Information mechanics: conservation and assimilation

Inference and learning are commonly cast in terms of optimisation, yet the 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.

Starting from the pointwise identity implied by Bayes' rule, minimal requirements of additivity, symmetry, and finite-resolution robustness select only two distinct additive projections, yielding conservation identities for Shannon entropy, governing global uncertainty, and Fisher information, encoding complementary local geometry.

Because part of Fisher information is fixed by entropy, the residual structure is captured by a non-additive, coordinate-scale-invariant state function, the information potential $\Phi$.

This yields a two-coordinate description that separates the entropic baseline from residual geometric complexity. $\Phi$ vanishes uniquely for isotropic Gaussian distributions and decreases under Gaussian coarse-graining.

In finite-resolution multimodal landscapes, $\Phi$ asymptotically scales with the logarithm of the effective number of local optima, linking information geometry to inference difficulty.

The same two-coordinate formalism extends to the Markov chain linking hidden states, observations, and internal representations, yielding assimilation inequalities that constrain faithful external-state inference.

Together, these results identify invariant constraints underlying inference, learning, and computation across biological and artificial systems.