The archimedean threshold of Weil positivity on odd functions: the ground-state energy of log |D| on an interval

Let A(g) = 1 π R |ĝ(r)| 2 θ ′ (r) dr − 2ĝ(i/2) 2 be the archimedean part of Weil's quadratic form for ζ on odd real g ∈ L 2 (−a, a), where θ is the Riemann-Siegel theta function. For a ≤ 1 2 log 2 no prime enters the explicit formula and Connes-Consani proved A ≥ 0. We show that the problem is governed by a single scale-free constant: A(g) ≥ (M 1 − log a − log 2π)∥g∥ 2 + G E − 2ĝ(i/2) 2 , where G is the autocorrelation of g, E an explicit bounded kernel with E(0) = − 1 4 , and M 1 = inf⟨g, log |D| g⟩/∥g∥ 2 over odd g supported in (−1, 1) is the ground-state energy of log |D| on the odd sector of the unit interval; numerically M 1 = 0.866461, and the minimiser of A is within 0.008 (in L 2) of that ground state. We prove M 1 ≥ 0.775 by an elementary bathtub argument, M 1 ≥ 0.7985 by a layer-cake with an explicit Hilbert-Schmidt majorant of Slepian's operator, and we show that every such concentration-based certificate is capped at the layer-cake value with the exact Slepian eigenvalue, numerically 0.8056. We then certify M 1 ≥ 0.866379 by Temple's inequality in exact arithmetic with an outward-rounded interval evaluation (a self-contained verifier is supplied), using a real-space formula for log |D| on the interval and the explicit lower bound λ 2 ≥ 1 2 log 9π 2 4 for the second odd eigenvalue. Combined with an interval-arithmetic bound for the kernel term this gives A(g) ≥ 0 for all odd g ∈ L 2 (−a, a) with a ≤ 0.360, beyond 1 2 log 2 = 0.3466; the numerical threshold is 0.3716. This is a statement about the archimedean part only: with the primes included, Weil positivity is now certified on much larger supports (Zhu, [−0.8, 0.8]), and our contribution is the identification and certification of the archimedean threshold and its constant, not an extension of Weil positivity. The even sector, where the global ground state lives and the form is nearly degenerate, is not treated.