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Information Physics of Intelligence: Unifying Logical Depth and Entropy under Thermodynamic Constraints

Jianfeng Xu, Zeyan Li

TL;DR

The paper presents an information-thermodynamics framework that unifies memory and computation through Derivation Entropy, linking logical depth with Shannon entropy under a fundamental energy–time–storage constraint. It introduces a phase-transition bound that determines when it is thermodynamically favorable to retrieve from memory versus derive from compressed laws, and it formalizes optimal hybrid strategies with a frequency-aware threshold. Key contributions include a formal system model, a derivation-depth based information metric, a Landauer-anchored triality bound, and practical guidelines for organizing knowledge systems (LLMs, knowledge graphs, edge devices) under energy budgets. The work provides a principled basis for designing next-generation AI architectures that balance parametric storage, retrieval, and on-demand computation to minimize total action and energy dissipation while maintaining performance across workloads.

Abstract

The rapid scaling of artificial intelligence models has revealed a fundamental tension between model capacity (storage) and inference efficiency (computation). While classical information theory focuses on transmission and storage limits, it lacks a unified physical framework to quantify the thermodynamic costs of generating information from compressed laws versus retrieving it from memory. In this paper, we propose a theoretical framework that treats information processing as an enabling mapping from ontological states to carrier states. We introduce a novel metric, Derivation Entropy, which quantifies the effective work required to compute a target state from a given logical depth. By analyzing the interplay between Shannon entropy (storage) and computational complexity (time/energy), we demonstrate the existence of a critical phase transition point. Below this threshold, memory retrieval is thermodynamically favorable; above it, generative computation becomes the optimal strategy. This "Energy-Time-Space" conservation law provides a physical explanation for the efficiency of generative models and offers a rigorous mathematical bound for designing next-generation, energy-efficient AI architectures. Our findings suggest that the minimization of Derivation Entropy is a governing principle for the evolution of both biological and artificial intelligence.

Information Physics of Intelligence: Unifying Logical Depth and Entropy under Thermodynamic Constraints

TL;DR

The paper presents an information-thermodynamics framework that unifies memory and computation through Derivation Entropy, linking logical depth with Shannon entropy under a fundamental energy–time–storage constraint. It introduces a phase-transition bound that determines when it is thermodynamically favorable to retrieve from memory versus derive from compressed laws, and it formalizes optimal hybrid strategies with a frequency-aware threshold. Key contributions include a formal system model, a derivation-depth based information metric, a Landauer-anchored triality bound, and practical guidelines for organizing knowledge systems (LLMs, knowledge graphs, edge devices) under energy budgets. The work provides a principled basis for designing next-generation AI architectures that balance parametric storage, retrieval, and on-demand computation to minimize total action and energy dissipation while maintaining performance across workloads.

Abstract

The rapid scaling of artificial intelligence models has revealed a fundamental tension between model capacity (storage) and inference efficiency (computation). While classical information theory focuses on transmission and storage limits, it lacks a unified physical framework to quantify the thermodynamic costs of generating information from compressed laws versus retrieving it from memory. In this paper, we propose a theoretical framework that treats information processing as an enabling mapping from ontological states to carrier states. We introduce a novel metric, Derivation Entropy, which quantifies the effective work required to compute a target state from a given logical depth. By analyzing the interplay between Shannon entropy (storage) and computational complexity (time/energy), we demonstrate the existence of a critical phase transition point. Below this threshold, memory retrieval is thermodynamically favorable; above it, generative computation becomes the optimal strategy. This "Energy-Time-Space" conservation law provides a physical explanation for the efficiency of generative models and offers a rigorous mathematical bound for designing next-generation, energy-efficient AI architectures. Our findings suggest that the minimization of Derivation Entropy is a governing principle for the evolution of both biological and artificial intelligence.

Paper Structure

This paper contains 37 sections, 21 theorems, 79 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. The Underlying Logical System $\mathcal{L}$
  4. Enabling Mappings and Information
  5. Information Cost of Energy and Time
  6. Atomic Information
  7. Semantic Inference Structure
  8. Derivation Depth
  9. Derivation Depth as an Information Metric
  10. Derivation Entropy and Information Content
  11. Frequency-weighted storage–computation tradeoff
  12. Mutual Information and Submodularity
  13. Thermodynamic Constraints and Physical Limits
  14. Landauer's Principle and Energy Bounds
  15. Energy-Time-Space Triality Bound
  16. ...and 22 more sections

Key Result

Theorem 1

Let $\mathcal{I} : S_O(O,T_o) \rightrightarrows S_C(C,T_r)$ be an Ideal Information instance, and let $I = \log_2 |S_O(O,T_o)|$ denote the amount of information transmitted (in bits) as in eq:information-amount. Then any physical implementation of $\mathcal{I}$ satisfies where $E$ is the energy resource, $\tau$ is the implementation time, and $\kappa$ is the universal constant from Assumption ass

Figures (4)

  • Figure 1: Phase transition in query latency. The figure shows: (top-left) latency vs. storage capacity, (top-right) gradient $\partial L/\partial M$ marking the phase transition signature, (bottom-left) cache hit rate, and (bottom-right) latency vs. storage in bits on log scale. The red dashed vertical lines mark the transition point at approximately 10% storage capacity, where latency improvement shifts from rapid to gradual, demonstrating the phase transition phenomenon predicted by theory.
  • Figure 2: Baseline strategy comparison across cache sizes (10-500). The three panels show: (left) cache hit rate, (center) average query latency, and (right) total computation cost. Our FreqDepth strategy (freq $\times$ depth heuristic) consistently outperforms LRU and LFU across all metrics and cache sizes. TrueMI (mutual information maximization) performs poorly due to ignoring access frequency.
  • Figure 3: Sensitivity analysis. Left pair: cost vs. storage fraction for different Zipf parameters $\alpha$ and optimal $\beta^*$ vs. $\alpha$. Right pair: cost vs. storage fraction for different logical depths and optimal $\beta^*$ vs. depth. Theoretical bounds hold across all configurations.
  • Figure 4: Scaling behavior across 50$\times$ entity range (100 to 5000). Observed/theoretical ratio remains stable, and $H(Q)$ shows logarithmic growth with knowledge base size.

Theorems & Definitions (37)

  • Definition 1: State Expressibility
  • Definition 2: Enabling Mapping
  • Remark 1
  • Definition 3: Information
  • Definition 4: Ideal Information
  • Remark 2
  • Theorem 1: Resource Bound for Ideal Enabling Mapping
  • Corollary 1: Landauer Principle for Ideal Information
  • Theorem 2: Existence and Computability of Semantic Atomic States
  • Definition 5: Atomic Information
  • ...and 27 more