What question did this study set out to answer?

The aim is to derive the weak mixing angle at the GUT scale using geometric properties of foam structures.

March 19, 2026Open Access

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

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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Luke Alexander Martin

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

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