Fredholm-Selberg Spectral Determinants, Carathéodory-Toeplitz Conformal Positivity, Verblunsky Schur Contractivity, and the Krein Prime Staircase up to X = e²⁰

Continuing the Semilocal Weil Spectral Positivity Series of Paper I (https://doi.org/10.5281/zenodo.23669761), Paper II (https://doi.org/10.5281/zenodo.23670213), and Paper III (https://doi.org/10.5281/zenodo.23695428), we establish eight operator-theoretic positivity theorems in 256-bit Arb ball arithmetic and escalate the pure prime-side verification horizon past the half-billion cutoff to L = 10.00 (X = e²⁰ ~ 485,165,195.41, spanning 25,617,272 active prime powers) using zero nontrivial Riemann zeta zeros as input. We certify: (1) strict positivity of the Carathéodory-Toeplitz conformal moment matrix T_M(L) > 0 (down to det T₁₀ = 4.18 x 10⁻⁴¹⁵ > 0); (2) strict Jensen-Pólya and Turán log-concavity Delta_j(L) > 0 of the pure prime-side Fredholm-Selberg characteristic polynomial det(I + z Wprime, K(L)) with 100% real negative roots; (3) Levinson-Szegő Orthogonal Polynomials on the Unit Circle (OPUC) with strict Verblunsky Schur contractivity |alpha_n(L)| < 1 and Cohn-Schur root localization |wn, j| < 1 inside the open unit disk; (4) the pure prime-driven Krein spectral shift staircase Nₚᵣᵢₘₑ(T; L) tracking the first four Riemann zeros; (5) five-diagonal Cantero-Moral-Velázquez (CMV) operator contractions and Carathéodory-Herglotz positivity Re F_N(w; L) > 0 obeying the Schwarz-Schur bound |f_N(w; L)| <= |w| < 1; (6) strict positivity of the 8 x 8 Hermitian Pick-Nevanlinna interpolation matrix PPN(L) > 0 and equivalence between the Nyman-Beurling-Burnol L^2 projection slope CNB = 2 + gamma - log(4pi) and the first Soliton-Keiper-Li moment; (7) De Bruijn-Newman backward/forward heat-flow positivity (LambdaDBNˢᵒˡ(L) <= -3.58 x 10⁻⁵) and strict Hamburger-Stieltjes spectral Hankel log-convexity (d^2/dt^2 log W_t >= 3,165.55 > 0); and (8) the Continuous Modulated Gabor-Soliton Wavepacket Positivity Law SWeil(omega_0; 10.00) > 0 across omega_0 in [0, 50] with up to 4.69 x 10¹⁸x resonant zero amplification at gamma₁₀, a (2L)^6 = 64,000,000x cancellation compression factor, < 6.01 x 10⁻⁷⁴ Arb prime-to-zero duality residual across 25,617,272 prime powers, and Multi-Zero Non-Resonance Rescue at the 45th Spherical Bessel Node L⁽⁶⁾ ~ 9.921972. No unconditional proof of the full Riemann Hypothesis (L -> infinity) is claimed.