Chapter 36 — Theoretical Boundaries: When Can the True State Be Identified?

The big picture

A tiny story: two honest maps

Two cartographers draw the same island. One turns the paper 90 degrees. Every distance and transition still agrees, but “up” no longer means north. Both maps can be correct.

That is why “identified up to an orthogonal transformation” is meaningful but limited. The learned coordinates are linearly related to the hidden coordinates through a rotation or reflection; individual axes need not receive human names. This is stronger than arbitrary nonlinear warping and weaker than recovering “x position” in coordinate 1.

Outside means “not directly covered by these theorems,” not “impossible to learn.”

The technical backpack

The passive result in When Does LeJEPA Learn a World Model? is a population/global-optimum statement for a specified class: observable state through the stipulated observation map, stationary additive-noise dynamics, Gaussian latent structure, exact Gaussian representation constraint, and matched latent/representation dimension. Under its separation conditions it gives linear recovery up to orthogonal coordinates. Its planning statement additionally needs transformed dynamics to agree and a cost that respects the orthogonal ambiguity. The encoder theorem does not by itself show that an action-conditioned transition was learned.

The controlled v2 paper has a different, stronger contract: invertible observation, stationary linear controlled latent dynamics with independent Gaussian noise, jointly Gaussian state/action behavior, exact Gaussian representation, matched dimension, a sufficiently expressive continuous predictor, population optimization, and positive state-conditioned action excitation. Under its conditions, it identifies the state and controlled conditional-mean transition up to a common orthogonal change. It does not recover every stochastic future.

Three common assumption gaps are different:

• Partial observability: two hidden states can make the same current image. History helps only when the hidden variable leaves a trace.
• Multimodal futures: squared-error point prediction may land between real modes; a conditional mean is not a full future distribution.
• Missing interventions: seeing Left and Right overall is insufficient if each action occurs in only one state. Counterfactual action branches remain unconstrained.

SIGReg in finite networks pressures finite minibatches, sampled projections, and numerical knots toward a Gaussian target. That is not exact population enforcement. The separate free-energy paper assumes constant encoder noise and successful exact isotropic-Gaussian enforcement; it is not an identifiability theorem and leaves empirical validation for later work.

Try to break the idea

Build two states. In State A, the behavior policy always chooses Right; in State B, it always chooses Left. The whole dataset contains both actions equally often, yet neither state contains both branches. A predictor can fit every recorded transition and invent anything for A+Left or B+Right.

Add a second dataset with both actions at both states while keeping observation, state distribution, model, and updates fixed. Include an oracle-state condition. If withheld-action error remains with oracle observations, the issue is transition coverage in this construction, not representation ambiguity. This test supports a conditional-excitation mechanism; it does not diagnose every LeWM failure.

Before citing a theorem, ask: Is observation invertible enough? Does the transition/noise family match? Are stationarity, dimension, Gaussian constraint, and global optimum actually available? After conditioning on state, does every needed action direction vary? Unknown answers mean “use as guidance,” not “claim coverage.”

Experiment receipt and evidence boundary

• When Does LeJEPA Learn a World Model? v1, released 2026-05-25; public snapshot de7503f.
• On the Identifiability of Controlled World Models v2, revised 2026-07-27; no author-linked implementation identified by the cutoff.
• The SIGReg Objective as Variational Free Energy v1, released 2026-07-15; and LeWorldModel v3, frozen code 8edfeb3, only for the practical baseline boundary.
• Evidence cutoff: 2026-08-20, Asia/Tokyo. Theoretical evidence is assumption-bounded; demonstrations are author-reported where present; independent reproduction is not established here. Never turn orthogonal recovery into named physical coordinates, a probe into identifiability, or a linear conditional-mean theorem into guaranteed nonlinear planning.

Three quick questions

1. What does orthogonal ambiguity preserve, and what semantic names can it change?
2. Why is marginal action diversity weaker than state-conditioned action excitation?
3. What is the correct conclusion when a benchmark lies outside a theorem’s assumptions?