Findings · 502 linear rules, three registered rounds

The linear atlas

Build a rule from rings of neighbours, called shells, and add mod 2. Every such rule is linear, so its period on a torus is decided by algebra. We asked Kepler for laws of 42 of them, registered the pattern behind those laws, and refined it twice, each time because a registered prediction failed.

Round	Registered	Predictions	Outcome	What the misses revealed
1	Kepler's laws for 42 rules, 38 never studied, from tori up to 60	1,930 pairs	held all 34 rules with a law kept one	Afterwards: rules that split into 1D passes inherit the 1D period, and the rest reach the full bound
2	That pattern, on 116 new rules; the 34 laws at harder sizes up to 400	113 + 624	refuted 7 of 113 failed; the laws held at all 624	Powers and dilations, then predicted exactly at 55 of 55 sizes; small fields at m = 13; one case at m = 37
3	The refined statement as a fixed procedure, on 344 new rules with nine new shells	11,358	refuted 43 failed (0.4%); every dilation and power held	The twisted power, whose expectation, written first, matched 34 of 34 sizes; small-field coincidences; m = 37 again

Why: eigenvalues in a finite field

On the torus (Z/m)² with m odd, a rule f acts diagonally. Its eigenvalues are f(ζ^j, ζ^k), where ζ is a primitive m-th root of unity in the field F(2^d) and d = ord_m(2). The period's odd part is the lcm of their multiplicative orders. That bridge is a theorem in our Lean library (period_eq_eigenLcm).

Each square of the map is one eigenvalue. Squares in the same Frobenius orbit, (j, k) → (2j, 2k), are conjugates and share an order. A symmetric rule's eigenvalues lie in F(2^sord(m)), so its period divides the full bound 2^sord(m) − 1. A rule falls short of it only if something keeps every eigenvalue inside a proper subgroup.

Eigenvalue map

computed in F(2^d) here

Rule

Odd part m

Colour by

field

full bound

odd part

verdict

eigenvalues

pointerpoint at a square

What holds a rule below the bound

Four mechanisms confine the eigenvalues, and each is a polynomial identity over GF(2) that you can multiply out below: splitting into one-dimensional passes, powers, dilations and twisted powers. The triangular lattice adds one more fact: each rule's 2 × 2 matrix squares to a plane rule s times the identity, so two triangle steps are one step of s.

Identities, multiplied out mod 2

Identity

Each square is a translation the rule reads, and the ring marks the origin. A product of rules is the sum of all pairwise translations, kept where it occurs an odd number of times.

Coincidences, and one that is not

In a small field a rule with few distinct eigenvalues can have all of them in a subgroup by chance. The census below takes all 1,274 shell rules of the atlas, on all three lattices, at every odd m where the cubes are a proper subgroup. It counts the rules whose every eigenvalue is a cube although no known mechanism accounts for it. The count falls away as the field grows, as chance would, except at m = 37.

Unexplained cube confinement, by m

the census the agent was given held-out sizes, registered first not yet computed

Point at a bar to read it.

answered in the fourth pilot

The cubes at m = 37

At m = 37 the full bound is 2¹⁸ − 1 = 262,143 = 3 × 87,381. Six rules that none of the four mechanisms touches have every eigenvalue a cube there, and period 87,381, which is rule 90's own period at 37:

square A4 + K8 + B4 + self
triangle VS + T9 + T13 + self
square K8 + M8 + E4
square C4 + L8 + M8
square C4 + L8 + M8 + E4
square A4 + B4 + C4 + L8 + E4 + self

Each has only 9 to 11 distinct eigenvalues up to Frobenius, so chance would do this about once in 20,000 to 180,000 rules. It happened six times among 1,274, and none of the six falls short at any other odd size from 9 to 161.

The fourth pilot set an agent on this question, in its own container, with the census above up to m = 161 and the Lean library. Nobody knew the answer. It found a mechanism, half-Frobenius confinement, that explains all six, and proved it in Lean. Its general condition made 633 claims at 94 sizes, 180 of them at sizes it never saw, and none was false.

Rule names are the atlas's neighbour shells: N4 the four nearest cells, D4 the diagonals, A4, B4, C4, K8, L8, M8 and E4 further square shells, H1 the hexagon's six neighbours, and E, VO, VS, T9 and T13 the triangle's.