Boundary-Layer Hardy-Sobolev Regularization, Slepian Prolate Spheroidal Wave Functions, and Macroscopic Kato-Rellich Contraction up to 𝑋 = 𝑒 20 X=e 20

We establish the macroscopic extension of the cleanroom-verified, zero-free Weil spectral positivity program to L=10.00L = 10.00L=10.00 (X=e20≈485.17×106X = e^{20} \approx 485.17 \times 10^6X=e20≈485.17×106, encompassing 25,617,27225,617,27225,617,272 active prime powers) and modal dimension M=12M = 12M=12 on the codimension-1 pole-free odd subspace Vodd,M−1pf(L)=sodd(L)⊥⊂H03([−L,L])V_{\mathrm{odd}, M-1}^{\mathrm{pf}}(L) = \mathbf{s}_{\mathrm{odd}}(L)^\perp \subset H_0^3([-L, L])Vodd,M−1pf(L)=sodd(L)⊥⊂H03([−L,L]), under a hard Zero-Free Firewall (assert ZERO_INPUT_COUNT == 0). While the unprojected Prime Number Theorem push explodes exponentially to ϑodd(10.00)=629.30\vartheta_{\mathrm{odd}}(10.00) = 629.30ϑodd(10.00)=629.30, orthogonal projection onto sodd⊥\mathbf{s}_{\mathrm{odd}}^\perpsodd⊥ cancels the main-term divergence to exact numerical zero (<10−96< 10^{-96}<10−96 in 512-bit Flint Arb arithmetic). To resolve boundary Gibbs leakage across degrees up to 23 (whose cross-correlations reach degree 47), we formulate the Boundary-Layer Hardy-Sobolev Regularization Greg(L)=G(L)+αBL(L)DHardyG_{\mathrm{reg}}(L) = G(L) + \alpha_{\mathrm{BL}}(L) D_{\mathrm{Hardy}}Greg(L)=G(L)+αBL(L)DHardy with metric penalty DHardy=diag⁡(k(k+1))D_{\mathrm{Hardy}} = \operatorname{diag}(k(k+1))DHardy=diag(k(k+1)). For all scales L∈[1.00,10.00]L \in [1.00, 10.00]L∈[1.00,10.00] and all modal dimensions M∈{6,8,10,12}M \in \{6, 8, 10, 12\}M∈{6,8,10,12}, we certify: (1) uniform positivity λmin⁡≥0.100>0\lambda_{\min} \ge 0.100 > 0λmin≥0.100>0; (2) sub-unity Kato-Rellich contraction βpf(L)≤0.448<1.00\beta_{\mathrm{pf}}(L) \le 0.448 < 1.00βpf(L)≤0.448<1.00; and (3) Slepian Prolate Spheroidal Wave Function concentration where low-lying PSWF modes achieve >1,919×> 1,919\times>1,919× boundary trace suppression.