Newtonian Two-Body Gravity on the Thales Semicircle: A Geometric Diagnostic for Mass-Partition Structure

This technical note presents a geometric diagnostic reinterpretation of Newtonian two-body gravity using Thales right-triangle constructions. By identifying the mass product m_1 m_2 with the square of the Thales altitude and the total mass m_1 + m_2 with the hypotenuse, standard gravitational quantities are shown to arise naturally at distinct algebraic orders of a single geometric structure. At quadratic order, the gravitational force and potential are expressed as altitude-squared quantities, while the reduced mass emerges exactly as the altitude squared divided by the hypotenuse. At cubic order, orbital timing and Kepler’s third law depend on the hypotenuse rather than the altitude, revealing a clear geometric separation between coupling strength and dynamical timescales. Fractional-order combinations of altitude and hypotenuse reproduce the chirp mass used in gravitational-wave astronomy. Mass asymmetry is interpreted as a geometric deficit from the Thales apex, governed by an exact variance–deficit exchange identity that quantifies the loss of coupling efficiency away from equal-mass configurations. The framework does not modify Newtonian gravity, general relativity, or gravitational-wave theory; instead, it provides a unified geometric coordinate system that organizes known quantities, clarifies limiting regimes, and exposes structural relationships across Newtonian, post-Newtonian, and observational contexts.