Whether your signal's geometry is locked in place or free to deform -- the difference between a crystal and a liquid.
Embeds byte triples into the 3D Poincaré ball, forms tetrahedra, and then asks: if you shake the data slightly, does the hyperbolic geometry change? Mostow's rigidity theorem says that in dimension 3 and above, the shape of a hyperbolic manifold is completely determined by its topology -- there are no continuous deformations. This geometry tests the discrete analog: whether small perturbations (5% Gaussian noise) alter the distance structure, volume distribution, and spectral properties.
Correlation between the original and perturbed pairwise distance matrices. Rule 30 (0.985) and Symbolic Lorenz (0.984) are nearly rigid: shaking the input barely changes the distances. This means their structure is determined by the combinatorial pattern, not the exact byte values. Constants score 0.0 (no distances to correlate).
Shannon entropy of the hyperbolic tetrahedron volume distribution. Beta Noise (4.52), XorShift32 (4.50), and Dice Rolls (4.50) produce the most varied volumes -- their points fill the ball uniformly, creating tetrahedra of many different sizes. Logistic Period-2 (0.0) produces identical tetrahedra.
How stable is the total hyperbolic volume under perturbation? AES Encrypted (0.97) and Gzip (0.96) are volume-rigid: noise barely changes their total volume. Lorenz and Rössler Attractors score 0.0 -- their attractor geometry is sensitive to perturbation, the volumes shift substantially.
Ratio of 5th-percentile to 95th-percentile pairwise distances. Measures the gap between thin and thick parts of the geometry. Logistic Period-2 (1.0) has perfectly uniform spacing. Pulse-Width Mod (0.91) is nearly uniform. Low values mean the signal has both tightly clustered and widely separated regions.
Correlation between Laplacian eigenvalues before and after perturbation. L-System Dragon (1.0), Morse Code (1.0), and Symbolic Henon (1.0) have perfectly stable spectra -- their graph connectivity pattern is insensitive to small perturbations. Logistic Period-3 (0.33) has a fragile spectrum.
Average angular change between consecutive tetrahedra in the hyperbolic embedding. Logistic period-3 (1.0) and constants (1.0) have maximal turn angles (the trajectory makes sharp turns between tetrahedra). Beta Noise (0.04) and Logistic Chaos (0.07) have near-zero turn angles (smooth, gradual trajectory through hyperbolic space). This measures the "jerkiness" of the hyperbolic path.
| Source | Domain | Value |
|---|---|---|
| Rule 30 | exotic | 0.9848 |
| Symbolic Lorenz | exotic | 0.9841 |
| Rule 110 | exotic | 0.9839 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Collatz Gap Lengths | number_theory | 0.0194 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Square Wave | waveform | 0.9801 |
| Pulse-Width Modulation | waveform | 0.9145 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Forest Fire | exotic | 0.0007 |
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 1.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 1.0000 |
| Square Wave | waveform | 0.9893 |
| ··· | ||
| Beta Noise | noise | 0.0409 |
| Continued Fractions | number_theory | 0.0631 |
| Logistic Chaos | chaos | 0.0651 |
| Source | Domain | Value |
|---|---|---|
| Morse Code | waveform | 1.0000 |
| L-System (Dragon Curve) | exotic | 1.0000 |
| Symbolic Henon | exotic | 0.9999 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.3386 |
| Constant 0x00 | noise | 0.3656 |
| Constant 0xFF | noise | 0.4172 |
| Source | Domain | Value |
|---|---|---|
| Beta Noise | noise | 4.5196 |
| Dice Rolls | exotic | 4.4975 |
| Benford's Law | number_theory | 4.4928 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| AES Encrypted | binary | 0.9699 |
| Gzip (level 1) | binary | 0.9611 |
| Dice Rolls | exotic | 0.9610 |
| ··· | ||
| Lorenz Attractor | chaos | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
Mostow Rigidity uniquely separates signals whose geometric structure is topologically determined from those that are metrically flexible. The volume_rigidity metric cleanly splits encrypted/compressed data (rigid, 0.96+) from dynamical attractors (flexible, near 0.0) -- a distinction no other geometry captures. The distance_rigidity metric at the top is dominated by cellular automata and symbolic dynamics, signals whose structure is entirely combinatorial.