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This research investigates how arithmetic properties influence the structure of the Higgs sector in the Standard Model.

April 5, 2026Open Access

Arithmetic Admissibility and the Higgs Sector

Key Points

This research investigates how arithmetic properties influence the structure of the Higgs sector in the Standard Model.
Analysis of the structural integers of the Standard Model regarding prime factors.
Application of the Goldstone mechanism for electroweak symmetry breaking.
Use of elementary modular arithmetic to derive conclusions.
The Higgs sector is completely determined with exactly one doublet.
No triplets can coexist with the doublet under the given constraints.
The number 5 is identified as a limiting factor for both Higgs doublets and SU(5) grand unification.

Abstract

I show that if the structural integers of the Standard Model have no prime factor exceeding 3 (arithmetic admissibility), then the Higgs sector is completely determined. The number of Higgs doublets is exactly one, excluding the minimal supersymmetric standard model. The only admissible representations of SU (2) L for a complex Higgs field are the doublet and the triplet, but no triplets can coexist with the doublet, and a lone triplet is excluded by the electroweak ρ parameter. The proofs use the Goldstone mechanism for electroweak symmetry breaking and elementary modular arithmetic. The same arithmetic obstruction (the prime 5) that excludes two Higgs doublets also excludes SU (5) grand unification and a fourth quark-lepton generation.

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