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.
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 directional bias means the path oscillates in the exponential direction.
Lag-1 autocorrelation of the per-step contribution to Sol path length (the local exponential-weighted increment in the z-direction). fBm Persistent (0.92), Perlin Noise (0.91), and Levy Flight (0.90) lead — their long-memory increments make consecutive Sol steps positively correlated. Pell Word (-0.23) and Logistic Period-2 (-0.20) sit at the antipersistent end (consecutive steps oppose). Replaces dz_persistence (autocorrelation of raw z-increments), which double-counted the exponential weighting; sol_step_persistence reports persistence of the actual path-length contributions. Replaced 2026-05-25 (z_drift dropped — its clamp at ±5 was a censoring artifact rather than structural signal).
| Source | Origin | Value |
|---|---|---|
| fBm (Persistent) | stochastic-process | 0.9179 |
| Perlin Noise | stochastic-process | 0.9127 |
| Levy Flight | stochastic-process | 0.9008 |
| ··· | ||
| Periodic Word | symbolic-dynamics | -0.2712 |
| Pell Word | symbolic-dynamics | -0.2310 |
| Logistic r=3.2 (Period-2) | iterated-maps | -0.2000 |
| Source | Origin | Value |
|---|---|---|
| Logistic Edge-of-Chaos | iterated-maps | 33301994.5346 |
| Sine Map (Feigenbaum) | iterated-maps | 32828213.2679 |
| Logistic r=3.5 (Period-4) | iterated-maps | 31729860.1845 |
| ··· | ||
| Stable Noise | stochastic-process | 320.1348 |
| Logistic r=3.2 (Period-2) | iterated-maps | 6350.4459 |
| Circle Map Quasiperiodic | iterated-maps | 8597.0078 |
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 with persistent exponential drift from oscillatory ones, while path_length integrates the magnitude. sol_step_persistence catches whether the exponential amplification builds coherently (persistent fBm) or self-cancels (alternating period-2).