A Mesoscopic Repulsion Bound for Zeros of the Riemann Zeta Function via the Explicit Formula

I establish a lower bound on the localized pair correlation of non-trivial zeros of the Riemann zeta function ζ(s) in a mesoscopic window [T, T+H] with H ≥ T^θ for a fixed θ > 0. Using only the classical explicit formula and the positivity of a Fourier-weighted sum over primes, we prove that the pair correlation measure R₂(u; T, H) satisfies a quadratic vanishing lower bound near the origin — a signature of zero repulsion. Specifically, we show that ∫|u| ≤ U R₂(u; T, H) du ≥ c·U³ for some absolute constant c > 0 and all sufficiently small U > 0.The argument requires no unproven hypotheses: neither the Riemann Hypothesis nor the GUE conjecture is assumed. A corollary provides a weak average spacing bound of order N(T,H)−1/3. We discuss the relationship with random matrix theory predictions and directions for future work.MSC 2020: 11M26, 11M06