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 identity 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).

What is conditional. The prime-side identity W(g*,g*) = Z(g*) − H_local(σ,κ) + O(ε), under standard remainder bookkeeping for the Weil explicit formula applied to the constructed test functions (Proposition 3.1). The assumption is named explicitly and concerns the Lagarias four-piece decomposition at fixed parameters.

What is numerical. The diagonal energy D ≈ 9.471 at reference parameters (Observation 4.3). Finite-grid stability: Z_ren − H_ren > 0 at every tested prime cutoff except κ = 20 (Observation 6.1). The bridge constant D/(2π) ≈ 1.507.

What remains open. The spectral inequality — does the zero sum grow at least as fast as H_local(½,κ) ~ 2(log κ)²? Establishing this unconditionally would be of RH strength. Off-diagonal control of the zero sum via Selberg-type asymptotics. Both are stated precisely in the paper; neither is used as a hypothesis.

Paper 1 established the local curvature divergence H_local(½,κ) ~ 2(log κ)² at the critical line. The present paper constructs the Weil functional framework that Papers 3–6 build upon.