Derivation of the Weinberg Angle from Superstring Geometry at the Planck Scale

We show that the tree-level weak mixing angle sin⁡2θW=3/8sin2θW=3/8 can be obtained without postulating an SU(5) grand-unified group, using an effective long-wavelength description of the quantised bosonic Nambu–Goto string as a Cosserat–Kirchhoff rod wound into a helix. The Principle of Maximum Stability fixes the wrap angle at ψ=45∘ψ=45∘; the Wittrick wave-speed ratio then maps to the gauge-coupling ratio via a lattice gauge-wave isomorphism. A purely topological identity reduces the coupling ratio to 1/(1+ν)1/(1+ν), where νν is a conformal lattice Poisson ratio. The PMS-selected srs‑network gives a cubic directional ratio ν[100]=0ν[100]=0 and a separate conformal closure νeff=2/3νeff=2/3, yielding sin⁡2θW=3/8sin2θW=3/8. The algebraic chain 2/3⇒3/5⇒3/82/3⇒3/5⇒3/8 is machine-verified in Lean 4 and independently checked with SymPy.