CALLIOPE · BCOM

This note develops a preliminary language for ``algorithmic mechanics'': a description of physical systems in terms of states, dynamics, coarse-grainings, and finite descriptions. The starting point is deliberately conservative. Physics defines systems by state spaces and dynamical laws. In closed classical Hamiltonian and closed quantum systems there is a precise sense in which microscopic information is conserved: Hamiltonian evolution is invertible and phase-space-volume preserving, while unitary quantum evolution preserves the spectrum of the density operator and hence the global von Neumann entropy. Information theory enters when one assigns probabilities to microstates or macrostates; algorithmic information theory enters when one asks for the description length of a particular state, macrostate, trajectory, or boundary record. The note uses the contrast between entropy as a typical index cost and Kolmogorov complexity as an individual description cost to motivate an algorithmic refinement of statistical mechanics. Its central proposal is then surface-local: relative to a partition, a coarse-graining, a time direction, and a finite observer, information flow through a surface can be defined as boundary-mediated conditional mutual algorithmic information, namely the amount by which a boundary history shortens the description of the future state on one side beyond that side's past. Markov blankets provide the probabilistic screening analogue; Kolmogorov blankets provide the corresponding algorithmic screening condition.

Algorithmic mechanics proposes a precise, observer-relative definition of "information flow through a surface" using the language of shortest descriptions rather than probabilities or conserved currents.

The core intuition is simple: physics already tells us that exact, reversible dynamics conserve information — Hamiltonian flow preserves phase-space volume, unitary quantum evolution preserves the spectrum of the density operator. But the moment you coarse-grain, project onto a subsystem, or restrict to what an observer can actually track, that conservation breaks. The paper's starting move is to sharpen this familiar story by replacing Shannon entropy (a cost averaged over an ensemble) with Kolmogorov complexity (the length of the shortest program that produces a particular state). Entropy tells you how many bits a typical microstate in a macrostate requires; complexity tells you how many bits this specific microstate requires. For most states the two agree, but for structured, law-generated, or historically selected states they can differ dramatically — and that difference is exactly where algorithmic mechanics becomes useful.

The central construction is surface-local. Fix a partition of a physical system into interior, exterior, and boundary variables; fix a coarse-graining, a time direction, and a finite observer. The boundary history over an interval carries "algorithmic information flow" into the interior if it shortens the shortest description of the interior's future state beyond what the interior's own past already provides. Formally this is a conditional mutual algorithmic information (MAI): how much does the boundary record compress the future interior description, given the interior's history? The paper also introduces the Kolmogorov blanket — the algorithmic analogue of a Markov blanket — as the condition under which the boundary approximately screens off exterior history from the interior's future. Without that screening condition, the raw cross-cut MAI could reflect common causes or shared initial conditions rather than genuine boundary-mediated exchange.

Causation is deliberately kept optional. The primitive is directed algorithmic relevance — one description compresses another, given a time ordering and a boundary — not causation. The paper develops two optional causal readings: an interventional one (clamp or scramble the boundary record and ask whether the interior future changes) and a separation-based one (the boundary screens off exterior from interior in the actual history). Whether these two readings agree is posed explicitly as an open problem, requiring assumptions about locality, completeness of boundary variables, and the availability of admissible counterfactuals. In globally constrained or superdeterministic worlds, counterfactuals may not exist, and only the structural relevance reading survives. This is the paper's sharpest philosophical point: relevance, not causation, is the right primitive because it is defined on the single realized history and requires no alternative worlds.

The paper is honest about what it is: a working note that isolates a minimal vocabulary — finite descriptions, coarse-grainings, Kolmogorov blankets, boundary-mediated MAI — from which a fuller mechanics could be built. It sketches three open directions: an algorithmic divergence theorem relating interior complexity change to boundary terms, practical estimators using compressors or neural models, and a link to thermodynamic cost via stochastic thermodynamics. The construction is intended as the elementary exchange primitive on which later accounts of regulation, agency, and persistence in Kolmogorov Theory can be grounded, without presupposing any of them.