How much your signal's structure looks like a tree -- branching, hierarchical, pushing toward the boundary of a curved disk.
Maps consecutive byte pairs into the Poincaré disk, a model of hyperbolic space where distances grow exponentially near the edge. Flat, uniform data spreads evenly across the disk. Hierarchical or heavy-tailed data gets shoved toward the boundary, because the disk has exponentially more room out there -- the same reason tree-like data embeds naturally in hyperbolic space. The geometry computes where the mass centroid sits, how far points are from the origin, and how far apart they are from each other under the curved metric.
Average distance of embedded points from the disk's origin. L-System Dragon (4.72), Pulse-Width Mod (4.72), and Morse Code (4.72) all push points to the boundary, where hyperbolic distances explode. Bearing Inner (0.22) stays close to the center. This metric separates signals with extreme byte-value concentrations from those with more centered distributions.
Ratio of local curvature estimates at different scales in the Poincaré disk embedding. Logistic Edge-of-Chaos (1470), μ-law Sine (392), and Banded Chaos (319) score highest — their tightly-clustered point clouds create extreme multi-scale curvature variation. Logistic Chaos (222), Tent Map (218), and Hénon (102) follow. DNA sequences and constants stay near 1.0 (flat curvature). This measures how much the hyperbolic curvature varies across the point cloud. Internally multiplies a capped k-nearest-neighbor scale ratio; the standalone ratio is atlas-excluded because the cap (added to keep the metric bounded under Phase-2 float precision) flattened its discrimination, but it remains useful as a component of curvature_structure.
Correlation between spatial proximity in the Poincaré disk and temporal proximity in the original signal. NYSE (0.75), Nikkei (0.74), and NASDAQ (0.74) score highest — temporally nearby points are also spatially close in hyperbolic space, reflecting persistent dynamics. Pomeau-Manneville (-0.55) scores most negative — temporal neighbors are spatially distant, the signature of intermittent switching between distant states.
Variance of the hyperbolic radius over time. Collatz Parity (18.9), Symbolic Henon (18.2), and Symbolic Lorenz (17.8) score highest — their binary dynamics create alternating near-center and near-boundary points. Constants and logistic period-2 score 0.0 (fixed radius). This captures how much the signal's "hierarchical depth" fluctuates over time.
| Source | Domain | Value |
|---|---|---|
| Sine Map (Feigenbaum) | chaos | 3276.8316 |
| Logistic Edge-of-Chaos | chaos | 1496.3932 |
| Quartic Map (Feigenbaum) | chaos | 959.2443 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Logistic r=3.5 (Period-4) | chaos | 1.0000 |
| Pell Word | exotic | 1.0683 |
| Source | Domain | Value |
|---|---|---|
| Fibonacci Word | exotic | 4.7174 |
| Mian-Chowla | number_theory | 4.7174 |
| Pell Word | exotic | 4.7174 |
| ··· | ||
| Intermittency Type-III | chaos | 0.1330 |
| MFPT Inner Loaded | bearing | 0.1499 |
| Intermittency Type-II | chaos | 0.1517 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.74 (Period-5 Window) | chaos | 0.9217 |
| Intermittency Type-III | chaos | 0.8765 |
| Gaussian Collatz Orbit | number_theory | 0.8467 |
| ··· | ||
| Pomeau-Manneville | chaos | -0.5625 |
| μ-law Sine | waveform | -0.4012 |
| Devil's Staircase | exotic | -0.3266 |
| Source | Domain | Value |
|---|---|---|
| Collatz Parity | number_theory | 18.9112 |
| Symbolic Henon | exotic | 18.2333 |
| Symbolic Lorenz | exotic | 17.7651 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| OTOC Growth | quantum | 0.0024 |
Hyperbolic geometry is most useful for separating hierarchical data from flat or periodic data. The spatio_temporal_corr metric uniquely identifies signals where temporal and geometric proximity coincide — financial returns cluster tightly, while intermittent dynamics show negative correlation. The curvature_structure metric provides a multi-scale geometric fingerprint that separates space-filling chaos from low-dimensional manifolds.