Canonical Continuous Double–Turán Inequalities and a Derivative-Shift Barrier for the Riemann Xi Kernel

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Original abstract

This preprint develops a continuous-parameter form of the double–Turán inequalities for a broad class of even admissible kernels. After introducing a canonical entire continuation of normalized Mellin coefficients, it proves strict positivity of the associated continuous double–Turán expression throughout the closed canonical parameter range beginning at negative one-half. The proof extends the source-determinant method of Csordas and Dimitrov beyond the ordinary moment range. It establishes a strict determinant sign on an ordered integration region and treats the boundary parameter separately through an Abel-confluent limiting argument. This endpoint analysis resolves the interaction between a vanishing gamma factor and a singular source moment without relying on a formal continuation of the original moment integral. For the standard Jacobi-theta kernel associated with the Riemann xi-function, a bordered Toda identity transfers the two required logarithmic concavity properties to the first derivative profile. The resulting Mellin derivative shift extends strict continuous double–Turán positivity to a larger parameter range beginning at negative three-halves. The paper also determines the limitation of this derivative-closure mechanism. An exact rational interval certificate shows that the second derivative profile has logarithmic curvature of the opposite sign at the origin. This provides a rigorous barrier to repeating the same argument beyond the first derivative shift, while leaving open the possibility of different continuation methods. The accompanying computation archive contains exact symbolic identity checks, rational interval certificates for the origin signs and curvature barrier, and the finite certificates used for the fourth derivative Hankel determinant input. It supports direct verification and complete reconstruction of the bundled certificates using integer, rational, and exact symbolic arithmetic. The results concern kernel inequalities and continuous coefficient hierarchies in the Laguerre–Turán program. They do not establish a zero-free region, prove hyperbolicity of all Jensen polynomials, or imply the Riemann hypothesis. 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.

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