The large-scale shape of the signal's state space.
Delay-embeds the signal into 5D, builds a k-nearest-neighbor graph on the resulting point cloud, and measures three properties of the graph that are invariants of the "Cayley graph" of the underlying dynamics: how curved it is (hyperbolicity), how fast it grows (dimension), and how connected it is (spectral gap).
How does the number of points within graph distance r scale? β ≈ 1 means the dynamics live on a curve (1D). β ≈ 2 means area-filling (2D manifold). β > 2 means volume-filling or branching. BTC returns score 2.07 (the highest in the atlas — financial returns fill a 2D manifold in delay space). Lotka-Volterra scores 0.56 (its limit cycles live on a 1D curve). DNA scores moderately (1.5-1.7), consistent with a branching structure in sequence space.
Gromov δ-hyperbolicity, normalized by diameter. δ ≈ 0 means the graph is tree-like (hyperbolic, negative curvature). δ/diam ≈ 0.25 means flat (Euclidean). Logistic chaos (0.26) and DNA (0.26) are the most "flat" — their delay embeddings fill space uniformly, without tree-like branching. Periodic orbits score 0.0 (trivially tree-like — just a cycle).
Fiedler eigenvalue of the graph Laplacian: how well-connected is the state space? Large gap means rapid mixing (the dynamics explore the full space quickly). BTC returns (0.035) and neural net dense (0.029) have the largest gaps — both are highly mixing processes. Symbolic Henon scores 0.0 (its state space has disconnected components in the graph).
Average fraction of variance explained by the first principal component in each k-NN neighborhood. Logistic Chaos (0.99) and Bernoulli Shift (0.99) score highest — their delay embeddings form locally 1D manifolds (the trajectory looks like a curve at every point). Fibonacci Word and Thue-Morse score 0.33 (= 1/embed_dim, isotropic — neighbors are equally spread in all directions). This measures whether the attractor is locally a curve (high) or a cloud (low). Evolved via ShinkaEvolve.
Normalized radius at which the growth exponent saturates. DNA Dog, Rule 110, and Thue-Morse all hit 1.0 (growth saturates immediately — the graph fills its available space at the smallest scale). BTC Returns (0.008) and financial returns score near zero (growth continues to increase across the full range of graph distances — the state space is large relative to the trajectory).
| Source | Origin | Value |
|---|---|---|
| Padovan Word | symbolic-dynamics | 0.2656 |
| Moebius Function | number-theory | 0.2628 |
| Look-and-Say | symbolic-dynamics | 0.2628 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Mian-Chowla | number-theory | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| NASDAQ Returns | financial | 2.1044 |
| BTC Returns | financial | 2.0778 |
| Nikkei Returns | financial | 2.0607 |
| ··· | ||
| Logistic Edge-of-Chaos | iterated-maps | 0.3907 |
| Phi-Squared β-Shift | symbolic-dynamics | 0.6794 |
| Quartic Map (Feigenbaum) | iterated-maps | 0.6832 |
| Source | Origin | Value |
|---|---|---|
| Phyllotaxis | number-theory | 1.0000 |
| Circle Map Quasiperiodic | iterated-maps | 1.0000 |
| LFSR (16-bit) | algorithmic-bytes | 1.0000 |
| ··· | ||
| Tribonacci Word | symbolic-dynamics | 0.3333 |
| Penrose Substitution | symbolic-dynamics | 0.3333 |
| Golden-Mean β-Shift | symbolic-dynamics | 0.3333 |
| Source | Origin | Value |
|---|---|---|
| DNA Dog | genomic | 1.0000 |
| DNA Human | genomic | 1.0000 |
| DNA Chimp | genomic | 1.0000 |
| ··· | ||
| S&P 500 Returns | financial | 0.0080 |
| NASDAQ Returns | financial | 0.0080 |
| BTC Returns | financial | 0.0080 |
| Source | Origin | Value |
|---|---|---|
| NASDAQ Returns | financial | 0.0377 |
| BTC Returns | financial | 0.0360 |
| Nikkei Returns | financial | 0.0352 |
| ··· | ||
| Rule 110 | symbolic-dynamics | 0.0000 |
| Logistic Edge-of-Chaos | iterated-maps | 0.0000 |
| Pell Word | symbolic-dynamics | 0.0000 |
Cayley's spectral_gap was a key discriminator in the negative re-evaluation study: it helped reclassify Arnold cat map and GARCH as positive detections. Growth exponent provides an intrinsic dimension estimate that complements the extrinsic dimension from fractal geometries. In the atlas, Cayley separates the distributional view's C3 (low growth, tree-like: symbolic dynamics and periodic orbits) from C2 (high growth, flat: continuous chaos and noise).