How much of the signal's structure is causal — does the future follow from the past in a timelike way, or do successive values jump acausally?
Embeds consecutive (time, value) pairs as events in 1+1 Minkowski spacetime with metric ds² = −dt² + dx². For each pair of events, the Minkowski interval classifies the separation: timelike (|Δx| < |Δt|, causally connected), spacelike (|Δx| > |Δt|, causally disconnected), or lightlike (|Δx| = |Δt|, on the light cone).
Fraction of event pairs that are timelike-separated (causally ordered). Tidal gauge, projectile, and damped pendulum score 1.0 — smooth, slow-varying signals where each value follows causally from the previous one. Logistic edge-of-chaos scores 0.0 (every successive value is an acausal jump). This is the framework's most direct "smoothness vs. jumpiness" metric.
Fraction of pairs that are acausally separated. Logistic period-3 (0.18) and Chirikov standard map (0.16) score highest — their dynamics make large jumps between successive values. Lorenz and Rössler score 0.0: despite being chaotic, their continuous-time dynamics produce smooth trajectories in which consecutive samples are always causally close.
Fraction of pairs exactly on the light cone (|Δx| = |Δt|). Almost always near zero because exact equality is measure-zero for continuous data. Prime gaps (0.006) score highest: many consecutive prime gaps differ by exactly 1, landing on the light cone. This is a curiosity more than a useful discriminator.
| Source | Domain | Value |
|---|---|---|
| Tidal Gauge (SF) | geophysics | 1.0000 |
| Projectile with Drag | motion | 1.0000 |
| Damped Pendulum | motion | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Prime Gaps | number_theory | 0.0058 |
| Rule 110 | exotic | 0.0042 |
| ARMA(2,1) | noise | 0.0041 |
| ··· | ||
| Lorenz Attractor | chaos | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 0.1804 |
| Chirikov Standard Map | chaos | 0.1575 |
| Dice Rolls | exotic | 0.1571 |
| ··· | ||
| Lorenz Attractor | chaos | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
Lorentzian geometry captures something subtler than amplitude variance or entropy: the "speed" of the signal relative to its sampling rate. A high-frequency oscillation sampled slowly appears spacelike (large jumps); the same oscillation sampled fast appears timelike (smooth evolution). This makes Lorentzian metrics sensitive to the relationship between the signal's characteristic timescale and the observation timescale — a property relevant to aliasing detection and sampling adequacy. In the atlas, causal_order_preserved separates the distributional view's smooth oscillator cluster (C1) from the chaotic/noise clusters (C4, C5) along a "causality axis" orthogonal to the entropy axis.