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

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.