The grammar of ups and downs.
Ignores the actual values entirely and looks only at their rank order. In a window of 5 consecutive values, there are 120 possible orderings (permutations). Which orderings appear? Which transitions between orderings are allowed? This is the most purely encoding-invariant geometry in the framework.
How predictable is the next rank pattern given the current one? Wind speed, BTC range, and Poisson spacings max out at 1.0 (any pattern can follow any other). Logistic period-4 scores 0.0 (each pattern has exactly one successor). Chaos lives in between: logistic r=3.9 scores 0.82.
What fraction of theoretically possible pattern transitions never occur? Square wave and Morse code score 0.88 (only 12% of transitions are allowed — the signal's grammar is highly constrained). White noise and prime gaps score 0.0 (all transitions observed). Deterministic chaos forbids specific transitions that noise allows — this is a zero-training chaos detector.
Does the signal look different played backwards? Measures the asymmetry between forward and reversed ordinal-pattern transitions. After Phase-2 native-float input (uint8 quantization was injecting artificial asymmetry), this metric became a clean continuous-time vs discrete-time discriminator. Smooth dynamical signals cluster near zero — Lorenz (0.012), Van der Pol (0.018), Chua's Circuit (0.014), Double Pendulum (0.016), Ocean Swell (0.012), Sine Wave (0.012) — because their short-time dynamics approximate ODEs with time-reversal symmetry. Dissipative iterated maps saturate near 1 (Logistic Chaos 1.0, Hénon 0.997). Phyllotaxis also scores 1.0: the golden-angle rotation has a definite direction. Notably, Arnold's Cat scores 0.047 despite being chaotic — it's a symplectic torus automorphism conjugate to its inverse, so its ordinal statistics are genuinely time-symmetric. (At uint8, Lorenz previously measured 0.253 — the quantization artifact was 20× the true signal.)
The sweet spot between order and disorder. Peaks at the edge of chaos. L-System Dragon Curve (0.104) and logistic r=3.9 (0.102) are the most complex signals in the atlas. Both constants and perfect noise score 0.0 — neither is complex, for opposite reasons.
Does the signal need 2-step Markov to model its ordinal patterns, or is 1-step sufficient? Log-likelihood ratio of order-2 vs order-1 transition models. The metric cleanly separates chaos by phase-space dimensionality: 1D logistic maps (every parameter regime) score ~0 because their ordinal patterns are first-order Markov. Genuine 2D maps score high — Arnold's Cat (0.75), Hénon (0.44), Hénon Near-Crisis (0.43), Standard Map K=0.5 (0.39), Ikeda (0.51). The only two non-2D-map sources in the top 8 are Stern-Brocot Walk (1.08, the maximum) and Fibonacci Quasicrystal (0.55) — both have algebraically-induced 2-step memory. Effectively a cheap embedding-dimension estimator that reveals chaotic phase-space dimensionality through ordinal-pattern requirements.
How quickly does the ordinal transition matrix mix? Based on the spectral gap of the transition probability matrix. L-System Dragon (1.0) and constants score 1.0 (instant mixing — the chain reaches equilibrium in one step). Logistic periodic orbits score 0.0 (the chain cycles deterministically, never mixing). High mixing means the ordinal dynamics are ergodic; low mixing means they are trapped in cycles or absorbing states.
| Source | Domain | Value |
|---|---|---|
| Square Wave | waveform | 0.8800 |
| Pulse-Width Modulation | waveform | 0.8800 |
| Morse Code | waveform | 0.8800 |
| ··· | ||
| Gzip (level 1) | binary | 0.0000 |
| Gzip (level 9) | binary | 0.0000 |
| Bzip2 (level 1) | binary | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 1.0000 |
| Mian-Chowla | number_theory | 1.0000 |
| L-System (Dragon Curve) | exotic | 0.9999 |
| ··· | ||
| Quartic Map (Feigenbaum) | chaos | 0.0000 |
| Sine Map (Feigenbaum) | chaos | 0.0000 |
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Heisenberg Walk | exotic | 1.3699 |
| Stern-Brocot Walk | number_theory | 1.0766 |
| Fibonacci Tight-Binding | quantum | 0.9870 |
| ··· | ||
| Mian-Chowla | number_theory | 0.0000 |
| Devil's Staircase | exotic | 0.0000 |
| Quartic Map (Feigenbaum) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Padovan Word | exotic | 0.1040 |
| Logistic r=3.9 (Near-Full Chaos) | chaos | 0.1038 |
| L-System (Dragon Curve) | exotic | 0.1037 |
| ··· | ||
| Pi Digits | number_theory | 0.0003 |
| Euler-Mascheroni γ Digits | number_theory | 0.0003 |
| glibc LCG | binary | 0.0003 |
| Source | Domain | Value |
|---|---|---|
| Logistic Chaos | chaos | 1.0000 |
| Tent Map | chaos | 1.0000 |
| Sawtooth Wave | waveform | 1.0000 |
| ··· | ||
| Stern-Brocot Walk | number_theory | 0.0006 |
| Fibonacci Tight-Binding | quantum | 0.0007 |
| Triangle Wave | waveform | 0.0030 |
| Source | Domain | Value |
|---|---|---|
| Prime Gaps | number_theory | 1.0000 |
| Wind Speed | climate | 1.0000 |
| Earthquake Magnitudes | geophysics | 1.0000 |
| ··· | ||
| Quartic Map (Feigenbaum) | chaos | 0.0000 |
| Sine Map (Feigenbaum) | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
Ordinal Partition's forbidden_transitions is the primary separator between deterministic and stochastic sources in the atlas. Combined with transition_entropy, it places every signal on a 2D map from "fully constrained" (periodic) through "partially constrained" (chaos) to "unconstrained" (noise) — without any training data or parameter tuning. Post-Phase-2, time_irreversibility and memory_order add two further axes that uint8 quantization had previously hidden: continuous-time vs discrete-time dynamics (time_irreversibility), and 1D vs ≥2D phase-space dimensionality of chaotic maps (memory_order). Together the six metrics span: ordinal grammar size (transition_entropy, forbidden_transitions), grammar complexity (statistical_complexity), grammar mixing (markov_mixing), grammar dimensionality (memory_order), and grammar symmetry (time_irreversibility).