Pictorial form

A fine cellular pattern is projected into a smaller coarse grid, showing which distinctions a picture retains.

The question

The condition

The census

The noise floor

Scored predictions

Post hoc

What this says about world models

What this says about noise and attribution

Style and physics

The consequence

Rejected: emergent physics from scale

Limit

What this says about "what is a computer"

Checkpoint

Sources

Pictorial form

A picture of a world has physics of its own only when the world's dynamics commute with the picture:

π ∘ F^N = G ∘ π

This piece checks the condition exhaustively. The worlds are all 256 elementary cellular automata, and the pictures are every one-bit camera over blocks of 2, 3 and 4 cells:

112 worlds have a picture with nontrivial physics of its own.
90 have only pictures that go blank.
54, among them Rule 30 and Rule 110, have no closed picture at all.

Among chaotic worlds, the only pictures with lasting physics belong to the additive (linear) rules, and those depend on the mesh. Rule 90 is exactly picturable in blocks of 2 and 4 cells, and 97% unpredictable in blocks of 3.

Where a picture does not close, any model of pictures has to invent part of every next frame. Rule 30's most predictable 3-cell picture leaves 0.85 of its 0.95 bits per cell unpredictable, and six frames of history change that by half a percent. Rule 110's picture becomes 93% predictable with six frames. The noise a video world model adds stands for the part of the world its picture cannot see.

Status, 28 September 2026. Of nine predictions written before the census existed, seven held and two failed (P4, P9).

The question

Many researchers believe that good latent-space world models can emerge from video. The contrary claim tested here is that complexity science and computer science say this cannot work in general. A second worry: animation fuses style, character and physics into one spatio-temporal latent space, because that is efficient, and then no clear boundary separates a style choice from a physical, causal one. Hair and fur are where style and world models collide.

The researchers' logic, in a recent survey (He, Guo, Lin and Yu, Video Generation Models as World Models, arXiv 2603.28489, p. 1), rests on three advantages it states without testing:

Emergent Physics: Large-scale training on diverse video data allows models to learn complex interactions, such as agent-environment interactions or fluid dynamics, which are difficult to model via traditional analytical engines.

Latent Imagination: Modern world models often operate in compact latent spaces, allowing the imagination of future scenarios to occur at a lower computational cost than high-resolution pixel rendering.

Unified Reasoning: … the model acts as a general-purpose simulator for decision-making.

The same survey's own ten-minute comparison of six methods reports that "most methods struggle with object permanence and structural consistency in later stages" (pp. 5–6).

The condition

A world F acts on states x. A picture π reports some function of the state. The picture has physics of its own when there is a rule G on pictures such that taking a picture and then running G gives the same result as running the world and then taking a picture. In that case the next picture is a function of the current picture, and a model that sees only pictures can in principle be exact. Otherwise two world states with the same picture have different next pictures, and nothing that sees only the picture can tell which will happen.

Wittgenstein describes the same object in Tractatus 6.341–6.342: a white surface with black spots, covered by a net of square meshes, each mesh reported black or white. The possibility of such a description tells us nothing about the surface, "but what does characterize the picture is that it can be described completely by a particular net with a particular size of mesh." The census asks, for every elementary world, which nets describe it completely.

Term	Here
World	one of the 256 elementary cellular automata; rule number in the standard numbering
Picture, or camera, (N, P)	blocks of N cells, one bit per block by a truth table P over the 2^N blocks; one picture step is N world steps, so a picture cell's future depends on exactly its two neighbours
Closed	some elementary rule g satisfies π ∘ F^N = G ∘ π; checked exactly over all 2^3N triples of blocks
Trivial	g is constant: the picture goes blank or black after one step
Noise floor h	bits per picture cell per picture step that no rule on pictures can supply: the conditional entropy of the next picture bit given the three picture bits around it, optionally over several past frames
Predictable share	(H − h) / H, where H is the picture bit's own entropy

The census

All 256 elementary worlds, framed by whether they have a picture with physics

	Blocks of 2	of 3	of 4	Any of these
Worlds with a closed picture	92	141	202	202
with a closed picture that does not go blank	72	79	108	112

An independent implementation in a computer-algebra system recomputes the closure test for ten worlds at blocks of 2 and 3 and agrees on every count.

Across the 88 classes of worlds that are equivalent under mirror image and complement:

42 classes have a picture with physics of its own.
27 classes have only pictures that go blank.
19 classes have no closed picture.

Rule 30 and Rule 110 are among the 19. Of the 256 worlds, 30 are surjective (every pattern has a predecessor), and as the argument predicts, none of them has a picture that goes blank.

What the closed pictures are:

Additive worlds picture themselves. Keep the first cell of every pair and Rules 60, 90, 102, 150, 153, 165 and 195 reappear exactly, because over GF(2) squaring a polynomial doubles its exponents. Rule 105 pictures as 150. This is the nested, Sierpinski self-similarity of these worlds, written as a commuting square.
Growth and erosion picture themselves. Rules 128, 136, 192, 238, 252 and 254 commute with "is the block all black?" and its opposite.
Frozen and moving worlds (identity and shifts) commute with every camera.
One chaotic, non-additive world has a nontrivial picture, and it is a picture of loss. Rule 146 through the camera "block is all black" (N = 3) is exactly Rule 128. Rule 128 erases any black cell that lacks black on both sides, so the picture shows the world destroying its own solid runs. Israeli and Goldenfeld reported the same transition (Physical Review E 73, 026203, 2006).

The mesh decides. Rules 90, 105, 150 and 165 are exactly picturable in blocks of 2 and 4, and their most predictable 3-cell pictures are 95–97% unpredictable. Eight classes (1, 13, 19, 23, 29, 72, 77, 232) acquire a picture with physics only at blocks of 4.

The price of compression. Israeli and Goldenfeld (2004, 2006) ran the same exhaustive search up to N = 4, but allowed the coarse alphabet to be any size. They found nontrivial coarse-grainings for 240 of 256 worlds. Only 16 resisted: Rules 30, 45, 106 and 154 with their mirror images and complements. Song and Grochow (2020) extended the search to N = 7 and found 20, adding the classes of 37 and 164. The coarse-grainings they report for Rule 110 use blocks of 6 cells projected onto 63 of the 64 possible block states: a picture that forgets almost nothing. One bit per block, the compression that makes a latent space cheap, leaves 19 classes open instead of 4.

The noise floor

For a picture that does not close, h measures what a model of pictures has to invent. The deep looks use a world of 196,608 cells.

A world, its picture, and the picture run by its own rule

How much of the next picture bit its own past can predict

World, 3-cell mesh	Predictable, 1 frame	Predictable, 6 frames	Reading
Rule 90	3.1%	3.4%	a closed world through the wrong mesh
Rule 30	11.3%	11.8%	history adds nothing
Rule 22	19.9%	27.0%	chaotic; history adds a little
Rules 18 and 146	51%	56%	chaotic, with domains the picture partly sees
Rule 54	78.5%	88.6%	persistent structures
Rule 110	49.3%	92.6%	persistent structures; history recovers most of it
Rule 184	99.9%	99.9%	traffic: predictable on the states it reaches

Each row uses the world's most predictable camera. Rule 30's lowest-noise camera ("the block is 000") shows the extreme case. Its next bit is certain in five of the eight contexts. Nine zeros in a row stay zeros, and in four other arrangements the middle block is certain not to be all zeros three steps later. But those contexts are rare. In the commonest, with no all-zero block nearby (two times in three), the next bit is a one-in-seven chance. Its predictable share stays at 5.7% with 6 frames of history or 17 cells of width.

Two regimes are easy to confuse. History-predictable pictures (Rules 110 and 54) have hidden state the picture reveals over time, because the world's structures persist; a model with long context recovers most of it. Noise pictures (Rules 30, 22 and 90 at the wrong mesh) have hidden state that the world keeps generating from scales the picture never sees, and no amount of context recovers it.

Scored predictions

The predictions were fixed before the census existed.

ID	Prediction	Verdict	Evidence
P1	identity and shifts close under every camera; additive worlds close at N = 2 under "keep the first cell" (105 → 150, the rest → themselves); no surjective world has a blank picture	held	every case; 30 surjective worlds found by preimage balance
P2	Rule 30 has no closed picture at N ≤ 4	held	0 closed cameras at N = 2, 3, 4
P3	200–248 worlds have some closed picture	held	202
P4	every world with no closed picture spreads damage	failed	23 of the 54 open worlds do not spread damage
P5	Rule 110 has no nontrivial closed picture	held	none, not even a trivial one
P6	some chaotic, non-additive world has a nontrivial picture whose rule does not spread damage	held, weakly	only Rules 146 and 182, whose pictures follow the erasing Rules 128 and 254
P7	Rule 30's best 3-cell picture: h₁ ≥ 0.5 bits and h₆ ≥ 0.25 bits	held	0.510 and 0.510
P8	a stochastic picture model keeps the density within 0.02 and is at chance by step 5	held	density within 0.004; agreement 0.781 against chance 0.781 at step 5, and 0.790 at step 1
P9	the most-likely-bit model agrees on under 90% after one step and under 60% after five	failed	0.878 and 0.876

P4 failed because openness is not only chaos. Many calm worlds keep information in phases and positions that a one-bit block summary throws away. Rule 154 is the clearest case. It is one of the four rules Israeli and Goldenfeld could not coarse-grain at any alphabet size, yet the damage from one flipped cell stays within about 3 cells after 100 steps. The damage measure was also flawed in two ways that do not change the verdict. It samples at step 128, a power of two, where the additive worlds' Sierpinski difference patterns are 2–3 cells wide, so they register as calm. And it counts differing cells rather than the width of the difference region. Measured by width instead, 28 open worlds still keep their damage local.

P9 failed on its selection rule. "Lowest noise" picked Rule 30's sparse camera ("block is 000"), whose picture is 87.6% zeros. The most-likely-bit model then predicts almost all zeros and scores 0.878 against a base rate of 0.874: no skill, but above the threshold. The thresholds assumed a balanced picture. This is a measurement-design error, the second in two pieces (margin's stop threshold was the first).

Post hoc

Written after scoring; none of this was predicted.

Relative measures. Ranking cameras by predictable share instead of absolute noise gives the table above. The best any one-bit camera does for Rule 30 is 27.7% at blocks of 2 and 11.3% at blocks of 3.
Right on the data, undetermined off it. In 272 (world, mesh) pairs, across 138 worlds, some camera is noise-free on the states the world actually reaches from a random start, yet does not close on all states. Rule 184 comes within 0.1% of this at blocks of 3, and closes on all states only trivially. A model trained on such a world's pictures is right on everything it saw. The data fixes nothing about the states it did not see; there, the model's own assumptions decide.
Nonstationary worlds. Rule 110's predictable share at one frame is 35% over world steps 96–288 and 49% over steps 96–1,296, because its structures thin out as it ages.

What this says about world models

Emergent physics. A picture has physics only if it closes. In this universe no chaotic world's picture closes, except for the linear worlds and one picture of erasure. Where a picture does not close, the unpredictable part is set by the world and the camera, not by the model: a larger model of the same pictures cannot shrink it. More context helps only in worlds with persistent structures. Rule 110 goes from 49% to 93% predictable with six frames, and Rule 30 does not move.

Latent imagination. Imagining in a compact latent space is exact only for closed pictures. The compression that makes a latent cheap is what makes it open: Rule 110 closes only when its picture keeps 63 of 64 block states.

Unified reasoning. One architecture for every domain presumes that closure is general. The census shows it is specific to the world and the mesh.

Rare events. World models are often offered as generators of rare events. Those are the states off the data, where the 272 right-on-the-data pictures above are undetermined.

World first, then render. A world model can instead build the world first and then render it. Rendering a simulated world is exact, because the renderer never predicts; it only projects. Predicting a rendered picture is exact only when the picture closes.

What this says about noise and attribution

This section connects the census to the exact optimal denoiser of a diffusion model; the connection is argued, not measured.

In the census, the noise a picture model must add is not ignorance of the model. It stands for the fine state of the world that the picture does not show. A generative model of such pictures replaces that fine state with random numbers. Its outputs then have a correct distribution (P8: density within 0.004) and no cause in the world (P8: agreement at chance from the first step).

This is the precise sense in which noise causes the attribution problem:

Two answers to "why this pixel". A closed picture has a causal answer: its own rule, applied to the neighbouring picture cells one step earlier. A picture that does not close has no causal answer inside the picture. The cause lies in the unseen fine state, and a model of pictures can only offer an associative answer: which training data its random choice resembled.
The optimal denoiser gives associative receipts. For a denoiser with a fixed local window, trained on N images under the schedule z = (1 − t)x + tε, the exact optimal noise prediction is a sum over the training images, weighted by their posterior probabilities, which sum to 1 for each position. Those weights are exact associative receipts: they say which training images the choice copied. They cannot say why the world made that pixel, because the cause was never in the picture. A window that depends on the image being denoised, the realistic case, is left open.
The starting noise picks the winner. In the formula, noise enters the weights through the t² in the Gaussian's variance. Early in sampling, when t is large, the weights are soft and the random start tips them. Late, they are sharp and only refine. Which training image a token ends up copying is decided by the random start, the same stand-in for unseen causes that the census measures.

Style and physics

For a picture that does not close, the unpredictable part must be supplied as texture: something with the right statistics and no particular cause. That is what an artist, a renderer and a diffusion model all do with hair, fur, smoke and water. Where a world's picture does not close at the artist's mesh, "style" is the name for the part that cannot be predicted and must be chosen. The boundary moves with the mesh (Rule 90 is all physics at blocks of 2 and almost all texture at blocks of 3). That is why the artist finds no clear line between a style choice and a physical one.

One constructive proposal is to train on prompts, interface actions, render logs and frames, and to diffuse in vector and film composition space. That chooses a state space that closes by construction. A scene's construction state, meaning its objects, rig, camera and render settings, evolves by the editing operations that made it, and the frame is a render of it. That is the census's closed case, reached by design rather than found.

Checkpoint

Question	Answer for this piece
What did the subject force to change?	The expectation that openness means chaos (P4). Worlds that keep information in phases and positions defeat one-bit cameras without spreading damage, and Rule 154 resists every coarse-graining while keeping damage local. It also forced a second look at the damage measure.
What does the compelling result leave unestablished?	Everything about real video. These are 256 one-dimensional worlds and one-bit cameras up to 4 cells. The census is exact; its bearing on video world models is an argument.
Can a stranger enter without background?	The page shows a world, its picture and the picture's own rule side by side, with wrong cells in red. It needs no background. It has not been shown to a stranger.
Can a participant change the terms?	Yes: any of the 256 worlds, any mesh up to 4, and any camera the viewer designs by toggling its truth table.
Achieved, failed and open, at one scale	Achieved: the exact census, the noise floors, the regimes, agreement with Israeli–Goldenfeld on 146 → 128, and the page. Failed: P4 (openness is not only chaos) and P9 (a selection rule that picked a sparse picture). Open: pictures with more than one bit, running the optimal denoiser on these worlds, and a real video model's noise measured against its world.

Sources

Wittgenstein, L. (1921). Tractatus Logico-Philosophicus, 2.1–2.18 and 6.341–6.342 (trans. Pears and McGuinness).
Israeli, N. and Goldenfeld, N. (2004). Computational irreducibility and the predictability of complex physical systems. Physical Review Letters 92, 074105.
Israeli, N. and Goldenfeld, N. (2006). Coarse-graining of cellular automata, emergence, and the predictability of complex systems. Physical Review E 73, 026203.
Song, Y. and Grochow, J. A. (2020). An improved algorithm for coarse-graining cellular automata. arXiv 2012.12153.
He, Guo, Lin and Yu (2026). Video Generation Models as World Models: Efficient Paradigms, Architectures and Algorithms. arXiv 2603.28489.
Kamb, M. and Ganguli, S. (2024). An analytic theory of creativity in convolutional diffusion models. arXiv 2412.20292 (related analytic theory for convolutional denoisers).

Pictorial form?

The question?

The condition?

The census?

The noise floor?

Scored predictions?

Post hoc?

What this says about world models?

What this says about noise and attribution?

Style and physics?

The consequence?

Rejected: emergent physics from scale?

Limit?

What this says about "what is a computer"?

Checkpoint?

Sources?

Pictorial form

The question

The condition

The census

The noise floor

Scored predictions

Post hoc

What this says about world models

What this says about noise and attribution

Style and physics

The consequence

Rejected: emergent physics from scale

Limit

What this says about "what is a computer"

Checkpoint

Sources