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.
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 | 0.3897 |
| Logistic Edge-of-Chaos | chaos | 0.3745 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| L-System (Dragon Curve) | exotic | 0.0000 |
| Earthquake Intervals | geophysics | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 4729.7977 |
| Logistic r=3.5 (Period-4) | chaos | 4618.0925 |
| Logistic Edge-of-Chaos | chaos | 4508.6136 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.8109 |
| Thue-Morse | exotic | 0.9197 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 0.9226 |
| Devil's Staircase | exotic | 0.9072 |
| Logistic r=3.5 (Period-4) | chaos | 0.9016 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.0417 |
| White Noise | noise | 0.0503 |
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).