{"id":"hexagon/2610.00721","pdf":"https://hexagonmath.org/pdf/2610.00721v1","url":"https://hexagonmath.org/abs/2610.00721v1","tags":["math.NT","msc:33E05","msc:15B48","msc:26C10","msc:11M26","Csordas-Varga","Jacobi theta function","Riemann Xi","sign regularity","total positivity"],"score":0,"title":"Sharp order-four sign regularity of a Jacobi theta kernel","domain":"math.CA","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CA","summary":"We prove that the Hankel kernel $\\Phi(\\sqrt{x+y})$, where $\\Phi$ is the Jacobi theta kernel whose cosine transform is the Riemann $\\Xi$-function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at $t=10^{-2}$ shows failure at order five. The third-order condition is Csordas's conjecture that $s'^2-ss''$ is strictly log-concave for $s(t)=\\Phi(\\sqrt t)$. The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for $t\\ge1$ and interval arithmetic on $[0,1]$. The…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}