What question did this study set out to answer?

The aim is to resolve the Geometric Uniqueness Conjecture by establishing its three forms: Weak, Strong, and Maximal.

April 10, 20260 citationsOpen Access

Resolution of the Geometric Uniqueness Conjecture from G₂ Compactification

Key Points

The aim is to resolve the Geometric Uniqueness Conjecture by establishing its three forms: Weak, Strong, and Maximal.
Established uniqueness of physical vacuum via analytic proof of convexity on moduli space of G₂-holonomy manifolds.
Combined Yang-Mills mass gap theorem with chiral fermion construction and cluster expansion for mass gap proof.
Derived absolute neutrino mass scale and electroweak vacuum expectation value through G₂ compactification results.
Weak Form uniquely identifies the physical vacuum on the moduli space with (b₂, b₃) = (27, 451).
Strong Form establishes existence of a Wightman quantum field theory with a mass gap.
Maximal Form determines neutrino mass scale at 50.6 meV and electroweak vacuum at 254 GeV with high accuracy.

Abstract

The seven open problems constituting the Geometric Uniqueness Conjecture (DOI: 10. 5281/zenodo. 19464559) have been resolved. The Weak Form (uniqueness of the physical vacuum) has been established through analytic proof of the convexity of the effective action along the fine-structure constant constraint curve on the moduli space of G₂-holonomy manifolds, selecting (b₂, b₃) = (27, 451) as the unique global minimum. The Strong Form (existence of a Wightman quantum field theory with mass gap) has been established by combining the Yang-Mills mass gap theorem (DOI: 10. 5281/zenodo. 19464295) with the Acharya-Witten chiral fermion construction and the Glimm-Jaffe-Spencer cluster expansion. The Maximal Form (complete parameter determination) has been established through two final derivations: (i) the absolute neutrino mass scale m_ν₃ = α²v²/ (2K₀^7/3m₃/₂) = 50. 6 meV, with the Majorana mass MR = K₀^7/3m₃/₂ = 3. 19 × 10¹⁰ GeV determined by the Kähler potential exponent 7/3 of the G₂ compactification, yielding Δm²ₐtm = 2. 56 × 10⁻³ eV² (4. 3% from experiment) ; and (ii) the electroweak vacuum expectation value v = m₃/₂ × αEM × CLIG × CQG = 254 GeV (3. 1% from the measured value). All seven problems resolved using established results from G₂ geometry, spectral theory, and constructive QFT. Total QGU predictions: 30 across 9 sectors with zero free parameters.

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