Polar Defect and 111-Sharpness for Symmetric Cubics Through Seven Variables, with a Complete Six-Variable Orbit Classification

Abstract

This paper studies concise symmetric cubic tensors of minimal border rank. It establishes a general polar-defect obstruction for tensors that are 111-abundant but not 111-sharp, and combines this obstruction with the low-dimensional geometry of cubic hypersurfaces with vanishing Hessian. As a result, every concise 111-abundant symmetric cubic in at most seven variables is proved to be 111-sharp. Consequently, ordinary tensor border rank and symmetric border rank coincide throughout the minimal-border-rank locus in these dimensions. In six variables, the paper gives a complete classification up to linear equivalence. The locus consists of twenty one-generic trace-cubic orbits arising from six-dimensional commutative Artin–Gorenstein algebras and two one-degenerate Perazzo orbits. The two Perazzo orbits are distinguished explicitly, their projective orbit dimensions are determined, and the lower-dimensional orbit is shown to be the unique concise codimension-one boundary orbit of the higher-dimensional one. Explicit symmetric degeneration families are also constructed. The accompanying computation package verifies the displayed trace cubics, the Perazzo reductions and degeneration identities, the Hessian calculations for reducible cubics, the centroid computations, and the projective stabilizer ranks. All finite calculations use exact arithmetic and include independent finite-field checks. 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.