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March 19, 20260 citationsOpen Access

The Weinberg Angle from Foam Geometry: sin²θW = 3/8 at the GUT Scale

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LMLuke Alexander Martin

Key Points

  • The aim is to derive the weak mixing angle at the GUT scale using geometric properties of foam structures.
  • Derived the Weinberg angle using the mode dimension ratio of the face Laplacian from a truncated octahedron.
  • Analyzed various sectors: T₂g for SU(3)_c, T₁u for SU(2)_L, and A₁g for U(1)_Y.
  • Performed a 2-loop self-consistent GUT calculation using foam-derived beta coefficients.
  • Calculated sin²θ_W as 3/8 at the GUT scale.
  • Determined GUT mass scale M_GUT = 1.50×10¹⁶ GeV and coupling constant α_GUT = 1/46.
  • Reproduced the strong coupling constant α_s(M_Z) exactly as 0.1179.

Abstract

The weak mixing angle sin²θW = 3/8 at the GUT scale is derived as an exact theorem from the mode dimension ratio of the face Laplacian of the truncated octahedron (Kelvin cell). The T₂g sector (dim=3) gives SU (3) c, the T₁u sector (dim=3) gives SU (2) L, and the A₁g sector (dim=1) gives U (1) Y. The ratio sin²θW = 3/8 follows from the hypercharge assignments of these sectors. The 2-loop self-consistent GUT calculation using all foam-derived beta coefficients gives MGUT = 1. 50×10¹⁶ GeV, αGUT = 1/46, and reproduces αₛ (MZ) = 0. 1179 exactly from the GUT chain.

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Cite This Study

Luke Alexander Martin (2026) studied this question.

synapsesocial.com/papers/69bb9336496e729e6298126ehttps://doi.org/10.5281/zenodo.19063822
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