Sharp Low-Coordinate Dynamics and Dimension-Eight Order-Sixteen Nonexistence for Normalized Solutions of the Araoka–Tokihiro Equation

Abstract

This work studies normalized bijective solutions of the Araoka–Tokihiro functional equation over finite binary vector spaces, with particular emphasis on translation kernels, low-coordinate dynamics, and the possible orders of the associated two-site sweep.A complete translation-kernel lift classification is developed in terms of unary cocycles, together with exact counting and conjugacy methods. The normalized five-dimensional solution set is determined exactly, and the maximal-filtration stratum is classified.The paper proves that every normalized solution through ambient dimension six has quartic two-site dynamics. It then constructs an explicit seven-dimensional normalized solution of exact order eight and proves that dimension seven is the first dimension in which this behavior can occur.The six-dimensional obstruction frontier is completely classified. It consists of one active conjugacy class with a two-dimensional first translation kernel and twenty-three reduced active scalar-shear moduli with a one-dimensional first kernel. These data yield a finite presentation ledger with 19,456 entries that maps surjectively onto the full seven-dimensional order-eight family.A uniform half-orbit certificate is then established for the entire ledger. As a consequence, every scalar cocycle over every normalized seven-dimensional order-eight solution has vanishing eight-step residual, and every further finite-dimensional translation-kernel lift remains of exact order eight.This proves that no normalized eight-dimensional solution can have order sixteen and that no normalized order-sixteen solution can have a seven-dimensional first-kernel quotient. Therefore, if a normalized order-sixteen solution exists, its ambient dimension is at least nine. The global existence of normalized order sixteen remains open.The accompanying computation files contain the programs, finite classification data, reference outputs, and reproducibility scripts supporting all computer-assisted statements in the paper.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. The final mathematical arguments are stated explicitly in the manuscript, and the computational supplement is provided for reproducibility.