Exponential anisotropy — whether one direction stretches while another contracts.
Builds a path through Sol space using the group law (x1 + e^{z1} x2, y1 + e^{-z1} y2, z1 + z2). Each step from a triple of byte values moves through a geometry where the x-direction is exponentially inflated and the y-direction exponentially compressed by the z-coordinate. Data with directional bias accumulates a strong z-drift; isotropic data wanders near z=0. The Sol metric amplifies any asymmetry in the data's local structure into exponentially different path lengths.
Total distance traveled in the Sol metric. Symbolic Henon (30.2M) scores highest: its chaotic dynamics create large steps that the exponential Sol metric amplifies enormously. Fibonacci word (28.3M) is close behind — its substitution structure produces systematic z-accumulation that inflates path length. Logistic period-3 scores only 82,909 — its tight orbit keeps z near zero, suppressing the exponential amplification. The 3-order-of-magnitude range across non-degenerate sources makes this a powerful separator.
Final z-coordinate after traversing the signal. Clamped to [-5, 5] to keep the exponential factors finite. Stern-Brocot walk, El Centro earthquake, and forest fire all hit the +5 ceiling — they have systematic upward drift in their byte-triple structure. Prime gaps, divisor count, and Collatz gap lengths hit -5 (systematic downward drift). Z-drift is a signed metric: its polarity reveals which direction the data's asymmetry favors.
Variance of the z-coordinate along the path. Van der Pol (23.4) and Hilbert walk (23.1) score highest: their oscillatory dynamics push z back and forth across its full range. Rainfall (0.018) barely moves z at all — the exponential distribution's near-zero values produce tiny increments. High z_variance without high z_drift means the path oscillates in the exponential direction; high drift with low variance means it ramps monotonically.
| Source | Domain | Value |
|---|---|---|
| Symbolic Henon | exotic | 30240756.7878 |
| Fibonacci Word | exotic | 28311153.2548 |
| Collatz Parity | number_theory | 24020010.3910 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 82909.0120 |
| Source | Domain | Value |
|---|---|---|
| Stern-Brocot Walk | number_theory | 5.0000 |
| El Centro 1940 | geophysics | 5.0000 |
| Forest Fire | exotic | 5.0000 |
| ··· | ||
| Divisor Count | number_theory | -5.0000 |
| Prime Gaps | number_theory | -5.0000 |
| Collatz Gap Lengths | number_theory | -5.0000 |
| Source | Domain | Value |
|---|---|---|
| Van der Pol Oscillator | exotic | 23.4155 |
| Hilbert Walk | exotic | 23.1458 |
| Rudin-Shapiro | number_theory | 22.5120 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.0176 |
Sol is the only Thurston geometry where the metric itself is exponential — the other seven (Euclidean, spherical, hyperbolic, Nil, the two products, SL(2,R)) all have at most polynomial distortion. This makes Sol path_length the framework's most sensitive amplifier of structural asymmetry. In the symmetry view, Sol separates sources that have directional z-bias (natural signals that saturate z_drift at ±5) from symmetric sources (chaos, noise) that wander without trend. The three metrics together form a concise dynamical portrait: drift tells you the direction, variance tells you the oscillation, and path_length integrates both through the exponential lens.