A Curvature Decomposition of the Explicit Formula for the Riemann Zeta Function

This paper is the first in a series developing an operator-theoretic framework for studying the distribution of the non-trivial zeros of the Riemann zeta function. It studies the second logarithmic derivative of the completed zeta function, H_ξ (σ, t) = ∂²_σ log |ξ (σ+it) |², interpreting it as a second-derivative specialisation of the classical explicit formula. This yields a prime-cutoff decomposition into an archimedean contribution ψ₁ (σ) and local curvature contributions Vₚ (σ) from the finite Euler factors, together with a spectral remainder encoding the nontrivial zeros. No hypothetical input is used: the Riemann Hypothesis, the GUE conjecture, and the Hilbert–Pólya postulate are explicitly avoided throughout. What is proved. Local positivity Vₚ (σ) ≥ 0 at every finite place, for all primes p and all σ > 0. Critical divergence Hₗocal (½, κ) ~ 2 (log κ) ² → ∞ of the truncated local curvature on the critical line. Convergence Hₗocal (σ, κ) → C (σ) ½. The critical line σ = ½ is the unique phase boundary: divergence below, convergence above. What is numerical. The asymptotic ratio Hₗocal (½, κ) / 2 (log κ) ² → 1 is verified for κ up to 10⁶. The σ-profile confirms the sharp transition at σ = ½. What remains open. Whether the singular curvature kernel underlying this formulation can be embedded into the admissible Weil test-function framework — a question addressed in Paper 2. The automorphic Rankin–Selberg analogue and a structural comparison with Li-type positivity criteria are discussed: local positivity does not imply global Li positivity. The present paper constructs the curvature framework that the entire series builds upon. Papers 2–6 develop the Weil functional, the Hilbert-space model, the dual operator, the spectral trace formula, and the curvature bias analysi