Sharp order-four sign regularity of a Jacobi theta kernel
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…
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