The Full 22-Dimensional Codimension-2 Parity-Split Pole-Free Hilbert Space, Stieltjes Integration-by-Parts Operator Bound, and Continuum Convergence up to đ = đ 20 X=e 20
Year: 2026
Type: preprint
Abstract: We establish the first unified 22-dimensional, full codimension-2 parity-split pole-free Hilbert space formulation of the Semilocal Weil Spectral Positivity Program across the full macroscopic horizon Lâ[1.00,10.00]L \in [1.00, 10.00]Lâ[1.00,10.00] (X=e20â485.17Ă106X = e^{20} \approx 485.17 \times 10^6X=e20â485.17Ă106, encompassing 25,617,27225,617,27225,617,272 active prime powers) under a strict Zero-Free Firewall (assert ZERO_INPUT_COUNT == 0). Prior volumes investigated the codimension-1 odd subspace soddâ„\mathbf{s}_{\mathrm{odd}}^\perpsoddâ„ (111111 modes) in isolation. Here, we prove the Even Pole Orthogonal Decomposition: the even-mode Weil operator admits an exact rank-1 pole factor Ppole,even=+0.5cevencevenTP_{\mathrm{pole, even}} = +0.5 \mathbf{c}_{\mathrm{even}} \mathbf{c}_{\mathrm{even}}^TPpole,even=+0.5cevencevenT, enabling projection onto the codimension-1 even subspace cevenâ„\mathbf{c}_{\mathrm{even}}^\perpcevenâ„ (111111 modes). Unifying both parities constructs the full 22-dimensional codimension-2 Hilbert space H0,22pf([âL,L]):=ceven(L)â„âsodd(L)â„\mathcal{H}_{0, 22}^{\mathrm{pf}}([-L, L]) := \mathbf{c}_{\mathrm{even}}(L)^\perp \oplus \mathbf{s}_{\mathrm{odd}}(L)^\perpH0,22pf([âL,L]):=ceven(L)â„âsodd(L)â„. Across all 25.6M25.6\text{M}25.6M prime powers in 512-bit Arb arithmetic: (1) the raw 800.02Ă800.02\times800.02Ă even and 629.30Ă629.30\times629.30Ă odd PNT explosions cancel to exact numerical zero (<10â150< 10^{-150}<10â150); (2) the unmodified 22-dimensional operator is strictly positive definite (λminâĄâ„0.0868>0\lambda_{\min} \ge 0.0868 > 0λminâ„0.0868>0); (3) we prove the Stieltjes Integration-by-Parts Operator Bound ÎČIBP(L)â€0.5102<1.00\beta_{\mathrm{IBP}}(L) \le 0.5102 < 1.00ÎČIBP(L)â€0.5102<1.00; and (4) verify continuum convergence as MââM \to \inftyMââ.
Author: Munawer, Qamar
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225000855