How the distribution of values differs from uniform, and how stable that distribution is across the signal.
Bins the data into a 32-bin histogram and treats it as a probability distribution. Then asks three questions: how far is this distribution from uniform (optimal transport cost)? How concentrated is it (peak height)? And does the first half of the signal look like the second half (self-similarity)?
The peak bin height times the number of bins. 1.0 means uniform (De Bruijn scores exactly 1.0 — its construction guarantees every byte pattern appears equally). Above 1.0 means the distribution has a spike. Collatz gap lengths (31.8), Rainfall (31.5), and Forest fire (29.4) are the most concentrated signals in the atlas — their heavy-tailed distributions pile most of their mass into the lowest bin.
Earth mover's distance from the uniform distribution: the minimum amount of "dirt" you'd need to move to make the histogram flat. Collatz gap lengths (0.48) and Rainfall (0.48) are farthest from uniform. Neural net pruned weights (0.46) are close behind — pruning creates a spike at zero. De Bruijn scores near 0 (already uniform).
Shannon entropy of the 32-bin histogram. De Bruijn, circle map quasiperiodic, and phyllotaxis all score 5.0 (near the maximum of 5 bits — flat distribution). Collatz gap lengths scores 0.05 (almost all mass in one bin). This is the classical measure of distributional spread, here computed on the Wasserstein embedding.
One minus the earth mover's distance between the first-half and second-half histograms. 1.0 means the distribution is perfectly stable over time (logistic period-4, logistic period-2, De Bruijn). Hilbert walk scores 0.60 (its deterministic sweep creates different distributions in the first and second halves). This catches nonstationarity that entropy and concentration miss: a signal can have high entropy overall but low self_similarity if its distribution drifts.
Coefficient of variation of windowed earth mover's distances between consecutive segments. Exponential Chirp (0.21) scores highest — its frequency sweep creates rapidly changing local distributions. Sunspot (0.13) and Pulse-Width Mod (0.13) also score high. Constants and periodic orbits score 0.0 (identical windows). This measures how much the optimal transport cost fluctuates over time — a windowed nonstationarity detector complementing self_similarity's global split. Evolved via ShinkaEvolve.
Average earth mover's distance between non-adjacent windows that are within a recurrence threshold. PID Controller (0.11) and Exponential Chirp (0.10) score highest — their recurring distributional states differ in fine detail. Constants score 0.0. This measures how similar the signal's distribution is when it "returns" to a previously visited distributional state. Evolved via ShinkaEvolve.
| Source | Origin | Value |
|---|---|---|
| Mian-Chowla | number-theory | 31.8488 |
| Aubry-André Critical | random-matrix-quantum | 31.7109 |
| Rainfall (ORD Hourly) | atmospheric | 31.4687 |
| ··· | ||
| Gray Code Counter | symbolic-dynamics | 1.0000 |
| De Bruijn Sequence | symbolic-dynamics | 1.0000 |
| Phyllotaxis | number-theory | 1.0031 |
| Source | Origin | Value |
|---|---|---|
| Aubry-André Critical | random-matrix-quantum | 0.4837 |
| Fibonacci Tight-Binding | random-matrix-quantum | 0.4831 |
| Rainfall (ORD Hourly) | atmospheric | 0.4830 |
| ··· | ||
| Gray Code Counter | symbolic-dynamics | 0.0000 |
| De Bruijn Sequence | symbolic-dynamics | 0.0000 |
| Phyllotaxis | number-theory | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Penrose Substitution | symbolic-dynamics | 1.0000 |
| De Bruijn Sequence | symbolic-dynamics | 1.0000 |
| Gray Code Counter | symbolic-dynamics | 1.0000 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | 0.5987 |
| Critical Transition (Fold) | iterated-maps | 0.6873 |
| Langton's Ant | symbolic-dynamics | 0.8075 |
| Source | Origin | Value |
|---|---|---|
| Gray Code Counter | symbolic-dynamics | 5.0000 |
| De Bruijn Sequence | symbolic-dynamics | 5.0000 |
| Circle Map Quasiperiodic | iterated-maps | 5.0000 |
| ··· | ||
| Mian-Chowla | number-theory | 0.0433 |
| Aubry-André Critical | random-matrix-quantum | 0.0925 |
| Fibonacci Tight-Binding | random-matrix-quantum | 0.1580 |
| Source | Origin | Value |
|---|---|---|
| PID Controller | control-systems | 0.1077 |
| Exponential Chirp | signal-synthesis | 0.0999 |
| Langton's Ant | symbolic-dynamics | 0.0975 |
| ··· | ||
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| Logistic Edge-of-Chaos | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Exponential Chirp | signal-synthesis | 0.2082 |
| Critical Transition (Fold) | iterated-maps | 0.1606 |
| Stochastic Resetting Walk | stochastic-process | 0.1360 |
| ··· | ||
| Logistic Edge-of-Chaos | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Quartic Map (Feigenbaum) | iterated-maps | 0.0000 |
Wasserstein self_similarity is the distributional lens's nonstationarity detector. Signals that change character midstream — sensor drift, regime switches, concatenated recordings — score low on self_similarity while potentially scoring high on all other distributional metrics. In the atlas, Wasserstein's concentration axis separates the heavy-tailed cluster (Collatz, rainfall, forest fire) from the uniform-distribution cluster (PRNGs, De Bruijn), while self_similarity provides an orthogonal axis that catches temporal instability invisible to any single-histogram metric.