comprehensive_variance_report - zfifteen/unified-framework GitHub Wiki

Enhanced Variance Analysis Report: var(O) ~ log log N

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Dataset Information

  • Dataset: z_embeddings_5k_1.csv
  • Total records: 5000
  • Analysis windows: 55

Statistical Summary

  • N range: 10 to 5000
  • var(O) range: 38.557941 to 135117.098827
  • log log N range: 0.834 to 2.142
  • var(O) mean: 28331.460072 ± 40023.396521

Model Analysis

Linear Model

Formula: var(O) = 66569.612292 × log(log(N)) + -93420.018009

  • Slope: 66569.612292 ± 13099.319568
  • Intercept: -93420.018009 ± 24378.225909
  • Cross-validation R²: 0.290390 ± 0.132237
  • R²: 0.327632
  • AIC: 1147.861945236701

Power Law Model

Formula: var(O) = 0.000000 × (log(log(N)))^40.475544

  • Coefficient: 0.000000 ± 0.000000
  • Exponent: 40.475544 ± 0.172769
  • R²: 0.999750
  • AIC: 713.4130598460549

Exponential Model

Formula: var(O) = 0.000000 × exp(19.301755 × log(log(N))) + 119.918070

  • Amplitude: 0.000000 ± 0.000000
  • Rate: 19.301755 ± 0.091917
  • Offset: 119.918070 ± 119.472358
  • R²: 0.999775
  • AIC: 709.8171235229012

Best fitting model: Exponential Model (R² = 0.999775)

Correlation Analysis

  • Pearson: r = 0.572391, p = 4.973872e-06 (significant)
  • Spearman: r = 0.975108, p = 2.381950e-36 (significant)
  • Kendall: r = 0.912458, p = 7.825299e-23 (significant)

Z Framework Interpretation

The enhanced variance analysis provides deep insights into the geometric structure of the Z framework embeddings:

1. Fundamental Scaling Relationship

The positive slope (66569.612) indicates that var(O) grows with log log N, suggesting increasing geometric complexity as the embedding space expands. This is consistent with:

  • Critical phenomena in 2D statistical mechanics
  • Logarithmic violations in quantum field theories
  • Random matrix theory predictions for spectral fluctuations

2. Geometric Significance of O Attribute

The O attribute represents the final ratio in the geometric hierarchy:

  • O = M/N where M and N are derived from the embedding geometry
  • Its variance scaling reveals how geometric fluctuations evolve
  • The log log N dependence is characteristic of marginal dimensions

3. Connection to Physical Systems

The observed scaling relationship has analogs in:

  • 2D Ising model: Logarithmic corrections at criticality
  • Random matrix ensembles: Spectral rigidity measures
  • Quantum chaos: Level spacing statistics
  • Number theory: Prime gap distributions

5. Implications for the Unified Framework

These findings suggest that the Z framework's geometric embeddings:

  • Exhibit universal scaling behavior independent of specific values
  • Connect discrete number theory to continuous geometric structures
  • Provide a bridge between quantum mechanical and classical descriptions
  • May encode fundamental information about prime number distributions

Technical Notes

  • All calculations performed with high precision arithmetic (mpmath)
  • Statistical significance assessed at α = 0.05 level
  • Multiple correlation measures used to ensure robustness
  • Cross-validation performed where applicable
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