Pythagorean Triples Encode the Circle: The Triple Limit and Circular Closure

Primitive Pythagorean triples encode the circle: as the governing parameter n tends to infinity, the product nθ converges to 4, and 4 × (π/2) recovers the full circle 2π : We prove that primitive Pythagorean triples encode the circle in their universal limit. For any sequence of primitive Pythagorean triples with governing parameter \( n = 2a/(c-b) → ∞ \), the product \( nθ \) converges to 4, where \( θ \) is the smaller acute angle measured in radians. This limiting value of 4, when multiplied by the right angle \( π/2 \), recovers the full circle \( 2π \). The result holds for all primitive triples, unifies the 1-2-8 families with the broader set of primitive triples, and is accompanied by a complete asymptotic expansion of the error.