CALLIOPE · BCOM

Thesis. We revisit Conant--Ashby’s Good Regulator Theorem (GRT)—that good regulators must contain a model of what they regulate {Conant--Ashby70}{https://doi.org/10.1080/00207727008920220}—and extend it in the Kolmogorov Theory (KT) of agents. Our central claim is observer‑dependent: relative to a chosen interface (coarse‑grained sensors/actuators) and a viability functional, any persistent pattern can be read as a raw agent—a program with a modeling engine, an objective, and a planning engine. From this starting point we stratify agency by action (causal loop‑closure, homeostasis, telehomeostasis) and by learning (automata vs.\ learners), and we propose a concrete definition of life.

Results. (i) We restate GRT in a finite, pedagogical form and show that successful regulation factors through a predictive quotient $M=f(S)$ ; (ii) we prove three elementary theorems: T1 persistence self‑regulation; T2 self‑regulation KT‑agent with a predictive internal model; T3 therefore persistence general (indifferent) agent. (iii) We add a loop‑closure (teletransport) test for causal agency using interventional semantics {Pearl09}{https://www.cambridge.org/core/books/causality/4D0CBAA2643D3F0D1B8B0D51C50C4EC6}, distinguish homeostatic agents as those that regulate variables necessary for their own survival, and define telehomeostatic agents as those acting to preserve a pattern across contexts. (iv) We prove an algorithmic Occam result: among behaviorally equivalent realizations, the coarsest predictive model is Kolmogorov‑minimal {Li--Vitanyi}{https://link.springer.com/book/10.1007/978-3-030-11298-1}, {GTV01}{https://homepages.cwi.nl/~paulv/papers/algorithmicstatistics.pdf}. (v) A POMDP appendix lifts the statements to sequential/partial‑information settings via belief states {Smallwood--Sondik73}{https://doi.org/10.1287/opre.21.5.1071}, {Kaelbling98}{https://doi.org/10.1016/S0004-3702(98)00023-X}.

Definition of life (KT). Life is observer‑relative telehomeostatic learning: a system is alive (at an interface and horizon) if (a) its outputs are causally indispensable for maintaining pattern‑level viability (telehomeostasis), and (b) it learns—its persistent memory accumulates predictive information that changes future input–output mappings and improves performance (measurable via growth in $K(M_t)$ or $I(M_t;history)$ , nonzero directed information from past inputs to future actions, and sublinear regret). This frame reconciles stones (general agents), thermostats (causal automata, non‑homeostatic), and organisms (telehomeostatic learners) under a single, observer‑dependent theory of agency and life, consistent with predictive minimal models {Shalizi--Crutchfield01}{https://arxiv.org/abs/cond-mat/9907176} and semantic/viability perspectives {KolchinskyWolpert18}{https://royalsocietypublishing.org/doi/10.1098/rsfs.2018.0041}.

Persistence is all you need to be an agent — everything else is just how much you do about it.

The core move here is surprisingly simple. If a pattern persists — a stone, a thermostat, a cat, a lineage — then by definition something is keeping it from falling apart. That "something" must be doing implicit regulation, and any successful regulator (by Conant and Ashby's 1970 theorem) must internally encode a model of what it's regulating. So persistence logically implies the existence of a modeling engine, an objective, and a planning engine. That's what the authors call a KT agent. The chain is: persistence → self-regulation → agent with a predictive internal model. Three theorems, each elementary, but the conclusion is striking: even a granite boulder qualifies as an agent under this definition.

That might sound like it inflates "agent" to meaninglessness, but the paper's real contribution is the stratification that follows. Not all agents are equal. The key question is whether the system's outputs are causally necessary for its own survival. You test this with what the authors call a "teletransport" operation: cut the system's outputs and route them to a sink. If viability drops, the outputs matter — the system is a causal agent. If the variables being regulated are specifically the ones the system needs to keep running (glucose, battery charge), it's homeostatic. And if it acts to preserve not just itself but the pattern it instantiates — across substrates, contexts, or generations — it's telehomeostatic. That last category is where the paper locates life. A second orthogonal axis separates automata (bounded memory, fixed behavior) from learners (memory that grows, improves predictions, and drives sublinear regret). Life, in this framework, is telehomeostatic learning: causally indispensable action to preserve a pattern, combined with memory that accumulates and improves.

There's also a clean Occam result. Among all internal models that produce the same input-output behavior, the coarsest one — the one that lumps together environment states that demand identical actions and produce identical outcomes — is Kolmogorov-minimal. The best regulator is not just any model of its environment; it's the simplest sufficient model. This connects the Conant-Ashby theorem to algorithmic information theory and to the ε-machine literature on minimal predictive representations.

The whole framework is explicitly observer-relative. Whether something counts as a causal agent, a homeostatic agent, or alive depends on the interface you choose to observe it through and the viability functional you assign. This isn't a bug — it's the point. The same physical system can be a stone at one granularity and a learner at another. The paper is pedagogical in intent, building a clean conceptual ladder from a 1970 cybernetics result up through Pearl-style causality, POMDP belief states, and algorithmic complexity, with worked examples (stone, thermostat, Roomba, organism, LLM) at each rung.