From Local Curvature to the Weil Functional: An Explicit Construction

This paper is the second in a series developing an operator-theoretic framework for studying the distribution of the non-trivial zeros of the Riemann zeta function. Building on the curvature decomposition of Paper 1, it constructs an explicit family of admissible Schwartz-class test functions for the Weil explicit formula and derives a prime-side decomposition connecting the zero sum to local curvature. The construction uses antisymmetrised Gaussian bumps adapted to the adelic zeta integral; no exotic test functions are needed. No hypothetical input is used: the Riemann Hypothesis, the GUE conjecture, and the Hilbert–Pólya postulate are explicitly avoided throughout.

What is proved. The admissibility of the test functions in Lagarias' formulation of the Weil criterion (Lemma 2.1). Strict positivity of the renormalized prime weights c_p^ren = f_p^{1/2} > 0, by an elementary algebraic argument (Lemma 4.1). Convergence of the diagonal energy series D = Σ_p (c_p^ren)² (Lemma 4.2). The normalised Gaussian correlation K_ε localises the diagonal first-prime contribution exactly to V_p(σ) by an algebraic identity (Observation 3.2).

What is numerical. The diagonal energy D ≈ 9.470 (Observation 4.3). Finite-grid stability: Z_ren − H_ren > 0 at every tested prime cutoff except κ = 20 (Observation 6.1).

What remains open. The Weil-normalised embedding: whether the diagonal localisation of Observation 3.2 can be implemented inside the Lagarias–Weil functional including archimedean and zero-side contributions. The spectral inequality — does the zero sum grow at least as fast as H_local(½,κ) ~ 2(log κ)²? Off-diagonal control of the zero sum. All three are stated precisely as open problems; none is used as a hypothesis.

Paper 1 established the local curvature divergence at the critical line; the present paper constructs the Weil-functional framework and the renormalised prime weights on which the later papers build. The other papers of the series are linked as Related works on this record. The accompanying verification scripts are available at https://github.com/utehrani/analysislab-nt. Part of a cumulative series; no paper in the series claims a proof of the Riemann Hypothesis.