Whether the signal looks statistically different played forward vs played backward.
Most framework metrics are time-direction-invariant by construction (entropy, PCA, distance distributions, spectral power, persistent homology). This geometry targets the orthogonal axis. Weiss (1975) proved that linear Gaussian stationary processes are time-reversible (TR); conversely, nonlinearity or non-Gaussianity generically breaks TR, dissipative chaos breaks TR (entropy increase), and Hamiltonian conservative dynamics preserve TR (Liouville). The axis therefore tracks conservative-vs-dissipative dynamics and is one of the open axes flagged in the multi-axis dynamical fingerprint investigation. Note: the discriminating null is the unshuffled linear-Gaussian process (AR(1), fGn, white noise) — shuffling destroys every time-dependent statistic and drives all three metrics to 0.
Half the L¹ distance between the length-4 ordinal-pattern distribution of x and of x reversed (Bandt-Pompe / Ramsey-Rothman TR test). Zero iff every pattern π occurs as often as its reverse π_rev. Collatz Trajectory, Sawtooth Wave, Logistic Edge-of-Chaos, Logistic Period-4, and Logistic Period-3 saturate at 1.0. Fibonacci Word, Prime Indicator, Devil's Staircase, and constants sit at 0 — binary/staircase patterns are symmetric under reversal. The only rank-only (encoding-invariant) metric in the geometry.
max over τ ∈ {1, 2, 4, 8, 16, 32, 64} of |skew(x[τ:] − x[:−τ])|, tanh-squashed to [0, 1]. The third-moment branch of the geometry. Sawtooth Wave, Forest Fire, ECG Fusion, and Collatz Trajectory all hit 1.0 — sawteeth have asymmetric ramps; bursty event streams have one-sided spike shapes. Logistic Period-2, Thue-Morse, and Devil's Staircase score near 0 (symmetric or piecewise-constant increments).
| Source | Domain | Value |
|---|---|---|
| Sawtooth Wave | waveform | 1.0000 |
| Forest Fire | exotic | 1.0000 |
| ECG Fusion | medical | 1.0000 |
| ··· | ||
| Devil's Staircase | exotic | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Thue-Morse | exotic | 0.0005 |
| Source | Domain | Value |
|---|---|---|
| Sawtooth Wave | waveform | 1.0000 |
| Logistic Edge-of-Chaos | chaos | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| ··· | ||
| Devil's Staircase | exotic | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Prime Indicator | number_theory | 0.0002 |
Time Reversibility is the only framework geometry that targets the time-arrow axis directly. The two atlas metrics catch the asymmetry through different mathematical channels: ordinal patterns (rank-based, encoding-invariant) and increment skewness (third moment of differences). Sawtooth Wave is the cleanest positive on both because its asymmetric ramp triggers every test simultaneously. The geometry is in active use as the candidate carrier for the conservative-vs-dissipative axis flagged in the multi-axis fingerprint, where the framework otherwise lacks an explicit phase-space-contraction signal. A sister metric directional_run_asymmetry (Wasserstein-1 distance between up-run and down-run length distributions) is class-only — kept in compute_metrics for direct callers but dropped from the atlas because its internal correlation with the two atlas metrics (r=+0.74 with increment_skewness_max, r=+0.63 with ordinal_reversal_distance) makes it the weakest of the three (F=2.72) while the remaining two contribute to PC5.