A Certified Negative Interval for the Ninth Derivative Laguerre Quantity of the Riemann Xi Kernel

Abstract

This revised preprint studies the derivative Laguerre quantities associated with the Jacobi theta kernel in the Fourier representation of the Riemann xi-function. It gives exact rational certificates showing that the ninth quantity is negative throughout a nontrivial interval around the symmetry point, with the certified range extended to absolute parameter value at most one fiftieth. It also verifies positivity at the symmetry point for levels one through eight and negativity at level nine. The proof uses explicit derivative polynomials, exact rational interval arithmetic, and elementary exponential bounds. A supplementary Python verifier reproduces the decisive sign computations using integer and rational arithmetic only. Ryan Kielhorn publicly deposited an exact level-nine counterexample at the symmetry point before the original Koide deposit. Brandon Yates later registered a Lean 4 formalization of the point counterexample. This revised version makes no priority claim for the point counterexample. Its distinct contribution is the certified interval of negativity, together with an exact and independently executable reproducibility certificate. 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.