Reversible Finite Quantization of Elliptic Curvature

The calculation of elliptical perimeters has historically relied on infinite series expansions or transcendental elliptic integrals, creating a fundamental tension between the continuous nature of Euclidean geometry and the finite precision of computational arithmetic. This paper introduces the Newman-Ramanujan Hybrid Seal, a rigorous protocol that resolves this tension by quantifying geometric curvature into a discrete, reversible scalar known as the Newman Coordinate (v). By utilizing the Inverse Omega Operator, this protocol encapsulates the nonlinear distortion factors of Ramanujan’s Second Approximation into a single, finite value. This process "seals" the infinite complexity of the shape into the linear Newman Domain, allowing for the subsequent retrieval of high-precision geometric parameters using strictly linear arithmetic via the Forward Omega Operator. We demonstrate that any elliptical shape, from a perfect circle to the theoretical infinite limit defined as the Newman Horizon, can be stored and scaled without recurring computational cost.