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

Abstract

For every finite field F of characteristic 2 and every bijection f of F satisfying f(x) + f(x + f(y)) = f(x + f(y + f(x))), the cellular automaton that runs the R-matrix R(x, y) = (y + f(x + y), x - f(x + y)) along a row of N cells with the helical boundary condition returns to its initial state after |F| steps, for every N. This proves the conjecture of A. Araoka and T. Tokihiro (arXiv:2602.17148), who proved it for |F| = 4 and 8.

Definition 1.1 (The R-matrix).

$$\mathrm{rmat}\left(f,x,y\right)=(y+f\left(x+y\right),x-f\left(x+y\right))$$

Formalization. D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.rmat (✓ std3).

Citation. Aoi Araoka; Tetsuji Tokihiro (2026). Integrable Cellular Automata on Finite Fields of Order 2^n. DOI: 10.1007/s11040-026-09569-9. URL: https://arxiv.org/abs/2602.17148v1.

Commentary.

The R-matrix of the paper, R(x, y) = (y + f(x + y), x - f(x + y)); its first output is the new cell value and its second output is passed to the next cell.

Definition 1.2 (The auxiliary values).

$$\mathrm{carry}\left(f,x,b,0\right)=b,\mathrm{carry}\left(f,x,b,i+1\right)=\mathrm{snd}\left(\mathrm{rmat}\left(f,x_i,\mathrm{carry}\left(f,x,b,i\right)\right)\right)(i<N)$$

Formalization. D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.carry (✓ std3).

Citation. Aoi Araoka; Tetsuji Tokihiro (2026). Integrable Cellular Automata on Finite Fields of Order 2^n. DOI: 10.1007/s11040-026-09569-9. URL: https://arxiv.org/abs/2602.17148v1.

Commentary.

The values passed along the row: y_0 = b is the boundary value, and y_(i+1) is the second output of R at cell i, for i < N.

Definition 1.3 (One time step).

$$\mathrm{step}\left(f,(x,b)\right)=((\mathrm{fst}\left(\mathrm{rmat}\left(f,x_i,\mathrm{carry}\left(f,x,b,i\right)\right)\right){)}_{i<N},\mathrm{carry}\left(f,x,b,N\right))$$

Formalization. D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.step (✓ std3).

Citation. Aoi Araoka; Tetsuji Tokihiro (2026). Integrable Cellular Automata on Finite Fields of Order 2^n. DOI: 10.1007/s11040-026-09569-9. URL: https://arxiv.org/abs/2602.17148v1.

Commentary.

One time step maps the cell values x_0, …, x_(N-1) and the boundary value b to the first outputs of R at each cell and, by the helical boundary condition, the new boundary value y_N.

Definition 1.4 (The conjecture).

$$claim\iff (\forall F\in \mathrm{FiniteField},(\mathrm{char}\left(F\right)=2)\Rightarrow \left(\forall f\in F\to F,(\mathrm{Bijective}\left(f\right))\Rightarrow \left((\forall x\in F,\forall y\in F,f\left(x\right)+f\left(x+f\left(y\right)\right)=f\left(x+f\left(y+f\left(x\right)\right)\right))\Rightarrow \left(\forall N\in \mathbb{N},\mathrm{step}{\left(f\right)}^{|F|}={\mathrm{id}}_{F^N\times F}\right)\right)\right))$$

Formalization. D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.claim (✓ std3).

Citation. Aoi Araoka; Tetsuji Tokihiro (2026). Integrable Cellular Automata on Finite Fields of Order 2^n. DOI: 10.1007/s11040-026-09569-9. URL: https://arxiv.org/abs/2602.17148v1.

Commentary.

The conjecture of the paper: over a finite field of characteristic 2, for every bijective solution f of the displayed equation, which the paper shows is implied by the Yang-Baxter equation for R, and every number N of cells, the |F|-th iterate of one time step is the identity, so the period divides |F|.

Theorem 1.5 (Proof of the conjecture).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.result (✓ std3). ∎

Resolves. Problems/araoka-2026-yang-baxter-automaton-period (proved) by D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.result.

Source. Repository-derived.

Acknowledgement. Aoi Araoka; Tetsuji Tokihiro (2026). Integrable Cellular Automata on Finite Fields of Order 2^n. DOI: 10.1007/s11040-026-09569-9. URL: https://arxiv.org/abs/2602.17148v1.

Commentary.

Only the additive group of F is used, and x + x = 0 for every x. Replacing f by x -> f(x + a) with f(a) = 0 gives a solution g with g(0) = 0, and then g(g(x)) = x. The involutions L_x(y) = x + g(x + y) fix x and satisfy L_(L_x(y)) L_x = L_(L_y(x)) L_y by the equation. For such a family the maps p -> p L_(p^(-1)(z)) are commuting involutions of the group P generated by the L_x, and the group they generate acts transitively on P, so P is a 2-group. The translations t_a(x) = x + a normalise P, so P together with the translations generates a 2-group Q. A nontrivial central element of Q commutes with every translation, so it is a translation t_u with u different from 0, and it commutes with g = L_0, so g(x + u) = g(x) + u and hence f(x + u) = f(x) + u. Then R commutes with adding elements of {0, u} to its inputs, so one time step commutes with adding states whose entries lie in {0, u}, and it descends to the quotient of F by {0, u}, whose order is |F|/2. By induction on the order, the |F|/2-th iterate G of one time step is the identity modulo {0, u}, so G(s) = s + d with d in {0, u} entrywise, and G(G(s)) = G(s) + d = s. Hence the |F|-th iterate is the identity.

References

• Truth anchor: D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.carry
• Truth anchor: D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.claim
• Truth anchor: D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.result
• Truth anchor: D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.rmat
• Truth anchor: D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod.step