Quadratic Normal Flatness and a Determinantal Cubic Jet for a Degree-Thirteen Covariant of Cubic Forms

This paper studies the local behavior of an explicit degree-thirteen, quartic-valued covariant of cubic forms near a direct-sum locus associated with two ternary cubics and an additional cube. The first main result proves complete quadratic vanishing on the fifteen-dimensional normal quotient determined by pure quadratic perturbations. Thus, although the ambient covariant has nontrivial first-order behavior, its restriction to the relevant normal directions has contact of order at least three. The second main result determines the first nonzero cubic term on the nine-dimensional mixed block. This term is a nonzero scalar multiple of the determinant of the mixed perturbation matrix, multiplied by the natural quartic invariants of the two ternary cubics and the transverse difference of the two cubic summands. Consequently, every invertible mixed perturbation has exact order-three contact. The proof combines invariant theory, apolar geometry, five-epsilon contraction formulas, block decompositions, plethysm calculations, and exact modular computation. The computational certificates use four independent primes, explicit coefficient bounds, and integer reconstruction. Source code, exact output data, and verification scripts are provided in the accompanying computation package. The paper does not claim a determination of the complete cubic initial ideal on the full normal space, the full normal cone, local irreducibility, radicality, or local multiplicity. 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.