The Positivity Gap: A Precise Localization of the Riemann Hypothesis Obstruction via de Branges Spaces and GL Vacuum Stability

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.