Thermodynamic Limits of Causal Inference: A Feedback-Corrected Uncertainty Relation for Hamiltonian Causal Models

We ask what it costs, thermodynamically, to measure a causal effect to a given precision. Causal inference from physical systems requires reconciling statistical notions of causality with thermodynamic irreversibility. Hamiltonian Causal Models (HCMs) address this by witnessing causal effects through entropy production in controlled nonequilibrium processes. While HCMs support adaptive interventions that dynamically steer systems to maximize inferential power, such feedback renders standard entropy production measures inapplicable: the time-reversed protocol becomes acausal, inflating apparent dissipation beyond genuine thermodynamic cost. Here we derive a feedback-corrected Thermodynamic Uncertainty Relation (TUR) for HCM causal inference, showing the relevant cost is marginal entropy production, comprising local dissipation, information-flow rate, and heat correction. Using Fokker-Planck analysis and Girsanov calculus, we demonstrate that the information-flow term quantifies the rate at which probability current explores cross-node correlations. Numerical experiments show: (i) the TUR holds across open-loop and adaptive protocols, (ii) adaptive control can raise the thermodynamic ceiling (Maxwell's demon gain), and (iii) the bound tightens as control policies diverge. The feedback-corrected TUR gives a lower bound on the thermodynamic cost of causal inference: resolving a causal contrast to precision, or relative uncertainty, $\varepsilon$ requires dissipating at least $2k_{\rm B}T/\varepsilon^2$ of energy, analogous to the Landauer's principle that sets the minimum energy required to erase information in physical computing machines.