A Nonsplit Special 2-Group Extension Governed by the 2×2 Matrix-Multiplication Tensor

Abstract

This paper determines the extension structure of the full linear stabilizer of an explicit configuration of 1,024 six-dimensional subspaces over the binary field. The configuration was introduced in the preceding paper Exact Linear Symmetry of a Four-Core Rank-Metric Sunflower Configuration. Its stabilizer maps onto a translation group extended by a macro-symmetry group, with a large pointwise flag kernel. The present paper proves that this extension is nonsplit. The obstruction is already detected over a single cyclic translation subgroup, whose nontrivial coset contains no involution. The inverse image of the full translation subgroup is identified as a special 2-group of order two to the forty-third power. The pointwise kernel on the 1,024 subspaces is a special group of order 128, and the corresponding faithful quotient is a special 2-group of order two to the thirty-sixth power. The structural mechanism is described by four copies of the two-by-two matrix-multiplication tensor. This tensor normal form explains the augmentation filtration, the exact rank distribution of the translation operators, the Segre geometry of the rank-two locus, and the left-right action of the macro symmetries. The paper also determines the involution-liftability distribution for all 65,536 translation cosets, computes the conjugacy-class number and commuting probability of the faithful quotient, and recovers the full stabilizer in the original twenty-three-dimensional ambient space. All finite calculations are exact over the binary field or over the integers and are reproduced by an accompanying deterministic verification 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.