Agency Requires a Coarse-Graining It Cannot Derive

Every agent runs on a compressed self-model it could never have computed from scratch.

The core problem is this: a living agent doesn't regulate its behavior by tracking every atom in its body. It tracks a small set of coarse variables — body temperature, hunger, threat level, a compressed self-description — and acts on those. This is not a limitation; it's the only way regulation is possible at all. The Algorithmic Regulator Theorem (an earlier BCOM result) already established that a successful regulator must carry a model of the world compressed into such variables. What this paper asks is the next question: can the agent compute which variables to use, given a complete description of its own microstate? The answer is no, and the paper proves it.

The proof strategy is clean. First, the paper defines a "regulation-sufficient projection" — a map from the agent's full microstate down to a coarse description that retains everything relevant to staying alive, up to a small tolerance. Then it introduces a new object called the targeted Kolmogorov structure function: the best model of bounded complexity that preserves information about a specific target (the persistence-relevant future), rather than about the string itself. The key lemma shows this object is uncomputable, by subsumption — the ordinary (untargeted) structure function is a special case, and Vereshchagin and Vitányi already proved that is uncomputable. The main theorem (Theorem B) then follows: no algorithm can return the shortest regulation-sufficient projection, because doing so would solve the uncomputable targeted structure function. The regulatory coarse-graining is therefore algorithmically emergent in a precise, formal sense.

Three corollaries sharpen the result. First, the optimal Kolmogorov blanket — the boundary that best separates "agent" from "world" — is uncomputable and generically non-unique. There is no fact of the matter about exactly where an agent ends; boundary fuzziness is a theorem, not a metaphor. Second, since the optimal projection cannot be derived but must exist (any persisting agent already carries one), it must have been acquired — through evolution, development, learning, or culture — never deduced from first principles. Third, agents find good-enough projections empirically, at "crack points" in the structure function where a small increase in model complexity buys a large drop in residual unpredictability. This search has no stopping certificate: self-modeling is open-ended by necessity.

The paper is careful about what it does and doesn't claim. The uncomputability is uniform — no single algorithm works for all agents — not a claim that every individual agent's projection is hard in isolation. And "cannot derive" does not mean "must inherit genetically"; the space of acquisition routes is wide. What is ruled out is derivation-from-substrate, full stop. The companion paper WP0192 handles the reflexive case: even the variables that constitute the self are subject to the same barrier. Anderson's "More is Different" has always been an intuition; this paper puts an algorithmic-information theorem underneath it.