Even sizes never vanish. An Evidence Press briefing. Some integrals oscillate so fast that ordinary numerical rules need thousands of points. One remedy moves the calculation into the complex plane and builds a Gaussian-type rule from polynomials that are orthogonal with respect to a complex weight. Here the weight is one minus t squared, raised to a power alpha, times a wave of frequency x, on the interval from minus one to one. With a complex weight there is a catch: the polynomial of a given degree may simply not exist. It exists exactly when a certain determinant of moments, called a Hankel determinant, is not zero. In 2022 Celsus, Deaño, Huybrechs and Iserles proved that for the flat weight, where alpha is zero, the even-size determinants never vanish. This paper proves the same for every alpha greater than minus one and every real frequency: every even-size determinant is strictly positive, so the even-degree polynomials always exist. Odd sizes behave differently: each odd-size determinant has infinitely many real zeros. The proof follows the determinants as alpha changes, using an identity showing that neighbouring determinants never vanish together, and a large-frequency expansion that holds uniformly in alpha. The same tools prove degree formulas conjectured in 2008, and give complete proofs of two recurrence-coefficient conjectures stated by Lyu and Zhou in 2026. An earlier preprint, posted on ResearchGate by J. M. Ashfaque in July 2026, claimed those recurrence formulas; its key determinant formula fails as printed, and this release claims the first complete proofs known to it, not the first claim. The limits are clear. The result is about existence only, not about distinct quadrature nodes or numerical stability, and one corollary depends on an unrefereed preprint. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on the fourth of November twenty twenty-six, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.