The Pythagorean Angle Lattice: Arithmetic Kernel, Theta Series, and the Bridge Identity

We study the arithmetic kernel arising from the Pythagorean Angle Lattice and the Omni-Metallic Framework. The arithmetic kernel is r(k) = ⌊(k−1)/2⌋, k ≥ 2. We prove the closed form r(k) = k/2 − 3/4 − (1/4)(−1)^k, and the generating function F(q) = ∑ₖ₌₂^∞ r(k) q^k = q³/((1−q)(1−q²)). The theta series of the lattice is defined as Θ_L(τ) = ∑ₖ₌₂^∞ r(k) qk²/2, q = e2πiτ. We establish the Lattice-Partition Connection r(k) = p1,2(k−3), where p1,2(n) counts partitions of n into 1s and 2s. The Bridge Identity is Z(s) = ∑ₖ₌₂^∞ r(k)/k^s = (1/2)ζ(s−1) − (1/2)(1 + 2⁻ˢ)ζ(s). We also present the Fourier expansion Φ(q) := 1 + 24 ∑ₙ₌₁^∞ (r(n) − σ_1(n)) q^n = 24 F(q) + E_2(τ), where E_2(τ) is the Eisenstein series of weight 2. All results are rigorously derived and numerically confirmed.