How rough or smooth the signal is at each point — and how much that roughness varies.
If you zoom into a smooth signal, it looks linear. If you zoom into a rough signal, it stays jagged no matter how far you zoom. The Holder exponent quantifies this: high = smooth, low = rough. This geometry computes it at every point and asks: is the roughness uniform (monofractal) or does it vary wildly from place to place (multifractal)?
The global roughness summary. Sine waves score 0.92 (very smooth, persistent). White noise scores ~0 (uncorrelated). Logistic maps in the period-doubling cascade score -2.0 (actively anti-persistent — each value fights the previous one). In the seismic P-wave investigation, earthquake arrivals scored dramatically lower than ambient noise (d = 9.06): P-waves are impulsive, ambient microseisms are smooth.
Average local regularity. Wigner semicircle (0.99) and triangle wave (0.98) are the smoothest signals in the atlas. Logistic period-2 (-3.2) is the roughest — it alternates between two values with no interpolation.
How much does roughness vary? English literature scores highest (1.87): some passages are smooth (common words), others are jagged (rare words, punctuation). Fibonacci word scores 0.0 (perfectly uniform roughness at every point — it's monofractal).
Minimum local Hölder exponent. Rudin-Shapiro (0.66) and Triangle Wave (0.56) have the smoothest worst-case points. Logistic period-2 (-3.2) and full Logistic Chaos (-2.0) have extremely rough minima. The smaller the minimum, the rougher the signal's worst-case point — the local analog of multifractal width.
Temporal persistence of the Hölder exponent sequence. Hodgkin-Huxley (0.77) and Van der Pol (0.76) score highest — their smooth oscillatory dynamics create slowly-varying local regularity. Number-theoretic sequences (Divisor Count, Prime Gaps) and anti-persistent fBm score 0.0 (roughness varies randomly from point to point).
Lag-1 autocorrelation of the increment (first-difference) sequence. Clipped Sine (0.98) and Lotka-Volterra (0.98) score highest — their smooth waveforms have highly predictable increments. Circle Map QP and Critical Circle Map score 0.0 (increments change unpredictably).
| Source | Domain | Value |
|---|---|---|
| Logistic Chaos | chaos | 0.4235 |
| Heisenberg Walk | exotic | 0.3748 |
| Tent Map | chaos | 0.3698 |
| ··· | ||
| Solar Wind IMF | astro | 0.0000 |
| Poker Hands | exotic | 0.0000 |
| Pulse-Width Modulation | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Gaussian Collatz Orbit | number_theory | 1.3452 |
| BTC Volatility | financial | 1.0391 |
| Sprott-B | chaos | 1.0081 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -3.3219 |
| Intermittency Type-III | chaos | -1.9631 |
| Heisenberg Walk | exotic | -1.8990 |
| Source | Domain | Value |
|---|---|---|
| Rudin-Shapiro | number_theory | 1.0000 |
| Sawtooth Wave | waveform | 1.0000 |
| Tank Drain Cascade | exotic | 0.9964 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -3.3219 |
| Logistic Chaos | chaos | -2.0000 |
| Tent Map | chaos | -2.0000 |
| Source | Domain | Value |
|---|---|---|
| Goldbach r(2n) | number_theory | 1.9736 |
| ECG Beat Conformity | medical | 1.9195 |
| Beta Noise | noise | 1.8933 |
| ··· | ||
| Mian-Chowla | number_theory | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Pell Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Magnetic Pendulum (3-Magnet) | motion | 0.9689 |
| OTOC Growth | quantum | 0.9538 |
| Sine Wave | waveform | 0.9334 |
| ··· | ||
| Logistic Edge-of-Chaos | chaos | -2.0000 |
| Logistic r=3.2 (Period-2) | chaos | -2.0000 |
| Logistic r=3.5 (Period-4) | chaos | -2.0000 |
| Source | Domain | Value |
|---|---|---|
| OTOC Growth | quantum | 0.9999 |
| Magnetic Pendulum (3-Magnet) | motion | 0.9998 |
| Sine Wave | waveform | 0.9995 |
| ··· | ||
| Circle Map Quasiperiodic | chaos | 0.0000 |
| Critical Circle Map (Silver Mean) | chaos | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
Holder Regularity was the #2 discriminator in the seismic P-wave investigation (Cohen's d = 9.06), detecting earthquake arrivals through the collapse of local smoothness. It separates the Distributional view's C1 (smooth oscillators) from C4 (anti-persistent chaos) along the persistence axis — the biggest gap in ordinal space.