PMNS Mixing Angles

New predictions for neutrino mixing angles derived.

Key Results

1. sin²θ₁₃ = φ⁻⁸ (improved from 1/15, error: 2.4% vs 206%)
2. sin²θ₂₃ = 1/2 + φ⁻⁵/2 (maximal mixing + φ correction, error: 0.02%)
3. sin²θ₁₂ = 1/3 (Diff3 triplet symmetry, existing)

Probability-Theoretic Interpretation

The mixing angles arise from transition probabilities:

• θ₁₃: Direct Diff2→Diff6 transition (8 = 6 + 2 steps)
• θ₂₃: Maximal mixing with Diff5 correction
• θ₁₂: Triplet symmetry from Diff3

Cosmological Density Parameters

The golden partition φ⁻¹ + φ⁻² = 1 constrains Ω_Λ + Ω_m = 1:

• Ω_Λ ≈ φ⁻¹ + correction
• Ω_m ≈ φ⁻² - correction

Section 1: PMNS Reactor Angle θ₁₃

Previous prediction: sin²θ₁₃ = C(2,2)/C(6,2) = 1/15 ≈ 0.0667 Experimental value: 0.0218 (error: 206%)

New prediction: sin²θ₁₃ = φ⁻⁸ ≈ 0.0213 Experimental value: 0.0218 (error: 2.4%)

Probability interpretation: 8 = D_max + D_min = 6 + 2

noncomputable def FUST.PMNSMixingAngles.sin2_theta13_new : ℝ

sin²θ₁₃ = φ⁻⁸

Equations

• FUST.PMNSMixingAngles.sin2_theta13_new = FUST.φ ^ (-8)

theorem FUST.PMNSMixingAngles.theta13_exponent_derivation : 6 + 2 = 8

The exponent 8 comes from D_max + D_min

theorem FUST.PMNSMixingAngles.sin2_theta13_pos : sin2_theta13_new > 0

φ⁻⁸ is positive

theorem FUST.PMNSMixingAngles.sin2_theta13_lt_one : sin2_theta13_new < 1

φ⁻⁸ is less than 1

theorem FUST.PMNSMixingAngles.old_theta13_prediction : 1 / 15 = 1 / ↑(Nat.choose 6 2)

Comparison with old prediction 1/15

theorem FUST.PMNSMixingAngles.phi_gt_three_halves : φ > 3 / 2

φ > 3/2 (used for numeric bounds)

theorem FUST.PMNSMixingAngles.new_theta13_smaller_than_old : sin2_theta13_new < 1 / 15

φ⁻⁸ < 1/15 (new prediction is closer to experiment)

Section 2: PMNS Atmospheric Angle θ₂₃

Experimental value: sin²θ₂₃ ≈ 0.545 (normal ordering)

Prediction: sin²θ₂₃ = 1/2 + φ⁻⁵/2 ≈ 0.5451 Error: 0.02% (remarkable agreement!)

Probability interpretation:

• Maximal mixing (1/2) as base
• φ⁻⁵ correction from 5 active D-levels

noncomputable def FUST.PMNSMixingAngles.sin2_theta23 : ℝ

sin²θ₂₃ = 1/2 + φ⁻⁵/2

Equations

• FUST.PMNSMixingAngles.sin2_theta23 = 1 / 2 + FUST.φ ^ (-5) / 2

theorem FUST.PMNSMixingAngles.theta23_correction_from_active_levels : 6 - 2 + 1 = 5

The correction exponent 5 = number of active D-levels

theorem FUST.PMNSMixingAngles.sin2_theta23_bounds : 0 < sin2_theta23 ∧ sin2_theta23 < 1

sin²θ₂₃ is in valid range (0, 1)

theorem FUST.PMNSMixingAngles.sin2_theta23_gt_half : sin2_theta23 > 1 / 2

sin²θ₂₃ > 1/2 (deviation from maximal mixing is positive)

noncomputable def FUST.PMNSMixingAngles.theta23_deviation : ℝ

The deviation from maximal mixing

Equations

• FUST.PMNSMixingAngles.theta23_deviation = FUST.φ ^ (-5) / 2

theorem FUST.PMNSMixingAngles.theta23_deviation_pos : theta23_deviation > 0

theorem FUST.PMNSMixingAngles.sin2_theta23_decomposition : sin2_theta23 = 1 / 2 + theta23_deviation

sin²θ₂₃ = 1/2 + deviation

Section 3: PMNS Solar Angle θ₁₂

Existing prediction: sin²θ₁₂ = 1/3 Experimental value: 0.307 (error: 8.6%)

def FUST.PMNSMixingAngles.sin2_theta12 : ℚ

Equations

• FUST.PMNSMixingAngles.sin2_theta12 = 1 / 3

theorem FUST.PMNSMixingAngles.theta12_from_triplet_symmetry : sin2_theta12 = 1 / ↑(Nat.choose 3 2)

Section 4: Cosmological Density Parameters

The golden partition: φ⁻¹ + φ⁻² = 1

Hypothesis: Ω_Λ + Ω_m = 1 follows the golden partition structure

• Ω_Λ ≈ φ⁻¹ (with small correction)
• Ω_m ≈ φ⁻² (with opposite correction)

Experimental (Planck 2018):

• Ω_Λ = 0.685
• Ω_m = 0.315

Golden values:

• φ⁻¹ ≈ 0.618
• φ⁻² ≈ 0.382

theorem FUST.PMNSMixingAngles.golden_partition : φ ^ (-1) + φ ^ (-2) = 1

Golden partition: φ⁻¹ + φ⁻² = 1

noncomputable def FUST.PMNSMixingAngles.Omega_Lambda_golden : ℝ

Dark energy density from φ⁻¹

Equations

• FUST.PMNSMixingAngles.Omega_Lambda_golden = FUST.φ ^ (-1)

noncomputable def FUST.PMNSMixingAngles.Omega_m_golden : ℝ

Matter density from φ⁻²

Equations

• FUST.PMNSMixingAngles.Omega_m_golden = FUST.φ ^ (-2)

theorem FUST.PMNSMixingAngles.density_sum_one : Omega_Lambda_golden + Omega_m_golden = 1

Densities sum to 1

theorem FUST.PMNSMixingAngles.dark_energy_dominates : Omega_Lambda_golden > Omega_m_golden

Ω_Λ > Ω_m (dark energy dominates)

noncomputable def FUST.PMNSMixingAngles.Omega_Lambda_refined : ℝ

Refined prediction with φ⁻⁵ correction

Equations

• FUST.PMNSMixingAngles.Omega_Lambda_refined = FUST.φ ^ (-1) + FUST.φ ^ (-5) / 2

noncomputable def FUST.PMNSMixingAngles.Omega_m_refined : ℝ

Equations

• FUST.PMNSMixingAngles.Omega_m_refined = FUST.φ ^ (-2) - FUST.φ ^ (-5) / 2

theorem FUST.PMNSMixingAngles.refined_density_sum_one : Omega_Lambda_refined + Omega_m_refined = 1

Refined densities still sum to 1

Section 5: CKM Matrix Elements (φ-power structure)

CKM matrix elements follow φ^(-3|i-j|) scaling:

• |V_us| ≈ φ⁻³ (1 generation gap)
• |V_cb| ≈ φ⁻⁶ (2 generation gap × 3)
• |V_ub| ≈ φ⁻⁹ (3 generation gap × 3)

noncomputable def FUST.PMNSMixingAngles.ckm_scaling (gen_diff : ℕ) : ℝ

CKM element scaling: |V_ij| ∝ φ^(-3|i-j|)

Equations

• FUST.PMNSMixingAngles.ckm_scaling gen_diff = FUST.φ ^ (-(3 * ↑gen_diff))

noncomputable def FUST.PMNSMixingAngles.V_us_theory : ℝ

|V_us| ≈ φ⁻³

Equations

• FUST.PMNSMixingAngles.V_us_theory = FUST.PMNSMixingAngles.ckm_scaling 1

noncomputable def FUST.PMNSMixingAngles.V_cb_theory : ℝ

|V_cb| ≈ φ⁻⁶

Equations

• FUST.PMNSMixingAngles.V_cb_theory = FUST.PMNSMixingAngles.ckm_scaling 2

noncomputable def FUST.PMNSMixingAngles.V_ub_theory : ℝ

|V_ub| ≈ φ⁻⁹

Equations

• FUST.PMNSMixingAngles.V_ub_theory = FUST.PMNSMixingAngles.ckm_scaling 3

theorem FUST.PMNSMixingAngles.ckm_elements_pos (n : ℕ) : ckm_scaling n > 0

CKM elements are positive

theorem FUST.PMNSMixingAngles.ckm_hierarchy : V_ub_theory < V_cb_theory ∧ V_cb_theory < V_us_theory

CKM hierarchy: larger generation gap → smaller mixing

theorem FUST.PMNSMixingAngles.V_us_is_phi_cubed_inv : V_us_theory = φ ^ (-3)

V_us relates to Cabibbo angle: |V_us| = φ⁻³

Section 6: Complete PMNS Predictions Summary

theorem FUST.PMNSMixingAngles.pmns_predictions_complete : sin2_theta13_new = φ ^ (-8) ∧ sin2_theta23 = 1 / 2 + φ ^ (-5) / 2 ∧ sin2_theta12 = 1 / 3 ∧ (0 < sin2_theta13_new ∧ sin2_theta13_new < 1) ∧ 0 < sin2_theta23 ∧ sin2_theta23 < 1

Complete PMNS mixing angle predictions

theorem FUST.PMNSMixingAngles.cosmological_predictions_complete : φ ^ (-1) + φ ^ (-2) = 1 ∧ Omega_Lambda_golden > Omega_m_golden ∧ Omega_Lambda_refined + Omega_m_refined = 1

Cosmological predictions from golden partition