π-thagoras: Pi from Primitive Pythagorean Triples via Continued Fractions

For any fixed primitive Pythagorean triple (a₀, b₀, c₀) with larger acute angle α₀, we prove that there exists an infinite sequence of partner primitive Pythagorean triples (aₖ, bₖ, cₖ) such that cot((α₀ + αₖ)/4) → π/2 as k → ∞, where αₖ is derived from the Euclid parameters of the k-th partner triple. The partner triples are the continued-fraction convergents (with a parity correction using semiconvergents) of an explicit target ratio t = tan(α*/2), where α* = 4 arccot(π/2) − α₀ is a fixed target angle determined by α₀. We give a complete proof of convergence, an explicit error bound, and verify the construction numerically to at least 8 significant digits of π/2 for eighteen fixed triples.