Information mechanics: conservation and assimilation

Abstract

Inference and learning are cast as optimisation, yet invariant constraints governing uncertainty reduction remain unclear. This work presents information mechanics (infomechanics), a first-principles framework that describes informational structure in two canonical state coordinates. From Bayes' pointwise identity, minimal requirements of additivity, symmetry, and finite-resolution robustness select two robust additive projections, yielding conservation identities for entropy and Fisher information, governing global uncertainty and local geometry. Their residual defines the non-additive, coordinate-scale-invariant information potential $\Phi$, separating the entropic baseline from residual geometric complexity. $\Phi$ vanishes uniquely for isotropic Gaussians, decreases under Gaussian coarse-graining, and in finite-resolution multimodal landscapes asymptotically approaches the logarithm of the effective number of local optima. Extending the formalism to the Markov chain linking hidden states, observations, and internal representations yields assimilation inequalities constraining faithful external-state inference. Small bidirectional information potentials constrain representations towards near-linear relations with hidden causes. Together, these results identify invariant constraints underlying inference, learning, and computation across biological and artificial systems, irrespective of implementation.