Bayesian Emergent Dissipative Structures (BEDS)

• Bayesian Emergent Dissipative Structures (BEDS) are a framework for continuous probabilistic inference that accounts for dynamic belief decay and energy constraints.
• It quantifies the trade-off between thermodynamic power and precision through exponential dissipation and Bayesian updates, establishing concrete energetic bounds.
• BEDS introduces new inferential problem classes beyond classical decidability and links closure-induced pathologies in logic, computation, and thermodynamics.

Bayesian Emergent Dissipative Structures (BEDS) constitute a formal framework for continuous probabilistic inference in the presence of unavoidable information loss (dissipation) and explicit energy constraints. Distinct from classical models assuming infallible memory and computation, BEDS addresses scenarios—such as brain-like cognition or embedded sensor networks—where beliefs are maintained dynamically and precision decays unless counteracted by active information refresh, incurring thermodynamic costs. It establishes a fundamental link between thermodynamic power, inference precision, and dissipation, introducing new problem classes beyond the scope of classical Turing-style decidability. The framework also posits the Gödel-Landauer-Prigogine (GLP) conjecture, relating closure-induced pathologies in logic, computation, and thermodynamics to a common structural origin (Caraffa, 5 Jan 2026).

1. Formal Definition and Framework

A BEDS system is defined as a tuple $$\mathcal{B} = (\Theta,\,q_0,\,\gamma,\,\varepsilon)$$ where:

• $\Theta\subseteq\mathbb{R}^d$ is the parameter space of hypotheses.
• $q_0(\theta)$ denotes the initial probability density on $\Theta$.
• $\gamma>0$ governs exponential dissipation (precision decay) rate.
• $\varepsilon>0$ is a “crystallization threshold” for variance.

Belief dynamics result from the interplay of:

• Dissipation: In absence of new data, precision $\tau(t) = 1/\sigma^2(t)$ of a Gaussian belief $q_t=N(\mu_t, \sigma_t^2)$ decays as

$$\frac{d\tau}{dt} = -\gamma\,\tau \implies \tau(t) = \tau_0 e^{-\gamma t}$$

• Bayesian Updates: Upon observing data of precision $\tau_D$, posterior precision jumps: $\tau^+ = \tau^- + \tau_D$.

Crystallization is defined as the event $\mathrm{Var}[q_T] < \varepsilon$, after which the system halts with estimate $\theta^* = \mathbb{E}[q_T]$. The BEDS framework captures the ongoing energetic and informational requirements to maintain and refine beliefs in dissipative systems (Caraffa, 5 Jan 2026).

2. Thermodynamic Trade-offs: The Energy-Precision-Dissipation Theorem

The central BEDS theorem defines the minimum continuous power required to sustain desired belief precision amid dissipation:

• At steady state, the system must satisfy $\lambda \tau_D = \gamma \tau^*$, with
  • $\lambda$: average rate of observations,
  • $\tau_D$: precision gain per observation,
  • $\tau^*$: maintained precision.

Each observation involves an energetic cost, derived from Landauer's principle,

$$E_{\mathrm{obs}} \geq k_B T \Delta H, \qquad \Delta H = \tfrac{1}{2} \ln \left( 1 + \frac{\tau_D}{\tau} \right)$$

where $k_B$ is Boltzmann’s constant and $T$ temperature.

The minimum continuous power to maintain steady precision is

$$P_{\min} = \lambda E_{\mathrm{obs}} = \frac{\gamma \tau^*}{\tau_D} k_B T \tfrac{1}{2} \ln \left( 1 + \frac{\tau_D}{\tau^*} \right)$$

In the regime $\tau_D \ll \tau^*$, this reduces to a universal bound

$$P \geq \frac{\gamma k_B T}{2}, \quad \text{so that} \quad P \propto \gamma \tau^*$$

Thus, the energetic cost of belief maintenance is fundamentally linked to both dissipation and desired precision. For variance $\sigma^{*2} = 1/\tau^*$, $P_{\min} \propto \gamma / \sigma^{*2}$, indicating that improving precision (reducing variance) is quadratically costly in power (Caraffa, 5 Jan 2026).

3. BEDS-Oriented Problem Classes and Classical Decidability

The BEDS framework delineates new classes of inference problems unaligned with classical Turing-decidability:

Class	Criterion	Energy Characterization
BEDS-Attainable	$\lim_{t\to\infty} \mathrm{KL}(q_t \| \pi^*) = 0$ (finite energy)	$\int_0^\infty P(t)dt < \infty$
BEDS-Maintainable	$\mathrm{KL}(q_t \| \pi^*) < \delta$ after $T_0$	$\sup_{t>T_0} P(t) < \infty$
BEDS-Crystallizable	At $T$, $\mathrm{Var}[q_T] < \varepsilon$ and $|\mathbb{E}[q_T] - \theta^*| < \delta$	Halts at finite energy

The hierarchy is strict: Crystallizable $\implies$ Attainable, but not conversely. For example, tracking a moving target can be BEDS-attainable without ever crystallizing. BEDS classes concern continuous accuracy under bounded energetic budgets, not just halting on finite inputs. Some classical decision tasks need unbounded memory (not BEDS-maintainable), and certain continuous tracking tasks are BEDS-maintainable without classical halting semantics (Caraffa, 5 Jan 2026).

4. The Gödel-Landauer-Prigogine (GLP) Conjecture: Closure Pathologies

The BEDS framework identifies analogous pathologies that arise in three domains under enforced closure:

• Formal logic: Gödel incompleteness when axiomatically closed.
• Computation: Irreversible bit-erasure energy cost with no heat export.
• Thermodynamics: Entropy increase in closed (isolated) systems.

The GLP Conjecture states: In any self-referential system $\mathcal{S}$, enforcing

• O: closed to external flux,
• D: no entropy export,
• R: no hierarchical recursion

produces characteristic pathology (e.g., incompleteness, diverging computation cost, increasing disorder). Restoration of openness, dissipation, and recursion resolves these, but with energetic cost per the Energy-Precision theorem.

Evidence cited includes:

• Mathematical practice as an open, dissipative, multi-level endeavor (meta-levels, communal interaction) avoids incompleteness.
• Biological brains’ open, dissipative, and hierarchical structure averts formal logical pathologies.
• AI models with “frozen” weights (no learning/forgetting) manifest “hallucinations” and systematic errors.

Testable predictions involve reduced hallucination under continual learning with structured forgetting, energy costs for logical consistency in proof assistants scaling with dissipation, and the measurable historical decay of unproductive mathematical branches (Caraffa, 5 Jan 2026).

5. Illustrative Scenarios

Key examples illustrate BEDS principles:

• Gaussian Belief Tracking: Precision decays exponentially due to dissipation and rebounds at discrete data arrivals; stationary accuracy is maintained by balancing $\gamma\tau$ against $\lambda\tau_D$.
• Drifting Target: When the target parameter follows $\theta^*(t)=t$, the system is BEDS-attainable (sustains tracking with $P \propto \gamma$), but never crystallizable since absolute accuracy is unattainable.
• Low-Power Sensor Networks: Power expenditures to maintain even minimal precision are governed by the universal lower bound $P \geq \gamma k_B T / 2$, irrespective of the specific desired accuracy, so long as updates are small.

These scenarios clarify the dissipation-driven nature of energetic cost in continuous inference and the resulting limits on memory and inference quality in practical systems (Caraffa, 5 Jan 2026).

6. Broader Implications and Open Problems

BEDS establishes a foundational bound on the power-precision trade-off for any embodied inference agent, with implications for:

• Continuous inference: First-principles bounds for energy-constrained inference devices.
• Thermodynamics of learning: Reversible computation cannot bypass the energetic cost imposed by ongoing belief maintenance.
• Machine learning architectures: Continuous forgetting and relearning (dissipative approaches) may enhance robustness to nonstationary environments.
• Neuroscience: Provides a quantitative rationale for the human brain’s $\sim$20 W power budget as expenditures to offset synaptic dissipation.
• Foundations: Suggests a unifying “closure cost” framework bridging logic, computation, and thermodynamics.

Open problems include generalization to non-Gaussian priors, accommodation of moving (time-varying) targets, the analysis of multi-agent inference, and the quantitative formulation of “logical entropy” central to the GLP conjecture (Caraffa, 5 Jan 2026).