One Ledger, Two Projections: The Thermodynamic Arrow and Pattern Persistence under Reversible Dynamics

Reversible physics conserves total information, but coarse descriptions can still grow more disordered — and persistent patterns can still survive — because these are answers to two completely different questions.

The paper's central move is to stop conflating three things that are easy to muddle: (1) total algorithmic information is conserved under reversible dynamics, (2) thermodynamic entropy increases, and (3) some macroscopic patterns persist through time. All three can be simultaneously true. The paper builds the precise accounting that keeps them from bleeding into each other.

The key tool is what the authors call the projection ledger. A projection is just a coarse-graining map — it takes a complete microstate and throws away most of it, retaining only some summary (a macrostate). The ledger says: the cost of describing the full microstate equals the cost of describing the macrostate, plus the cost of specifying which microstate within that macrostate you're in, minus a correction called randomness deficiency. Deficiency measures how atypical the realized microstate is relative to a random draw from its current macrostate — how much simpler it is than a generic member of its fiber. Under reversible dynamics, total complexity is conserved, so changes in the two-part coarse-grained cost track changes in deficiency exactly. This is the thermodynamic arrow reread: as entropy grows, the current microstate becomes increasingly atypical relative to its present macrostate, because it carries fine-grained correlations with a special low-entropy past that the macrodescription cannot see. The paper is careful to say this is not a derivation of the second law — it's the algorithmic bookkeeping identity that holds once you've already accepted the Boltzmannian setup.

Pattern persistence is a separate, horizontal question. Instead of comparing a microstate to its current fiber, you compare an observer's compressed description of a pattern at one time to that same description at a later time. Persistence is defined as the normalized mutual algorithmic information between these two descriptions — roughly, what fraction of the description is reusable across the time gap. The paper proves that high persistence directly bounds how many bits the observer needs to update their description: if persistence is θ, the update costs at most (1−θ)K₀ bits. A nearly-persistent pattern is one whose identity-bearing description is almost entirely recycled from one moment to the next, even as the underlying microstate changes completely.

The logical independence of these two criteria is demonstrated by explicit reversible register constructions — simple bit-swap machines — showing all four combinations: deficiency growing with persistence high, deficiency flat with persistence high, deficiency growing with persistence vanishing, and so on. This rules out any shortcut argument that entropy increase threatens persistent patterns, or that persistent patterns somehow resist entropy increase. They operate on orthogonal axes. The paper closes by naming the genuinely hard open problem: formation — not whether a persistent representation can exist in principle, but which dynamics actually drive a system into a region where such a representation becomes reachable in the first place.