A Geometric Framework for the Inference–Action Boundary: Wasserstein Dynamics, Bures Covariance Geometry, and Thermodynamic Constraint

The transition from distributed probabilistic inference to committed action can be posed as a problem of changing state-space geometry. In broad inferential regimes, belief states are naturally described as probability distributions and their evolution is governed by transport geometry. In low-uncertainty regimes, by contrast, the local structure is more naturally described in terms of Gaussian covariance. This paper develops a formal framework for that transition and proposes a biologically constrained thermodynamic closure. We introduce an extended state space M = X Z, where X is a hypothesis space and z [0, ) is a scalar uncertainty coordinate. The variable z is interpreted both as a coarse precision parameter and as a candidate hidden coordinate through which reduced non-Markovian dynamics on X may be embedded into a Markovian dynamics on M. Within this setting, a stochastic-resetting process drives the system intermittently from a distributed inferential regime toward a low-uncertainty boundary. Under standard Laplace asymptotics, the bulk Wasserstein geometry reduces locally to Bures geometry on the covariance sector of a Gaussian boundary regime. Analysis of that boundary regime using Williamson’s symplectic decomposition yields a lower bound on independent single-mode covariance compression and motivates the study of cross-mode covariance reorganisation in the deep-boundary limit. Under an additional architectural assumption, this reorganisation can be interpreted as cross-partition covariance structure. The paper then proposes a constitutive closure linking the metric norm of free-energy descent to haemodynamic demand, with cerebral autoregulation represented as a substrate-level constraint on admissible gradient-flow dynamics. This produces a self-consistency condition in which the effective metric depends on thermodynamic support. The resulting relation is not a tensorial field equation and is not claimed to be equivalent to the Einstein equation; the comparison is only structural, in the limited sense that geometry is treated as constrained by the processes it supports. Finally, the paper discusses reported cardiac-phase-linked spin-coherence measurements as a possible empirical point of contact with the proposed boundary regime.