Predictions, written before any census ran

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

The claim under test

Definitions

Predictions

What would count against the claim

Limits known in advance

Term

Definition

World

an elementary cellular automaton f (one of 256), global map F

View (N, P)

cut the line into blocks of N cells; report one bit per block by a Boolean function P of the block's N cells. P is non-constant; there are 14, 254 and 65,534 such P for N = 2, 3, 4

Picture

the coarse spacetime record y_t = π(F^Nt(x)): one coarse step is N fine steps, so a fine light cone of radius N matches a coarse neighbourhood of radius 1

Closed view

some elementary rule g on pictures satisfies π ∘ F^N = G ∘ π for every configuration. Checked exactly: for all 2^3N triples of blocks (a, b, c), P(Φ_N(a, b, c)) = g(P(a), P(b), P(c)), where Φ_N gives the middle block after N steps

Trivial closed view

g is constant: after one coarse step the picture is blank or full whatever the world was

Nontrivial closed view

g is not constant

Spreads damage

ring of 256 cells, random start, one cell flipped; after 128 steps more than 5% of cells differ, averaged over 64 trials

Informative view

the picture's own bits carry at least 0.5 bits of entropy per cell

Noise floor h_k

conditional entropy, in bits per coarse cell per coarse step, of the next picture bit given the 3 picture bits around it over the last k coarse frames (3k bits of context); measured on a ring of 12,288 cells from a uniform random start after 96 fine steps

Best view

the informative view at a given N with the lowest h₁

Prediction

Confidence

Fails if

Certain cases (checks on the code). Identity (204) and the shifts (170, 240) close under every P. Every additive rule closes at N = 2 under "keep the first cell of each block", because over GF(2) p(z)² = p(z²): Rules 60, 90, 102, 150, 153, 165 and 195 are then their own pictures (g = f), and Rule 105 pictures as 150 (its complement terms cancel in pairs). No surjective rule has a trivial closed view.

97%

any case violated

Rule 30 has no closed view at N = 2, 3 or 4, trivial or not.

75%

any closed view found

Most worlds have some closed picture. Between 200 and 248 of the 256 rules have at least one closed view (trivial or not) at some N ≤ 4.

50%

fewer than 200 or more than 248

Openness means chaos. Every rule with no closed view at any N ≤ 4 spreads damage.

70%

a rule with no closed view that does not spread damage

Rule 110 has no nontrivial closed view at N ≤ 4.

50%

one is found

Pockets of reducibility. At least one rule that spreads damage and is not additive has a nontrivial closed view whose coarse rule g does not spread damage: a chaotic world with a simple picture.

60%

none

The picture of Rule 30 is noisy at any history length. Rule 30's best view at N = 3 has h₁ ≥ 0.5 bits, and six frames of history (18 bits of context) leave h₆ ≥ 0.25 bits.

55%

h₁ < 0.5 or h₆ < 0.25

Right statistics, wrong world. A stochastic picture model that samples each next bit from Rule 30's measured conditional distribution (best view, N = 3) keeps the picture's density of 1s within 0.02 of the true picture's over 50 coarse steps, while its cell-by-cell agreement with the true picture is within 0.05 of chance (0.5 for a balanced picture) by coarse step 5.

65%

density off by more than 0.02, or agreement above chance + 0.05 at step 5

The best deterministic picture model forgets fast. For the same view, the rule g* that always predicts the most likely next bit agrees with the true picture on under 90% of cells after one coarse step, and on under 60% after five.

60%

90% or more at step 1, or 60% or more at step 5

Predictions, written before any census ran?

The claim under test?

Definitions?

Predictions?