Positivity of the k=1 Toeplitz Minors of the Riemann ξ-Coefficients

Abstract

We prove strict positivity of the complete (k=1) family of consecutive Toeplitz minors associated with the Taylor coefficients of the Riemann (\xi)-function: D_{r,1} > 0 for every r >= 1. The proof rewrites (D_{r,1}) as a coefficient of the reciprocal generating function a_0 / G(-z), G(z) = (1/8) xi(1/2 + sqrt(z)/2), and then separates the canonical product into a positive core generated by zeros that have been rigorously verified to lie on the critical line and an unverified high-zero tail. Using the verified Riemann Hypothesis up to height: H_0 = 3 x 10^12, together with an explicit zero-counting bound, we show that the total transformed mass of all remaining zeros is too small to change the sign of any reciprocal coefficient. This yields strict positivity in every order without assuming the Riemann Hypothesis beyond the verified range. The result settles the entire (k=1) line of the Toeplitz-minor positivity problem associated with the Riemann (\xi)-function. It does not prove the Riemann Hypothesis, which would require positivity of the full two-parameter family of minors.