A Scaling-Cancellation Principle for Chord-Based Geometric Approximations

This paper develops a Richardson-type scaling-cancellation framework for chord-based geometric approximations with even-power asymptotic expansions. It proves the two-scale cancellation theorem, derives the unique multi-scale Lagrange weights, gives the surviving-error law for geometric scale nodes, and records the equivalent recursive fixed-base extrapolation ladder. Applications include polygonal approximation of pi and an explicitly cited ellipse-factorization input. The paper also distinguishes fixed-resolution order improvement from the nonlinear proportional-subtended cubic residual law for approximate angle division.