Nonnegative Multiweight Smith Profiles and Contracted Strata of Shared-Socle Jordan Degenerations

Abstract

This article studies one-parameter degenerations of chains of nilpotent Jordan blocks joined along their socle vectors. It gives a complete Smith-normal-form description of the associated self-extension torsion for arbitrary chain length and all nonnegative edge valuations. The result includes an explicit path-matching formula, a sharp finite reduction in the block-size parameters, a classification of the Jordan types created by zero-valued couplings, and an equality between the number of positive Smith factors and the codimension of the corresponding nilpotent-orbit degeneration. The article also identifies the precise size gaps that cause failure of the full type-A interval profile, derives exact torsion-length deficit formulas, and packages the profile through Fitting ideals and transverse-slice dimensions. Exact verification scripts and machine-readable summaries accompany the paper. 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.