Towards a Geometry and Analysis for Bayesian Mechanics

The paper explicitly asserts that, for a certain class of controlled non‑equilibrium steady states (NESS), maximising constrained entropy and minimising variational free energy are equivalent; it gives the key factorisation F(µ,b)=Eq[ln q(η|µ)−ln p(η|µ,b)]−ln p(µ,b) and develops the max‑entropy setup S[p;J] with constraints, connecting potentials J to −ln p at steady state (see eq. (4) for the factorisation and surrounding discussion; also the constrained entropy functional (2); and Theorem 6.2 on NESS with J(x)=E(x−x̂) and the E[E(x−x̂)]=0 constraint) . However, the NESS argument relies on coarse‑graining and near‑equilibrium assumptions (e.g., effective equilibrium, local detailed balance), which the paper acknowledges rather than fully proving in general . The model’s solution mirrors the paper’s logic: (i) a standard Jaynes variational derivation producing p*(x)∝e^{−λJ(x)} and identifying J with −ln p(x;κ)+c, and (ii) the same free‑energy factorisation with the expected minimiser q=p(η|µ,b). That said, the model contains a sign error when choosing the additive constant c: to enforce E_p[J]=0 with J=−ln p+c, one needs c=−H(p), not c=H(p). The model also omits the paper’s caveats on near‑equilibrium/LD balance that underpin the effective‑equilibrium step. Because the paper’s core claims are provided at a sketch/constructive level and the model contains a concrete (though correctable) error and elides some assumptions, we judge both as incomplete rather than fully rigorous or fully correct .