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).
| Source | Domain | Value |
|---|---|---|
| Logistic Chaos | chaos | 0.2611 |
| DNA Centromere | bio | 0.2557 |
| DNA Plasmodium | bio | 0.2557 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| BTC Returns | financial | 2.0668 |
| Rainfall (ORD Hourly) | climate | 1.9916 |
| Neural Net (Dense) | binary | 1.9710 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Lotka-Volterra | bio | 0.5559 |
| Source | Domain | Value |
|---|---|---|
| DNA Plasmodium | bio | 1.0000 |
| DNA Thermus | bio | 1.0000 |
| DNA Centromere | bio | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| BTC Returns | financial | 0.0080 |
| Source | Domain | Value |
|---|---|---|
| BTC Returns | financial | 0.0354 |
| Neural Net (Dense) | binary | 0.0288 |
| Gaussian Noise | noise | 0.0287 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Symbolic Henon | exotic | 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).