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. 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). This is a directional statistic: 1.0 means the data has a single preferred direction, 0.0 means it covers the sphere uniformly.
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.
| Source | Domain | Value |
|---|---|---|
| Rule 30 | exotic | 1.0000 |
| Wigner Semicircle | quantum | 0.8824 |
| Forest Fire | exotic | 0.8290 |
| ··· | ||
| Morse Code | waveform | -1.0000 |
| Pulse-Width Modulation | waveform | -0.8000 |
| ECG Fusion | medical | -0.6043 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.2500 |
| Square Wave | waveform | 0.2500 |
| Rule 30 | exotic | 0.2500 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 1.0000 |
| Collatz Gap Lengths | number_theory | 0.9999 |
| Rainfall (ORD Hourly) | climate | 0.9988 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| L-System (Dragon Curve) | exotic | 0.0006 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.5001 |
| Thue-Morse | exotic | 0.3232 |
| Ikeda Map | chaos | 0.2949 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -1.0000 |
| Double Pendulum | motion | -0.9949 |
| Van der Pol Oscillator | exotic | -0.9948 |
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.