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.
Standard deviation of the per-scale Gini coefficient of |detail| across the first 8 wavelet levels. Captures cross-scale heterogeneity of burst sparsity: self-similar signals (Sine, Berry Random Wave, Human Proteome) have ~identical Gini at every scale (→ ≈0), while scale-localized bursty signals have Gini that swings across scales. Fibonacci Word (0.302), Logistic Edge-of-Chaos (0.294), Pulse-Width Modulation (0.241), Random Steps (0.209), and Square Wave (0.207) top the list. Constants, Period-2, and Period-3 collapse to 0. Orthogonal in construction to cascade_coupling (cross-scale linear correlation) and intermittency_slope (kurtosis trend) — Gini is a within-scale distributional concentration, dispersion is its cross-scale spread. Integrated via the Wavelet Cascade pilot.
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 0.9899 |
| Aubry-André Critical | random-matrix-quantum | 0.9350 |
| Takagi Function | special-functions | 0.8737 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | iterated-maps | -1.0000 |
| Thue-Morse | symbolic-dynamics | -0.6000 |
| Pell Word | symbolic-dynamics | -0.5556 |
| Source | Origin | Value |
|---|---|---|
| Sine Map (Feigenbaum) | iterated-maps | 1.4313 |
| Logistic Edge-of-Chaos | iterated-maps | 1.4015 |
| Quartic Map (Feigenbaum) | iterated-maps | 0.8176 |
| ··· | ||
| Stochastic Resetting Walk | stochastic-process | -2.8137 |
| Forest Fire | self-organized-criticality | -2.7407 |
| Sawtooth Wave | signal-synthesis | -2.6338 |
| Source | Origin | Value |
|---|---|---|
| Sine Map (Feigenbaum) | iterated-maps | 0.4416 |
| Fibonacci Word | symbolic-dynamics | 0.3023 |
| Penrose Substitution | symbolic-dynamics | 0.3021 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.0002 |
| OTOC Growth | random-matrix-quantum | 0.0074 |
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), whether extreme-event concentration grows with scale (intermittency_slope), and whether burst sparsity is itself scale-localized (scale_gini_dispersion). These 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), cellular automata with emergent structure (Rule 110), and quasiperiodic substitution sequences (Fibonacci Word). The geometry was inherited from earlier multi-scale work; the surviving three metrics target three orthogonal cross-scale signatures.