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.
How far the hyperbolic centroid drifts from the disk's origin. Collatz Gap Lengths (4.71) and Rainfall (4.69) push the centroid far off-center, reflecting asymmetric, heavy-tailed distributions. Prime Gaps (4.52) does the same. Constants score 0.0 (collapsed to a single point). L-System Dragon scores 0.0002 despite being complex -- its byte distribution is too symmetric to shift the centroid.
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.
Average hyperbolic distance between sampled point pairs. L-System Dragon (6.72) and Rule 30 (6.65) have the widest spread -- their embedded points scatter across the disk far from each other. Logistic Period-2 scores 0.0 because its two-valued signal collapses to a single pair of points in the embedding.
| Source | Domain | Value |
|---|---|---|
| Collatz Gap Lengths | number_theory | 4.7089 |
| Rainfall (ORD Hourly) | climate | 4.6880 |
| Prime Gaps | number_theory | 4.5179 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| L-System (Dragon Curve) | exotic | 0.0002 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 4.7174 |
| Pulse-Width Modulation | waveform | 4.7174 |
| Morse Code | waveform | 4.7174 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Bearing Inner | bearing | 0.2166 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 6.7168 |
| Rule 30 | exotic | 6.6536 |
| Rule 110 | exotic | 6.4050 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
Hyperbolic geometry is most useful for separating hierarchical data from flat or periodic data. The centroid_offset metric uniquely identifies signals with heavy tails or asymmetric distributions -- Collatz gaps, rainfall, and prime gaps form a tight cluster at the top that no other geometry highlights the same way. Periodic signals collapse to near-zero on all three metrics.