Radical Base-Change Failure and a Nilpotence-Index Jump in a Determinantal Degeneration of Segre Projections

This paper studies a determinantal degeneration arising from low-rank projections of a Segre variety. A generic rank-two projection degenerates to a rank-one projection together with an additional linear component, producing a family whose geometric and scheme-theoretic behavior changes sharply at the special fiber. The total family is analyzed through an explicit primary decomposition. Its horizontal components and its unique vertical determinantal component are identified separately, allowing the two principal degeneration mechanisms to be distinguished. The vertical component accounts for the jump in dimension, while the collision of horizontal primary components produces a failure of radical base change. At the special fiber, the nilpotence index increases from two to three. This jump is explained by an explicit Loewy filtration, together with descriptions of the new defect layer and the nonzero square layer. The resulting structure shows that the dimension jump and the nilpotence jump arise from different geometric mechanisms, even though they occur in the same degeneration. The arguments are valid over arbitrary fields and include special treatment of small characteristics. All symbolic calculations are exact and are supported by an accompanying reproducibility package containing the computational scripts, reference outputs, and an isolated verification runner. 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.