example_basic_geodesic_clustering_report - zfifteen/unified-framework GitHub Wiki

Geodesic Clustering Analysis Report

Overview

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.

Methodology

Geodesic Coordinate Generation

  • 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

Clustering Analysis

  • Algorithms: KMeans, DBSCAN, Agglomerative Clustering
  • Metrics: Silhouette score, Calinski-Harabasz index
  • Preprocessing: StandardScaler normalization

Statistical Measures

  • Kolmogorov-Smirnov tests on distance distributions
  • Normality tests on coordinate distributions
  • Geometric measures: coordinate ranges, variances, mean distances

Results

Dataset Summary

Dataset Points Dimensions Mean Distance Distance Variance
Primes 99 3 255.4676 32425.9150
Zeta_Zeros 99 3 34.0935 515.7539
Random_Uniform 99 3 280.2382 16024.2893
Random_Gaussian 99 3 289.0526 19868.5139

Clustering Results

Kmeans Clustering

Dataset Clusters Silhouette Score Calinski-Harabasz
Primes 4 0.567 290.9
Zeta_Zeros 4 0.328 51.2
Random_Uniform 4 0.328 48.8
Random_Gaussian 4 0.251 33.9

Dbscan Clustering

Dataset Clusters Silhouette Score Calinski-Harabasz
Primes 2 0.588 174.3
Zeta_Zeros 1 -1.000 0.0
Random_Uniform 1 -1.000 0.0
Random_Gaussian 1 -1.000 0.0

Agglomerative Clustering

Dataset Clusters Silhouette Score Calinski-Harabasz
Primes 4 0.567 264.8
Zeta_Zeros 4 0.312 48.2
Random_Uniform 4 0.283 42.0
Random_Gaussian 4 0.219 25.6

Statistical Comparisons

Zeta_Zeros vs Primes

  • Kolmogorov-Smirnov Test: statistic = 0.7660, p-value = 0.0000
  • Mean Distance: 34.0935
  • Distance Variance: 515.7539
  • Coordinate Ranges: ['8.824', '8.974', '98.000']
  • Coordinate Variances: ['6.754', '7.165', '816.667']

Random_Uniform vs Primes

  • Kolmogorov-Smirnov Test: statistic = 0.1779, p-value = 0.0000
  • Mean Distance: 280.2382
  • Distance Variance: 16024.2893
  • Coordinate Ranges: ['480.642', '516.815', '94.370']
  • Coordinate Variances: ['19323.109', '26697.510', '780.676']

Random_Gaussian vs Primes

  • Kolmogorov-Smirnov Test: statistic = 0.1899, p-value = 0.0000
  • Mean Distance: 289.0526
  • Distance Variance: 19868.5139
  • Coordinate Ranges: ['724.832', '773.849', '162.826']
  • Coordinate Variances: ['23920.867', '26148.653', '1118.109']

Key Findings

  1. Clustering Quality: Primes show better clustering than random distributions

    • Average prime silhouette score: 0.574
    • Average random silhouette score: -0.153
  2. Distance Distributions: KS test vs uniform random p-value = 0.0000

    • Significant difference detected
  3. Geometric Structure: Prime and zeta zero geodesics exhibit distinct geometric patterns compared to random distributions

Conclusions

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.

Files Generated

  • 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 09:20:13 Z Framework Geodesic Clustering Analysis

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