Non-crystallographic symmetry in 4-byte windows.
Projects each group of 4 consecutive bytes onto the 120 roots of H4 — the vertices of a 600-cell, the most complex regular polytope in 4D. The roots come in three families: 8 axis-aligned, 16 half-integer (all coordinates ±1/2), and 96 "golden" vectors built from even permutations of (0, 1/2, 1/2phi, phi/2). H4 is the largest non-crystallographic Coxeter group, governing the symmetry of 4D polytopes with icosahedral cross-sections.
Fraction of the 120 roots actually used. Sunspot, Kepler non-planet, and speech "zero" all score 0.233 (28 of 120 roots). Logistic period-2 uses a single root (0.008). With 120 roots available, the diversity ceiling is much lower than for E8 (240 roots) — most data concentrates on a small fraction of the 600-cell's directions.
Uniformity of root usage. BTC returns (0.640) and neural net dense weights (0.638) have the most uniform distributions — their high-entropy byte structure genuinely samples the 4D root system. Periodic orbits and Morse code score near 0.0. The entropy range (0 to 0.64) is wider than H3's, reflecting the larger root system's greater capacity to differentiate sources.
How closely does the trajectory through H4 root space return to its starting point? Fibonacci QC (1.0) and Logistic Period-2 (1.0) close perfectly. Random Steps (0.0) never returns. Evolved via ShinkaEvolve.
Fraction of consecutive root transitions that follow edges of the 600-cell graph. Fibonacci QC (1.0) and Logistic Period-2 (1.0) always follow edges. Random Steps (0.0) never does. High fractions mean the dynamics respect the polytope's adjacency structure. Evolved via ShinkaEvolve.
| Source | Domain | Value |
|---|---|---|
| Sine Map (Feigenbaum) | chaos | 0.7000 |
| Quartic Map (Feigenbaum) | chaos | 0.6750 |
| Logistic Edge-of-Chaos | chaos | 0.6687 |
| ··· | ||
| Thue-Morse | exotic | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Critical Circle Map | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Euler Totient Ratio | number_theory | 1.0000 |
| Logistic r=3.68 (Banded Chaos) | chaos | 1.0000 |
| Aliquot Orbit Lengths | number_theory | 0.9995 |
| ··· | ||
| Lorenz Attractor | chaos | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Sawtooth Wave | waveform | 0.0000 |
H4 shares its 4-byte window size with D4 Triality but probes a completely different symmetry: D4's 24 roots are crystallographic (they tile via lattice translations), while H4's 120 roots are non-crystallographic (they cannot). This means H4 detects structural preferences that D4 misses — specifically, whether the data's 4-byte patterns prefer golden-ratio-related directions. In practice, H4's two metrics provide a coarse separation: high normalized_entropy signals (financial, neural network weights, noise) vs. low (periodic, symbolic). Its unique role is completing the non-crystallographic Coxeter family alongside H3.