  In this paper, a simple case of Bayesian mechanics under the free energy
principle is formulated in axiomatic terms. We argue that any dynamical system
with constraints on its dynamics necessarily looks as though it is performing
inference against these constraints, and that in a non-isolated system, such
constraints imply external environmental variables embedding the system. Using
aspects of classical dynamical systems theory in statistical mechanics, we show
that this inference is equivalent to a gradient ascent on the Shannon entropy
functional, recovering an approximate Bayesian inference under a locally
ergodic probability measure on the state space. We also use some geometric
notions from dynamical systems theory$\unicode{x2014}$namely, that the
constraints constitute a gauge degree of freedom$\unicode{x2014}$to elaborate
on how the desire to stay self-organised can be read as a gauge force acting on
the system. In doing so, a number of results of independent interest are given.
Overall, we provide a related, but alternative, formalism to those driven
purely by descriptions of random dynamical systems, and take a further step
towards a comprehensive statement of the physics of self-organisation in formal
mathematical language.
