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.
Average run length of consecutive same-sign area increments. Constants hit 16,382 (maximum — flux never changes sign). Square Wave (149) has moderate persistence. Logistic periodic orbits score 1.0 (sign alternates every step). Long runs of same-sign flux mean the phase-space trajectory consistently sweeps area in one direction — the signature of sustained rotation or drift in phase space.
Fraction of windowed phase-space areas that are within a threshold of a previously observed value. Constants (0.995) and Collatz Gap Lengths (0.995) have near-perfect recurrence — their phase-space areas repeat almost exactly. Dice Rolls (0.027) and PRNGs score lowest (each window's area is unique). This measures how repetitive the phase-space dynamics are across the signal.
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 16382.0000 |
| Square Wave | waveform | 148.9725 |
| Middle-Square (von Neumann) | binary | 33.8194 |
| ··· | ||
| Logistic r=3.5 (Period-4) | chaos | 1.0000 |
| Logistic Edge-of-Chaos | chaos | 1.0000 |
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| 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 |
| Collatz Gap Lengths | number_theory | 0.0054 |
| Poisson Counts | exotic | 0.0094 |
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 0.9950 |
| Collatz Gap Lengths | number_theory | 0.9950 |
| Poisson Counts | exotic | 0.9949 |
| ··· | ||
| Dice Rolls | exotic | 0.0273 |
| glibc LCG | binary | 0.0275 |
| Gzip (level 1) | binary | 0.0277 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 3071.5000 |
| Thue-Morse | exotic | 2730.3000 |
| Rule 110 | exotic | 2290.5500 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| fBm (Persistent) | noise | 0.0350 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 11.1482 |
| Accel Sit | motion | 5.5747 |
| Rainfall (ORD Hourly) | climate | 5.5291 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | 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.