How the signal's distribution changes across scales — is it self-similar, drifting, or scale-free?
Splits the signal into blocks at multiple scales (2, 4, 8, 16, ... blocks), computes the histogram of each block, and measures the Wasserstein (earth mover's) distance between adjacent blocks at each scale. Self-similar signals have constant Wasserstein distance across scales. Non-stationary signals show drift. Scale-free processes follow a power law.
Mean Wasserstein distance averaged across all scales. Captures the overall level of distributional heterogeneity regardless of scale structure.
Standard deviation of Wasserstein distances across scales. High values mean the signal's distributional contrast is strongly scale-dependent; low values mean it's roughly constant.
Average Wasserstein distance between adjacent blocks at the finest scale. Morse Code (0.394) dominates: its on/off structure creates maximally different local distributions between signal and silence segments. Symbolic Lorenz (0.260) and Sprott-B (0.194) are next — chaotic dynamics with occasional regime switches produce large local distributional variation. Periodic orbits and constants score 0.0 (adjacent blocks have identical distributions).
Dynamic range of Wasserstein distances across scales (max/min). Triangle Wave scores 1521: at the finest scale, adjacent blocks have similar distributions (both are local ramps), but at the coarsest scale, one half is ascending and the other descending, creating maximum distributional contrast. Standard Map Mixed (1269) and Logistic Period-3 (1118) have similarly extreme ratios because their periodic structure creates scale-dependent distributional asymmetry.
Log-log slope of Wasserstein distance vs. number of blocks. Positive slope means distributional differences grow at finer scales (more blocks = more local contrast between adjacent windows). Negative slope means they grow at coarser scales (fewer, larger blocks reveal structure that fine divisions miss). De Bruijn (6.38) has the steepest positive slope: adjacent small windows differ sharply because the de Bruijn sequence cycles through all byte patterns locally. Devil's Staircase (-0.74) has the most negative slope: its long constant plateaus make adjacent fine-scale windows identical, but coarse-scale blocks straddle plateau boundaries, creating distributional contrast.
| Source | Domain | Value |
|---|---|---|
| Morse Code | waveform | 0.3942 |
| Symbolic Lorenz | exotic | 0.2599 |
| Sprott-B | chaos | 0.1939 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Triangle Wave | waveform | 1520.9765 |
| Standard Map K=0.5 (Mixed) | chaos | 1268.6816 |
| Logistic r=3.83 (Period-3 Window) | chaos | 1118.2270 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Pulse-Width Modulation | waveform | 0.1872 |
| Square Wave | waveform | 0.1483 |
| μ-law Sine | waveform | 0.1450 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| De Bruijn Sequence | number_theory | 6.3834 |
| Logistic Edge-of-Chaos | chaos | 3.4461 |
| Fibonacci Word | exotic | 2.2934 |
| ··· | ||
| Devil's Staircase | exotic | -0.7396 |
| fBm (Persistent) | noise | -0.6030 |
| ETH/BTC Ratio | financial | -0.5976 |
| Source | Domain | Value |
|---|---|---|
| Square Wave | waveform | 0.1674 |
| Clipped Sine | waveform | 0.1634 |
| μ-law Sine | waveform | 0.1622 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
Multi-Scale Wasserstein measures distributional self-similarity — something no entropy metric captures directly. Two signals with the same entropy can have completely different scale profiles: one might have constant distributions at all scales (self-similar), while the other has scale-dependent drift. In the atlas, w_slope separates sources with fine-scale local variation (De Bruijn, Fibonacci: positive slope, adjacent small windows differ) from sources with coarse-scale structure (Devil's Staircase, fBm: negative slope, distributional contrast only emerges in large blocks).
# Quasicrystal Geometries