The Pythagorean Angle Lattice: Arithmetic, Angular, and Spectral Structure Triangular Lattice Counting, Half-Angle Geometry, Arithmetic Zeta Functions, and Spectral Scaling

The arithmetic kernel r(k) is revealed as the slice-counting function of a triangular lattice, yielding an exact arithmetic–angular–spectral dictionary (tan²(θ/2) = N/k²) that links integer right triangles, half-angle geometry, zeta functions, heat kernels, and two distinct spectral systems: We study the arithmetic kernel r(k) = floor((k-1)/2) arising from the Pythagorean Angle Lattice and the Omni-Metallic Framework. We identify r(k) as the slice-counting function of the triangular lattice T = {(k, n) ∈ N² : 2n < k}, yielding the exact cumulative counting law sum_{k≤K} r(k) = floor((K-1)²/4). The quadratic Omni-Metallic equation x² − 2nx − N = 0 gives N = k(k − 2n); when N = q², this is exactly the Pythagorean locus n² + q² = (k − n)². The corresponding angle satisfies tan²(θ/2) = N/k², establishing an exact arithmetic–angular–spectral dictionary. We construct an Arithmetic Model Operator with spectrum k² and multiplicity r(k), compute its Dirichlet series, spectral zeta function, heat kernel, and vacuum energy. We also formulate an intrinsic angular Sturm–Liouville problem with polynomial eigenvalues n(n − 1), and compare the two spectral systems. The results establish a coherent arithmetic–angular dictionary rooted in the geometry of integer right triangles.