Whether 3-byte patterns prefer icosahedral directions.
Projects each triple of consecutive bytes onto the 30 roots of H3 — the vertices of an icosidodecahedron. These roots split into 6 axis-aligned vectors (permutations of (1,0,0)) and 24 "golden" vectors involving the golden ratio (even permutations of (1/2, phi/2, 1/2phi)). H3 is the symmetry group of the icosahedron, featuring 5-fold rotational symmetry that no crystallographic lattice can achieve. The geometry asks whether the data's 3-byte structure has any preference for these non-crystallographic directions.
Nearest-neighbor enrichment: how much more likely are consecutive windows to share a root than expected by chance. Measures temporal locality in root space. Evolved via ShinkaEvolve.
Autocorrelation of the root assignment sequence. High for smooth signals where 3-byte windows change gradually; low for noise and chaos.
How often the root-space trajectory returns near its starting point over various time horizons. Captures periodicity and recurrence in 3D root coordinates.
Uniformity of root usage across all 30 H3 roots. Periodic orbits score near 0 (all weight on one root); high-entropy sources spread across many roots.
Fraction of the 30 H3 roots actually used. Complements normalized_entropy: diversity_ratio measures how many roots appear at all, while entropy measures how uniformly they are used.
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 5.8535 |
| Fibonacci Word | exotic | 3.9837 |
| English Literature | speech | 2.6667 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.0000 |
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Ikeda Map | chaos | 0.6953 |
| Henon Map | chaos | 0.6843 |
| Logistic Chaos | chaos | 0.6800 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -0.0000 |
| Constant 0xFF | noise | -0.0000 |
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 0.6408 |
| Fibonacci Word | exotic | 0.3077 |
| Mertens Function | number_theory | 0.2663 |
| ··· | ||
| Henon Map | chaos | 0.0000 |
| Tent Map | chaos | 0.0000 |
| Lorenz Attractor | chaos | 0.0000 |
H3 is the only geometry in the framework probing 5-fold icosahedral symmetry at the 3-byte scale, between G2 (pairs) and D4/H4 (quadruples). The evolved metrics (nn_enrichment, temporal_coherence, path_closure) shift focus from static root statistics to temporal dynamics in root space — how the signal's trajectory through the 30 icosahedral directions evolves over time.