Thales Sphere as Geometric Diagnostic for Three-Channel Variance Partition in Binary Black Hole Inspirals

We introduce the Thales sphere, a geometric diagnostic framework for analyzing the three-channel variance partition in binary black hole inspirals. After normalization, the component mass contributions and orbital momentum define a point on the positive octant of a unit sphere, which decomposes hierarchically into two orthogonal Thales semicircles: a mass-partition face encoding the mass ratio and an energy-partition face encoding the balance between rest mass and orbital kinetic energy. The inspiral trajectory on this sphere is shown to be meridional: the azimuthal coordinate, determined solely by the mass ratio, is geometrically frozen, while the polar coordinate evolves dynamically under radiation reaction. The mass ratio emerges as a geometric invariant, equal to the ratio of cross-face coupling altitudes at every instant. Applying the variance–deficit exchange identity on each face reveals two distinct regimes. The mass-partition face operates in a small-deficit (quadratic) regime with near-maximal coupling efficiency, while the energy-partition face operates in a large-deficit (nonlinear) regime throughout the observed inspiral. Application to the binary black hole merger GW231123 confirms these predictions and shows that the symmetric apex of the energy-partition face maps to an unstable Schwarzschild orbit at r = 5M, lying inside the ISCO. Stable orbits therefore carry a mandatory nonzero energy-partition deficit. The Thales sphere provides a unified, model-agnostic geometric coordinate system for mass ratio characterization, energy partitioning, and coupling efficiency in binary black hole systems, without modifying equations of motion or waveform models.