Pythagorean Angle Lattice: Ramified Gaussian–Pell Geometry, Primitive Spectra, and Dihedral Maaß Modes

A single arithmetic–geometric–spectral framework that starts from primitive Pythagorean triples and the ramified Gaussian prime 1+i, recovers the silver mean and Pell dynamics, and reformulates the Riemann hypothesis as the location of poles of a primitive spectral zeta function while producing explicit dihedral Maaß eigenvalues: We develop a unified arithmetic–geometric–spectral framework generated by primitive Pythagorean triples, the Pythagorean Angle Lattice, and the Omni-Metallic Framework. The starting angular quantity is Nθ = 2N arctan(2/N), lim N→∞ Nθ = 4. The Gaussian half-angle transform z = u + iv ↦→ (1 + i)z̄ produces the integral matrix J = [[1,1],[1,−1]], J² = 2I, |det J| = 2, whose projective action is the involution r ↦→ (1 − r)/(1 + r), with unique positive fixed point r∗ = √2 − 1, the reciprocal silver mean. This gives the structural factorization 4 = 2 NZ[i](1 + i), where one factor 2 is the angular doubling from Gaussian squaring and the other is the norm of the ramified Gaussian prime 1 + i. An integral change of coordinates reveals the hidden quadratic form Q(u, v) = u² + 2uv − v², which equals the sum of the two Euclidean Pythagorean legs. The Pell matrix A = [[0,1],[1,2]] and the Gaussian fold J generate an infinite-dihedral projective group ⟨[A],[J]⟩ ≅ D∞. On the spectral side, primitive lattice sieving gives the primitive spectral zeta function ζprim(s) = ζ(2s − 1)/(2(1 − 2^{−2s})ζ(2s)) − 1/2, yielding an exact reformulation of the Riemann hypothesis in terms of the polar locus of ζprim: RH ⇔ all nontrivial-zero-induced poles lie on ℜs = 1/4. Pell-orbit decomposition gives 1 + 2O(s) = ζ(s)L(s, χ₈)/((1 + 2^{−s})ζ(2s)). Finally, Fourier decomposition along the Pell-unit direction yields the family of real-quadratic Hecke characters ψₙ and the twisted primitive orbit functions Gₙ(s) = L(s, ψₙ)/((1 + (−1)ⁿ 2^{−s})ζ(2s)). The corresponding Pell Dirac operator has spectral ladder tₙ = πn/(2 log(1 + √2)), and the associated dihedral Maaß eigenvalues are λ_Maaßⁿ = 1/4 + tₙ². The resulting structure links primitive Pythagorean geometry, Gaussian ramification, the silver mean, Pell dynamics, Barnes cone counting, Möbius inversion, Riemann-zero spectral poles, and real-quadratic Hecke L-functions within one arithmetic framework.