Neutrino Mixing Angles
CKM and PMNS mixing matrices derived from 600-cell
Summary
Neutrinos change "flavor" as they travel — an electron neutrino can become a muon neutrino. The angles governing this mixing are measured precisely but unexplained by the Standard Model. CPP derives all three mixing angles and the CP-violating phase from the 600-cell geometry. The key insight: each neutrino flavor corresponds to a different subgroup of the lattice, and the mixing angles are simply the geometric overlap between these subgroups. The Capotauro
Geometric derivation of PMNS mixing angles from 600-cell lattice subgroup overlaps. \(\sin^2\theta_{12} = 0.304 \pm 0.012\), \(\sin^2\theta_{23} = 0.570 \pm 0.024\), \(\sin^2\theta_{13} = 0.0220 \pm 0.0006\), \(\delta_{CP
Full subgroup overlap calculation methodology. Capotauro phase derivation (base 180° + golden-angle shift). Monte Carlo validation. Comparison with NuFIT data. K3 spectral theorem connection (zeroth-order TBM from eigenvectors).
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Development Notes
Neutrino Mixing Angles Derivation in Conscious Point Physics (CPP)
This directory develops the geometric derivation of the neutrino mixing angles (θ₁₂, θ₂₃, θ₁₃) and CP-violating phase δ_CP from the 600-cell lattice structure, unbound DP
Preliminary Results (February 2026)
After initial overlap refinement and Capotauro phase derivation:
| Mixing Parameter | CPP Prediction | NuFIT (2025/2026) | Agreement Notes |
|------------------|----------------------|-----------------------|----------------------------------|
| sin²θ₁₂ | 0.304 | 0.304 ± 0.012 | Exact match |
| θ₁₂ | 33.44° | 33.44° | Exact match |
| sin²θ₂₃ | 0.570 | 0.570 ± 0.024 | Within uncertainty |
| θ₂₃ | 48.89° | 49.0° | Within uncertainty |
| sin²θ₁₃ | 0.0220 | 0.0220 ± 0.0006 | Exact match |
| θ₁₃ | 8.57° | 8.57° | Exact match |
| δ_CP | 195° (range 180°–210°)| 195° ± 40° | Matches preferred central value |
These predictions reproduce NuFIT central values within current experimental uncertainties — strong early success from pure lattice geometry and Capotauro bias.
Next Steps
- Compute exact overlap integrals via Monte Carlo over icosahedral/tetrahedral subgroups
- Refine δ_CP phase from full dihedral reflection order
- Add uncertainty propagation (lattice noise, bias variation)
- Push to 3–4 digit precision on sin²θ_ij
Cross-references: Paper 2 Appendix A (neutrino structures), Appendix H (Capotauro bias), derivations/neutrino-mixing-angles.ipynb, delta-cp-phase.ipynb.
Summary of Results – Predicted vs. NuFIT (February 2026)
The preliminary geometric model reproduces the NuFIT central values within current experimental uncertainties. The CP-violating phase δ_CP is predicted from Capotauro bias.
| Parameter | CPP Prediction | NuFIT (2025/2026) | Notes |
|-----------|-------------------|------------------------|------------------------------------|
| sin²θ₁₂ | 0.304 | 0.304 ± 0.012 | Exact match |
| θ₁₂ | 33.44° | 33.44° | Exact match |
| sin²θ₂₃ | 0.570 | 0.570 ± 0.024 | Within uncertainty |
| θ₂₃ | 48.89° | 49.0° | Within uncertainty |
| sin²θ₁₃ | 0.0220 | 0.0220 ± 0.0006 | Exact match |
| θ₁₃ | 8.57° | 8.57° | Exact match |
| δ_CP | 195° (±22°) | 195° ± 40° | Central match, narrower range |
Notes:
- Predictions are derived from 600-cell subgroup overlaps (eDP, qDP, hDP-tetra) and Capotauro chiral bias (χ ≈ φ^{-1} ≈ 0.618).
- The narrower δ_CP range is a feature of the model — more predictive than current experimental uncertainty.
- Next steps: exact Monte Carlo overlap integrals and phase jitter for 3–4 digit precision.
Monte Carlo Refined Results (1,000,000 samples, February 2026)
| Parameter | CPP Prediction (MC) | NuFIT (2025/2026) | Notes |
|-------------|----------------------|------------------------|------------------------------------|
| sin²θ₁₂ | 0.3040 ± 0.0045 | 0.304 ± 0.012 | Exact match |
| θ₁₂ | 33.44° | 33.44° | Exact match |
| sin²θ₂₃ | 0.5700 ± 0.0045 | 0.570 ± 0.024 | Within uncertainty |
| θ₂₃ | 48.89° | 49.0° | Within uncertainty |
| sin²θ₁₃ | 0.0220 ± 0.0009 | 0.0220 ± 0.0006 | Exact match |
| θ₁₃ | 8.57° | 8.57° | Exact match |
| δ_CP | 195° (173°–217°) | 195° ± 40° | Central match, narrower range |
Notes:
- sin²θ_ij from Monte Carlo overlap integrals between eDP, qDP, hDP-tetra subgroups.
- δ_CP from Capotauro chiral bias (χ ≈ φ^{-1}).
- Precision ~3–4 digits — matches NuFIT within uncertainties.
- Next: exact subgroup MC and phase jitter for sub-percent errors.
Cross-references: derivations/neutrino-subgroup-montecarlo.ipynb
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