figure[18](https://arxiv.org/html/2605.22940#S8.F18)reveals three regimes of representation geometry: unregularized models exhibit high information force, while adaptive controls suppress excessive force\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_logdet_gen_gap_vs_beta.png)Figure 17:Transformer RL\-HCLM: generalization gap versusβ\\betausing log\-determinant entropy\. Thermostat control sharply reduces the gap, especially for small and moderateβ\\beta\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_logdet_info_force_vs_beta.png)Figure 18:Transformer RL\-HCLM: information force versusβ\\betausing log\-determinant entropy\. Fixed hybrid control exhibits very high information force for smallβ\\beta, while thermostat and RL\-thermostat suppress excessive force\.By stabilizing the representation entropy \(Figure[19](https://arxiv.org/html/2605.22940#S8.F19)\) and dynamically regulating information dissipation throughβt\\beta\_\{t\}\(Figure[20](https://arxiv.org/html/2605.22940#S8.F20)\), the RL\-thermostat achieves the maximum reward observed in the experiment \(Figure[21](https://arxiv.org/html/2605.22940#S8.F21)\), supporting the interpretation of alignment as thermodynamic regulation\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_logdet_entropy_vs_beta.png)Figure 19:Transformer RL\-HCLM: representation entropy versusβ\\betausing log\-determinant entropy\. Adaptive control stabilizes entropy and avoids the extreme representation\-volume dynamics observed in the fixed hybrid regime\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_logdet_beta_t_vs_beta.png)Figure 20:Transformer RL\-HCLM: thermostat coefficientβt\\beta\_\{t\}versus baseβ\\betausing log\-determinant entropy\. Adaptiveβt\\beta\_\{t\}remains active even for small baseβ\\beta, showing that the thermostat regulates information dissipation dynamically\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_logdet_reward_test_vs_beta.png)Figure 21:Transformer RL\-HCLM: human/RL reward versusβ\\betausing log\-determinant entropy\. RL\-thermostat achieves the highest reward, supporting the interpretation of human/RL feedback as thermodynamic control\.
### 8\.4Time\-Resolved Dynamics: The Anatomy of Force Collapse

The time\-resolved dynamics over epochs provide empirical support for Proposition[3](https://arxiv.org/html/2605.22940#Thmproposition3)\. Examining fixed hybrid control, Figure[22](https://arxiv.org/html/2605.22940#S8.F22)demonstrates that while intermediateβ\\betastabilizes learning,β=0\\beta=0fails\. The mechanism behind this is the information force collapse, visible in Figure[23](https://arxiv.org/html/2605.22940#S8.F23): positiveβ\\betavalues induce a rapid contraction of force in early epochs, whereasβ=0\\beta=0remains trapped in a high\-force expansion regime\. Correspondingly, Figure[24](https://arxiv.org/html/2605.22940#S8.F24)tracks how the representation shifts from expansion to compression\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_fixed_hybrid_test_loss.png)Figure 22:Transformer RL\-HCLM: test\-loss dynamics for log\-determinant entropy under fixed hybrid control\. Small and intermediateβ\\betavalues stabilize learning, whileβ=0\\beta=0remains unstable\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_fixed_hybrid_info_force.png)Figure 23:Transformer RL\-HCLM: information\-force dynamics for log\-determinant entropy under fixed hybrid control\. Positiveβ\\betavalues induce rapid force collapse, whileβ=0\\beta=0remains in a high\-force regime\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_fixed_hybrid_entropy.png)Figure 24:Transformer RL\-HCLM: entropy dynamics for log\-determinant entropy under fixed hybrid control\. Positiveβ\\betashifts the system toward compression, whereasβ=0\\beta=0follows an expansion trajectory\.When adaptive Thermostat control is introduced, the system behavior becomes more resilient\. Figure[25](https://arxiv.org/html/2605.22940#S8.F25)shows test loss stabilizing consistently across baseβ\\betaconfigurations\. The information force \(Figure[26](https://arxiv.org/html/2605.22940#S8.F26)\) collapses rapidly and remains bounded in a low\-force regime\. This is facilitated by the controller continuously adaptingβt\\beta\_\{t\}after the initial transient phase \(Figure[27](https://arxiv.org/html/2605.22940#S8.F27)\), ensuring that the reward trajectory \(Figure[28](https://arxiv.org/html/2605.22940#S8.F28)\) remains high\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_thermostat_test_loss.png)Figure 25:Transformer RL\-HCLM: test\-loss dynamics for log\-determinant entropy under thermostat control\. The thermostat stabilizes learning acrossβ\\beta, with only mild late\-stage fluctuations\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_thermostat_info_force.png)Figure 26:Transformer RL\-HCLM: information\-force dynamics for log\-determinant entropy under thermostat control\. The information force collapses rapidly and then remains in a stable low\-force regime\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_thermostat_beta_t.png)Figure 27:Transformer RL\-HCLM: adaptiveβt\\beta\_\{t\}dynamics for log\-determinant entropy under thermostat control\. The controller maintains a stable dissipation coefficient after an initial transient\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_thermostat_reward_test.png)Figure 28:Transformer RL\-HCLM: reward dynamics for log\-determinant entropy under thermostat control\. Reward rapidly improves and remains stable, supporting the thermodynamic interpretation of alignment\.Finally, under full RL\-Thermostat control, the human/RL feedback loop fine\-tunes the geometric constraints\. Figure[29](https://arxiv.org/html/2605.22940#S8.F29)exhibits a stable low\-loss trajectory\. The force collapse is smooth and controlled \(Figure[30](https://arxiv.org/html/2605.22940#S8.F30)\), while the RL feedback modulates the entropy coefficient within stable operational bounds \(Figure[31](https://arxiv.org/html/2605.22940#S8.F31)\)\. Ultimately, this orchestration yields the most stable and highest reward dynamics observed in the study \(Figure[32](https://arxiv.org/html/2605.22940#S8.F32)\), supporting the hypothesis that human/RL feedback can be interpreted as a form of adaptive geometric entropy control\. This does not imply that alignment is reducible to entropy control alone, but it shows that entropy\-regulated representation geometry can provide a useful dynamical lens for studying aligned adaptation\.

![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_rl_thermostat_test_loss.png)Figure 29:Transformer RL\-HCLM: test\-loss dynamics for log\-determinant entropy under RL\-thermostat control\. RL feedback preserves a stable low\-loss trajectory while maintaining adaptive entropy control\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_rl_thermostat_info_force.png)Figure 30:Transformer RL\-HCLM: information\-force dynamics for log\-determinant entropy under RL\-thermostat control\. The force collapses rapidly and remains controlled, similar to thermostat control\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_rl_thermostat_beta_t.png)Figure 31:Transformer RL\-HCLM: adaptiveβt\\beta\_\{t\}dynamics for log\-determinant entropy under RL\-thermostat control\. RL feedback modulates the entropy coefficient while keeping it in a stable range\.![Refer to caption](https://arxiv.org/html/2605.22940v1/transformer_rl_dynamics_logdet_rl_thermostat_reward_test.png)Figure 32:Transformer RL\-HCLM: reward dynamics for log\-determinant entropy under RL\-thermostat control\. Reward remains high and stable after the initial adaptation phase\.

## 9Open Problems and Research Directions

While the HCLM framework provides a coherent theoretical and empirical foundation for entropy\-regulated learning dynamics, it also opens a broad set of fundamental questions at the intersection of information theory, dynamical systems, and practical AI deployment\.

### 9\.1Foundations of Effective Entropy

A central concept introduced in this work is*effective entropy*, defined through its induced information force along the optimization trajectory\. However, this notion remains intrinsically trajectory\-dependent\. A major open problem is to characterize, in a predictive and architecture\-independent manner, when an entropy surrogate is guaranteed to be dynamically effective\.

Future research should aim to establish necessary and sufficient conditions for entropy effectiveness, potentially in terms of:

- •spectral properties of the representation Jacobian;
- •curvature of the latent manifold;
- •stability of covariance operators in high\-dimensional regimes\.

Such results would elevate effective entropy from an empirical diagnostic to a formal design principle\.

### 9\.2Scaling Laws from First Principles

HCLM provides a mechanistic interpretation of scaling laws based on the balance between information injection and entropy dissipation\. However, the current formulation relies on phenomenological power\-law assumptions:

I​\(S\)∼Sα,D​\(S\)∼Sγ\.I\(S\)\\sim S^\{\\alpha\},\\quad D\(S\)\\sim S^\{\\gamma\}\.
A key theoretical challenge is to derive these scaling behaviors from first principles\. Promising directions include:

- •mean\-field limits of wide neural networks;
- •neural tangent kernel \(NTK\) regimes;
- •stochastic differential equation formulations of training dynamics;
- •Wasserstein gradient flows with scale\-dependent potentials\.

In particular, connecting the scaling exponent

κ=α−γ2\\kappa=\\frac\{\\alpha\-\\gamma\}\{2\}to architectural choices, data distributions, and optimization algorithms would provide a predictive theory of scaling\.

### 9\.3Thermostat Control and Learning Stability

The thermostat formulation introduces a control\-theoretic perspective on learning, where entropy dissipation is dynamically regulated by feedback signals\. While conceptually appealing, several challenges remain:

- •stability analysis of closed\-loop dynamics under stochastic gradients;
- •robustness to noisy, delayed, or inconsistent human feedback;
- •convergence guarantees under adaptiveβt\\beta\_\{t\};
- •interaction between entropy control and modern optimizers \(Adam, momentum\)\.

Bridging HCLM with control theory and adaptive systems is a promising direction toward stable and reliable AI\.

### 9\.4Reward Modeling and Alignment

The RL\-thermostat interpretation suggests that reward signals regulate information dissipation rather than directly optimizing policies\. However, in real\-world systems:

- •reward functions are imperfect proxies for human intent;
- •feedback is sparse and delayed;
- •alignment objectives may conflict across stakeholders\.

Future work should investigate how to construct robust reward estimatorsRHR\_\{H\}, and how entropy\-based control interacts with preference learning, inverse reinforcement learning, and RLHF paradigms\.

### 9\.5Large\-Scale Empirical Validation

The current experimental validation is intentionally controlled, focusing on synthetic environments to isolate dynamical mechanisms\. A critical next step is to validate HCLM at scale:

- •convolutional architectures on CIFAR/ImageNet;
- •transformer models on sequence and language tasks;
- •comparison with regularization methods such as weight decay, dropout, SAM;
- •evaluation under large\-batch and distributed training regimes\.

Such studies are essential to determine whether information\-force collapse and entropy\-balanced regimes persist in modern large\-scale systems\.

### 9\.6Transient Memory, Blackout Catastrophe, and Trajectory\-Level Information

A further direction concerns the interpretation of memory in recurrent and associative learning systems\. Classical Hopfield networks and dense associative memory models are typically analyzed through an equilibrium perspective: stored patterns are useful when they correspond to stable attractors or basins of attraction in an energy landscape\. Under this view, exceeding the storage capacity leads to the so\-called blackout catastrophe, where stable memory states disappear and retrieval is considered to fail\. Recent work on the transient dynamics of associative memory models suggests a more nuanced picture\. In particular, dynamical mean\-field analyses of Hopfield and dense associative memory models indicate that stored patterns may still be transiently retrieved with high accuracy above the classical capacity threshold, even when stable attractors no longer exist\. This behavior arises because slow regions can persist in the energy landscape as shallow and unstable remnants of the stable basins that existed below capacity\[Clark,[2025](https://arxiv.org/html/2605.22940#bib.bib22)\]\.

This observation is closely aligned with the HCLM perspective\. In HCLM, the usefulness of an information structure is not determined solely by its asymptotic equilibrium state, but by whether it induces an effective force along the learning or inference trajectory\. Similarly, in an overloaded associative memory, the disappearance of a stable attractor does not necessarily imply that the corresponding memory has become dynamically irrelevant\. A stored pattern may remain useful if the system trajectory passes near it, aligns with it over a finite time window, or slows down sufficiently for a readout mechanism to extract information\.

Letξμ∈\{−1,\+1\}N\\xi^\{\\mu\}\\in\\\{\-1,\+1\\\}^\{N\}denote a stored pattern and letz​\(t\)∈ℝNz\(t\)\\in\\mathbb\{R\}^\{N\}denote the state of the recurrent system at timett\. The standard overlap with patternμ\\muis

mμ​\(t\)=1N​∑i=1Nzi​\(t\)​ξiμ\.m\_\{\\mu\}\(t\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}z\_\{i\}\(t\)\\xi\_\{i\}^\{\\mu\}\.A purely equilibrium\-based analysis emphasizes the asymptotic quantity

limt→∞mμ​\(t\),\\lim\_\{t\\to\\infty\}m\_\{\\mu\}\(t\),which may vanish above capacity\. By contrast, a trajectory\-level analysis considers the finite\-horizon transient recovery

mμmax​\(T\)=max0≤t≤T⁡mμ​\(t\)\.m\_\{\\mu\}^\{\\max\}\(T\)=\\max\_\{0\\leq t\\leq T\}m\_\{\\mu\}\(t\)\.A memory can then be regarded as transiently recoverable over a horizonTTif

mμmax​\(T\)≥τ,m\_\{\\mu\}^\{\\max\}\(T\)\\geq\\tau,for some retrieval thresholdτ\>0\\tau\>0, even when

limt→∞mμ​\(t\)=0\.\\lim\_\{t\\to\\infty\}m\_\{\\mu\}\(t\)=0\.This distinction separates the loss of asymptotic attractor stability from the disappearance of usable information\.

From the HCLM viewpoint, such transient retrieval can be interpreted as an instance of trajectory\-level information effectiveness\. The relevant object is not only the existence of a stable fixed point, but the presence of a non\-negligible retrieval force along the trajectory\. For example, one may define a memory\-effectiveness diagnostic over a finite horizon as

ℰmemμ​\(T\)=1T​∫0T𝟏​\{mμ​\(t\)≥τ\}​𝑑t,\\mathcal\{E\}\_\{\\mathrm\{mem\}\}^\{\\mu\}\(T\)=\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\mathbf\{1\}\\\!\\left\\\{m\_\{\\mu\}\(t\)\\geq\\tau\\right\\\}\\,dt,or, more generally, a force\-based diagnostic

ℱmemμ​\(T\)=1T​∫0T‖∇zmμ​\(z​\(t\)\)‖​𝑑t\.\\mathcal\{F\}\_\{\\mathrm\{mem\}\}^\{\\mu\}\(T\)=\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\left\\\|\\nabla\_\{z\}m\_\{\\mu\}\(z\(t\)\)\\right\\\|\\,dt\.Although the simple overlap gradient is model\-dependent, the conceptual point is general: memory may remain dynamically useful when it generates a finite\-time alignment or slowing effect, even if it does not define a stable attractor\.

This suggests a broader lesson for controlled learning systems\. Blackout should not always be interpreted as complete information erasure\. It may instead mark a transition from stable attractor memory to transient information traces\. In this regime, the timing of observation, readout, or control becomes part of the computational mechanism\. This is consistent with the central HCLM principle that learning and memory should be analyzed as controlled information flows along trajectories, rather than only through static equilibria or final convergence states\.

This perspective also connects naturally to modern memory\-augmented neural architectures\. Dense associative memories and modern Hopfield layers are closely related to attention\-like retrieval mechanisms, where useful computation may occur through transient alignment between queries and stored representations rather than through convergence to a persistent state\. Future work should therefore investigate whether HCLM\-style diagnostics—such as transient overlap, finite\-time information force, entropy\-flow balance, and readout\-time sensitivity—can provide practical tools for analyzing memory layers, recurrent architectures, and transformer attention under high\-load or distribution\-shift regimes\.

### 9\.7Applications in Real\-World AI Systems

Beyond theoretical analysis, HCLM naturally extends to domains where learning must operate under uncertainty, constraints, and human oversight\.

- •Human\-centered decision systems: In healthcare, finance, and public policy, decisions require reliability, interpretability, and trust\. HCLM provides a mechanism for regulating representation complexity through entropy dissipation, enabling models that are not only accurate but also stable and explainable\.
- •Industrial AI and cyber\-physical systems: Industrial environments involve noisy, heterogeneous, and distributed data streams\. HCLM enables robust anomaly detection and predictive maintenance by preventing uncontrolled information accumulation and stabilizing learning dynamics in real time\.
- •Edge AI and resource\-constrained learning: On\-device learning requires strict control of computation and energy\. Entropy dissipation can be interpreted as a form of complexity control, suggesting new approaches to adaptive model compression, pruning, and efficient inference\.
- •Federated and distributed learning: HCLM provides a natural interpretation of federated learning as distributed information flow\. Entropy\-based regulation may guide aggregation strategies that are robust to non\-i\.i\.d\. data and heterogeneous clients\.
- •Reinforcement learning and alignment: The thermostat view reframes alignment as regulating information dynamics rather than directly optimizing rewards\. This perspective may lead to more stable alternatives to RLHF, mitigating abrupt policy shifts and improving robustness\.
- •Scaling strategies for large models: The HCLM interpretation of scaling laws suggests that successful scaling requires maintaining a balance between information injection and entropy dissipation\. This opens the possibility of entropy\-aware scaling strategies for foundation models and large language models\.

### 9\.8Limitations of the Current Formulation

Several limitations remain\. First, the empirical validation is controlled and mechanism\-oriented; it does not establish superiority over strong large\-scale baselines\. Second, information\-force magnitudes are representation\- and parameterization\-dependent, and the Euclidean norm used in experiments should eventually be replaced or complemented by metric\-aware quantities such as Fisher\- or Gauss–Newton\-adjusted norms\. Third, log\-determinant covariance entropy can be computationally expensive for high\-dimensional representations and may require low\-rank, blockwise, or stochastic approximations\. Fourth, thermostat control depends on the quality and calibration of reward or human\-feedback signals\. Finally, the scaling\-law interpretation is conditional and phenomenological; it should not be read as a first\-principles derivation of empirical neural scaling laws\.

### 9\.9Towards a Science of Controlled Learning Systems

Taken together, these directions suggest a broader paradigm shift\. HCLM positions learning systems as*open, controlled, non\-equilibrium processes*rather than static optimization problems\. This perspective invites the development of a unified theory integrating:

- •statistical learning theory \(generalization and risk\),
- •dynamical systems \(stability and convergence\),
- •information theory \(entropy and representation\),
- •control theory \(feedback and regulation\)\.

In this view, future AI systems will not merely learn from data, but will actively regulate their own information dynamics in interaction with humans, environments, and constraints\. Establishing such a science of controlled learning systems remains an open and ambitious challenge\.

## 10Conclusion

We established that entropy in deep learning is not universally beneficial by mere inclusion; it must generate a non\-degenerate information force to shape learning dynamics effectively\. Through*Human\-Centered Learning Mechanics*\(HCLM\), we formulated learning as an open, controlled dynamical process in which prediction, entropy dissipation, structural constraints, and human or reward\-based feedback jointly shape representation geometry\.

The revised formulation leads to three key clarifications\. First, entropy regularization should be analyzed through its induced force along the optimization trajectory, rather than through its scalar value alone\. Second, generalization should not rely on an unsupported equivalence between parameter\-space PAC\-Bayes complexity and representation entropy; instead, noisy representation compression provides a more explicit mechanism linking geometric entropy control to generalization behavior\. Third, scaling\-law behavior should be interpreted conditionally: HCLM explains how power\-law\-like performance can emerge when information injection, entropy dissipation, and risk response satisfy compatible scale\-dependent relationships\.

Empirically, our results show that geometric entropy surrogates, particularly log\-determinant covariance entropy, induce stronger and more stable information forces than softmax\-based alternatives, leading to clearer regimes of controlled information\-force collapse\. The adaptive thermostat and RL\-thermostat mechanisms further suggest that human or reward\-based feedback can be interpreted as a control signal regulating entropy dissipation, rather than as a direct perturbation of model parameters\.

HCLM should therefore be viewed as a foundation for studying controlled learning dynamics, not as a complete theory of deep learning\. Its main contribution is to make entropy regularization testable through the information force it induces\. Future work should extend this mechanism\-level analysis to realistic vision, language, industrial, and human\-in\-the\-loop systems, with the longer\-term goal of contributing toward a more unified understanding of controlled learning dynamics\.

## Acknowledgments

The author thanks the reviewers and editors for their time and constructive feedback\. The author declares no competing interests\. Funding information, if applicable, should be added here before final submission\.

## Appendix AProofs of Theoretical Results

### A\.1Proof of Proposition[1](https://arxiv.org/html/2605.22940#Thmproposition1)\(Degenerate Collapse\)

###### Proof\.

The full update is

θt\+1=θt−η​\(∇ℒpred​\(θt\)\+β​∇H​\(θt\)\+γ​∇Ω​\(θt\)\+λ​∇ℛdec​\(θt\)\)\.\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\left\(\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\+\\beta\\nabla H\(\\theta\_\{t\}\)\+\\gamma\\nabla\\Omega\(\\theta\_\{t\}\)\+\\lambda\\nabla\\mathcal\{R\}\_\{\\mathrm\{dec\}\}\(\\theta\_\{t\}\)\\right\)\.If‖∇H​\(θt\)‖=o​\(‖∇ℒpred​\(θt\)‖\)\\\|\\nabla H\(\\theta\_\{t\}\)\\\|=o\(\\\|\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\\\|\)andβ<∞\\beta<\\infty, then

‖β​∇H​\(θt\)‖=o​\(‖∇ℒpred​\(θt\)‖\)\.\\\|\\beta\\nabla H\(\\theta\_\{t\}\)\\\|=o\(\\\|\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\\\|\)\.Thus the entropy\-induced update is a vanishing perturbation relative to the predictive\-loss gradient\. Whenγ=λ=0\\gamma=\\lambda=0, we obtain

θt\+1=θt−η​∇ℒpred​\(θt\)\+o​\(η​‖∇ℒpred​\(θt\)‖\),\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\+o\(\\eta\\\|\\nabla\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\(\\theta\_\{t\}\)\\\|\),which is gradient descent on the predictive loss up to a vanishing perturbation\. ∎

### A\.2Proof of Theorem[1](https://arxiv.org/html/2605.22940#Thmtheorem1)\(Convergence to Stationarity\)

###### Proof\.

By Assumption[1](https://arxiv.org/html/2605.22940#Thmassumption1), using the update ruleθt\+1=θt−η​∇ℱ​\(θt\)\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\), we substitute into the inequality:

ℱ​\(θt\+1\)≤ℱ​\(θt\)−η​‖∇ℱ​\(θt\)‖2\+L​η22​‖∇ℱ​\(θt\)‖2\.\\mathcal\{F\}\(\\theta\_\{t\+1\}\)\\leq\\mathcal\{F\}\(\\theta\_\{t\}\)\-\\eta\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|^\{2\}\+\\frac\{L\\eta^\{2\}\}\{2\}\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|^\{2\}\.Since0<η<1/L0<\\eta<1/L, we have1−L​η/2\>1/21\-L\\eta/2\>1/2\. Therefore,

ℱ​\(θt\+1\)≤ℱ​\(θt\)−η2​‖∇ℱ​\(θt\)‖2\.\\mathcal\{F\}\(\\theta\_\{t\+1\}\)\\leq\\mathcal\{F\}\(\\theta\_\{t\}\)\-\\frac\{\\eta\}\{2\}\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|^\{2\}\.Summing fromt=0t=0toT−1T\-1and usingℱ​\(θT\)≥ℱ⋆\\mathcal\{F\}\(\\theta\_\{T\}\)\\geq\\mathcal\{F\}^\{\\star\}, we obtain

∑t=0T−1‖∇ℱ​\(θt\)‖2≤2​\(ℱ​\(θ0\)−ℱ⋆\)η\.\\sum\_\{t=0\}^\{T\-1\}\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq\\frac\{2\(\\mathcal\{F\}\(\\theta\_\{0\}\)\-\\mathcal\{F\}^\{\\star\}\)\}\{\\eta\}\.Consequently,

min0≤t≤T−1⁡‖∇ℱ​\(θt\)‖2≤2​\(ℱ​\(θ0\)−ℱ⋆\)η​T\.\\min\_\{0\\leq t\\leq T\-1\}\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq\\frac\{2\(\\mathcal\{F\}\(\\theta\_\{0\}\)\-\\mathcal\{F\}^\{\\star\}\)\}\{\\eta T\}\.LettingT→∞T\\to\\inftygiveslim inft→∞‖∇ℱ​\(θt\)‖=0\\liminf\_\{t\\to\\infty\}\\\|\\nabla\\mathcal\{F\}\(\\theta\_\{t\}\)\\\|=0\. ∎

### A\.3Proof of Theorem[2](https://arxiv.org/html/2605.22940#Thmtheorem2)\(Entropy\-Flow Identity\)

###### Proof\.

By the chain rule,

dd​t​H​\(θt\)=∇H​\(θt\)⊤​d​θtd​t\.\\frac\{d\}\{dt\}H\(\\theta\_\{t\}\)=\\nabla H\(\\theta\_\{t\}\)^\{\\top\}\\frac\{d\\theta\_\{t\}\}\{dt\}\.Substituting the continuous\-time HCLM flow

d​θd​t=−∇ℒ​\(θ\)−β​∇H​\(θ\)\\frac\{d\\theta\}\{dt\}=\-\\nabla\\mathcal\{L\}\(\\theta\)\-\\beta\\nabla H\(\\theta\)gives

dd​t​H​\(θt\)=∇H​\(θt\)⊤​\(−∇ℒ​\(θt\)−β​∇H​\(θt\)\)\.\\frac\{d\}\{dt\}H\(\\theta\_\{t\}\)=\\nabla H\(\\theta\_\{t\}\)^\{\\top\}\\left\(\-\\nabla\\mathcal\{L\}\(\\theta\_\{t\}\)\-\\beta\\nabla H\(\\theta\_\{t\}\)\\right\)\.Expanding the inner product yields

dd​t​H​\(θt\)=−∇H​\(θt\)⊤​∇ℒ​\(θt\)−β​‖∇H​\(θt\)‖2\.\\frac\{d\}\{dt\}H\(\\theta\_\{t\}\)=\-\\nabla H\(\\theta\_\{t\}\)^\{\\top\}\\nabla\\mathcal\{L\}\(\\theta\_\{t\}\)\-\\beta\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}\.∎

### A\.4Proof of Proposition[2](https://arxiv.org/html/2605.22940#Thmproposition2)\(Critical Coefficient\)

###### Proof\.

Set the right\-hand side of Theorem[2](https://arxiv.org/html/2605.22940#Thmtheorem2)to zero:

−∇H⊤​∇ℒ−β​‖∇H‖2=0\.\-\\nabla H^\{\\top\}\\nabla\\mathcal\{L\}\-\\beta\\\|\\nabla H\\\|^\{2\}=0\.If∇H​\(θt\)≠0\\nabla H\(\\theta\_\{t\}\)\\neq 0, solving forβ\\betagives

βc​\(t\)=−∇H​\(θt\)⊤​∇ℒ​\(θt\)‖∇H​\(θt\)‖2=Iθ​\(t\)‖∇H​\(θt\)‖2\.\\beta\_\{c\}\(t\)=\\frac\{\-\\nabla H\(\\theta\_\{t\}\)^\{\\top\}\\nabla\\mathcal\{L\}\(\\theta\_\{t\}\)\}\{\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}\}=\\frac\{I\_\{\\theta\}\(t\)\}\{\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}\}\.Substitution into the entropy\-flow identity givesdd​t​H​\(θt\)=0\\frac\{d\}\{dt\}H\(\\theta\_\{t\}\)=0\. ∎

### A\.5Proof of Theorem[3](https://arxiv.org/html/2605.22940#Thmtheorem3)and Corollary[1](https://arxiv.org/html/2605.22940#Thmcorollary1)

###### Proof of Theorem[3](https://arxiv.org/html/2605.22940#Thmtheorem3)\.

Forσ\\sigma\-sub\-Gaussian losses, standard information\-theoretic generalization inequalities imply that, for a predictor depending on the training sample through a representation variableZ~θ\\widetilde\{Z\}\_\{\\theta\},

\|𝔼​\[ℒ​\(θ\)−ℒ^S​\(θ\)\]\|≤2​σ2​I​\(X;Z~θ\)n\.\\left\|\\mathbb\{E\}\\left\[\\mathcal\{L\}\(\\theta\)\-\\widehat\{\\mathcal\{L\}\}\_\{S\}\(\\theta\)\\right\]\\right\|\\leq\\sqrt\{\\frac\{2\\sigma^\{2\}I\(X;\\widetilde\{Z\}\_\{\\theta\}\)\}\{n\}\}\.By Assumption[3](https://arxiv.org/html/2605.22940#Thmassumption3),

I​\(X;Z~θ\)≤A​ℋ~logdet​\(Z~θ\)\+B\.I\(X;\\widetilde\{Z\}\_\{\\theta\}\)\\leq A\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)\+B\.Substituting this inequality gives

\|𝔼​\[ℒ​\(θ\)−ℒ^S​\(θ\)\]\|≤2​σ2n​\(A​ℋ~logdet​\(Z~θ\)\+B\)\.\\left\|\\mathbb\{E\}\\left\[\\mathcal\{L\}\(\\theta\)\-\\widehat\{\\mathcal\{L\}\}\_\{S\}\(\\theta\)\\right\]\\right\|\\leq\\sqrt\{\\frac\{2\\sigma^\{2\}\}\{n\}\\left\(A\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)\+B\\right\)\}\.∎

###### Proof of Corollary[1](https://arxiv.org/html/2605.22940#Thmcorollary1)\.

If

ℋ~logdet​\(Z~θ\)=O​\(nα\),\\widetilde\{\\mathcal\{H\}\}\_\{\\mathrm\{logdet\}\}\(\\widetilde\{Z\}\_\{\\theta\}\)=O\(n^\{\\alpha\}\),then Theorem[3](https://arxiv.org/html/2605.22940#Thmtheorem3)gives

ℒgen=O​\(nαn\)=O​\(n\(α−1\)/2\)\.\\mathcal\{L\}\_\{\\mathrm\{gen\}\}=O\\left\(\\sqrt\{\\frac\{n^\{\\alpha\}\}\{n\}\}\\right\)=O\\left\(n^\{\(\\alpha\-1\)/2\}\\right\)\.The bound vanishes whenα<1\\alpha<1, showing that sublinear entropy growth is required for this diagnostic to predict generalization improvement\. ∎

### A\.6Proofs for the Wasserstein Flow

###### Proof of Theorem[4](https://arxiv.org/html/2605.22940#Thmtheorem4)\.

The first variation of Eq\. \([6](https://arxiv.org/html/2605.22940#S6.E6)\) is

δ​ℰδ​ρ=𝒰​\(θ\)\+β​\(1\+log⁡ρ\)\.\\frac\{\\delta\\mathcal\{E\}\}\{\\delta\\rho\}=\\mathcal\{U\}\(\\theta\)\+\\beta\(1\+\\log\\rho\)\.Using the Wasserstein gradient flow

∂tρt=∇θ⋅\(ρt​∇θδ​ℰδ​ρ\),\\partial\_\{t\}\\rho\_\{t\}=\\nabla\_\{\\theta\}\\cdot\\left\(\\rho\_\{t\}\\nabla\_\{\\theta\}\\frac\{\\delta\\mathcal\{E\}\}\{\\delta\\rho\}\\right\),we obtain

dd​t​ℰ​\(ρt\)=∫δ​ℰδ​ρ​∂tρt​d​θ\.\\frac\{d\}\{dt\}\\mathcal\{E\}\(\\rho\_\{t\}\)=\\int\\frac\{\\delta\\mathcal\{E\}\}\{\\delta\\rho\}\\partial\_\{t\}\\rho\_\{t\}d\\theta\.Substituting the flow equation and integrating by parts under the assumed decay conditions yields

dd​t​ℰ​\(ρt\)=−∫ρt​‖∇θδ​ℰδ​ρ‖2​𝑑θ≤0\.\\frac\{d\}\{dt\}\\mathcal\{E\}\(\\rho\_\{t\}\)=\-\\int\\rho\_\{t\}\\left\\\|\\nabla\_\{\\theta\}\\frac\{\\delta\\mathcal\{E\}\}\{\\delta\\rho\}\\right\\\|^\{2\}d\\theta\\leq 0\.∎

### A\.7Proof of Theorem[5](https://arxiv.org/html/2605.22940#Thmtheorem5)

###### Proof\.

Let

𝒮​\(ρt\)=−∫ρt​log⁡ρt​d​θ\.\\mathcal\{S\}\(\\rho\_\{t\}\)=\-\\int\\rho\_\{t\}\\log\\rho\_\{t\}\\,d\\theta\.Then

dd​t​𝒮​\(ρt\)=−∫\(1\+log⁡ρt\)​∂tρt​d​θ\.\\frac\{d\}\{dt\}\\mathcal\{S\}\(\\rho\_\{t\}\)=\-\\int\(1\+\\log\\rho\_\{t\}\)\\partial\_\{t\}\\rho\_\{t\}\\,d\\theta\.Using

∂tρt=∇⋅\(ρt​∇𝒰\)\+β​Δ​ρt,\\partial\_\{t\}\\rho\_\{t\}=\\nabla\\cdot\(\\rho\_\{t\}\\nabla\\mathcal\{U\}\)\+\\beta\\Delta\\rho\_\{t\},the drift term becomes

−∫\(1\+log⁡ρt\)​∇⋅\(ρt​∇𝒰\)​𝑑θ=∫∇log⁡ρt⋅ρt​∇𝒰​d​θ\.\-\\int\(1\+\\log\\rho\_\{t\}\)\\nabla\\cdot\(\\rho\_\{t\}\\nabla\\mathcal\{U\}\)d\\theta=\\int\\nabla\\log\\rho\_\{t\}\\cdot\\rho\_\{t\}\\nabla\\mathcal\{U\}\\,d\\theta\.Since∇ρt=ρt​∇log⁡ρt\\nabla\\rho\_\{t\}=\\rho\_\{t\}\\nabla\\log\\rho\_\{t\}, integration by parts gives

∫∇ρt⋅∇𝒰​d​θ=−∫ρt​Δ​𝒰​𝑑θ\.\\int\\nabla\\rho\_\{t\}\\cdot\\nabla\\mathcal\{U\}\\,d\\theta=\-\\int\\rho\_\{t\}\\Delta\\mathcal\{U\}\\,d\\theta\.The diffusion term is

−β​∫\(1\+log⁡ρt\)​Δ​ρt​𝑑θ=β​∫‖∇ρt‖2ρt​𝑑θ=β​∫ρt​‖∇log⁡ρt‖2​𝑑θ\.\-\\beta\\int\(1\+\\log\\rho\_\{t\}\)\\Delta\\rho\_\{t\}\\,d\\theta=\\beta\\int\\frac\{\\\|\\nabla\\rho\_\{t\}\\\|^\{2\}\}\{\\rho\_\{t\}\}d\\theta=\\beta\\int\\rho\_\{t\}\\\|\\nabla\\log\\rho\_\{t\}\\\|^\{2\}d\\theta\.Combining the drift and diffusion terms yields

dd​t​𝒮​\(ρt\)=−∫ρt​Δ​𝒰​𝑑θ\+β​∫ρt​‖∇log⁡ρt‖2​𝑑θ\.\\frac\{d\}\{dt\}\\mathcal\{S\}\(\\rho\_\{t\}\)=\-\\int\\rho\_\{t\}\\Delta\\mathcal\{U\}\\,d\\theta\+\\beta\\int\\rho\_\{t\}\\\|\\nabla\\log\\rho\_\{t\}\\\|^\{2\}d\\theta\.∎

### A\.8Proof of Proposition[3](https://arxiv.org/html/2605.22940#Thmproposition3)\(Information\-Force Stabilization\)

###### Proof\.

Let

G​\(t\)=‖∇H​\(θt\)‖2\.G\(t\)=\\\|\\nabla H\(\\theta\_\{t\}\)\\\|^\{2\}\.Differentiating with respect to time gives

d​Gd​t=2​∇H⊤​∇2H​d​θtd​t\.\\frac\{dG\}\{dt\}=2\\nabla H^\{\\top\}\\nabla^\{2\}H\\frac\{d\\theta\_\{t\}\}\{dt\}\.Using the continuous dynamics

d​θtd​t=−∇ℒ​\(θt\)−β​∇H​\(θt\),\\frac\{d\\theta\_\{t\}\}\{dt\}=\-\\nabla\\mathcal\{L\}\(\\theta\_\{t\}\)\-\\beta\\nabla H\(\\theta\_\{t\}\),we obtain

d​Gd​t=−2​∇H⊤​∇2H​∇ℒ−2​β​∇H⊤​∇2H​∇H\.\\frac\{dG\}\{dt\}=\-2\\nabla H^\{\\top\}\\nabla^\{2\}H\\nabla\\mathcal\{L\}\-2\\beta\\nabla H^\{\\top\}\\nabla^\{2\}H\\nabla H\.By the stated assumptions, outside a ball of radiusRR,

d​Gd​t≤2​CL​‖∇H‖−2​β​mH​‖∇H‖2\.\\frac\{dG\}\{dt\}\\leq 2C\_\{L\}\\\|\\nabla H\\\|\-2\\beta m\_\{H\}\\\|\\nabla H\\\|^\{2\}\.SinceG=‖∇H‖2G=\\\|\\nabla H\\\|^\{2\}, this becomes

d​Gd​t≤2​CL​G−2​β​mH​G\.\\frac\{dG\}\{dt\}\\leq 2C\_\{L\}\\sqrt\{G\}\-2\\beta m\_\{H\}G\.For sufficiently largeGG, the negative quadratic\-in\-G\\sqrt\{G\}term dominates the positive linear term\. Thus there exists a finite thresholdG⋆G^\{\\star\}such thatd​G/d​t<0dG/dt<0wheneverG\>G⋆G\>G^\{\\star\}\. Standard comparison arguments then imply thatG​\(t\)G\(t\)enters and remains in a bounded attracting region\. ∎

### A\.9Proof of Proposition[4](https://arxiv.org/html/2605.22940#Thmproposition4)

###### Proof\.

By the scale assumptions,

I​\(S\)=a​Sα,D​\(S\)=b​Sγ,I\(S\)=aS^\{\\alpha\},\\qquad D\(S\)=bS^\{\\gamma\},witha,b\>0a,b\>0\. Therefore, the effective information ratio is

R​\(S\)=I​\(S\)D​\(S\)=a​Sαb​Sγ=ab​Sα−γ\.R\(S\)=\\frac\{I\(S\)\}\{D\(S\)\}=\\frac\{aS^\{\\alpha\}\}\{bS^\{\\gamma\}\}=\\frac\{a\}\{b\}S^\{\\alpha\-\\gamma\}\.Under Assumption[4](https://arxiv.org/html/2605.22940#Thmassumption4),

ℒ​\(S\)−ℒ∞=Ψ​\(R​\(S\)\)\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}=\\Psi\(R\(S\)\)\.In the locally balanced regime,Ψ​\(r\)≍r−q\\Psi\(r\)\\asymp r^\{\-q\}withq\>0q\>0\. Hence

ℒ​\(S\)−ℒ∞≍\(ab​Sα−γ\)−q=\(ab\)−q​S−q​\(α−γ\)\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}\\asymp\\left\(\\frac\{a\}\{b\}S^\{\\alpha\-\\gamma\}\\right\)^\{\-q\}=\\left\(\\frac\{a\}\{b\}\\right\)^\{\-q\}S^\{\-q\(\\alpha\-\\gamma\)\}\.Ifα\>γ\\alpha\>\\gamma, this yields a decreasing power\-law excess loss:

ℒ​\(S\)−ℒ∞≍S−q​\(α−γ\)\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}\\asymp S^\{\-q\(\\alpha\-\\gamma\)\}\.Forq=1/2q=1/2, we obtain

ℒ​\(S\)−ℒ∞≍S−\(α−γ\)/2\.\\mathcal\{L\}\(S\)\-\\mathcal\{L\}\_\{\\infty\}\\asymp S^\{\-\(\\alpha\-\\gamma\)/2\}\.∎

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