bibkey: araoka2026integrable authors: Aoi Araoka; Tetsuji Tokihiro year: 2026 title: “Integrable Cellular Automata on Finite Fields of Order 2^n” doi: 10.1007/s11040-026-09569-9 url: https://arxiv.org/abs/2602.17148v1 claim: “For a map f of a finite field of characteristic 2, the R-matrix R(x,y) = (y + f(x+y), x - f(x+y)) satisfies the Yang-Baxter equation exactly when f(x) + f(x + f(y)) = f(x + f(y + f(x))); the cellular automaton that runs R along N cells with the helical boundary condition b(t+1) = y_N(t) is conjectured, for bijective f, to have period dividing the order of the field (proved in the paper for orders 4 and 8).” strata_touched:

• D5/S3/StatisticalMechanics/CellularAutomata/YangBaxterAutomatonPeriod license: citation-only triage: anchor

Integrable Cellular Automata on Finite Fields of Order 2^n

A. Araoka and T. Tokihiro, arXiv:2602.17148 (v1 2026-02); Math. Phys. Anal. Geom. 29, 30 (2026). Subject: nlin.SI.

The paper builds R-matrices on a finite field from a single map f, R: (x, y) ↦ (y + f(x + y), x − f(x + y)), shows that in characteristic 2 the Yang–Baxter equation reduces to f(x) + f(x + f(y)) = f(x + f(y + f(x))), and counts the bijective solutions (16, 736 and 269,056 for orders 4, 8 and 16). It constructs a cellular automaton by running R along a row, R: (x_i(t), y_{i−1}(t)) ↦ (x_i(t+1), y_i(t)) with y_0 = b(t), and the helical boundary condition b(t+1) = y_N(t). It 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.

Verified locator

• DOI: https://doi.org/10.1007/s11040-026-09569-9 (publication metadata from the Springer page; the journal text was not read).
• URL: https://arxiv.org/abs/2602.17148v1 (source retrieved 2026-09-30): sn-article_tokihiro.tex, the R-matrix (eq. eq:finiteYB), the reduced equation (eq. FYB_eq), the construction of the automaton (§3.1) and the conjecture (§3.2).