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? Phyllotaxis scores 1.0 (strongly irreversible — the golden-angle rotation has a definite direction). Stern-Brocot walk scores 0.008 (nearly reversible). Financial returns, being famously irreversible (crashes are fast, recoveries are slow), score 0.3-0.5.
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 have hidden higher-order dependencies? Stern-Brocot walk scores highest (1.08): its next value depends on the last two values, not just the last one. This detects Markov order that transition_entropy misses.
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 |
|---|---|---|
| Pulse-Width Modulation | waveform | 0.8800 |
| Morse Code | waveform | 0.8800 |
| Square Wave | waveform | 0.8800 |
| ··· | ||
| Prime Gaps | number_theory | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
| Pi Digits | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 1.0000 |
| L-System (Dragon Curve) | exotic | 0.9999 |
| Neural Net (Pruned 90%) | binary | 0.9945 |
| ··· | ||
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Stern-Brocot Walk | number_theory | 1.0772 |
| Arnold Cat Map | chaos | 0.7519 |
| Fibonacci Quasicrystal | number_theory | 0.5888 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.1037 |
| Noisy Period-2 | chaos | 0.1028 |
| Logistic r=3.9 (Near-Full Chaos) | chaos | 0.1023 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Pi Digits | number_theory | 0.0003 |
| glibc LCG | binary | 0.0003 |
| Source | Domain | Value |
|---|---|---|
| Wigner Semicircle | quantum | 1.0000 |
| Forest Fire | exotic | 1.0000 |
| Square Wave | waveform | 1.0000 |
| ··· | ||
| Stern-Brocot Walk | number_theory | 0.0080 |
| Triangle Wave | waveform | 0.0087 |
| Noisy Period-2 | chaos | 0.0308 |
| Source | Domain | Value |
|---|---|---|
| Wind Speed | climate | 1.0000 |
| Geometric Waiting Times | exotic | 1.0000 |
| BTC Volume | financial | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | 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.
# Topological Lens