Coherence horizon

A Rule 30 cellular automaton forms a dark triangular history crossed by an orange horizontal horizon.

The question

Setup

Results

The exchange rate between accuracy and coherence

What this says about "what is a computer"

Checkpoint

Coherence horizon

A diffusion model trained on exact pictures of Rule 30 paints pictures that pass for Rule 30: their 3 × 3 patterns are 0.07 from the rule's in total variation, against 0.88 for coin flips. Yet 2.2% of its cells break the rule. Given the true first row, it runs the rule for about six rows and loses the first row by row 14. That loss is explained by its local mistakes and the rule's own sensitivity: the exact rule, making mistakes at the painter's measured per-row rate, loses the first row at the same row. A row-by-row runner trained on the same pictures is exact, and on Rule 4 the painter is exact too. The painter's mistakes concentrate where Rule 30 computes a parity (left ⊕ right).

How much coherence a lower mistake rate buys is a property of the world, not the model. The horizon scales as ε^−1/D, where D is the growth exponent of the set of earlier cells that can influence a cell. Measured slopes are −0.53, −0.33 and −0.26 for chaotic rules in one, two and three dimensions (predicted −1/2, −1/3, −1/4). Rule 90 matches a closed form exactly.

Status, 28 September 2026. Done: the Rule 30 pilot, Rule 4 of the six-rule ladder, and the model-free exchange-rate law. The ladder stopped during Rule 184. Rules 184, 170, 110, 90 and 30 (seed 1), the sampling-steps sweep, the long-training run and the flat-painter test are open.

The question

The claim under test: the more a visual phenomenon behaves like a complex, noise-like system in the real world, the worse a noise model (a diffusion model) does at keeping it coherent. Smoke is the standard hard case.

Smoke has no single correct pixel to check. A one-dimensional cellular automaton does. Its spacetime picture is a video of a one-dimensional world: row 0 is the first frame and each row is computed from the row above by a fixed rule. Asking a model to continue a picture from its true first row is image-to-video with an exact answer key. Rule 30 stands in for the smoke because it is the standard example of a simple rule whose evolution looks random and admits no known shortcut.

Two models learn from the same exact pictures:

The painter is a diffusion model, the kind that makes images and video. It starts from Gaussian noise and refines every cell of every row together over 256 steps.
The runner models row t + 1 given row t and writes the rows in order, the way a language model writes words.

Setup


World	ring of 40 cells (periodic), 40 rows, row 0 a fair-coin row; 40 and not 32 because linear rules degenerate on power-of-two rings
Rules	4 (frozen), 184 (traffic), 170 (shift), 110 (class 4), 90 (linear), 30 (chaotic)
Painter	U-Net, 3.8 M parameters, circular padding in space, one attention block at 10 × 10; Gaussian diffusion on ±1 bits with a cosine schedule; predicts the clean bits (cross-entropy); shown the true row 0 in half of its training examples; ancestral sampling
Runner	width-5 convolution over row t, then two 1 × 1 layers; it has to find which neighbours matter
Training	fresh exact batches every step (no fixed dataset): painter 3,000 steps × 64 pictures, runner 1,500 steps × 64 pictures, same for every rule
Evaluation	256 held-out first rows per rule; every measurement against an exact reference automaton, which matches an independent implementation bit for bit

Measure	Meaning
v	fraction of cells in rows 1–39 that differ from the rule applied to their actual parents; 0 is lawful
TV	total variation distance between the 3 × 3 window distributions of samples and of exact pictures
acc(t)	fraction of row t equal to the exact evolution from the same row 0; 1 is right, ½ is no information
H	the horizon: first row with acc(t) < ¾
noisy runner	the exact rule with independent flips at the painter's own measured per-row rate v(t)

Eight predictions were fixed before any model was trained.

Results

Two runs are complete: the Rule 30 pilot (seed 0) and Rule 4 from the ladder (seed 1). Both use the setup above.

Rule	lzma bits/cell (exact)	damage at row 39	painter TV / coin TV	painter v, unconditional	painter v, from row 0	H painter	H exact rule + painter's v(t)	H runner
4	0.022	0.019	0.031	0.001%	0.00%	> 39	> 39	> 39
30	1.000	0.497	0.082	2.73%	2.20%	14	14	> 39

On Rule 4 the painter is exact from the true first row: all 256 pictures are lawful at every row. On Rule 30, none of 256 pictures is lawful, in either setting.

Exact evolution, painter, their difference and runner for one held-out first row

The painter runs the rule for about six rows, then paints. Its per-row mistake rate v(t) is near zero through row 6 and 3–4% after. At step 1 of 256 its guess already holds the first eight rows; the rest is committed as texture everywhere at once. The runner decides one row at a time.

The painter's guess over sampling steps, and the runner's rows

The painter's loss of coherence is its mistakes plus the rule's sensitivity. The exact rule, flipping cells at the painter's own per-row rate, reproduces the painter's accuracy curve to within 0.023 at every row, with the same horizon.

Row accuracy for the painter, the noisy runner and the runner

The painter's mistakes follow the rule's logic. Rule 30 is l ⊕ (c ∨ r). Where the centre is 1 the new cell is ¬l, a one-cell dependency; where the centre is 0 it is l ⊕ r, a parity of two cells.

Parent neighbourhood	Rule 30 gives	Painter error rate, from row 0
111	0	0.66%
110	0	1.59%
011	1	0.44%
010	1	0.79%
101	0	3.00%
100	1	4.48%
001	1	2.57%
000	0	4.12%

The parity cases carry three to six times the error rate. P9, written before the ladder reached Rules 170 and 90, turns this into a test: v(90) > v(30) > v(170).

Sampling steps (pilot). 32 steps: v 2.49%, H 14. 256 steps: v 2.20%, H 14. Eight times the sequential passes did not move the horizon.

Scored predictions

ID	Verdict so far	Evidence
P1	holds for the two rules run	TV as a fraction of coin-flip TV: Rule 4 0.031, Rule 30 0.082 (must be < 0.2); four rules not run
P2	undecided	needs Rules 170 and 110
P3	undecided	Rule 4 has no horizon within 39 rows, as predicted; five rules not run
P4	held	H(30) = 14 ≤ 20
P5	holds for Rule 30	maximum gap 0.023 ≤ 0.1; Rules 90 and 110 not run
P6	undecided	needs Rule 90
P7	undecided	the sweep (32 to 512 steps) did not run; 32 → 256 steps left H at 14
P8	holds for the two rules run	runner v = 0 and acc(t) = 1 on Rules 4 and 30
P9	undecided	needs Rules 170 and 90

Open

Finish the ladder (Rules 184, 170, 110, 90, 30), then the steps sweep. This scores P1–P3 and P5–P9.
Long training (4×) on Rule 30: does the painter's v fall, and does H follow ε^−1/2 as the exchange rate says?
Flat painter (all layers at full resolution): is the six-row exact zone set by how much of the U-Net runs at full resolution? Write its prediction before running it.

The exchange rate between accuracy and coherence

This part needs no model. Run the exact rule and flip each new cell with probability ε, then find the horizon. The slope of log H against log ε is how much coherence a lower mistake rate buys in that world.

A geometric argument predicts the slope. A cell in row t is corrupted if any mistake falls among the earlier cells that can influence it. If that influence set grows like t^D, then H ∝ ε^−1/D. A mistake that spreads through a filled cone in one dimension gives D = 2. A mistake that persists without spreading gives D = 1. A filled cone in d space dimensions gives D = d + 1.

Horizon against mistake rate, log-log, with predicted slopes

World	Measured slope	Predicted −1/D	What D counts
Rule 4, frozen	−1.15	−1.00	a mistake stays where it fell
Rule 170, shift	−0.99	−1.00	a mistake rides one diagonal
Rule 184, traffic	−0.86	—	mistakes mostly travel as particles
Rule 110, class 4	−0.60	—	irregular spreading
Rule 90, linear	−0.63	−0.63	a Sierpinski gasket, D = log₂3
Rule 30, chaotic	−0.53	−0.50	a filled cone, D = 2
2D chaotic rule	−0.33	−0.33	a filled cone in two dimensions, D = 3
3D chaotic rule	−0.26	−0.25	a filled cone in three dimensions, D = 4

One-dimensional rules: ring of 256 cells, 400 rows, 200 runs per point. The 2D rule is W ⊕ (C ∨ E) ⊕ (N ∧ S) on a 160 × 160 torus; the 3D rule adds ⊕ (U ∧ D) on a 64 × 64 × 64 torus. Both fill about 38% of their light cones with differences.

Rule 90 is exact. It is linear over GF(2), so a cell is wrong exactly when an odd number of mistakes fall in its influence set, and row k of its single-seed pattern has 2^popcount(k) cells (Glaisher's count of odd binomial coefficients). With S(t) the sum of 2^popcount(k) over k < t, which is 3ⁿ at t = 2ⁿ,

acc(t) = (1 + (1 − 2ε)^S(t)) / 2.

The closed form predicts every measured Rule 90 horizon to within one row and the whole accuracy curve to within 0.002 over 120 rows.

Halving the mistake rate multiplies the horizon by 2^1/D:

D	World	Horizon gain from halving ε
1	a mistake persists without spreading	× 2.00
log₂3	Rule 90	× 1.55
2	Rule 30	× 1.41
3	2D chaotic rule	× 1.26
4	3D chaotic rule	× 1.19

A volume of smoke is a three-dimensional world whose disturbances spread through a light cone. If it behaves like the 3D rule, a model needs sixteen times fewer local mistakes to double how long its smoke stays the same smoke. The 3D rule is a cellular automaton, not fluid dynamics: the smoke is an analogy standing on a measured 3D case, not a measurement of smoke.

What this says about "what is a computer"

The week's question is what a computer is. This piece gives one operational answer and its limits. The runner and the painter learned from the same pictures. The runner applies what it learned once per row and feeds each row back in, so its sequence of operations follows the world's own order of steps; it keeps every rule exactly for as long as it runs. The painter also iterates, 256 times, but its iterations sharpen the whole picture at a shared noise level instead of advancing time. One pass of its network carries Rule 30 about six rows, and eight times as many passes did not carry it further (see Sampling steps above). In the sense Rule 30 tests, a computer is a system whose operations follow the order of what it computes.

Rule 90 complicates the usual story. It has a closed form, so it is reducible, yet a model that makes local mistakes loses it about as fast as Rule 30. A shortcut helps only a system that computes exactly. What sets the horizon is exactness and sensitivity; the absence of a shortcut determines only that exact computation has to take every step, which is why simulated smoke has to be computed frame by frame.

This is one sense of "computer". It says nothing about computing as a final good, which is a question about what the running is for.

Checkpoint

Five questions asked of every piece this week.

Question	Answer for this piece
What did the subject force to change?	The expected explanation was computational irreducibility. The measurements point to sensitivity and exactness instead: the horizon is set by D and by the painter's local mistake rate, and the mistakes concentrate where the rule computes a parity.
What does the compelling result still leave unestablished?	Nothing here measures a production video model, fluid dynamics or whether a picture is good to look at. The 3D rule is an automaton. The painter has 3.8 M parameters and trained for 11 minutes per rule.
Can a stranger enter without background?	The page shows a rule as its eight cases and lets the viewer make mistakes; no background in cellular automata is needed. It has not yet been shown to a stranger.
Can a participant change the terms?	On the page: any of the 256 rules, mistakes placed anywhere, a mistake rate, a new first row. The painter's samples are fixed.
Achieved, failed and open, at one scale	Achieved: the model-free law in 1D, 2D and 3D, the Rule 90 closed form, the Rule 30 mechanism. Failed: nothing yet. Open: five of six ladder rules, so seven of nine predictions are unscored.

Coherence horizon?

The question?

Setup?

Results?

The exchange rate between accuracy and coherence?

What this says about "what is a computer"?

Checkpoint?

Coherence horizon

The question

Setup

Results

The exchange rate between accuracy and coherence

What this says about "what is a computer"

Checkpoint