The Universal Pythagorean Angle Lattice: Converging to π, e, γ, ζ(3), Catalan's Constant, and the Golden Ratio φ

The Pythagorean angle lattice provides a single universal geometric framework that generates explicit sequences of primitive Pythagorean triples converging to any positive real constant T — including π, e, γ, ζ(3), Catalan’s constant G, and exactly recovering the golden ratio φ. We introduce the Pythagorean angle lattice, a countable set of real numbers obtained by applying the cotangent function to rational divisions of finite sums of acute angles of primitive Pythagorean triples. We prove that a distinguished single-partner-angle sublattice of this structure is constructively universal: for every positive real constant T > 0, and for any starting primitive Pythagorean triple with larger acute angle α0, there exists an explicit sequence of primitive Pythagorean triples such that cot((α0 + ϕ(uk, vk))/D) → T, where D is an integer denominator determined by T and α0, and ϕ(uk, vk) is the Euclid angle of the partner triple. The sequence is generated by the continued fraction convergents (with a parity correction using semiconvergents) of an explicit target ratio t = tan((D arccot(T) − α0)/2). The convergence rate is O(1/u²_k) for subsequences of true convergents, and O(1/u_k) for semiconvergents. We demonstrate the universality of the construction with five pillar constants: π, Euler’s number e, the Euler–Mascheroni constant γ, Apéry’s constant ζ(3), and Catalan’s constant G, and we establish the golden ratio φ as an exact special case. The framework unifies algebraic constants (metallic means and the golden ratio φ), transcendental constants (π, e), and constants whose irrationality remains open (γ, ζ(3), G), within a single geometric construction arising from integer right triangles.