The Sharp Normalized Two-Site Exponent in Dimension Twelve for the Araoka–Tokihiro Equation

Abstract

This preprint determines the sharp universal exponent of the ordered two-site sweep associated with normalized bijective solutions of the Araoka–Tokihiro functional equation in dimension twelve over the binary field.It proves that every such two-site sweep has period dividing thirty-two and that this bound is attained. The proof combines a general algebraic saturation theorem for three successive scalar translation lifts with exact computer-assisted closure of the remaining finite branches.The algebraic part eliminates one of the three possible dimension-ten configurations without requiring a classification of the corresponding dimension-nine quotients. The other two configurations are resolved by exact binary linear algebra, component-voltage reductions, source-equation certificates, verified transports between presentations, and explicit lift-back checks.The accompanying computation archive contains the source data, finite certificates, reproducibility scripts, independent source-level verification for the largest finite branch, and the low-dimensional verifier suite used by the proof. All decisive finite calculations use exact arithmetic rather than floating-point approximations.The result concerns normalized solutions and the ordered two-site sweep. It does not claim the full many-site period conjecture for the Araoka–Tokihiro cellular automaton.Research methodology and AI assistance: This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology combining persistent research archives, literature and prior-art investigation, iterative proof exploration, and exact verification procedures. Generative AI was used for mathematical exploration, computational reasoning, literature research, proof development, and manuscript preparation.