Whether hierarchical depth co-varies with a linear trend.
Maps each byte triple into the Poincare disk (first two bytes) crossed with the real line (third byte). The Poincare disk distorts distances: points near the boundary are exponentially far from the center in hyperbolic terms. Data that clusters near the disk's edge has large hyperbolic variance; the depth_height_corr metric then asks whether this hierarchical depth is coupled to the linear drift.
Variance of the hyperbolic distance from the disk's origin. ECG ventricular tachycardia (3.45), damped pendulum (3.35), and ECG supraventricular tachycardia (3.34) dominate. These signals oscillate between near-center and near-boundary in the Poincare disk — the heartbeat's QRS complex pushes points to the boundary, while the baseline returns them to center. Logistic period-2 scores near 0 (its two-value alternation maps to a fixed radius). The ECG dominance is notable: pathological heart rhythms explore the full radial range of the disk.
Mean Poincaré disk distance d(z, w) = 2·arctanh(|z − w| / |1 − z̄·w|) between consecutive points. The canonical Möbius-invariant hyperbolic metric, not a chord through the disk. Logistic Period-2 (9.43) and Period-4 (8.28) dominate because their alternating values map to widely separated disk points; Symbolic Hénon (7.73), Rule 110 (7.62), and Logistic Edge-of-Chaos (7.42) follow. Partition Function and Minkowski ?(x) (~0.005) sit near zero — these monotone number-theoretic sources trace tight neighborhoods in disk coordinates. Constants and Period-3 saturate at 0.
Mean angle-deficit area A = π − (α + β + γ) of consecutive disk triples (p_i, p_{i+1}, p_{i+2}), computed via the hyperbolic law of cosines on side lengths. By Gauss-Bonnet on a curvature-(−1) surface, the angle deficit IS the hyperbolic area, bounded in [0, π]. Rotation-invariant. Logistic Period-4 (2.37) tops the list — its 4-cycle traces non-degenerate triangles every step — followed by Logistic Edge-of-Chaos (2.18), DNA Centromere (2.05), Beta Noise (1.99), and DNA SARS-CoV-2 (1.91). Period-2 and Period-3 collapse to 0 because their cycles produce degenerate (collinear) triangles. Distinct from hyperbolic_step_dispersion: that metric tracks how far each step jumps, while triangle area tracks how much the path bends.
Correlation between hyperbolic distance from the origin and the height coordinate. Forest fire (0.68) and humidity (0.62) have the strongest positive correlation: when they move toward the boundary (high hierarchical depth), their height increases. Wind speed (-0.79) and wave height (-0.81) have strong negative correlation: high hierarchical depth coincides with decreasing height. Intermittent silence scores -0.83, the strongest negative value — its bursts of silence (near-center) alternate with active periods (boundary) in a way that anti-correlates with the height trend.
Rate of change of hyperbolic radius over time — how fast does the signal move toward or away from the disk boundary? BTC Close (0.14) and ETH/BTC (0.12) score highest — financial data makes rapid excursions between center and boundary. DNA and logistic period-2 score 0.0 (fixed radial position). This captures the temporal dynamics of hierarchical depth that the static hyperbolic_variance misses.
Autocorrelation of the hyperbolic radius sequence — does the signal's hierarchical depth persist over time? Devil's Staircase (4.98) and Champernowne (4.97) score highest — their radial positions are strongly correlated over time. L-System Dragon and Fibonacci Word score 0.0 (no radial memory — the depth changes randomly from step to step).
| Source | Domain | Value |
|---|---|---|
| BTC Close Price | financial | 0.1462 |
| Takagi Function | exotic | 0.1364 |
| Spectral Form Factor | quantum | 0.1332 |
| ··· | ||
| Fibonacci Word | exotic | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Forest Fire | exotic | 0.6801 |
| Tank Drain Cascade | exotic | 0.6782 |
| Spectral Form Factor | quantum | 0.6722 |
| ··· | ||
| Wave Height (Buoy) | geophysics | -0.8331 |
| Intermittent Silence | exotic | -0.8275 |
| Surface Wind (ORD 5-min) | climate | -0.7936 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 9.4349 |
| Logistic r=3.5 (Period-4) | chaos | 8.2805 |
| Symbolic Henon | exotic | 7.7290 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| OTOC Growth | quantum | 0.0057 |
| Spectral Form Factor | quantum | 0.0306 |
| Source | Domain | Value |
|---|---|---|
| Quartic Map (Feigenbaum) | chaos | 2.5129 |
| Logistic r=3.5 (Period-4) | chaos | 2.3647 |
| Sine Map (Feigenbaum) | chaos | 2.2428 |
| ··· | ||
| Pell Word | exotic | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| ECG Ventricular | medical | 3.4537 |
| Damped Pendulum | motion | 3.3435 |
| ECG Supraventr. | medical | 3.3422 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Penrose Substitution | exotic | 0.0000 |
| Pell Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Champernowne | number_theory | 4.9771 |
| Spectral Form Factor | quantum | 4.9732 |
| OTOC Growth | quantum | 4.9718 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Kolakoski Sequence | exotic | 0.0000 |
H² x R is the hyperbolic counterpart to S² x R: where S² x R decomposes into direction + scalar, H² x R decomposes into hierarchical depth + scalar. The hyperbolic_variance metric's ECG dominance (top 3 are all cardiac signals) suggests it captures a specific physical phenomenon: the heartbeat's voltage range maps naturally to radial excursion in the Poincare disk, and pathological rhythms (tachycardia) explore this range more than normal sinus rhythm. The depth_height_corr metric then adds temporal coupling: environmental signals (humidity, forest fire) show positive coupling while ocean/atmospheric signals (wind, waves) show negative.