We present BEDS (Bayesian Emergent Dissipative Structures), a theoretical framework that unifies concepts from non-equilibrium thermodynamics, Bayesian inference, information geometry, and machine learning. The central thesis proposes that learning, across physical, biological, and computational systems, fundamentally constitutes the conversion of flux into structure through entropy export. Building on Prigogine's theory of dissipative structures, we establish a formal isomorphism between thermodynamic processes and Bayesian updating, demonstrating that sustainable learning systems must follow dissipative patterns where crystallized posteriors become priors for subsequent levels of emergence. We derive fundamental mathematical constants (e, π, φ) as fixed points of Bayesian inference under minimal axioms, suggesting these constants emerge necessarily from any system capable of representing and updating uncertainty. Furthermore, we propose a conjecture linking Gödel's incompleteness theorems to thermodynamic constraints, hypothesizing that pathologies of formal systems (incompleteness, undecidability) are structurally analogous to dissipation deficits in physical systems. As practical validation, we present a peer-to-peer network architecture implementing BEDS principles, achieving six orders of magnitude improvement in energy efficiency compared to existing distributed consensus systems while enabling continuous learning. This work bridges fundamental physics, mathematical logic, and practical system design, offering both theoretical insights into the nature of learning and computation, and a concrete pathway toward sustainable artificial intelligence.

翻译：我们提出BEDS（贝叶斯涌现耗散结构）理论框架，该框架统一了非平衡态热力学、贝叶斯推断、信息几何与机器学习中的核心概念。核心论点指出：跨越物理、生物与计算系统的学习过程，本质上是通过熵输出将通量转化为结构的过程。基于普里高津的耗散结构理论，我们建立了热力学过程与贝叶斯更新的形式同构，证明可持续的学习系统必须遵循耗散模式——其中结晶化的后验分布将作为后续涌现层级的先验分布。我们从最小公理出发推导出基本数学常数（e, π, φ）作为贝叶斯推断的不动点，表明这些常数必然从任何能够表征和更新不确定性的系统中涌现。此外，我们提出一个将哥德尔不完备性定理与热力学约束相联系的猜想，假设形式系统的病理特征（不完备性、不可判定性）在结构上类似于物理系统中的耗散缺陷。作为实践验证，我们展示了一种实现BEDS原理的对等网络架构，在支持持续学习的同时，相比现有分布式共识系统实现了六个数量级的能效提升。这项工作连接了基础物理学、数理逻辑与实用系统设计，既为学习与计算的本质提供了理论洞见，也为实现可持续人工智能提供了具体路径。