How energy flows between dyadic scales of a Haar wavelet decomposition — does a coarse-scale burst imply a fine-scale burst (cascade), and does extreme-event concentration grow with scale (intermittency)?
Decomposes the signal via a manual Haar wavelet transform into log₂(N) levels of detail coefficients. Each level j has its own sequence of wavelet coefficients; level 0 captures the finest fluctuations, level N-1 captures the coarsest. This gives a scale-by-scale picture of the signal's fluctuation energy. The two metrics then probe two aspects of how those scales talk to each other. Inspired by Richardson's turbulence cascade ("big eddies have little eddies that feed on their velocity") and self-organized criticality (avalanches that propagate across scales).
Temporal correlation of |detail_j| energy between adjacent scales. For each pair of adjacent scales, align them (coarse coefficient k covers fine coefficients 2k and 2k+1), then correlate the fine and coarse |detail| sequences. Positive values mean fine and coarse scales co-activate at the same time locations — the signature of a cascade. Quantum Walk (0.82), PID Controller (0.81), Devil's Staircase (0.78), SIR Epidemic (0.77), Rainfall (0.75), and Lotka-Volterra (0.75) score highest — all signals with event-driven dynamics where large-scale bursts recruit fine-scale activity. Correlates with Symplectic:windowed_area_cv (r=+0.878).
log-log slope of kurtosis versus scale. Flat for Gaussian processes (kurtosis doesn't depend on scale). Increasing for intermittent or multifractal signals where rare extreme events concentrate at particular scales, pushing kurtosis up there. Logistic Edge-of-Chaos (1.40) dominates — the chaotic-threshold regime produces extreme events whose size distribution broadens strongly at coarse scales. Rule 110 (0.75), Quantum Walk (0.61), Phyllotaxis (0.39), and Circle Map Quasiperiodic (0.38) follow. Internally anti-correlated with cascade_coupling (r=-0.608): signals that broaden kurtosis with scale tend to have weaker point-to-point cross-scale coupling.
| Source | Domain | Value |
|---|---|---|
| Partition Function | number_theory | 1.0000 |
| Minkowski Question Mark | exotic | 0.9563 |
| Takagi Function | exotic | 0.8737 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -1.0000 |
| Thue-Morse | exotic | -0.4500 |
| Morse Code | waveform | -0.1784 |
| Source | Domain | Value |
|---|---|---|
| Logistic Edge-of-Chaos | chaos | 1.4015 |
| Rule 110 | exotic | 0.7520 |
| Quantum Walk | quantum | 0.6142 |
| ··· | ||
| Forest Fire | exotic | -2.7504 |
| Sawtooth Wave | waveform | -2.6338 |
| Rainfall (ORD Hourly) | climate | -2.5849 |
| Source | Domain | Value |
|---|---|---|
| Fibonacci Word | exotic | 0.3022 |
| Logistic Edge-of-Chaos | chaos | 0.2944 |
| Pulse-Width Modulation | waveform | 0.2414 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Partition Function | number_theory | 0.0002 |
Wavelet Cascade occupies a specific niche in the scale lens: most scale metrics measure the amplitude of fluctuations at each scale independently (Hölder exponent, multi-scale Wasserstein, p-variation). Wavelet Cascade instead asks whether the scales are correlated — whether a coarse-scale anomaly temporally coincides with fine-scale anomalies (cascade_coupling) and whether extreme-event concentration grows with scale (intermittency_slope). Both signatures show up in signals with event-driven or threshold-driven dynamics: epidemiology models (Lotka-Volterra, SIR), chaotic attractors near crisis (Logistic Edge-of-Chaos), and cellular automata with emergent structure (Rule 110). The geometry was inherited from earlier multi-scale work; four candidate metrics were pruned down to these two during atlas curation because the rest were either too noisy or too redundant with Symplectic and spectral metrics.
# Quasicrystal Lens