From Local Curvature to the Weil Functional: An Explicit Construction

Building on the curvature decomposition of the companion note we construct an explicit family of admissible test functions for the Weil explicit formula and derive a prime-side identity connecting the zero sum to the local curvature studied in Paper 1. We prove that the renormalized prime weights are strictly positive by an elementary algebraic argument (numerator 2p−1 > 0 for all primes p), and that the associated energy series converges to D ≈ 9.471. The Mertens constant emerges as the natural scale of the renormalized curvature remainder, with a structural connection to the Euler–Mascheroni constant and the first Li coefficient. The open problem is isolated as a single spectral inequality: does the zero sum grow at least as fast as the local curvature H_local(1/2, κ) ~ 2(log κ)²? Numerical verification code is available at github (in preparation)

v3:

• Selberg diagonal term identified as unconditional;
• off-diagonal control isolated as the open step.
• Connection to Montgomery pair correlation clarified.