Comprehensive Addendum: First-Principles Axiomatic Proof of Reciprocal Duality in Inverse-Square Fields and the Mathematical Uniqueness of f(x) = x^-2 - x^2

Abstract

This comprehensive addendum establishes a complete axiomatic framework grounded in First Principles to resolve the long-standing singularity and divergence crises in inverse-square physical fields. By synthesizing the cognitive bias of "linear inertia," the metric distortion produced by plotting reciprocal fields on Cartesian grids, and the anti-symmetric constraint of the reciprocal duality operator T: x -> 1/x, this work rigorously proves the mathematical uniqueness of the dimensionless potential matrix f(x) = x^-2 - x^2. Furthermore, via the dual point transformation r(x) = x / sqrt(x^4 - 1), we demonstrate how the microscopic interval r in (0, 1] is topologically unfolded into an infinite hyperbolic manifold, proving that r=0 is strictly a Coordinate Pseudo-Singularity induced by Euclidean linear coordinates.