The effective dimensionality your data's point cloud presents to a diffusing particle -- and whether that dimensionality follows known physical laws.
Delay-embeds the signal into 5 dimensions, builds an epsilon-neighborhood graph on the resulting point cloud, and computes the Laplacian spectrum. Two numbers come out: the spectral dimension (how fast heat spreads on the graph) and the Weyl exponent (how eigenvalues grow with index). For a regular d-dimensional lattice, the spectral dimension equals d and the Weyl exponent equals 2/d. Deviations from these relationships signal anomalous geometry -- fractals, clustered manifolds, or degenerate embeddings.
The effective dimensionality seen by diffusion on the data manifold. Logistic periodic orbits (Period-2, Period-3, Period-4) all score 2.004 -- their delay embeddings land on clean low-dimensional manifolds (circles, figure-eights) where heat diffuses in exactly 2 dimensions. Tidal Gauge (0.10) has nearly zero spectral dimension, meaning diffusion is almost completely blocked -- the point cloud is fragmented or one-dimensional. The gap between 2.0 and 0.1 is the gap between a smooth manifold and a dust-like point set.
The eigenvalue growth rate. Wigner Semicircle (2.15) and Circle Map QP (2.01) have exponents near 2, matching the Weyl law prediction for a 1D manifold. Henon-Heiles (1.97) is close behind. Periodic logistic maps score near zero because their spectra have too few distinct eigenvalues for a meaningful power-law fit. High Weyl exponent means the Laplacian spectrum is rich and well-structured.
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | chaos | 2.0045 |
| Logistic r=3.2 (Period-2) | chaos | 2.0044 |
| Logistic r=3.83 (Period-3 Window) | chaos | 2.0039 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Tidal Gauge (SF) | geophysics | 0.1034 |
| Source | Domain | Value |
|---|---|---|
| Wigner Semicircle | quantum | 2.1480 |
| Circle Map Quasiperiodic | chaos | 2.0066 |
| Hénon-Heiles | chaos | 1.9701 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
Spectral Graph is the atlas's cleanest test for manifold dimensionality. It separates smooth delay embeddings (spectral_dim near 2.0) from fragmented point clouds (near 0) with almost no overlap. The spectral_dim and weyl_exponent pair together provide a two-parameter characterization that other topological geometries lack -- Persistent Homology counts features, Cayley measures graph growth, but Spectral Graph directly measures how space "feels" to diffusion.