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

Regulation can make a system’s behavior much simpler despite changing disturbances. We show when this simplification requires the regulator to contain a model of its world. If the simplification is large and little additional information is needed to reconstruct what would have happened without regulation, then substantial world information must already be present in the regulator. This is model content in a precise descriptive sense: knowing the regulator helps describe the world more concisely. We compare output records with and without regulation, holding the initial world, disturbances, measurement rule, and horizon fixed. The Good Algorithmic Regulator Theorem bounds their difference in prefix Kolmogorov complexity by the information initially shared between the world and the regulator, plus a counterfactual residual, up to logarithmic coding overhead. The residual is the additional description needed to reconstruct the uncontrolled output from the regulated output and the initial regulator. An information-conserving realization explains where the suppressed information can reside: in the regulator, its actions, or elsewhere in the world. Sufficient episode records make this account quantitative. A retained generative model can guide control and reconstruct the uncontrolled record; reactive feedback and clamps illustrate other ways to obtain the same simple output. Initial model content can include stored data; reuse is assessed through regulation across specified changes in conditions. Conditioning on a bounded residual gives a probabilistic regulator theorem under any probability law on admissible explanations. Independent initial sampling from fixed computable laws also gives an exponential bound on obtaining a large reduction with a small residual. These results identify when successful regulation supports the model-based interpretation underlying classical regulator theorems and algorithmic accounts of agency.