The parity laws
A parity rule sets each cell to the sum, mod 2, of the cells it reads. On a finite torus it must eventually repeat, and how long its cycle is follows exact laws set by finite fields. Our loop found those laws from simulation, tested them in advance, got one wrong, corrected it, and proved most of what it found.
One live cell
exact period, computed hereThe rule is linear and the same everywhere, so the orbit of one cell has the period of the whole rule. Your browser finds it by taking powers of the rule in the group ring F2[(Z/n)²], and dividing primes out of a bound it must repeat at.
The square law
Write the side as n = 2vm with m odd, and let sord(m) be the least j with 2j ≡ ±1 (mod m). For the square parity rule B1357/S1357 the period is
P(n) = 2v · (2sord(m) − 1),
and at n = 2v the torus dies. Kepler found this from simulations. Its predictions were registered before the sizes were computed, and held at every n from 3 to 200. The rival law, rule 90's period, falls short of it at ten of those sizes, starting at 37, and was refuted at each one. The 2-part is proved in Lean at every size. The odd part is checked by machine up to 999, except at ten sizes where the check is incomplete.
Measured periods, n = 3 to 200
The hexagon: a registered law that failed
On the hexagonal lattice Kepler proposed the same form with an extra factor of 2, P(n) = 2 · 2v · (2sord(m) − 1). It was registered at 70 sizes and refuted at three: 74, 148 and 190, each off by a factor of 3. All three have m = 37 or 95, where rule 90's period falls short of the full bound.
That pointed at the fix. Two hexagon generations are three passes of rule 90, one along each lattice direction, so the hexagon inherits rule 90's period R(m) exactly: P(n) = 2 · 2v · R(m). The corrected law was registered at 100 new sizes from 202 to 400 and held at all of them. It is now proved for every even n, and its 2-part was proved in Lean by an AI agent in the first pilot.
Hexagon periods against the two laws
Triangles, and what is new
Both triangle parity rules, B13/S13 and B1,3,…,11/S1,3,…,11, follow P(n) = 2 · 2v · (2sord(m) − 1). They escape rule 90's shortfalls, as the square rule does. The law held at all 154 sizes registered, and its 2-part was proved in Lean by the second pilot's agent.
Some of this was known. McLaughlin published the square rule's odd sizes up to 23; the structure of these laws is Martin, Odlyzko and Wolfram's for rule 90 (1984); and Vadali and Turner (2025) proved that the four-neighbour square rule has rule 90's cycle lengths. A brief search did not find the rest: the eight-neighbour square rule escapes rule 90's shortfalls, the hexagonal rule shares them exactly, and the triangular rules escape them. The identity behind the hexagon result is elementary, and a specialist may know it.
Why the laws take this form, and why some rules fall short of it, is the subject of the linear atlas.