A Numerical Study of the Explicit Formula Error Term: N-Zero Approximations to ψ(x) and Consistency with the Riemann Hypothesis

We construct explicit N-zero approximations ψ_N(x) to the Chebyshev prime-counting function ψ(x) using the first N non-trivial zeros of the Riemann zeta function ζ(s), all verified to lie on the critical line Re(s) = 1/2. The approximation is derived directly from Riemann's explicit formula under the stated assumption that all zeros lie on the critical line. We compute the error term E_N(x) = ψ_N(x) − ψ(x) for N = 30, 1,000, and 100,000 across four orders of magnitude from x = 10^5 to x = 10^8. The mean normalised drift — the primary measure of interior approximation quality — remains stable at 0.005 ± 0.0004 across all four scales with N = 100,000, while the absolute error grows as √x with observed decade ratios of 3.27 — within 3% of the theoretical √10 ≈ 3.162. The global maximum normalised error of 0.3678 and interior maximum of 0.1089 are anchored to fixed small-x points (x = 7 and x = 1,013 respectively), a known property of the slow convergence of the explicit formula at small arguments. The truncation error E_N(x) is shown to be small relative to the full PNT error |ψ(x) − x|; the Schoenfeld bound applies to |ψ(x) − x|, not to E_N(x) directly, and these quantities are carefully distinguished throughout. We emphasise that numerical consistency with a conditional bound does not constitute a proof of the Riemann Hypothesis. Full code and data are available at https://github.com/rmathers502-byte/rhymes (Zenodo: 10.5281/zenodo.20113604).