E = mc² : Geometric Limits of Quadratic Relativity

This note examines the relativistic energy–momentum relation E^2 = (pc)^2 + (mc^2)^2 as an exact geometric construction. When normalized by the rest energy, the relation forms a right triangle that admits a precise Thales-circle embedding. Within this construction, the altitude of the triangle is shown to equal the dimensionless velocity parameter \beta = v/c, demonstrating that velocity arises as a geometric projection rather than as a primitive quantity. The analysis extends beyond quadratic structure by examining squared and cubed quantities associated with the triangle. While the quadratic (Pythagorean) geometry is self-consistent and closed, cubic and sixth-power quantities—such as the \gamma^3 proper-acceleration factor and the \gamma^6 self-force scaling familiar from radiation reaction—are shown to escape the Thales geometry. The “cubed triangle” does not close for any relativistic velocity, providing a geometric explanation for why higher-order dynamical effects cannot be contained within the same quadratic framework. A distinguished configuration occurs at \gamma = \sqrt{2}, where the triangle becomes isosceles and the right-angle vertex reaches the apex of the Thales semicircle. This point corresponds to quadratic equipartition between momentum and rest-mass contributions, but it does not represent maximal dynamical stability. The analysis clarifies why renormalization and truncation procedures arise naturally when cubic and higher-order effects are included. The work does not modify special relativity or propose new dynamics. Instead, it provides a geometric diagnostic that distinguishes between the closed structure of quadratic relativistic kinematics and the unavoidable overflow of cubic and sixth-power phenomena, offering a unified geometric perspective on proper acceleration and radiation reaction.