The Dirac Equation on the Thales Sphere: A Geometric Variance–Deficit Interpretation of Relativistic Spinors

This paper presents a geometric reinterpretation of the Dirac equation using the Thales semicircle as a diagnostic construction for variance partition in relativistic spinors. The normalized upper and lower components of the Dirac spinor define an exact binary partition whose geometry yields closed-form expressions for the altitude, variance deficit, and exchange rate, all expressed purely in terms of the velocity parameter β. These identities are algebraic consequences of the Dirac spinor normalization and involve no approximation. The analysis shows that the Dirac semicircle has altitude h = β/2 and deficit δ = 1 − β, exactly half the corresponding Klein–Gordon altitude, reflecting the square-root structure of the Dirac equation. The Thales apex—corresponding to zero deficit and equal upper/lower partition—is identified with known instability thresholds: the massless Weyl limit for free fermions and the supercritical nuclear charge Z = 1/α for the Dirac hydrogen atom. In both cases, stable configurations carry mandatory nonzero deficit. The construction extends naturally to antiparticle states, completing the semicircle to a full circle with particle and antiparticle arcs meeting at the massless apex. Finally, structural parallels with Thales-based diagnostics in relativistic orbital dynamics are noted, emphasizing the role of the Thales construction as a cross-domain geometric coordinate system rather than a physical theory. No equations of motion are modified, and no new dynamics are proposed; the framework is purely kinematic and interpretive.