π-thagoras Crown Sum: A Structural Bridge Between the Pythagorean Angle Lattice and the Omni-Metallic Framework

We present a combinatorial expression for the constant π that emerges from the Pythagorean angle lattice. The Crown Sum is defined as a Dirichlet series involving the divisor function τ(n²+4) evaluated at the metallic means Mₙ. Using Hooley's theorem on the average order of τ(n²+4) and partial summation, we show that the Crown Sum has a double pole at s=1 with residue 9/(4π). This yields the expression π = 9 / (4 · lim_{s→1} (s−1)² Σ_{n=5}^∞ τ(n²+4)/(n+√(n²+4))^s). The formula is a reformulation of Leibniz's series for π/4, viewed through the arithmetic of primitive Pythagorean triples and divisor sums. The Crown Sum serves as a structural bridge between the Pythagorean angle lattice and the Omni-Metallic framework. The derivation reveals connections between the geometry of right triangles, metallic means, and Dirichlet L-functions.