The Good Algorithmic Regulator Theorem: Model It, Transmit It, or Leave It in the World

The claim that successful regulation requires an internal world model motivates predictive brain theories and the algorithmic agent of Kolmogorov Theory. Conant and Ashby's good regulator theorem and the internal model principle establish model relations under specific assumptions. The Algorithmic Regulator Theorem (ART) extended the question to individual finite outputs, deriving a probabilistic bound that relates the posterior weight of a specified world--regulator explanation to the output complexity reduction, regulator complexity, and shared algorithmic information. Its proposed aggregate concentration on regulators sharing substantial world information fails: a family of complex worlds regulated by one fixed clamp retains nonvanishing posterior mass. We derive a deterministic balance organized by global information conservation: under computable reversible dynamics, the complete regulated episode retains the information needed to recover the initial world and replay its uncontrolled output. Let $W,R$ describe the initial world and regulator, and let $O_{W,\varnothing}^{(N)},O_{W,R}^{(N)}$ be the matched uncontrolled and regulated outputs over $N$ steps. With $K$ denoting prefix Kolmogorov complexity and $C$ the fixed comparison frame, the Good Algorithmic Regulator Theorem (GART) bounds the gap $\Delta_N=K(O_{W,\varnothing}^{(N)}\mid C)-K(O_{W,R}^{(N)}\mid C)$ by $\Delta_N\leq I_K(W{:}R\mid C)+L_{\mathrm{cf}}^{(N)}+O(\log N)$. Here $I_K$ is mutual algorithmic information and the residual $L_{\mathrm{cf}}^{(N)}$ is the description still needed to recover the uncontrolled output given the regulated output and initial regulator. A large gap and a small residual therefore force substantial initial information shared with the world. When the selected episode records suffice for reconstruction, they account for the residual through actions, final memory, and complementary world records: model it, transmit it, or leave it in the world. A retained generative model makes the output constant while preserving its information, with the gap bounded by the model's complexity; feedback and clamps can produce a large gap while leaving a large residual. GART's deterministic balance concerns individual finite records and requires neither a disturbance distribution nor a prior over programs. Its probabilistic reading follows: conditioning on a residual bound repairs ART's aggregate inference under any probability law on admissible world--regulator pairs, so shared information cannot fall below the gap by more than that bound and the coding allowance. For worlds and regulators drawn independently from computable laws, an exponential bound limits the probability of a large gap with a small residual.