Cumulative Prime Sums and Riemann Zeros: An Empirical Correspondence

We establish a generative formula connecting Riemann zeta zeros to cumulative prime sums. For each zero ρₙ = 1/2 + itₙ, there exists a critical prime Nₙ such that tₙ = log(Sₙ) + O(e^{-αt}), where Sₙ = 1 + Σ(p≤Nₙ) p is the cumulative sum of primes up to Nₙ, and α ≈ 0.46. The convergence is exponentially fast: errors decrease from 10⁻³ for the first zero to 10⁻⁷ for the fifth. We derive the scaling law tₙ ≈ 2·log(Nₙ), explain the factor of 2 from quadratic growth of prime sums, and establish the Halved Exponent Law governing transitions between critical primes. This provides a computational interpretation: Riemann zeros encode the logarithmic scale of prime accumulation.