This paper introduces the Thales sphere construction as a cross-domain geometric diagnostic for constrained variance partitions arising in quantum mechanics and general relativity. Rather than proposing a unifying theory or new dynamics, the work demonstrates that both domains impose normalization or partition constraints that can be represented within a shared geometric framework. In quantum mechanics, the upper/lower component normalization of the Dirac spinor defines an exact Thales semicircle with altitude h = \beta/2 and deficit \delta = 1 - \beta, where \beta = v/c. In general relativity, the relativistic energy–momentum partition of a particle in a Schwarzschild orbit defines a distinct Thales semicircle with altitude h = \beta/\gamma and a different deficit function. Although both constructions share the same kinematic parameter \beta, they measure different physical quantities and operate on different “faces” of the Thales sphere. The Thales apex (zero deficit, maximum altitude) is shown to coincide with known instability thresholds in each domain independently: the massless Weyl limit and supercritical bound states in quantum mechanics, and unstable circular orbits in Schwarzschild spacetime. For a massive particle in a stable orbit, both quantum-mechanical and gravitational deficits are simultaneously nonzero, vanishing together only in the massless limit. An appendix demonstrates that classical statistical partitions (e.g., PCA) share the same constraint geometry as relativistic quantum systems but traverse the semicircle in the opposite direction, clarifying how identical geometry can underlie entropic and energetic systems with opposite physical arrows. The Thales construction is presented as a translation and classification tool: a shared coordinate system that makes constraint geometry visible across domains, enabling meaningful comparison without modifying, unifying, or extending existing physical theories.