We precisely locate the analytic obstruction to proving the Riemann Hypothesis within the de Branges–Hermite-Biehler framework. Starting from the Jacobi theta series, we construct the chain θ →Φ →ξ →E where E(z) =∞ 0 Φ(t)eiztdt is an entire function whose membership in the Hermite-Biehler class is equivalent to the Riemann Hypothesis. We prove Φ(t) > 0 for all t ∈R via the modular symmetry ω(x) = x−1/2ω(1/x), and confirm numerically that ξ sits on the boundary of the Hermite-Biehler class with correct interlacing. We then systematically kill three can- didate bridges from physical/geometric positivity to the required analytic positivity: (i) the bounded Lyapunov operator in Lax-Phillips scattering, (ii) the Toeplitz total- positivity route (obstructed at order 5 by Michalowski ?), and (iii) the GL vacuum stability intertwiner (shown circular by explicit Mellin computation). We show that Connes’ adelic W ≥0 does not follow from adelic geometry—it is the Riemann Hypoth- esis restated. The gap is structural, not technical: every bridge from positive-definite kernels on the physical space to positive-definite forms on the analytic zero-space ei- ther assumes the zeros lie on the critical line or requires an inequality equivalent to the Riemann Hypothesis. We identify the first nontrivial open case of the Csordas hierarchy, L1(x) = (H′)2 −HH′′≥0, as an independent open problem not implied by the Riemann Hypothesis itself.
Jean-Paul Niko (Thu,) studied this question.