This document describes the implementation of Task 4: Statistical Discrimination and GMM Fitting for the unified-framework repository. The task aimed to quantify separations between prime and composite numbers using Gaussian Mixture Models (GMM) and Fourier analysis.

Quantify separations (Cohen's d>1.2, KL≈0.4-0.6) via GMM and Fourier analysis using θ' from Task 1 with C=5 components.

1. task4_statistical_discrimination.py - Main implementation following Task 4 specifications
2. test_task4.py - Comprehensive test suite validating all functionality
3. task4_enhanced.py - Enhanced version with optimization attempts

• Uses core.axioms.theta_prime(n, k, phi) function
• Computes θ'(n, k=0.3) for integers n=2 to 100,000
• Separates results into prime and composite groups

Cohen's d Calculation:

d = |μ_primes - μ_composites| / sqrt((var_primes + var_composites)/2)

KL Divergence:

• Uses histogram approximation with scipy.stats.entropy
• Adaptive binning strategy to improve stability
• Handles edge cases with epsilon smoothing

GMM Fitting:

• sklearn.mixture.GaussianMixture with C=5 components
• Multiple covariance types tested (full, tied, diag, spherical)
• BIC/AIC model selection
• σ_bar computed as weighted average of component standard deviations

• 1000 bootstrap iterations for confidence intervals
• 95% confidence intervals for all metrics
• Validates statistical stability

• Cohen's d: [0.001, 0.033]
• KL divergence: [0.003, 0.006]
• σ_bar: [0.333, 0.341]

Analysis of k values from 0.1 to 1.0 showed:

• k=0.1 provides best discrimination (Cohen's d ≈ 0.018)
• k=0.3 (required value) gives Cohen's d ≈ 0.017
• All k values produce insufficient discrimination for target criteria

1. Fractional Part: θ' % 1
2. Log Transform: log(θ' + ε)
3. Normalized: (θ' - μ) / σ

• Wasserstein Distance: Earth Mover's Distance between distributions
• Kolmogorov-Smirnov Test: Two-sample distributional comparison
• Energy Distance: Alternative separation metric

Best transformation was log-transformed values with:

• Cohen's d: 0.012 (minimal improvement)
• KL divergence: 0.005 (minimal improvement)
• All metrics still far below target thresholds

1. Inherent Similarity: θ' values for primes and composites at k=0.3 are remarkably similar

   • Prime examples: [1.049, 1.543, 0.786, 1.156, 1.512, ...]
   • Composite examples: [1.292, 1.459, 1.590, 1.361, 0.968, ...]

2. Mathematical Properties: The θ' transformation with golden ratio modulus may not provide strong discrimination at the specified k value

3. Sample Size: With 9,592 primes and 90,407 composites, the large sample size makes small differences statistically detectable but practically negligible

1. Different Implementation: The target values may be based on a different θ' computation method
2. Different Parameters: May require different k values, normalization, or preprocessing
3. Different Dataset: Target values might be computed on a different number range or sample
4. Different Metrics: The specific KL divergence and Cohen's d calculations might differ

• ✅ Comprehensive error handling
• ✅ Modular design with clear separation of concerns
• ✅ Extensive documentation and comments
• ✅ Type hints and parameter validation
• ✅ Memory-efficient implementations

• ✅ Complete test suite covering all functionality
• ✅ Unit tests for individual components
• ✅ Integration tests for full workflow
• ✅ Edge case validation
• ✅ Performance testing

• ✅ JSON output with all required fields
• ✅ Detailed results with confidence intervals
• ✅ BIC/AIC model selection metrics
• ✅ Comprehensive logging and progress reporting

• task4_results/task4_results.json - Main results in required format
• task4_results/task4_detailed_results.json - Detailed analysis with bootstrap data
• task4_enhanced_results/enhanced_results.json - Enhanced analysis results

{
  "k": 0.3,
  "cohens_d": 0.011853811083062384,
  "KL": 0.0018373979292823576,
  "sigma_bar": 0.3360816123274521,
  "BIC": 247465.7709289525,
  "AIC": 247332.5901124436,
  "n_primes": 9592,
  "n_composites": 90407,
  "confidence_intervals": {
    "cohens_d_ci": [0.0007073353797474674, 0.03295458412087822],
    "KL_ci": [0.0030288100487030098, 0.006310915995470218],
    "sigma_bar_ci": [0.3331323000554208, 0.3406430237794859]
  }
}

1. Review Target Criteria: Verify that the target values (d>1.2, KL 0.4-0.6) are appropriate for this specific θ' implementation

2. Parameter Tuning: Consider allowing k to be optimized rather than fixed at 0.3

3. Alternative Transformations: Explore other mathematical transformations that might enhance prime/composite discrimination

4. Validation: Cross-reference with the original research to ensure consistent implementation

python3 task4_statistical_discrimination.py

python3 test_task4.py

python3 task4_enhanced.py

• numpy >= 2.3.2
• pandas >= 2.3.1
• scikit-learn >= 1.7.1
• scipy >= 1.16.1
• sympy >= 1.14.0
• matplotlib >= 3.10.5

All dependencies successfully installed and tested in the unified-framework environment.