Certified Weil Positivity Beyond the First Prime Threshold: A Five-Rung Coercivity Ladder up to 1.12 log 2

We prove five explicit coercive lower bounds for the full Riemann–Weil quadratic functional on real test functions supported in intervals of length up to (1.12log 2). This range lies strictly beyond the prime-free endpoint: the displacement (log 2) is active, while no further prime-power displacement occurs. The proof combines an exact real-space identity, a clipped free-kernel minorant, a Galerkin compression in which the prime shift is represented exactly, a Fourier-symbol estimate for the off-span term, and interval Cholesky certification using Arb arithmetic. At the largest certified support length,[D=1.12log 2,]the result gives[Qfull(ψ)≥3.28015×10⁻⁵|ψ|_2^2.] Equivalently, in the standard symmetric normalization,[Wstd(ψ)≥6.56030×10⁻⁵|ψ|_2^2.] The accompanying reproducibility package contains the LaTeX source, fixed certificate data, verification scripts, numerical cross-checks, and the interval-arithmetic verification log. 2020 Mathematics Subject Classification:11M26, 11Y35, 47B35, 65G20.