Choose what the centre column should say: the primes, the Fibonacci numbers, π, your own word. The start is built one square at a time, leftwards, so that the centre says it. Each square decides the centre at one more step, without disturbing the steps before. So a start of a given number of squares makes the centre say the sequence for that many steps, and then, almost always, it breaks: the first square the start lacks reaches the centre, one step per square of distance. To say it for ever, the start would need infinitely many squares.
Rule 30 makes a square black when its upper-left neighbour differs from (the square above or its upper-right neighbour). Follow that back from the centre at step t, and the square t places to the left of the centre in the start decides it outright: flip that one square and the centre flips at step t, and nothing earlier changes, because its influence takes t steps to arrive. So choose the centre's square for step 0, the next square left for step 1, and so on. Every sequence can be made this way, with the right half left white. But about half the squares come out black, for ever: the start never ends.
Cut the start to a number of squares and the centre follows the sequence for at least that many steps, then breaks at the first square it needed and lacks. Building it this way is not the cleverest finite start, so the page checked every one. Over every start of up to 18 squares, in every position around the centre, the best follows the sequence only a few steps longer than its width:
| squares | 6 | 10 | 14 | 18 |
|---|---|---|---|---|
| primes | 7 | 11 | 16 | 18 |
| Fibonacci numbers | 8 | 11 | 15 | 20 |
| Fibonacci odd or even | 7 | 11 | 15 | 21 |
| π in binary | 6 | 10 | 14 | 19 |
| 0101… | 7 | 15 | 15 | 18 |
About one step per square, as the record found on 5 October for every pattern it tried, random ones included.
A finite start can say a sequence for ever only if the sequence is one Rule 30 makes by itself, like the single black square's own centre column (press Rule 30's own: one square, and it never breaks). Problem 1 asks whether any repeating pattern is among those. For a pattern that repeats every step the answer is no (Condrey). For 0101…, repeating every two, it is open, and it is the case this record works on. The Fibonacci numbers' odd and even, 0, 1, 1, 0, 1, 1, repeat every three, so saying them for ever is the same question at period 3. The primes and π never repeat, so Problem 1 does not cover them; nothing proved rules them out, and every measurement says a finite start cannot keep them up. π has a further echo: its digits look random, and whether they are balanced in the long run is unproved, just as Problem 2 asks of Rule 30's own centre column.
The table and the chart's ochre dots come from tests/probes/lexicon/rule30_cloud_centre_sequences.py
(CS, 10 October 2026, predictions written before it ran; every start of total width w = 1 to 18 tried, horizon 70
steps). The law it confirms is RULE30-PRIZE.md §8.42 (rule30_uniform.py), and the square-by-square
construction is the left-permutive inverse of rule30_periodic.py. These are measurements on small
starts, not theorems. The page checks itself: it rebuilds the centre from the start it shows, and confirms that
each break falls exactly at the first square the start lacks.