The Pythagorean Angle Lattice: Product Formulas and Spectral Data

From Pythagorean triples and metallic means to zeta-regularized products, spectral data, and an S_3 Galois classification of the cubic lattice branch—complete with an explicit density-zero exceptional family of cubes in the Gaussian integers: We study the arithmetic kernel arising from the Pythagorean Angle Lattice and the Omni-Metallic Framework: varrho(k) = floor((k-1)/2), k >= 2. We present product formulas derived from the Bridge Identity: Z(s) = sumₖ₌₂^infty varrho(k)/k^s = (1/2)zeta(s-1) - (1/2)(1 + 2⁻ˢ)zeta(s), which converges absolutely for Re(s)>2 and extends meromorphically to C with simple poles at s=2 (residue 1/2) and s=1 (residue -3/4). For Re(s) > 2 and epsilon = ±1: prodₖ₌₂^infty (1 - epsilon/k^s)ᵛᵃʳʳʰᵒ⁽ᵏ⁾ = exp(-sumₘ₌₁^infty (epsilon^m / m) Z(sm)). We define the Lattice Multiplicity Operator Delta_L with spectrum E_k = k^2 and multiplicity varrho(k) for k >= 3. Its spectral zeta function is: zetaDelta_L(s) = Z(2s) = (1/2)zeta(2s-1) - (1/2)(1 + 2⁻²ˢ)zeta(2s). Special values are: zetaDelta_L(0) = 11/24, zetaDelta_L(-1) = 1/240, zetaDelta_L(-2) = -1/504, zetaDelta_L(-3) = 1/480. The zeta-regularized product is: prodₖ₌₂^infty kᵛᵃʳʳʰᵒ⁽ᵏ⁾ = e-1/24 A1/2 2-1/4 pi-1/2, where A is the Glaisher–Kinkelin constant. We also establish the Lattice Spectral Duality: Z(s) + Z^*(s) = zeta(s-1) - zeta(s), where Z^*(s) = sumₖ₌₂^infty floor(k/2) k⁻ˢ. In addition, we establish the complete S_3 Galois locking classification of the cubic projection branch of the lattice: Gal(K_W / Q(i)) ≅ S_3 if W is not a cube in Z[i], and ≅ Z/2 if W is a cube in Z[i]. We identify the exceptional family explicitly and discuss the corresponding dihedral Galois representation. The exceptional family has a natural Pythagorean interpretation: it consists precisely of those W whose ray is the cube of the ray of another primitive X, i.e., arg(W^2) ≡ 3 arg(X^2) (mod 2π).