CALLIOPE · BCOM

We propose an Algorithmic Information Theory (AIT) framework to formalize pattern, stable pattern, and persistence in an ``algorithmic soup'' of interacting dynamical structures. The core tool is mutual algorithmic information (MAI): persistence is operationalized as sustained shared algorithmic information across time, while patterns are defined as (near-)minimal algorithmic sufficient statistics that compress observations. We argue that boundaries of patterns are inherently fuzzy because multiple near-optimal model--noise decompositions exist under different coarse-grainings and coding choices. Finally, we connect this view to the intuition that living beings are fundamentally dynamical patterns, and interpret death as an irreversible loss of persistence (collapse of time-separated MAI below viability thresholds).

Compression is identity: a thing persists as long as its future can be explained by its past.

Kolmogorov complexity — the length of the shortest program that produces a string — gives us a way to talk about regularity without invoking probability distributions or ensembles. This paper uses that tool to define what a "pattern" is for a single trajectory through time, not a statistical average over many. A pattern is the minimal model that compresses an observation: the part of the shortest two-part description (model + residual noise) that does the actual work. The residual is just randomness; the model is the pattern. This is clean and precise, and it grounds a word that usually floats free of any formal anchor.

Persistence is then defined as sustained mutual algorithmic information (MAI) between a system's description at time t and its description at time t+Δ. MAI measures how much knowing one object shortens the description of the other. If that quantity stays high across a range of time horizons, the system is genuinely persisting — its future is algorithmically continuous with its past. The paper normalizes this by the complexity of the smaller snapshot, so you're not just rewarding large, complicated systems for being large. A persistent pattern is one where this normalized score stays above a threshold for all horizons in some set. That's the formal reading of "the same thing is still there."

The boundary fuzziness result is the most philosophically interesting piece. Where does a pattern end and its environment begin? The paper shows this question has no unique answer: multiple different ways of carving up the world (different coarse-grainings, different extractors g) can yield near-equally-good compressions that nonetheless disagree substantially about what the subsystem looks like. Boundaries are ε-fuzzy by construction, not by ignorance. This isn't a bug — it's a theorem about the limits of compression-based individuation, and it has real consequences for any theory that tries to draw sharp lines around organisms or agents.

The paper then applies this machinery to life and death. A living system is one with sustained, robust, non-trivial persistence across many horizons — not mere periodicity, but historically-structured organization (invoking Bennett's logical depth). Death is the irreversible collapse of that MAI: the future no longer compresses from the past, and Landauer's principle explains why you can't get it back cheaply. The connection to autopoiesis (Maturana and Varela) and to Friston's Markov blankets is noted but left as open territory. The paper is honest that Kolmogorov complexity is uncomputable, and points to normalized compression distance as the practical proxy for empirical work.