Exact Linear Symmetry of a Four-Core Rank-Metric Sunflower Configuration

Abstract

This paper constructs and analyzes an explicit configuration of 1,024 six-dimensional subspaces over the binary field. The configuration is assembled from four tensor cores, a four-dimensional rank-metric center code, and two affine parameters. The intersection structure determines four connected components and sixty-four sunflower blocks, each consisting of sixteen subspaces with a common five-dimensional center. Every sunflower intrinsically recovers a nested flag of dimensions five, nine, and ten. The endomorphisms preserving all sixty-four flags form a twenty-eight-dimensional local algebra. Its Jacobson radical has dimension twenty-seven, its square has dimension two, and its cube is zero. The center and commutator structure of this algebra are also determined exactly. The corresponding flag-fixing unit group decomposes into an elementary abelian group of rank sixteen and a special group of order 2,048. The action induced on the sixty-four flags is determined from rank-colored center automorphisms and compatibility conditions between the four cores. Combining this quotient action with the flag-fixing kernel gives an exact description of the full ambient linear stabilizer, whose order is 422,212,465,065,984. No splitting of the resulting group extension is asserted. All computations are exact over the binary field. An accompanying deterministic verifier reconstructs the complete configuration, checks every pairwise intersection, recovers the sunflower and flag structures without using their labels, rebuilds the local algebra, verifies the group-theoretic invariants, and confirms the full stabilizer calculation without floating-point arithmetic, randomness, network access, or external computer algebra software. 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.