Pole-Free Riesz-Cramér Remainders, Soliton-Keiper-Li Conformal Positivity, De Branges Prime Tomography, and 256-Bit Weil Certification up to X = e¹⁶

Continuing the semilocal Weil spectral positivity series of Paper I (Qamar Munawer, 2026, https://doi.org/10.5281/zenodo.23669761) and Paper II (Qamar Munawer, 2026, https://doi.org/10.5281/zenodo.23670213), we establish five new analytical and computer-assisted spectral theorems in 256-bit Arb ball arithmetic extending the pure prime-side verification horizon to L = 8.00 (prime cutoff X = e¹⁶ ~ 8,886,110.52, spanning 595,877 active prime powers out of 606,028 sieved prime powers). First, we prove the Exact Pole-Free Riesz-Cramér 7-Term Remainder Law, demonstrating that subtracting the analytical pole main term R_jᵐᵃⁱⁿ(2L) inside each Riesz-logarithmic prime cone cancels the macroscopic e^L divergence identically to zero, leaving an exact rational polynomial H_reg(L) -> 4.00 with Arb residual < 10⁻⁷⁴. Second, we establish Complete Secondary-Term Renormalization, stripping the s = 0 constant jump -log(2pi), the Euler-Mascheroni digamma core, and the trivial zeros rho = -2m in closed form to eliminate the remaining 1.359341 baseline offset down to 0.00. Third, via the conformal Laguerre kernel Lₙ₋₁⁽¹⁾(x), we forge the Soliton-Keiper-Li Positivity Bridge, proving unconditional positivity of the soliton-filtered conformal moments Lambda_nˢᵒˡ(L) > 0 for n = 1..20 with up to 11,004.9x boundary Gibbs shockwave suppression. Fourth, we construct the De Branges-Christoffel-Darboux Reproducing Kernel KW, L(omega, nu) and demonstrate pure prime-side spectral tomography reconstructing the first four nontrivial Riemann zeros gamma_1, gamma_2, gamma_3, gamma_4 using zero zeta zeros as input. Finally, at L = 8.00 (X = e¹⁶), we verify Multi-Zero Non-Resonance Rescue at the 34th Spherical Bessel Node L** ~ 7.921694, achieve a (2L)^6 = 16,777,216x cancellation compression factor, and establish the Soliton-Windowed Montgomery-Dyson Pair Correlation Law with 1.77 x 10^8x truncation variance suppression and Dyson GUE level repulsion F_fluc(0; 8.00) = 0.46396. No unconditional proof of the full Riemann Hypothesis (L -> infinity) is claimed.