Reversible Finite Quantization of Elliptic Curvature

The perimeter of an ellipse is commonly expressed through elliptic integrals or through high-accuracy approximations such as Ramanujan’s Second Approximation. This document introduces the Newman-Ramanujan Ellipse Coordinate, a finite-precision coordinate method for storing and reusing the curvature multiplier produced by Ramanujan’s approximation. The method maps the Ramanujan multiplier (w) into a scalar coordinate (v) using the inverse Omega operator, then reconstructs the same multiplier through the forward Omega operator under a stated 15-decimal truncation policy. The result is not a new elementary closed form for the exact elliptic integral. It is a deterministic coordinate representation: the nonlinear square-root term is evaluated once, recorded as a finite coordinate, and later reused through affine arithmetic. This revision removes earlier interpretive terminology and narrows the claim boundary. The exactness claimed is the exactness of the stated coordinate transform under the stated finite-precision policy. The perimeter source remains Ramanujan’s approximation unless another source multiplier is explicitly substituted.