Exact Moment-Robust Hopf Thresholds through Order Twelve: Alternating Gauss–Radau Extremizers and Factorial Convergence

This paper determines exact robust stability thresholds for a scalar distributed-delay equation when the unknown delay distribution agrees with the uniform distribution on the unit interval through finitely many moments. Five consecutive even moment orders are solved. The unique worst-case distributions are alternating right and left Gauss–Radau measures, the boundary characteristic roots are simple and cross transversally, and the thresholds form a strictly increasing sequence converging to the uniform-delay threshold. A factorial convergence estimate valid for all sufficiently high moment orders is also proved. Every numerical bound used in the main results is certified by exact rational interval arithmetic, and a public reproduction package is provided. 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.