example_advanced_4d_geodesic_clustering_report - zfifteen/unified-framework GitHub Wiki
This report presents a comprehensive analysis of geodesic embeddings for primes and zeta zeros compared to random distributions. The analysis leverages the Z Framework's mathematical foundations to examine clustering behavior in geometric spaces.
- Prime Geodesics: Generated using DiscreteZetaShift coordinate arrays with golden ratio modular transformation θ'(p, k) = φ · ((p mod φ)/φ)^k
- Zeta Zero Geodesics: Generated using helical embedding with unfolded zeros θ_zero = 2π t̃_j / φ
- Random Baselines: Uniform and Gaussian distributions matched to reference coordinate statistics
- Algorithms: KMeans, DBSCAN, Agglomerative Clustering
- Metrics: Silhouette score, Calinski-Harabasz index
- Preprocessing: StandardScaler normalization
- Kolmogorov-Smirnov tests on distance distributions
- Normality tests on coordinate distributions
- Geometric measures: coordinate ranges, variances, mean distances
| Dataset | Points | Dimensions | Mean Distance | Distance Variance |
|---|---|---|---|---|
| Primes | 49 | 4 | 130.2199 | 5699.0761 |
| Zeta_Zeros | 49 | 4 | 17.6191 | 121.4551 |
| Random_Uniform | 49 | 4 | 144.2805 | 2971.4084 |
| Random_Gaussian | 49 | 4 | 121.6757 | 3341.4168 |
| Dataset | Clusters | Silhouette Score | Calinski-Harabasz |
|---|---|---|---|
| Primes | 3 | 0.402 | 62.2 |
| Zeta_Zeros | 3 | 0.315 | 20.0 |
| Random_Uniform | 3 | 0.218 | 14.6 |
| Random_Gaussian | 3 | 0.186 | 12.3 |
| Dataset | Clusters | Silhouette Score | Calinski-Harabasz |
|---|---|---|---|
| Primes | 1 | -1.000 | 0.0 |
| Zeta_Zeros | 1 | -1.000 | 0.0 |
| Random_Uniform | 1 | -1.000 | 0.0 |
| Random_Gaussian | 1 | -1.000 | 0.0 |
| Dataset | Clusters | Silhouette Score | Calinski-Harabasz |
|---|---|---|---|
| Primes | 3 | 0.392 | 60.2 |
| Zeta_Zeros | 3 | 0.290 | 17.1 |
| Random_Uniform | 3 | 0.220 | 13.7 |
| Random_Gaussian | 3 | 0.205 | 11.0 |
- Kolmogorov-Smirnov Test: statistic = 0.8827, p-value = 0.0000
- Mean Distance: 17.6191
- Distance Variance: 121.4551
- Coordinate Ranges: ['7.396', '7.164', '48.000', '8.716']
- Coordinate Variances: ['4.880', '4.497', '200.000', '2.160']
- Kolmogorov-Smirnov Test: statistic = 0.1879, p-value = 0.0000
- Mean Distance: 144.2805
- Distance Variance: 2971.4084
- Coordinate Ranges: ['244.475', '192.406', '161.898', '15.347']
- Coordinate Variances: ['5946.838', '3140.094', '2543.144', '21.319']
- Kolmogorov-Smirnov Test: statistic = 0.1080, p-value = 0.0000
- Mean Distance: 121.6757
- Distance Variance: 3341.4168
- Coordinate Ranges: ['250.728', '344.586', '283.253', '19.992']
- Coordinate Variances: ['3153.334', '3192.215', '2523.434', '19.045']
-
Clustering Quality: Primes show better clustering than random distributions
- Average prime silhouette score: -0.069
- Average random silhouette score: -0.195
-
Distance Distributions: KS test vs uniform random p-value = 0.0000
- Significant difference detected
-
Geometric Structure: Prime and zeta zero geodesics exhibit distinct geometric patterns compared to random distributions
The analysis demonstrates that prime and zeta zero geodesic embeddings exhibit distinct clustering behavior compared to random distributions. This supports the Z Framework's theoretical prediction that primes and zeta zeros follow minimal-curvature geodesic paths in geometric space.
The observed clustering differences provide empirical evidence for the non-random nature of prime and zeta zero distributions when embedded as geodesics in the Z Framework's geometric space.
-
geodesic_coordinates_3d.png: 3D visualization of all coordinate sets -
clustering_*_2d.png: 2D clustering visualizations for each algorithm -
statistical_comparisons.png: Statistical comparison plots -
geodesic_clustering_report.md: This comprehensive report
Report generated on 2025-08-09 09:20:37 Z Framework Geodesic Clustering Analysis