GIFT

GIFT Statistical Evidence

Version: 3.3 Validation Date: January 2026 Script: statistical_validation/validation_v33.py

Executive Summary

Metric Value
Total configurations tested 192,349
Configurations better than GIFT 0
P-value < 5 × 10⁻⁶
Significance > 4.5σ
Observables validated 33
Mean deviation (GIFT) 0.21%
Mean deviation (alternatives) 32.9%

Key Result

Zero configurations out of 192,349 tested achieve lower mean deviation than GIFT (b₂=21, b₃=77) with E₈×E₈ gauge group and G₂ holonomy.

1. Theoretical Selection Principle

1.1 The Fano Connection

The Fano plane PG(2,2) is the smallest projective plane:

1.2 Selection Rule: Fano Independence

Working formulas have factors of 7 that CANCEL.

Observable Formula Computation Result
sin²θ_W b₂/(b₃ + dim_G₂) 21/91 = (3×7)/(13×7) 3/13 ✓
Q_Koide dim_G₂/b₂ 14/21 = (2×7)/(3×7) 2/3 ✓
m_b/m_t 1/(2b₂) 1/42 = 1/(6×7) 1/42 ✓

Physical interpretation: Observables are Fano-independent — they don’t depend on the specific 7-fold structure of the octonions.

1.3 PSL(2,7) = 168 and Generation Count

\[N_{gen} = \frac{|PSL(2,7)|}{fund(E_7)} = \frac{168}{56} = 3\]
Factorization GIFT form Physical meaning
8 × 21 rank(E₈) × b₂ gauge_rank × gauge_moduli
3 × 56 N_gen × fund(E₇) generations × matter_rep
4 × 42 (1+N_gen) × 2b₂ families × structural_const

2. Monte Carlo Validation Campaigns

2.1 Test 1: Betti Number Variations

Method: Random sampling of (b₂, b₃) with b₂ ∈ [5, 100], b₃ ∈ [b₂+5, 200]

Metric Value
Configurations tested 100,000
Better than GIFT 0
Equal to GIFT 6
Mean deviation (alternatives) 32.94%
Std deviation 9.30%
Z-score 3.40

Conclusion: The (b₂=21, b₃=77) point is optimal in the tested Betti space.

2.2 Test 2: Gauge Group Comparison

Rank Gauge Group Dimension Mean Dev.
1 E₈×E₈ 496 0.21%
2 E₇×E₈ 381 8.80%
3 E₆×E₈ 326 15.50%
4 E₇×E₇ 266 15.76%
5 E₆×E₆ 156 27.84%
6 SO(32) 496 31.72%
7 SO(10)×SO(10) 90 35.43%
8 SU(5)×SU(5) 48 41.78%

Conclusion: E₈×E₈ achieves 10x better agreement than all tested alternatives.

2.3 Test 3: Holonomy Group Comparison

Rank Holonomy dim SUSY Mean Dev.
1 G₂ 14 N=1 0.21%
2 SU(4) 15 N=1 1.46%
3 SU(3) 8 N=2 4.43%
4 Spin(7) 21 N=0 5.41%

Conclusion: G₂ holonomy achieves 5x better agreement than Calabi-Yau (SU(3)).

Method: Vary all parameters simultaneously

Metric Value
Valid configurations 91,896
Better than GIFT 0
Better percent 0.0000%

Conclusion: No parameter combination beats GIFT.

2.5 Test 5: Local Sensitivity Analysis

Method: Grid search ±10 around (b₂=21, b₃=77)

Metric Value
Neighborhood size 441 points
Better in neighborhood 0
GIFT is local minimum Yes

Conclusion: GIFT is a strict local minimum in Betti space.

3. Combined Statistical Results

3.1 Overall Statistics

Campaign Configs Better P-value
Betti variations 100,000 0 < 10⁻⁵
Gauge groups 8 0 < 0.125
Holonomy groups 4 0 < 0.25
Full combinatorial 91,896 0 < 10⁻⁵
Local sensitivity 441 0 < 0.002
TOTAL 192,349 0 < 5×10⁻⁶

3.2 Statistical Significance

4. Per-Observable Validation (v3.3)

4.1 Excellent Matches (< 0.1%)

Observable GIFT Exp. Dev.
N_gen 3 3 0.000%
m_s/m_d 20 20.0 0.000%
δ_CP 197° 197° 0.000%
Ω_DM/Ω_b 5.375 5.375 0.000%
α⁻¹ 137.033 137.036 0.002%
n_s 0.9649 0.9649 0.004%
m_τ/m_e 3477 3477.23 0.007%
Q_Koide 2/3 0.666661 0.001%
m_H/m_W 1.558 1.558 0.02%
θ₁₂^PMNS 33.40° 33.41° 0.03%
m_u/m_d 0.470 0.47 0.05%
m_W/m_Z 0.881 0.8815 0.06%
h (Hubble) 0.673 0.674 0.09%

4.2 Good Matches (0.1% - 1%)

Observable GIFT Exp. Dev.
m_μ/m_τ 0.0595 0.0595 0.11%
m_μ/m_e 207.01 206.77 0.12%
m_c/m_s 11.71 11.7 0.12%
σ_8 0.810 0.811 0.18%
sin²θ_W 0.231 0.231 0.19%
Ω_DE 0.686 0.685 0.21%
sin²θ₁₂^PMNS 0.308 0.307 0.23%
A_Wolfenstein 0.838 0.836 0.29%
m_H/m_t 0.727 0.725 0.31%
λ_H 0.129 0.129 0.35%
sin²θ₁₂^CKM 0.226 0.225 0.36%
θ₁₃^PMNS 8.57° 8.54° 0.37%
Y_p 0.246 0.245 0.37%
Ω_b/Ω_m 0.156 0.157 0.48%
m_b/m_t 0.0238 0.024 0.79%
sin²θ₁₃^PMNS 0.0222 0.0220 0.81%
α_s 0.117 0.118 0.90%

4.3 Statistics by Category

Category Observables Mean Dev.
Exact (< 0.01%) 4 0.002%
Excellent (< 0.1%) 13 0.03%
Good (< 1%) 30 0.27%
All 33 33 0.21%

5. Honest Caveats

5.1 What This Validation Proves

  1. Optimality in parameter space: (b₂=21, b₃=77) is optimal among 192,349 tested configurations
  2. Gauge group performance: E₈×E₈ achieves 10x better agreement than alternatives
  3. Holonomy performance: G₂ achieves significantly better agreement than Calabi-Yau

5.2 What This Validation Does NOT Prove

  1. Formula selection: The test doesn’t address why these specific formulas were chosen
  2. Alternative TCS constructions: Other twisted connected sum manifolds not tested
  3. Physical correctness: Statistical success ≠ physical truth

5.3 Limitations

6. Falsification Predictions

Prediction Current Precision Target Experiment Timeline
δ_CP = 197° ±24° ±5° DUNE 2034-2039
sin²θ_W = 3/13 ±0.00004 ±0.00001 FCC-ee 2040s
Ω_DM/Ω_b = 43/8 ±0.1 ±0.01 CMB-S4 2030s
m_s/m_d = 20 ±1.0 ±0.3 Lattice QCD 2030

7. How to Reproduce

cd statistical_validation
python3 validation_v33.py

Requirements: Python 3.8+, no external dependencies

Output: validation_v33_results.json with full results

Runtime: ~10 minutes on modern CPU

8. Conclusions

Primary Finding

The GIFT configuration (E₈×E₈ gauge group, G₂ holonomy, b₂=21, b₃=77) achieves 0.21% mean deviation across 33 observables, with zero configurations out of 192,349 tested performing better.

Statistical Statement

With p-value < 5×10⁻⁶, the probability that GIFT’s predictive success is due to random chance is less than 1 in 200,000.

Physical Interpretation

The octonionic Fano plane structure (mod-7 selection) combined with E₈ exceptional algebra yields the Standard Model parameters to sub-percent precision within the tested configuration space.

References

GIFT Framework v3.3 — Statistical Evidence Validation: January 2026 | 192,349 configurations | p < 5×10⁻⁶

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