The Dirac Equation from Bipolar Recursion Geometry

The Schr¨odinger equation was established as the non-relativistic limit of the massive
recursion field in the Cohesion Unified Field Theory. This paper derives the relativistic
quantum limit: the Dirac equation. The derivation proceeds by factoring the KleinGordon equation into first-order operators acting on a spinor field, and then identifying
the 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 state
vector of a bipolar recursion whose two components encode Phase A and Phase B
of 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 the
meson structure established in the quark confinement paper. The gamma matrices
encode the geometry of torsion phase rotations: γ
0
encodes the structural time direction
and γ
i
encode the three spatial recursion directions, with the anticommutation relation
{γ
µ
, γν} = 2g
µν expressing the orthogonality of torsion phase operations. The Dirac
equation automatically predicts gs = 2 for the electron, confirming the Cohesion UFT
result that two torque injections per bipolar cycle produce a gyromagnetic ratio exactly
twice the classical value. This is the first identification of the Dirac spinor with the
bipolar recursion state vector within the Cohesion UFT framework.