Rank-One Syndrome Fibers, Minimum Orbit Separation, and Reconfiguration Geometry for a Binary Tensor Code

This preprint studies a binary quadratic tensor on a twenty-dimensional vector space and the orbit structure induced by its automorphism group. Rank-one contraction fibers are assembled into a complete invariant that distinguishes all 3,568 dual orbits. After identifying equivalent probe columns, the resulting classification problem has 3,039 effective probes. An explicit system of five probes separates every orbit, and a finite blocking-core certificate proves that four probes can never suffice. The paper then gives a complete classification of all minimum five-probe systems. There are exactly 2,078,105 effective systems and 3,173,771 corresponding raw lifts. The one-exchange graph on the effective minimum systems is also reconstructed in full. It has 11,946,710 edges and 177,469 connected components. A unique giant component contains 1,824,449 systems, while every other component has at most 59 systems. The accompanying computational supplement contains the exact input data, source code, finite certificates, and independent verification procedures used in the classification. It reconstructs the orbit-fiber data, verifies the optimality of five probes, checks exchange saturation, exhausts all 198 residual search branches, reproduces the complete list of minimum systems, and verifies the reconfiguration-graph census. 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.