The Yang–Baxter cellular automata over F_{2^n} have period dividing 2^n

A. Araoka and T. Tokihiro (arXiv:2602.17148, nlin.SI; Math. Phys. Anal. Geom. 29, 30 (2026)) build a cellular automaton over a finite field F from the R-matrix R(x, y) = (y + f(x + y), x − f(x + y)). In characteristic 2 the Yang–Baxter equation for R reduces to f(x) + f(x + f(y)) = f(x + f(y + f(x))). The automaton runs R along a row of N cells, R(x_i, y_{i−1}) = (x_i', y_i) with y_0 = b, and the helical boundary condition feeds y_N back as the next boundary value. The paper states:

Conjecture. The cellular automaton thus constructed over a finite field of order 2^n has a period that is a divisor of the order of the field.

and proves it for orders 4 and 8. Issue #11405 fixes the reading: every finite field of characteristic 2, every bijective solution f of the reduced equation, every N, and "period divides q" as step^q = id on the pair of cell values and boundary value.

The paper's numerical tables for orders 8 and 16 show fundamental periods 1, 2, 4, 8 and 16 and no others, and the authors single out this strict periodicity as a feature of the integrable automata: an R-matrix violating the Yang–Baxter equation gives other periods. The frozen declaration D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.result proves the conjecture for every order.

Issue #11405 preregisters the reading, the route and the literature check. The paper proves only the orders 4 and 8. INSPIRE has no record of it, and a web search finds no follow-up. The group-theoretic step of the route is a known fact about class-2 cycle sets (Dehornoy, Adv. Math. 282 (2015), Prop. 5.2; Feingesicht, IJAC 34 (2024), Prop. 2.9); its application to this automaton was not found in the searched sources.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent answer.

1. Only the additive group is used, and x + x = 0. Translating f to g(x) = f(x + a) with f(a) = 0 keeps the equation, and then g(g(x)) = x.
2. The involutions L_x(y) = x + g(x + y) fix x and satisfy the cycle identity L_{L_x(y)} L_x = L_{L_y(x)} L_y. The maps p ↦ p L_{p⁻¹(z)} are commuting involutions of the group P they generate, and they act transitively on P, so P is a 2-group.
3. The translations normalise P, so P and the translations generate a 2-group. A nontrivial central element is a translation t_u with u ≠ 0 that commutes with g = L_0, so f(x + u) = f(x) + u.
4. R commutes with adding elements of {0, u}, so one time step commutes with adding {0, u}-valued states and descends to F/{0, u}. By induction on the order, G = step^{q/2} satisfies G(s) − s ∈ {0, u} entrywise, hence G(G(s)) = s and step^q = id.

The statement would change if the automaton were read with a non-bijective f, for which the paper exhibits other periods, or with an R-matrix that does not satisfy the reduced equation. The proof uses only the reduced equation and bijectivity, not the full Yang–Baxter equation, so it covers every automaton of the paper.

All 16816 bijective solutions of order 16 with f(0) = 0, and 300 random solutions of order 32, have a nonzero u with f(x + u) = f(x) + u; sampled automata return to their initial states after q steps (issue #11405).

The canonical source is D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.lean. Its public declarations are rmat, carry, step, claim and result. The frozen module state has statement identity sha256:f947a618fac58f588ae535ee8bb8df20a6f8475e0d407bbbc9b287c344a5161a. The result declaration has statement identity sha256:b10b29174002230d2cfc73e10370c05becae4aeed3e10a5dcb1fa3e8d3ff9c04. The Freeze event is sha256:2c5a6318dc69062e26c4aa324abe4f486714f260314c137df967ab137f9d1968 and has no project-level frozen prerequisites. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Tier 1 conjecture from a published nlin.SI paper, preregistered in issue #11405 before any Lean. theorem; resolution proved. The public theorem has proof_shape: content; its escape witness is the translation lemma (form (1)). Its admission basis is open-problem-resolution. Utility none.

The published journal text was not read, so it is unverified whether the conjecture was edited there. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent answer. The Lean kernel does not authenticate the external source or its version history.