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.
Total accumulated twist after traversing the entire signal. Logistic period-2 scores 33.5 million — each alternation adds massive twist in the same direction. L-System Dragon Curve scores 1.0 (the path barely twists because the symbolic dynamics has no consistent correlation direction). The 7-order-of-magnitude range makes this the framework's most dynamic metric.
Standard deviation of the path in the x-y plane. Measures how widely the signal explores its amplitude space. Logistic period-2 again dominates (4,730): the two alternating values create maximum spread.
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 33550335.9998 |
| Logistic r=3.5 (Period-4) | chaos | 31984523.2810 |
| Logistic Edge-of-Chaos | chaos | 30485776.8321 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| L-System (Dragon Curve) | exotic | 1.0231 |
| 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 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.8014 |
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). In the ordinal view, the Heisenberg axis (PC3) separates sources by their correlation polarity: strongly anti-correlated chaos at one extreme, positively correlated drift at the other.