From Local Curvature to the Weil Functional: An Explicit Construction

Building on the curvature decomposition of the companion note (doi:10.5281/zenodo.19025598), 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 renormalized curvature remainder exhibits a sawtooth structure analogous to the prime-counting function pi(x). Numerical experiments show Z_ren > H_ren for all kappa >= 50.
The open problem is isolated as a single spectral inequality: does the zero sum grow at least as fast as H_local(1/2, kappa) ~ 2(log kappa)^2?
Code: github.com/utehrani/analysislab-nt
v4: Table values corrected; c_p^ren simplified; sawtooth structure identified; numerical table added.
v3: Selberg diagonal unconditional; Montgomery clarified.