Operator-Valued Cubic Moments after First-Order Cancellation: Sharp Rectangular Bounds and an Open Two-by-Two Problem

Abstract We examine the cubic operator-valued loop functional arising after exact cancellation of the first-order transition block of a finite-dimensional off-diagonal evolution. Direct expansion yields the cubic Dyson coefficient with a uniform fifth-order remainder, and a length argument gives the universal estimate ||J_3[F]|| ≤ L[F]^3/12. Using the previously published sharp scalar Martinet inequality, we derive a dimension-independent sharp bound for closed complex column and row paths by a Gaussian fourth-moment argument. For two-by-two matrix paths, we give a trace-determinant decomposition, establish the sharp scalar bound on a restricted rank-one-derivative subclass, and exhibit full-rank paths attaining the scalar benchmark. The global two-by-two extremal constant remains undetermined; neither equality with nor strict improvement over the scalar benchmark is claimed. AI Assistance and Research Provenance Use of generative AI. During preparation of this manuscript, the author used OpenAI ChatGPT to assist with manuscript organization, drafting and restructuring of text, language editing, and consistency checks based on author-supplied research materials and independently verifiable sources. The author reviewed, revised, and verified all AI-assisted content, mathematical arguments, references, interpretations, and conclusions, and takes full responsibility for the final manuscript. Publication status: Preprint v1.0. Not peer reviewed. DOI: 10.5281/zenodo.23239014.