BEDS: Bayesian Emergent Dissipative Structures: A Formal Framework for Sustainable Digital Twins and Continual Learning Systems

This document presents a theoretical framework called **BEDS** (Bayesian Emergent Dissipative Structures). The central observation: what we call "learning" in machine learning, "dissipation" in thermodynamics, "evolution" in biology, and "proof" in mathematics exhibit similar structural patterns. We propose a unified formalism to describe these patterns.The document is structured in five parts:1. **Analogy** — Building intuition through the river metaphor2. **Conjecture** — Connecting Gödel, Landauer, and Prigogine3. **Formal Results** — Mathematical constants as fixed points of inference4. **Implementation** — A sustainable P2P network architecture5. **Discussion** — Implications and open questions**Disclaimer.** The formalism presented here is deliberately simple — perhaps too simple, and I may have reinvented existing results. This is speculative work born from stepping back and reflecting on my research. It does not include a systematic literature review and does not reflect the rigor of my peer-reviewed academic work. Most of it was developed with Claude Opus 4.5 (claude.ai). However, I find this angle of attack compelling enough to share — hopefully others will find something valuable in it too.