Tracking Bound Conjecture

Determine the scaling law for the minimum power required for a BEDS system to maintain a belief about a target parameter that moves with velocity v in the parameter space, and show that the minimum power satisfies P_min proportional to γ τ* + v^2 τ*, where γ is the dissipation rate and τ* is the maintained precision.

Background

The Energy-Precision Theorem characterizes power requirements to maintain a fixed belief under dissipation. The authors extend this to moving targets and conjecture an additional tracking term that scales with the square of the target’s velocity.

This conjecture aims to quantify the cost of continuous tracking in dynamic environments, augmenting the dissipation-dependent term with a velocity-dependent term.

References

Conjecture [Tracking Bound] For a target moving with velocity $v$ in parameter space, the minimum power scales as: \begin{equation} P_{\min} \propto \gamma \tau* + v2 \tau*\end{equation}

— BEDS : Bayesian Emergent Dissipative Structures : A Formal Framework for Continuous Inference Under Energy Constraints  (2601.02329 - Caraffa, 5 Jan 2026) in Section 7.3 (Open Problems)