Whether the signal's values cluster around a preferred direction.
Maps each pair of consecutive values to a point on the 2-sphere via spherical coordinates: the first value becomes the polar angle (north-south), the second becomes the azimuthal angle (around the equator). The resulting point cloud on the sphere reveals directional bias that a histogram would miss — two signals with identical value distributions can have completely different spherical profiles if their consecutive-pair correlations differ.
Standard deviation of the angles between each point and the cloud's mean direction. Thue-Morse, L-System Dragon, and Rule 30 all score 1.57 (maximum spread — points uniformly scattered, no preferred direction). Constants score 0.0 (all points at one pole). This is the spherical analogue of standard deviation.
Resultant length of the mean direction vector: how tightly do the points cluster? 1.0 means all points at the same location (Logistic period-2, Collatz gap lengths, Rainfall). Near 0.0 means perfectly diffuse (Thue-Morse at 0.0001). A signal can have high entropy in the Torus geometry but high concentration on the sphere if its consecutive pairs always point in the same angular direction.
How evenly are points split between the northern and southern hemispheres? 1.0 means perfect 50/50 (Thue-Morse, L-System Dragon, Clipped Sine). 0.0 means all points on one side (constants, logistic period-2). This detects asymmetry in the polar angle distribution — signals that spend more time in one "half" of their range.
Gini coefficient of the power spectrum of the sequence of (theta, phi) angles on the sphere. Near 1.0 when the angular trajectory is dominated by a handful of Fourier modes (sparse spectrum); near 0 when spectral power is spread uniformly (white spectrum). Smooth and periodic signals saturate at ~1.0: Sine Wave (0.998), Triangle Wave (0.998), Damped Pendulum (0.998), plus the period-locked logistic orbits and fBm Persistent. PRNGs and uniform byte sources bottom out at ~0.33 (RANDU, AES, MINSTD, Wichmann-Hill, BSL Residues, Arnold Cat Map, White Noise — all within 0.005 of each other). Constants and Logistic Period-2 are exact 0.0 (degenerate spectrum on the constant trajectory). Correlates +0.93 with Gottwald-Melbourne:radial_spectral_structure and -0.87 with Spectral Analysis:spectral_entropy — it is the spherical-trajectory analogue of the global spectral-concentration axis.
| Source | Origin | Value |
|---|---|---|
| Thue-Morse | symbolic-dynamics | 1.5708 |
| L-System (Dragon Curve) | symbolic-dynamics | 1.5708 |
| Kolakoski Sequence | symbolic-dynamics | 1.5708 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Aubry-André Critical | random-matrix-quantum | 0.0453 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Aubry-André Critical | random-matrix-quantum | 0.9992 |
| Rainfall (ORD Hourly) | atmospheric | 0.9987 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | 0.0001 |
| L-System (Dragon Curve) | symbolic-dynamics | 0.0001 |
| Kolakoski Sequence | symbolic-dynamics | 0.0015 |
| Source | Origin | Value |
|---|---|---|
| Gray Code Counter | symbolic-dynamics | 1.0000 |
| Riemann-Hardy-Littlewood | number-theory | 0.9999 |
| Thue-Morse | symbolic-dynamics | 0.9999 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic Edge-of-Chaos | iterated-maps | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
Spherical geometry separates sources by their directional structure in a way that scalar statistics cannot. Two signals with the same mean and variance can have opposite mean_z values if one is right-skewed and the other is left-skewed. In the atlas, the concentration axis separates noise-like sources (diffuse, concentration near 0) from spike-dominated sources (concentrated, Collatz gaps and rainfall near 1.0). The hemisphere_balance axis is orthogonal to this, catching symmetry vs. asymmetry regardless of concentration.