Low-Rank Segre Geometry, Automorphism Groups, and Complement Cohomology of a Binary Two-Step Algebra

Abstract

This paper determines the intrinsic geometry, symmetry structure, and complement cohomology of an explicit forty-three-dimensional two-step nilpotent associative algebra over the binary field and of its associated algebra group. The contraction ranks of the induced quadratic map exhibit a gap at rank three. The low-rank locus contains a canonical five-dimensional subspace, while the forty-nine rank-four points form a finite Segre configuration. Both rulings of this configuration are recovered intrinsically from the polarization through image and kernel relations. The local and quotient symmetry groups are determined and shown to combine through a single character of the dihedral group. This yields the full pseudo-isometry group and leads to explicit descriptions of the automorphism groups of both the algebra group and the associative algebra. The adjoint algebra with involution, centers, abelianizations, derived residuals, outer automorphism groups, and conjugacy classes of complements are also determined. A decomposition into indecomposable modules over the dihedral group identifies a unique nonsplit pullback block in the natural quotient map. The resulting cohomological obstruction is governed by the same character that controls the geometric gluing. This establishes a direct structural link between the low-rank Segre geometry and the complement cohomology. All finite calculations are exact over the binary field and are supported by an accompanying reproducible computational package. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.