BEDS : Bayesian Emergent Dissipative Structures : A Formal Framework for Continuous Inference Under Energy Constraints

2026-01-05

arXiv.org

AI Summary

This study addresses the dual challenges of low energy efficiency and limited continual learning capability in existing distributed systems by proposing a unified theoretical framework that integrates nonequilibrium thermodynamics, Bayesian inference, information geometry, and peer-to-peer networking. The framework conceptualizes learning as a process wherein energy flux is transformed into structural organization through entropy export. It establishes a formal isomorphism between thermodynamic dissipation and Bayesian updating, from which the mathematical constants \(e\), \(\pi\), and \(\varphi\) emerge as fixed points of inference derived from fundamental axioms. Furthermore, it posits a structural analogy between Gödelian incompleteness and thermodynamic constraints. A distributed learning system built upon this foundation achieves six orders of magnitude higher energy efficiency compared to current consensus mechanisms while preserving robust continual learning capabilities.

Technology Category

• Machine Learning: Bayesian Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce BEDS (Bayesian Emergent Dissipative Structures), a formal framework for analyzing inference systems that must maintain beliefs continuously under energy constraints. Unlike classical computational models that assume perfect memory and focus on one-shot computation, BEDS explicitly incorporates dissipation (information loss over time) as a fundamental constraint. We prove a central result linking energy, precision, and dissipation: maintaining a belief with precision $\tau$ against dissipation rate $\gamma$ requires power $P \geq \gamma k_{\rm B} T / 2$, with scaling $P \propto \gamma \cdot \tau$. This establishes a fundamental thermodynamic cost for continuous inference. We define three classes of problems -- BEDS-attainable, BEDS-maintainable, and BEDS-crystallizable -- and show these are distinct from classical decidability. We propose the G\"odel-Landauer-Prigogine conjecture, suggesting that closure pathologies across formal systems, computation, and thermodynamics share a common structure.

Problem

• dissipative structures
• Bayesian inference
• non-equilibrium thermodynamics
• learning systems
• Gödel's incompleteness

Innovation

• Bayesian inference
• dissipative structures
• non-equilibrium thermodynamics
• information geometry
• sustainable AI