Arithmetic Spectral Theory from the Pythagorean Angle Lattice: Heat Kernel, Zeta Function, and Theta Inversion

From the geometry of integer right triangles comes a complete arithmetic spectral theory with an explicit heat kernel, zeta-regularised vacuum energy of 1/16, and a precise modularity obstruction in the theta inversion: We study the arithmetic kernel r(k) = floor((k-1)/2) arising from the Pythagorean Angle Lattice and the Omni-Metallic Framework. The Dirichlet series of this kernel yields the Bridge Identity: Z(s) = sum r(k)/k^s = (1/2)zeta(s-1) - (1/2)(1 + 2⁻ˢ)zeta(s). We define an Arithmetic Laplacian with spectrum E_k = k^2 and multiplicity r(k). Its spectral zeta function is zeta_Delta(s) = Z(2s). The associated heat kernel has the small-t asymptotic expansion: K(t) = 1/(4t) - (3 sqrt(pi)/8) t-1/2 + 11/24 - t/240 + O(t^2). We identify the zeta-regularised vacuum energy E_vac = 1/16, and prove that the spectral zeta function decays as zeta_Delta(sigma) \~ 3⁻²ˢⁱᵍᵐᵃ as sigma -> infinity. The heat kernel satisfies the inversion formula: K(1/t) = t/4 - (3 sqrt(pi)/8) sqrt(t) + 11/24 - 1/(240t) + O(t⁻²), t -> infinity. We further establish the exact theta inversion of the pure theta part of K, identifying the linear-weight sum T_1(t) = sum k e-t k^2 as the obstruction to modularity. These results form a coherent arithmetic spectral theory rooted in the geometry of integer right triangles.