he Weinberg Angle from SO(10) Recursion Closure Geometry

The Weinberg angle θW has been an open problem in the Cohesion UFT series since theelectroweak paper identified it geometrically as θW = arctan(ω6/ω2) at the electroweakdensity threshold Dacc. This paper derives sin2θW in two steps. First, at the GUTdensity DGUT, the value sin2θW = 3/8 is derived from the closure geometry of theSO(10) 5 representation: it is the ratio of the n = 2 bipolar mode’s T3 content to thetotal electric charge content of the unified representation, which contains three fractionalclosures (quarks) and one SU(2) doublet (lepton). The derivation is Tr(T23)/Tr(Q2) =(1/2)/(4/3) = 3/8, with 1/2 coming from the two bipolar (n = 2) torsion phases of thelepton doublet and 4/3 from the fractional slip-phase charges of the quarks plus thefull charge of the lepton. Second, the observed value sin2θW = 0.2312 at the Z scalefollows from the one-loop renormalisation group running established in the runningcouplings paper of this series. The two steps together close Open Problem 5. Thegeometric origin of sin2θW = 3/8 is identified as the ratio of bipolar mode content tototal closure charge content in the unified SO(10) representation — a statement aboutthe torsion phase geometry of the n = 2 and n = 6 modes within the SO(10) multiplet.