How much consecutive values twist around each other.
Lifts pairs of successive values into 3D Heisenberg group coordinates, where the z-axis accumulates the signed area swept by the (x, y) path. Correlated data twists the path into a helix; uncorrelated data stays flat. The centered version subtracts the mean first, so it measures correlation, not bias.
Standard deviation of the path in the x-y plane. Measures how widely the signal explores its amplitude space. Logistic period-2 dominates (4,730): the two alternating values create maximum spread.
Log-compressed ratio of standard deviations along the two principal axes of the (x, y) path: log1p(std_v / std_u). Near zero means the path is isotropic in the xy plane (no preferred direction); large values mean the path is stretched along one principal axis. De Bruijn Sequence (3.83) and Collatz Stopping Times (1.74) score highest — their dynamics produce strongly anisotropic phase-space orbits. Periodic and binary-symbolic sources cluster near zero (isotropic short orbits). The log1p compression was added in the 2026-05-18 audit; without it, kurtosis=127 from extreme outliers (range [0, 38,226]) drove F-stat to 0.67. Compressed form lifts to F=3.69.
Ratio of accumulated z-area to total path length. Measures how efficiently the path converts length into twist. Logistic period-2 (0.41) and period-4 (0.39) score highest — their alternating dynamics are almost purely twist-generating. L-System Dragon and constants score 0.0 (path length without twist). This normalizes the raw twist by path effort, making it comparable across signals of different amplitudes.
Spectral entropy of the z-coordinate's rate of change. Measures how many frequencies contribute to the twist dynamics. Logistic period-2 (0.92) and Devil's Staircase (0.91) score highest — their z-rate has rich spectral content. Rainfall (0.04) scores near zero (the twist rate is dominated by a single frequency). This captures the temporal complexity of the Heisenberg twist that static z-accumulation misses.
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 0.4054 |
| Logistic r=3.5 (Period-4) | chaos | 0.3899 |
| Sine Map (Feigenbaum) | chaos | 0.3746 |
| ··· | ||
| LIGO Hanford | astro | 0.0000 |
| LIGO Livingston | astro | 0.0000 |
| L-System (Dragon Curve) | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | chaos | 10.4882 |
| Logistic Edge-of-Chaos | chaos | 10.1699 |
| Sine Map (Feigenbaum) | chaos | 10.1276 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Morse Code | waveform | 0.0000 |
| fBm (Persistent) | noise | 0.0002 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 4729.7977 |
| Logistic r=3.5 (Period-4) | chaos | 4618.9876 |
| Sine Map (Feigenbaum) | chaos | 4509.4061 |
| ··· | ||
| LIGO Livingston | astro | 0.0000 |
| LIGO Hanford | astro | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.8014 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 0.9226 |
| Logistic r=3.5 (Period-4) | chaos | 0.9020 |
| Sine Map (Feigenbaum) | chaos | 0.8879 |
| ··· | ||
| LIGO Livingston | astro | 0.0000 |
| LIGO Hanford | astro | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.0461 |
Heisenberg twist is mathematically identical to the lag-1 autocorrelation — but computed through group multiplication rather than arithmetic. Its value is structural: it connects correlation detection to the geometry of 3-manifolds (Nil geometry is one of Thurston's eight). The area_length_ratio normalizes twist by path effort, and z_rate_spectral_entropy adds a temporal dimension — together they separate signals that twist efficiently at one frequency (periodic) from those that twist chaotically at many frequencies (complex dynamics).