Phase-space area and trajectory stationarity.
Pairs each value with its discrete derivative (the difference to the next value) to construct a phase portrait — the (position, momentum) representation that Hamiltonian mechanics uses. The symplectic area form (shoelace formula on the phase portrait) measures how much area the trajectory encloses. Windowed area CV then checks whether this area is stable across the signal or varies in bursts.
Absolute area enclosed by the phase-space trajectory (shoelace formula, then abs). L-System Dragon (3,072) and Thue-Morse (2,730) score highest: their binary dynamics create phase portraits with large, non-self-canceling loops. Rule 110 (2,291) is close behind. Logistic period-2 scores 0.0 (its phase portrait collapses to a line segment — alternating values have constant derivative, so q and p are locked together). High total area means the trajectory genuinely sweeps through phase space; zero means it's confined to a lower-dimensional manifold.
Coefficient of variation of the shoelace area computed over non-overlapping windows. Devil's staircase (11.1) dominates: its long constant plateaus (zero area) punctuated by sudden jumps (large area) create extreme variability. Accelerometer sitting (5.57) and rainfall (5.57) tie for second — both are bursty signals with long quiet intervals. Logistic period-2 scores near 0 (constant zero area in every window). This is the framework's most direct stationarity test: low CV means the dynamics look the same throughout; high CV means there are transient events or regime changes.
Standard deviation of the position coordinate in phase space. Thue-Morse and L-System Dragon both score 0.50 (maximum spread for binary data — their values span the full [0,1] range uniformly). Collatz gap lengths scores 0.005 (nearly zero — the gap lengths are confined to a narrow range). This is a simple amplitude measure, but it gains meaning in the symplectic context: data with high q_spread but low total_area has oscillatory dynamics (the trajectory retraces itself), while data with both high q_spread and high total_area has genuinely 2D phase-space dynamics.
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 0.5000 |
| L-System (Dragon Curve) | exotic | 0.5000 |
| Square Wave | waveform | 0.5000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Collatz Gap Lengths | number_theory | 0.0054 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 3071.5000 |
| Thue-Morse | exotic | 2730.2500 |
| Rule 110 | exotic | 2290.5500 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 11.1482 |
| Accel Sit | motion | 5.5747 |
| Rainfall (ORD Hourly) | climate | 5.5714 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
Symplectic's windowed_area_cv was one of the key metrics in the negative re-evaluation study (2026-03-05), contributing to the reclassification of Standard Map and Arnold Cat as positive detections. Its power is in detecting non-stationarity: bursty signals (devil's staircase, rainfall, earthquake) have high CV because their quiet periods produce near-zero windowed area while their active periods produce large area. This is a different kind of non-stationarity detection than Nonstationarity geometry's vol_of_vol (which measures descriptor trajectory burstiness) — Symplectic's version is rooted in the physical concept of phase-space area, making it particularly natural for signals with Hamiltonian or conservative dynamics.