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 consecutive event pairs that are timelike-separated (causally ordered). Smooth, slow-varying signals score near 1.0 (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 sampled pairs (at log-spaced separations) that are acausally separated. High for chaotic maps that make large jumps between values; zero for continuous-time attractors (Lorenz, Rössler) whose smooth trajectories keep consecutive samples causally close.
How often consecutive steps switch between timelike (subluminal) and spacelike (superluminal). Measures burstiness relative to the lightcone boundary c=1. Rule 30 scores ~0.5 (maximally bursty, random-looking transitions), Collatz Parity ~0.32 (structured runs of same causal character), smooth signals score 0.0 (always subluminal). Evolved via ShinkaEvolve atlas v1.
Lag-1 autocorrelation of the timelike/spacelike binary sequence, scaled to [0,1]. High values indicate long runs of the same causal character; 0.5 indicates random alternation. Collatz Parity scores ~0.62 (correlated causal runs), Rule 30 ~0.49 (uncorrelated). Degenerate when causal_order is near 0 or 1. Evolved via ShinkaEvolve atlas v1.
| Source | Domain | Value |
|---|---|---|
| Lorenz Attractor | chaos | 1.0000 |
| Rossler Attractor | chaos | 1.0000 |
| Critical Transition (Fold) | chaos | 1.0000 |
| ··· | ||
| Logistic r=3.74 (Period-5 Window) | chaos | 0.0000 |
| Critical Circle Map (Bronze Mean) | chaos | 0.0000 |
| Critical Circle Map (Silver Mean) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Heisenberg Walk | exotic | 0.9867 |
| Intermittency Type-III | chaos | 0.9691 |
| Pomeau-Manneville | chaos | 0.9085 |
| ··· | ||
| Tidal Gauge (SF) | geophysics | 0.0000 |
| Brownian Walk | noise | 0.0000 |
| fBm (Persistent) | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.7500 |
| Kolakoski Sequence | exotic | 0.6668 |
| Thue-Morse | exotic | 0.6667 |
| ··· | ||
| Tidal Gauge (SF) | geophysics | 0.0000 |
| Brownian Walk | noise | 0.0000 |
| fBm (Persistent) | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 0.1888 |
| Logistic Edge-of-Chaos | chaos | 0.1824 |
| Sine Map (Feigenbaum) | chaos | 0.1812 |
| ··· | ||
| Tidal Gauge (SF) | geophysics | 0.0000 |
| Brownian Walk | noise | 0.0000 |
| fBm (Antipersistent) | noise | 0.0000 |
Lorentzian geometry captures 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). The evolved metrics (crossing_density and causal_persistence) add temporal dynamics to the static causal fraction: two signals with the same causal_order can differ in whether their lightcone crossings are bursty (low persistence, high crossing) or structured (high persistence, moderate crossing). In the atlas, these four metrics separate smooth oscillators from chaotic/noise sources along a "causality axis" orthogonal to the entropy axis.