What question did this study set out to answer?

To establish the exact neutrino mass-squared ratio Δm²₃₁/Δm²₂₁ using geometric methods in unified foam field theory.

April 11, 2026Open Access

The Neutrino Mass-Squared Ratio from T₁u Geometry: Δm²₃₁/Δm²₂₁ = 33

Key Points

To establish the exact neutrino mass-squared ratio Δm²₃₁/Δm²₂₁ using geometric methods in unified foam field theory.
Deriving mass relationships from the master equation λ²−9λ+16=0
Using algebra to simplify the roots into r₁ and r₂
Correcting previous notation errors regarding mass-squared ratios
Found the ratio Δm²₃₁/Δm²₂₁ = 33 as exact and consistent with observational data
Observed value aligns closely with theoretical predictions at 33.57 ± 0.7
All calculations yield zero free parameters, confirming algebraic completeness in the neutrino sector.

Abstract

Proves that Δm²₃₁/Δm²₂₁ = 33 as a Tier 1 algebraic theorem of the Unified Foam Field Theory. The result follows from ‖M−I‖²F = (r₁−1) ²+ (r₂−1) ² = 49−18+2 = 33, where r₁ = (9−√17) /2 and r₂ = (9+√17) /2 are the roots of the master equation λ²−9λ+16=0. Given m₁ = 0 (proven theorem) and m₂ = m₃/√33, the ratio Δm²₃₁/Δm²₂₁ = m₃²/m₂² = 33 exactly. Observed value 33. 57 ± 0. 7 → 0. 8σ from prediction. Corrects a notation error in earlier UFFT documents: Δm²₃₂/Δm²₂₁ = 32 (also exact), not 33. The neutrino sector is now algebraically complete: m₁ = 0, m₂ = m₃/√33, m₃ = mₑ exp (− (11+13√17) /4), tan²θ₁₂ = √17/9, sin²θ₂₃ = 1/2+√17/81, sin²θ₁₃ = (√17/27) ² (1−√17/162) ², all Tier 1 or Tier 2. Zero free parameters.

The Neutrino Mass-Squared Ratio from T₁u Geometry: Δm²₃₁/Δm²₂₁ = 33

Key Points

Abstract

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