Positive-Definite Correlation Hierarchies, Zero Strips, and Derivative de Bruijn–Newman Constants

This paper develops a general theory of polynomially weighted correlation kernels and their connection with the Laguerre–Pólya class. A central hierarchy of correlations is shown to encode the generalized Laguerre inequalities through Fourier transformation, yielding an exact criterion for membership in the Laguerre–Pólya class in terms of simultaneous positive definiteness. An off-axis hierarchy is also constructed, providing a direct method for detecting the width of nonreal zero strips. For the de Bruijn–Newman deformation of the Riemann Xi kernel, the correlation hierarchy admits a three-direction transport structure involving correlation order, polynomial weight, and heat-flow time. This leads to strengthened resolutions of Csordas’ Open Problems 4.10 and 4.11, including simultaneous positive definiteness across all correlation orders beyond an appropriate derivative threshold. The paper also studies derivative de Bruijn–Newman constants. It proves that every fixed derivative order retains infinitely many nonreal zeros at every negative time, with such zeros occurring arbitrarily far from the critical line. Consequently, all derivative Newman constants are nonnegative, form a nonincreasing sequence, and converge to zero for the Riemann Xi function. The negative-time persistence argument extends to positive-degree members of the extended Selberg class. 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.