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.

Reversible physics conserves information exactly; the interesting question is what happens when you can only see part of the picture — and this paper proposes a precise way to measure that.

The core idea is simple to state. In classical and quantum mechanics, exact closed-system dynamics never destroys information: Hamiltonian flow preserves phase-space volume, unitary evolution preserves the spectrum of the density operator. But the moment you coarse-grain — group microstates into macrostates, trace out an environment, restrict to what an observer can actually see — that conservation breaks. Statistical mechanics handles this with entropy, which measures the typical cost of specifying a microstate given a macrostate. Algorithmic information theory (AIT) offers a sharper tool: Kolmogorov complexity, the length of the shortest program that produces a particular state. For most states in a large macrostate these two agree, but for structured, law-generated, or historically selected states, the shortest description can be far smaller than the entropy would suggest. That gap — the "randomness deficiency" — is where algorithmic mechanics lives.

The paper's central construction is surface-local. Fix a partition of a physical system into an interior region A, an exterior E, and a boundary B. The boundary history over some time interval carries "algorithmic information flow" into A if it shortens the shortest description of A's future state beyond what A's own past already provides. Formally this is a conditional mutual algorithmic information (MAI): how many bits does the boundary record save when describing the future interior, after conditioning on the interior's history? The paper also introduces a "Kolmogorov blanket" — the algorithmic counterpart of a Markov blanket — which is the condition that the boundary approximately screens off the exterior from the interior in description-length terms. Without that screening condition, the cross-cut MAI could reflect common causes or shared initial conditions rather than genuine boundary-mediated exchange.

Causation is treated carefully and kept optional. The paper distinguishes two readings of the directed flow: an interventional one (clamp or scramble the boundary and see what changes in A's future) and a separation-based structural one (does the boundary screen off exterior from interior in the actual realized history?). These two readings need not agree, and the paper states their equivalence as an open problem requiring locality, completeness of boundary variables, and a fixed time orientation. More pointedly, in globally constrained or "superdeterministic" worlds, the interventional reading may have no admissible counterfactuals at all — the fiber of globally consistent histories with a different boundary record may be empty. The structural, relevance-based reading survives this; interventional causation does not. This is why the paper takes directed algorithmic relevance — which description compresses which other, given a boundary, time ordering, and observer frame — as the primitive, with causation as an optional interpretive layer on top.

Three open directions are flagged: an algorithmic divergence theorem (decomposing changes in interior complexity into boundary terms plus internal production), practical estimators (Lempel-Ziv compressors, MDL models, neural compressors as computable upper bounds on exact K), and a link to thermodynamic cost via stochastic thermodynamics. The paper is explicitly a working note — it isolates the minimal objects needed and states what is definition versus conjecture, rather than delivering a finished mechanics.