Strict Total Positivity of Order Four for the de Bruijn--Newman Kernel

Abstract

We prove that the de Bruijn--Newman kernel associated with the Riemann Xi function is strictly totally positive of order four. Let \[K(x,y)=\Phi(x-y),\] where \(\Phi\) is the classical positive even Riemann kernel. We establish that for every \[1\le m\le4,\] and every pair of strictly ordered real \(m\)-tuples \[x_1<\cdots<x_m,\qquady_1<\cdots<y_m,\] one has \[\det\left[\Phi(x_i-y_j)\right]_{i,j=1}^{m}>0.\] Thus the associated Toeplitz kernel is strictly totally positive of order four and, in particular, \[\Phi\in\mathrm{PF}_4.\] The proof proceeds through global positivity of the third- and fourth-order confluent Toeplitz determinants. For the fourth-order determinant, the first three theta modes are separated from the infinite remainder. After the substitution \[q=\pi e^{4x},\] the principal part is reduced to a finite polynomial problem in \[d=4e^{-3q},\qquade=9e^{-8q}.\] The coefficient signs are certified by exact integer-polynomial expansions, Sturm root isolation, and elementary rational exponential bounds. The remaining infinite theta tail is controlled by explicit absolute derivative majorants and determinant multilinearity. No floating-point interval arithmetic is required in the proof. The confluent inequalities are then converted into genuine total positivity by an extended Chebyshev-system argument. Writing \[p=L',\qquadL=\log\Phi,\] and \[Q(p(x))=\frac{\Phi''(x)}{\Phi(x)},\qquadR(p(x))=\frac{\Phi'''(x)}{\Phi(x)},\] the order-three and order-four confluent determinants yield positivity of the initial Wronskians of \[1,\quad p,\quad Q,\quad R.\] This gives strict positivity of all ordered Toeplitz minors through order four. Together with the recently established failure of \(\mathrm{PF}_5\) for the same kernel, the result identifies the exact finite Pólya-frequency order of the de Bruijn--Newman kernel as \[\boxed{4}.\] The result is a finite-order total-positivity theorem and does not constitute a proof of the Riemann Hypothesis. The development of the proof was AI-assisted. ChatGPT(OpenAI) was used extensively for symbolic exploration, high-order algebraic calculations, proof-strategy search, construction of tail estimates, and independent consistency checks. Two conceptual ideas that directed the research originated with the human author: the “beat-wave integers” viewpoint, which helped identify the research object, and the later proposal to examine the Fourier structure transverse to the critical line, which helped identify the geometric direction of the problem and led to the total-positivity route.