# The Algorithmic Regulator

arXiv: https://arxiv.org/abs/2510.10300 · HTML: https://mlsanity.com/paper/2510.10300/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 75 references that are in the map.
- First posted on arXiv 11 Oct 2025.

## Builds on (most cited first, 13 of 13)

- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Variational image compression with a scale hyperprior](https://mlsanity.com/paper/1802.01436/) (2018)
- [SoundStream: An End-to-End Neural Audio Codec](https://mlsanity.com/paper/2107.03312/) (2021)
- [Joint Autoregressive and Hierarchical Priors for Learned Image Compression](https://mlsanity.com/paper/1809.02736/) (2018)
- [Language Modeling Is Compression](https://mlsanity.com/paper/2309.10668/) (2023)
- [Learned Image Compression with Mixed Transformer-CNN Architectures](https://mlsanity.com/paper/2303.14978/) (2023)
- [Video Compression With Rate-Distortion Autoencoders](https://mlsanity.com/paper/1908.05717/) (2019)
- [Practical Lossless Compression with Latent Variables using Bits Back Coding](https://mlsanity.com/paper/1901.04866/) (2019)
- [Transformer-based Image Compression](https://mlsanity.com/paper/2111.06707/) (2021)
- [LLMZip: Lossless Text Compression using Large Language Models](https://mlsanity.com/paper/2306.04050/) (2023)
- [Lenia - Biology of Artificial Life](https://mlsanity.com/paper/1812.05433/) (2018)
- [Compression with Flows via Local Bits-Back Coding](https://mlsanity.com/paper/1905.08500/) (2019)
- [Lossless data compression by large models](https://mlsanity.com/paper/2407.07723/) (2024)

## Abstract (arXiv)

The regulator theorem states that, under certain conditions, any optimal controller must embody a model of the system it regulates, grounding the idea that controllers embed, explicitly or implicitly, internal models of the controlled. This principle underpins neuroscience and predictive brain theories like the Free-Energy Principle or Kolmogorov/Algorithmic Agent theory. However, the theorem is only proven in limited settings. Here, we treat the deterministic, closed, coupled world-regulator system (W,R) as a single self-delimiting program p via a constant-size wrapper that produces the world output string~x fed to the regulator. We analyze regulation from the viewpoint of the algorithmic complexity of the output, K(x). We define R to be a good algorithmic regulator if it reduces the algorithmic complexity of the readout relative to a null (unregulated) baseline varnothing, i.e., [ Delta = Kbig(OW,varnothingbig) - Kbig(OW,Rbig) > 0. ] We then prove that the larger Delta is, the more world-regulator pairs with high mutual algorithmic information are favored. More precisely, a complexity gap Delta > 0 yields [ Prbig((W,R)mid xbig) ≤ C,2^,M(W:R),2-Delta, ] making low M(W:R) exponentially unlikely as Delta grows. This is an AIT version of the idea that ``the regulator contains a model of the world.'' The framework is distribution-free, applies to individual sequences, and complements the Internal Model Principle. Beyond this necessity claim, the same coding-theorem calculus singles out a canonical scalar objective and implicates a planner. On the realized episode, a regulator behaves as if it minimized the conditional description length of the readout.