Phase-portrait filling, sweep stationarity, orbital coherence, and recurrence.
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 metrics then quantify how the trajectory occupies that portrait: whether it spreads across the bounding box (Hamiltonian conservative flow ergodic on its energy surface) or collapses to a low-dimensional subset (dissipative attractor, periodic orbit, or symbolic max-jump cloud).
Fraction of (q, p) bounding-box cells visited in a 32×32 histogram. Hénon-Heiles (~0.63), FPUT N=16 (~0.72), and the multi-torus quasiperiodic sources (~0.78–0.83) all score high — their phase portraits densely fill a 2D region. Dissipative limit cycles (Van der Pol ~0.12, Lotka-Volterra ~0.12) and symbolic max-jump signals (L-System Dragon, Kolakoski, Thue-Morse, Rule 110 all ~0.004) score at the floor: the trajectory visits only a handful of cells. Replaced total_area 2026-05-26 (sweep-4 misnomer audit): the prior shoelace-amplitude metric had Hamiltonian sources at the floor and symbolic max-jump at the ceiling, inverted from what "Symplectic" should mean.
Pearson correlation between the (q, p) histogram and its mirror across p=0 (on an explicit symmetric p-bin grid). Hamiltonian conservative flow is time-reversal symmetric ((q, p) → (q, -p) under t → -t) so the portrait should mirror cleanly — Spring Pendulum (~0.93), Double Pendulum (~0.88), 4-Torus quasiperiodic (~0.90) all score near 1. Sine Wave (~0.69) is symmetric for the same reason. Van der Pol limit cycle (~0.01) has a preferred rotation direction and fails the test; Sawtooth Wave (~-0.03) is anti-symmetric (ramp-up then drop). Symbolic max-jump signals score at zero (random sparse cloud). Caveat: this is a necessary but not sufficient Hamiltonian test — dissipative systems with built-in equation symmetry score high too: Lorenz (~0.85, σ-symmetry), Aizawa (~0.93), Halvorsen (~0.64). Independent of phase_volume_explored: Hénon-Heiles fills (0.63) but its scalar-projected portrait is asymmetric (0.17); Spring Pendulum is the inverse (0.40 fill, 0.93 symmetric).
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.
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 | Origin | Value |
|---|---|---|
| Van der Pol Oscillator | continuous-flows | 345.3302 |
| Exponential Chirp | signal-synthesis | 320.7402 |
| Sawtooth Wave | signal-synthesis | 292.0574 |
| ··· | ||
| Sine Map (Feigenbaum) | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Logistic Edge-of-Chaos | iterated-maps | 1.0000 |
| Source | Origin | Value |
|---|---|---|
| Aubry-André Critical | random-matrix-quantum | 0.9999 |
| Fibonacci Tight-Binding | random-matrix-quantum | 0.9994 |
| Rudin-Shapiro | symbolic-dynamics | 0.9983 |
| ··· | ||
| Noisy Period-2 | iterated-maps | -0.0992 |
| Euler Totient Ratio | number-theory | -0.0545 |
| Clifford Attractor | iterated-maps | -0.0466 |
| Source | Origin | Value |
|---|---|---|
| Weierstrass | special-functions | 0.8749 |
| Exponential Chirp | signal-synthesis | 0.8310 |
| 3-Torus Quasiperiodic | coupled-oscillators | 0.8276 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0020 |
| Gap-Word β-Shift | symbolic-dynamics | 0.0029 |
| Collatz Cycle Word | number-theory | 0.0029 |
| Source | Origin | Value |
|---|---|---|
| Aubry-André Critical | random-matrix-quantum | 0.9831 |
| Rainfall (ORD Hourly) | atmospheric | 0.9782 |
| Mian-Chowla | number-theory | 0.9772 |
| ··· | ||
| Dice Rolls | stochastic-process | 0.0270 |
| XorShift32 | algorithmic-bytes | 0.0273 |
| Gzip (level 1) | algorithmic-bytes | 0.0277 |
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 10.3673 |
| Aubry-André Critical | random-matrix-quantum | 9.2084 |
| Fibonacci Tight-Binding | random-matrix-quantum | 9.0044 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Periodic Word | symbolic-dynamics | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 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.