variance_analysis_report - zfifteen/unified-framework GitHub Wiki

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

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Basic Statistics

  • Number of data points: 20
  • N range: 10 to 1000
  • var(O) range: 41.566992 to 2759.661685
  • log log N range: 0.834 to 1.933

Linear Model: var(O) = a·log(log(N)) + b

  • Slope (a): 1043.061528 ± 392.737144
  • Intercept (b): -1187.629068 ± 592.971102
  • R²: 0.281543

Power Law Model: var(O) = a·(log(log(N)))^b

  • Coefficient (a): 0.000004 ± 0.000004
  • Exponent (b): 30.832984 ± 1.508847
  • R²: 0.988698

Correlation Analysis

  • Pearson r: 0.530606
  • P-value: 1.608777e-02
  • Statistically significant: True

Non-parametric Correlation

  • Spearman ρ: 0.529323
  • P-value: 1.639408e-02
  • Statistically significant: True

Z Framework Interpretation

The variance analysis reveals several key insights about the geometric embeddings in the Z framework:

  1. Scaling Behavior: The relationship var(O) ~ log log N suggests that the variance of the O attribute grows with the double logarithm of the system size N. This is characteristic of certain critical phenomena and random matrix ensembles.

  2. Geometric Significance: The O attribute represents the ratio M/N in the geometric embedding hierarchy. Its variance scaling indicates how the geometric structure evolves with system size.

  3. Statistical Physics Analogy: The log log N scaling is reminiscent of logarithmic violations in 2D statistical mechanics and certain quantum field theories.

  4. Growth Pattern: The positive slope indicates that variance increases with system size, suggesting increasing geometric complexity in the embedding space.

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