What Flows When Information Is Conserved? Algorithmic Localization, Materialization, and Persistence

When a reversible system evolves, its total algorithmic complexity stays fixed — but the way that complexity is distributed across its parts can change dramatically. This paper builds a clean accounting framework for exactly that redistribution.

The core idea is simple. Take a system split into two parts, A and W. Their joint description length is conserved under reversible dynamics. But A's individual description length can grow, W's can shrink, or they can become more algorithmically entangled — as long as the three-way budget balances. The paper introduces "localization coordinates" (L_A, mutual algorithmic information, L_W) as a reparameterization of the standard complexity profile, chosen so the three numbers always sum to the joint complexity. This makes the conservation constraint visually obvious: any change in one coordinate must be offset by changes in the others.

The XOR copy gate is the paper's running example and earns its keep. Start with (x, 0): x is fully localized in A, W is blank. Apply XOR: you get (x, x). W now has a full copy of x. Yet the joint complexity hasn't budged — it was n bits before and n bits after. What changed is that the n bits of "A-specific" description became n bits of cross-partition mutual information. The receiver gained a copy without the sender losing one, and without the system gaining new information. The bookkeeping identity K(A,W) = K(A) + K(W) − I_K(A:W) absorbs it all.

A crucial distinction runs through the paper: generic mutual algorithmic information is not the same as a literal common string sitting in both subsystems. Two strings can have large mutual information with no extractable common file — this is a known result in algorithmic information theory. So the paper carefully separates "profile-level localization" (always defined, measures description savings) from "materialized localization" (a named description D is actually recoverable from a subsystem with few extra bits). The canonical XOR/SWAP register constructions live in the materialized regime; arbitrary pairs do not. This distinction matters because it determines when you can literally say "the same information moved" versus merely "the dependence profile changed."

The paper then connects this to persistence across time. If the same identity-bearing description D_P can be recovered cheaply from a pattern at two different times, then those two pattern states must share at least that much algorithmic information — a lower bound on cross-time persistence follows directly from the same recoverable-description overlap lemma used for the spatial case. Recurrent recoverability of a named description is thus a sufficient mechanism for persistence, not its definition. Causal direction — which subsystem drove which — requires something extra: an intervention, a mechanism, a declared counterfactual. The localization profile is symmetric; causation is not. The paper stops deliberately at this "kinematic boundary," leaving the directed causal and regulatory machinery to companion papers.