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).
Maximum local Hölder exponent across the signal. Beta Noise, Entanglement Entropy, and GOE Spacings all hit the cap at 4.0 (extremely smooth local regions). Logistic period-2 scores -3.2 (the roughest maximum — even its smoothest point is anti-persistent).
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 gap between holder_max and holder_min is 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 |
|---|---|---|
| Hodgkin-Huxley | bio | 0.7598 |
| Van der Pol Oscillator | exotic | 0.7581 |
| μ-law Sine | waveform | 0.5962 |
| ··· | ||
| Solar Wind IMF | astro | 0.0000 |
| Sandpile | exotic | 0.0000 |
| Sunspot Number | astro | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| x86-64 Machine Code | binary | 4.0000 |
| VLF Radio (Eclipse) | geophysics | 4.0000 |
| Earthquake Intervals | geophysics | 4.0000 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -3.2016 |
| Logistic r=3.5 (Period-4) | chaos | -1.6781 |
| Logistic Edge-of-Chaos | chaos | -0.8163 |
| Source | Domain | Value |
|---|---|---|
| Wigner Semicircle | quantum | 1.0121 |
| Triangle Wave | waveform | 0.9848 |
| Clipped Sine | waveform | 0.9746 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -3.2016 |
| Logistic r=3.5 (Period-4) | chaos | -1.8978 |
| Noisy Period-2 | chaos | -1.7357 |
| Source | Domain | Value |
|---|---|---|
| Rudin-Shapiro | number_theory | 0.6755 |
| Triangle Wave | waveform | 0.5581 |
| Mertens Function | number_theory | 0.0000 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -3.2016 |
| Sandpile | exotic | -2.0000 |
| Continued Fractions | number_theory | -2.0000 |
| Source | Domain | Value |
|---|---|---|
| English Literature | speech | 1.8741 |
| Beta Noise | noise | 1.8121 |
| Gaussian Collatz Orbit | number_theory | 1.7242 |
| ··· | ||
| Rule 30 | exotic | 0.0000 |
| Rule 110 | exotic | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 0.9219 |
| Chua's Circuit | exotic | 0.8889 |
| Van der Pol Oscillator | exotic | 0.8717 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -2.0000 |
| Logistic r=3.5 (Period-4) | chaos | -2.0000 |
| De Bruijn Sequence | number_theory | -2.0000 |
| Source | Domain | Value |
|---|---|---|
| Lotka-Volterra | bio | 0.9802 |
| Hodgkin-Huxley | bio | 0.9722 |
| Clipped Sine | waveform | 0.9679 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Phyllotaxis | bio | 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.