Whether the data has directional structure that drifts over time.
Maps each byte triple to a point on the sphere (first two bytes give polar angle and azimuth) plus a height on the real line (third byte). This is the Thurston geometry of spherical layers: data with stable directional concentration has high sphere_concentration, while the height coordinate tracks temporal drift in the third-byte channel.
Norm of the mean direction vector on S² — the von-Mises-Fisher resultant length. Logistic period-3 and Collatz gap lengths score 1.0 (all points cluster at a single direction on the sphere). L-System Dragon scores 0.0006 (nearly uniform coverage of the sphere — the binary symbolic dynamics maps to antipodal directions that cancel). Rainfall also scores 1.0 (its near-zero values all map to the same latitude). Measures DIRECTIONAL bias: 1.0 means one preferred pole, 0.0 means uniform OR antipodally cancelled.
Largest eigenvalue of the normalized scatter matrix Σpᵢpᵢᵀ/N. 1/3 ≈ uniform on the sphere, 1/2 ≈ concentrated on a great circle, → 1 concentrated on an axis. The Bingham distribution is the canonical model for axial data on S². Independent of sphere_concentration — Bingham measures AXIAL concentration regardless of direction, so a source whose points cluster along a single axis but split evenly between the two poles scores 1.0 here while sphere_concentration reads 0. Constants, Thue-Morse, Fibonacci Word, and Kolakoski Sequence saturate at 1.0; Wigner Semicircle (0.39) and Ikeda Map (0.45) sit at the diffuse end.
Mean geodesic distance arccos(pᵢ·pᵢ₊₁) between consecutive sphere points — the actual sphere metric, not Euclidean chord length. Logistic Period-2 (3.14 ≈ π) tops the list because consecutive values map to antipodal points (the diameter is the geodesic max). Period-4 (2.67), Logistic Edge-of-Chaos (2.55), and Noisy Period-2 (2.49) follow. Primes (0.001), Partition Function (0.001), Period-3, and constants collapse to 0 — their consecutive values barely move on the sphere. Distinct from bingham_concentration: that measures global axial structure; this measures step-by-step movement along the sphere.
Difference between final and initial height values. Rule 30 scores +1.0 (maximal upward drift) and Morse code scores -1.0 (maximal downward drift). ECG fusion scores -0.60. This captures systematic trends in the third-byte channel that the spherical components do not see.
Correlation between the z-coordinate on S² and the height in the ℝ component. L-System Dragon (0.50) has the strongest positive correlation: when its sphere position moves poleward, its height increases. Logistic period-2 (-1.0) has perfect negative correlation — its alternating values create a strict z-height anti-relationship. Double pendulum (-0.99) and Van der Pol (-0.99) also show strong negative correlation, reflecting their oscillatory dynamics coupling the spherical and linear components.
Variance of the height (ℝ) component. L-System Dragon, Square Wave, and Rule 30 all score 0.25 (maximum for binary data — the height alternates across its full range). Constants score 0.0 (fixed height). This measures how actively the linear component varies, independent of the spherical dynamics.
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 1.0000 |
| Penrose Substitution | exotic | 1.0000 |
| Kolakoski Sequence | exotic | 1.0000 |
| ··· | ||
| Wigner Semicircle | quantum | 0.3922 |
| Ramanujan Tau | number_theory | 0.4100 |
| Ikeda Map | chaos | 0.4500 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 3.1416 |
| Logistic r=3.5 (Period-4) | chaos | 2.6714 |
| Sine Map (Feigenbaum) | chaos | 2.5537 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| OTOC Growth | quantum | 0.0020 |
| Takagi Function | exotic | 0.0068 |
| Source | Domain | Value |
|---|---|---|
| Rule 30 | exotic | 1.0000 |
| Critical Transition (Fold) | chaos | 0.9718 |
| Forest Fire | exotic | 0.8272 |
| ··· | ||
| Morse Code | waveform | -1.0000 |
| Pulse-Width Modulation | waveform | -0.8000 |
| ECG Fusion | medical | -0.6043 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 1.0000 |
| Aubry-André Critical | quantum | 0.9994 |
| Rainfall (ORD Hourly) | climate | 0.9988 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0002 |
| L-System (Dragon Curve) | exotic | 0.0006 |
| De Bruijn Sequence | number_theory | 0.0013 |
S² x R is the only geometry that decomposes the signal into a directional and a scalar component simultaneously. The sphere_height_corr metric captures coupling between these two degrees of freedom: positive correlation means the signal's direction and amplitude co-vary; negative correlation means they oppose. This is distinct from what Heisenberg measures (pure correlation twist) and from what Sol measures (exponential anisotropy). In the symmetry view, sphere_concentration separates concentrated signals (periodic, near-constant) from diffuse ones (chaos, symbolic dynamics), while sphere_height_corr adds a second axis that distinguishes coupled from decoupled dynamics.