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 >= 26, with the ratio Z/H growing from 1.5 at kappa=29 to over 10 at kappa=1000.

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? Establishing this inequality unconditionally would be of RH strength.

Code: github.com/utehrani/analysislab-nt

**v5:** Framing of Weil positivity goal clarified; spectral inequality explicitly marked as open and of RH strength. All formal proofs and numerical results unchanged. v4: Table values corrected; c_p^ren simplified to f_p^{1/2}; sawtooth structure identified; numerical table with kappa=29 added. v3: Selberg diagonal unconditional; Montgomery clarified.