A Universal Scaling Law for Ellipse Perimeters

This paper establishes an exact scaling law for equal-parameter inscribed polygonal approximations of ellipse perimeters.For an ellipse parameterized by q(θ) = (a cos θ, b sin θ), the N-gon perimeter P_N factors exactly asP_N = sinc(π/N) M_N,where M_N is the composite midpoint approximation to the exact ellipse perimeter. Because the ellipse speed is analytic and periodic, the midpoint remainder is exponentially small for every fixed axis ratio.This factorization separates a universal algebraic error spectrum from a shape-dependent exponential onset. In particular,(P - P_N)/P = π²/(6N²) - π⁴/(120N⁴) + π⁶/(5040N⁶) - ... + O(e^(-ρN)),for every 0 < ρ < ρ*, with ρ* = artanh(b/a).The algebraic coefficients are independent of the ellipse axis ratio; eccentricity controls only how quickly the asymptotic regime is reached. A scale-tripling Richardson-type correction cancels the N^(-2) term exactly and raises the algebraic order from N^(-2) to N^(-4).The accompanying reproducibility package includes the LaTeX source, compiled manuscript, direct chord-sum verification code, numerical tables, and SHA-256 checksums.