From Boundaries to Models

The Necessity of Regulation Under Uncertainty

Once a boundary exists, it must be maintained—the interior held distinct from the exterior despite perturbation, degradation, and environmental fluctuation. This maintenance problem has a specific structure.

Let the interior state be $\mathbf{s}^{\text{in}} \in \R^m$ and the exterior state be $\mathbf{s}^{\text{out}} \in \R^k$. The boundary mediates interactions through observations $\mathbf{o}_t = g(\mathbf{s}^{\text{out}}_t, \mathbf{s}^{\text{in}}_t) + \bm{\epsilon}_t$ and actions $\mathbf{a}_t \in \mathcal{A}$ (boundary permeabilities, active transport). Persistence requires maintaining $\mathbf{s}^{\text{in}}$ within a viable region $\viable^{\text{in}}$ despite incomplete observation of $\mathbf{s}^{\text{out}}$, stochastic perturbation, degradation of the boundary itself, and finite resources.

The consequence runs deep: regulation requires modeling. Let $\mathcal{S}$ be a bounded system that must maintain $\mathbf{s}^{\text{in}} \in \viable^{\text{in}}$ under partial observability of $\mathbf{s}^{\text{out}}$. Any policy $\policy: \mathcal{O}^* \to \mathcal{A}$ achieving viability with probability $p > p_{\text{random}}$ implicitly computes a function $f: \mathcal{O}^* \to \mathcal{Z}$ where $\mathcal{Z}$ is a sufficient statistic for predicting future observations and viability-relevant outcomes.

POMDP Formalization

A bounded system under uncertainty admits precise formalization as a Partially Observable Markov Decision Process (POMDP): a tuple $(\mathcal{X}, \mathcal{A}, \mathcal{O}, T, O, R, \gamma)$ of state space, action space, observation space, transition kernel, observation kernel, reward function, and discount factor.

The agent never observes $\mathbf{x}_t$ directly, only $\mathbf{o}_t \sim O(\cdot | \mathbf{x}_t)$. The sufficient statistic for decision-making is the belief state—the posterior over world states given the history:

The belief state updates by Bayes’ rule, and a classical result establishes that $\belief_t$ is a sufficient statistic for optimal decision-making: any optimal policy $\policy^*$ can be written as $\policy^*: \Delta(\mathcal{X}) \to \mathcal{A}$, mapping belief states to actions.

It is tempting to conclude that any system beating random under partial observability is maintaining a belief state, and therefore has a model of the world. The conclusion is too strong in a way that matters later. What the result establishes is that successful regulation implies some sufficient statistic — some mechanism preserving viability-relevant information from history — not that it implies a complete Bayesian model. A reflex exploits one regularity without representing environmental dynamics at all, and calling its trigger threshold a "world model" makes the term do no work.

The World Model

In complex environments the full belief state is computationally intractable. Real systems maintain compressed representations.

A world model is a parameterized family of distributions $\worldmodel_\theta = \{p_\theta(\mathbf{o}_{t+1:t+H} | \mathbf{h}_t, \mathbf{a}_{t:t+H-1})\}$ that predicts future observations given history $\mathbf{h}_t$ and planned actions, for some horizon $H$. Modern implementations in machine learning typically use recurrent latent state-space models:

So the honest statement of the requirement is graded rather than categorical. Control under partial observability requires some mechanism for preserving viability-relevant information from history; how rich that mechanism must be scales with uncertainty, delay, and branching complexity. At the low end sits a threshold on a single sensed variable. At the high end sits a full generative model over latent dynamics. Nothing forces a system up that range. What moves it up is an environment in which the cheap mechanisms stop paying.

The same structure is what engineers arrive at when they build capable artificial agents — Dreamer (Hafner et al., 2020), MuZero (Schrittwieser et al., 2020), and JEPA (LeCun, 2022) differ in what they reconstruct and share the latent-dynamics core. The convergence is not coincidence; it reflects the mathematical structure of the control-under-uncertainty problem.

The Necessity of Compression

The world model is not merely convenient—it is constitutively necessary. This follows from an asymmetry between the world and any bounded system embedded within it, which the information bottleneck (Tishby et al., 1999) makes precise. Let $\mathcal{W}$ be the world state space and $\mathcal{S}$ a bounded system with finite computational capacity $C_\mathcal{S}$. Then:

The compression ratio $\chi = \dim(\mathcal{W}_{\text{relevant}}) / \dim(\latent)$ (written $\chi$ rather than $\kappa$, which Part II reserves for mode coupling), where $\mathcal{W}_{\text{relevant}}$ is the subspace of world states affecting viability, measures how much the system must discard to exist. The implication: compression determines ontology. What a system can perceive, respond to, and value is fixed by what survives compression. The world model’s structure—which distinctions it keeps, which it collapses—is the system’s effective ontology. The information bottleneck principle formalizes the tradeoff: the optimal representation maximizes information about viability-relevant outcomes while minimizing complexity,

with $\beta$ trading compression against fidelity. Different values yield different creatures: high $\beta$ gives coarse world models, low $\beta$ rich ones. The ratio is not a parameter to minimize but a constitutive feature of finite existence. What survives compression is what the system is.

This has a precise architectural consequence the experiments confirm (Experiment Catalog, V22–V27). A linear prediction head compresses hidden state to output through a single weight matrix—and a single matrix always decomposes into independent columns, each serving a separate target dimension. The result is a factored ontology: internal states channeled into independent streams, no pressure to coordinate. Replace the linear map with two layers and the compression changes. The chain rule through two weight matrices makes every hidden dimension's gradient depend on every other dimension's activation at the intermediate layer. Now compression demands coordination. What survives is not a collection of independent features but a coupled representation—parts that cannot be understood without the whole. Compression does not merely fix what the system perceives. It fixes whether its internal states are unified or factored.

The distinction runs deeper, and deserves a name. The environment has its own modes of variation—temperature cycles, predator patterns, resource fluctuations, social dynamics—and these modes are not independent: predator presence couples to resource availability, weather couples to everything. Call this pattern of mode couplings the environment’s eigenskeleton. The agent cannot track all of it; its representation is smaller than the world. Compression therefore selects not only which modes to preserve but which couplings between modes to preserve. A compression that keeps modes independently yields a flat eigenskeleton: the system tracks several aspects of the world but not how they relate. A compression that preserves couplings yields a curved one: the system tracks not just what varies but how variation in one dimension twists into another. The decomposability wall confirmed by V22–V27 is the wall between flat and curved. The formal treatment — what curvature means here, and how to measure it — comes with the integration measures in the forcing functions section.

Attention as Leverage

Compression determines what can be perceived. A second operation determines what is: attention. Even within the compressed representation, the system cannot respond to all viability-relevant features at once. It must allocate processing. Attention is that allocation.

The claim: attention is a high-leverage control variable. Let a system $\mathcal{S}$ inhabit a chaotic environment where small differences in observation lead to divergent action sequences. Its attention pattern $\omega: \mathcal{O} \to [0,1]$ weights which observations are processed at high fidelity and which are compressed or discarded. (The symbol $\omega$ is reserved for attention throughout; $\alpha$ denotes ascription, an unrelated perceptual axis introduced in Part II.) Since actions depend on processed observations and actions shape future states, a small change in $\omega$ steers the system onto a markedly different trajectory. Attention is not where a sovereign chooser stands above physics and picks. It is a variable the trajectory is unusually sensitive to — how much leverage it actually carries depends on the surrounding dynamics, not on attention alone.

The mechanism admits a precise formulation. Let $p_0(\mathbf{x})$ be the a priori distribution over states—the probability of finding the environment in state $\mathbf{x}$, governed by physics. Let $\omega(\mathbf{x})$ be the measurement distribution—the probability the system attends to, and registers, a perturbation at $\mathbf{x}$. The effective distribution it becomes correlated with is:

The system does not control $p_0$—that is physics. The leverage lives in $\omega$. Sharply peaked (narrow attention), the effective distribution concentrates on a small region regardless of the prior; broad (diffuse attention), it approximates the prior. The trajectory follows from the sequence of effective distributions, each conditioned on the previous—and each $\omega$ shapes the next, since what a system attends to now sets which features dominate the state that fixes its next attention. Attention is recursive: current settings bias future settings, and the loop compounds.

The consequence for agency requires care. The attention pattern is itself a product of the system’s internal dynamics — its world model, its self-model, its policy — so there is no uncaused chooser standing behind $\omega$ picking among branches. Attention therefore does not introduce freedom into a deterministic world; it identifies a point of high leverage within one. Agency is real in exactly that sense: the system’s internal states are the variables through which large changes in outcome are levered. Which is why attentional sovereignty is worth defending — not because attention is free, but because it is the lever. Capture a system’s attention and you capture its trajectory. What is at stake is control-capture, not a violation of free will.

The leverage also has temporal depth, though the mechanism is epistemic rather than physical and it is worth being exact about which. Registering an observation conditions the belief state, and the policy is conditioned on the belief state; sustained attention is repeated conditioning, holding a feature in the state that drives action and so stabilizing the trajectory near states consistent with it. Let the information degrade — forgotten, overwritten, lost to noise — and the policy is released from that conditioning. What re-expands is the system’s uncertainty, not the space of physically accessible futures, which was never contracted by anyone looking.