Pythagorean Angle Lattice: The Spectral Theory

The first spectral theory of the Pythagorean Angle Lattice yields an exact zeta-regularized product involving the Glaisher-Kinkelin constant, a closed spectral determinant, and a precise heat-kernel asymptotic for the angle-halving hierarchy. We develop the spectral theory of the Pythagorean Angle Lattice, a countable set of real numbers defined by: \[ L = \{ ((Z)/D) : Z = W^2 or iW^2, \ W ∈ Z[i]ₚᵣᵢₘ', \ D ∈ N,\ D ≥ 1 \}, \] where the broadened primitive Gaussian integer set excludes the degenerate real cases \((u,v)=(±1,0)\). The Omni-Metallic Family \(m,r(n,N)\), defined as the unique positive root of \(x^m - 2nxᵐ⁻¹ - N^r = 0\), provides the algebraic backbone through the Bridge Identity: \[ Z(s) = ∑ₖ₌₂∞ r(k)/k^s = 1/2ζ(s-1) - 1/2(1+2⁻ˢ)ζ(s), \] where \(r(k) = (k-1)/2 \) is the arithmetic kernel. We present four main results: 1. The zeta-regularized product: \[ ∏ₖ₌₂∞ kʳ⁽ᵏ⁾ = e-1/24 A1/2 2-1/4 π-1/2, \] where \(A\) is the Glaisher-Kinkelin constant. 2. The spectral zeta function of the formal spectral sequence: \[ ζ_{ΔL}(s) = Z(2s). \] 3. The exact growth law for the angle-halving sequence: \[ E_k = 2^k/θ - θ/3 2⁻ᵏ - θ^3/45 2⁻³ᵏ + O(2⁻⁵ᵏ), θ = arccot(C). \] 4. The zeta-regularized determinant: \[ (ΔL) = e-1/12 A · 2-1/2 · π⁻¹. \] These results lead to spectral consequences including pole structure, residues, and the heat-kernel asymptotic: \[ K(t) ~ -log t + log θ - γ/log 2 + 1/2 + O(t). \] The formal spectral sequence \(ΔL\) is a notational device defined by its spectrum; it is not a geometric operator on the lattice \(L\). Its determinant is understood in the zeta-regularized sense.