π-thagoras Counting: π from the Pythagorean Angle Lattice

We study the counting function P(X) that enumerates primitive, opposite-parity integer pairs (u, v) with u > v > 0 and u ≤ X. Using Möbius inversion, we prove P(X) = 2/π² X² + O(X log X). This reveals π as the natural scaling constant of the Pythagorean angle lattice: π = limX→∞ √(2X / P(X)). The evaluation of the constant relies on the classical Basel result ζ(2) = π²/6. The result contributes to a unified perspective within the author’s Omni-Metallic framework, complementing earlier work on metallic means and the Crown Sum.