The Dirac Equation from Bipolar Recursion Geometry

The Schr¨odinger equation was established as the non-relativistic limit of the massiverecursion field in the Cohesion Unified Field Theory. This paper derives the relativisticquantum limit: the Dirac equation. The derivation proceeds by factoring the KleinGordon equation into first-order operators acting on a spinor field, and then identifyingthe spinor components with the two torsion phases of the electron’s bipolar (n = 2)recursion. The Dirac spinor is not an abstract mathematical object: it is the statevector of a bipolar recursion whose two components encode Phase A and Phase Bof the n = 2 torsion cycle. The four-component structure arises from the particleantiparticle doubling, where the antiparticle is the phase complement of the particle —the bipolar recursion running in the opposite torsion orientation, consistent with themeson structure established in the quark confinement paper. The gamma matricesencode the geometry of torsion phase rotations: γ0encodes the structural time directionand γiencode the three spatial recursion directions, with the anticommutation relation{γµ, γν} = 2gµν expressing the orthogonality of torsion phase operations. The Diracequation automatically predicts gs = 2 for the electron, confirming the Cohesion UFTresult that two torque injections per bipolar cycle produce a gyromagnetic ratio exactlytwice the classical value. This is the first identification of the Dirac spinor with thebipolar recursion state vector within the Cohesion UFT framework.