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.
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.
Number of distinct points in the S² × ℝ embedding (after deduplication). Not yet in the atlas — a diagnostic metric.
| 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 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Collatz Gap Lengths | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 1.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 1.0000 |
| Collatz Gap Lengths | number_theory | 0.9999 |
| ··· | ||
| L-System (Dragon Curve) | exotic | 0.0005 |
| De Bruijn Sequence | number_theory | 0.0017 |
| Rule 30 | exotic | 0.0057 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.4998 |
| Thue-Morse | exotic | 0.3264 |
| Ikeda Map | chaos | 0.2952 |
| ··· | ||
| 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.