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) = A_arith(d) + A_HL(d) + Λ where Λ ≈ 0.535 is a universal additive constant matching CV_GUE ≈ 0.5350 to within 0.5%. We prove five results. (A) The primorial singular series kernel K_d — 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) K_d 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) = C_2 · 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) C_2 · H(1) = 2 exactly (the Bombieri–Davenport average). (E) The trace moment identity Tr(K_d)/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 K_d 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(K_d^n) to explicit sums over Riemann zeros.