Strict Laguerre Concavity and Double Turán Inequalities Throughout the de Bruijn–Newman Flow

This preprint studies Turán-type inequalities associated with the Riemann xi-function and their behavior under the de Bruijn–Newman heat flow. It proves the strict log-concavity of the first Laguerre quantity for the standard Riemann xi kernel, resolving Csordas’ Open Problem 4.14 and the earlier Problem 3.3 of Csordas and Dimitrov. The argument combines an exact determinant criterion with rigorous analytic estimates and an exact Bernstein-polynomial certificate. A Gaussian covariance principle then extends the result to every real heat parameter, showing that both the ordinary and double Turán inequalities hold strictly throughout the entire de Bruijn–Newman flow. Consequently, the heat-deformation question posed by Csordas and Dimitrov has an empty failure set. The result also shows that these Turán sign conditions do not detect the de Bruijn–Newman real-zero transition: they remain strictly valid even in parameter ranges where nonreal zeros occur. An ancillary verification archive is provided for the exact finite computations and symbolic identities used in the proof. The results concern necessary-condition structures related to the Riemann hypothesis and do not constitute a proof of the Riemann hypothesis.