Primorial Lane Geometry and the Abdullah Constant: Spectral Structure, Euler Product Identity, and Empirical Constants in Prime Gap Statistics

We study the coefficient of variation of Cramér-normalized prime gaps within residue lanes defined by the primorial wheel hierarchy, identifying a leading-term constant A (d) via the decay law ε (N, d) ~ A (d) /log N. We establish a three-way decomposition A (d) = Aₐrith (d) + AHL (d) + Λ where Λ ≈ 0. 535 is a universal additive constant matching CVGUE ≈ 0. 5350 to within 0. 5%. We prove five results. (A) The primorial singular series kernel Kd — a φ (d) × φ (d) matrix whose entries are Hardy–Littlewood singular series values — is real symmetric for all primorials d, hence has real spectrum. (B) Kd commutes exactly with the coprime involution P: r → d−r, mirroring the functional equation ξ (s) = ξ (1−s). (C) The singular series Euler product identity F (s) = C₂ · 2^ (−s) · ζ (s) · H (s), where H (s) is analytic and non-vanishing for Re (s) > 0, so F (s) inherits the non-trivial zeros of ζ (s) exactly. (D) C₂ · H (1) = 2 exactly (the Bombieri–Davenport average). (E) The trace moment identity Tr (Kd) /φ (d) = 1 exactly, with higher moments connecting to F (s) ⁿ and hence to ζ (s) ⁿ through Perron's formula. Numerical computation to d = 30030 (φ = 5760, run in 15 seconds) reveals that the eigenvectors of Kd are asymptotically Dirichlet characters mod d, with 100% character overlap at d = 210. The eigenvalues therefore approximate L-function values L (1, χ) — not zeros. A crucial distinction emerges: individual eigenvalues encode L-function values (log-normal distributed, Granville–Soundararajan), while trace moments factor through ζ (s) ⁿ and encode zero locations (GUE distributed, Montgomery–Odlyzko). The GUE connection lives in moments, not in raw eigenvalue spacings. The central open problem is establishing a rigorous primorial trace formula connecting Tr (Kdⁿ) to explicit sums over Riemann zeros.