The Universal Polygon Boundary Law: A Monotonic Boundary Convergence for Equal - Area Polygons and Circles

The Universal Polygon Boundary Law (UPBL) introduces a new geometric invariant describing the ratio between the circumference of a circle and the perimeter of a regular polygon of equal area. When restricted to polygons with n≥5, this ratio becomes a smooth, strictly monotonic function converging to unity as n→∞. The triangle and square act as geometric anomalies that obscure this behaviour, but once removed, the underlying monotonic structure becomes clear. The invariant : F(n)=π /ntan⁡(π/n) reveals that regular polygons with five or more sides lie on a continuous geometric path toward the circle, exhibiting natural boundary contraction under equal‑area constraints. This monotonic convergence has not been formalised in classical geometry or isoperimetric literature and constitutes an original mathematical law.