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 V_p(σ) from the finite Euler factors, together with a bookkeeping remainder, defined as the finite-cutoff difference, which absorbs the zero side. 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 V_p(σ) ≥ 0 at every finite place, for all primes p and all σ > 0. Critical divergence H_local(½,κ) ≍ (log κ)² → ∞ of the truncated local curvature on the critical line. Convergence H_local(σ,κ) → C(σ) < ∞ for all σ > ½. The critical line σ = ½ is the unique phase boundary: divergence below, convergence above. What is numerical. The asymptotic ratio H_local(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. 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 on which the series builds: the prime-cutoff decomposition, local positivity of the Euler-factor curvature, and the phase boundary at σ = ½. 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.