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.
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 | Origin | Value |
|---|---|---|
| Sine Map (Feigenbaum) | iterated-maps | 0.7000 |
| Quartic Map (Feigenbaum) | iterated-maps | 0.6750 |
| Logistic Edge-of-Chaos | iterated-maps | 0.6687 |
| ··· | ||
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Circle Map Quasiperiodic | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Euler Totient Ratio | number-theory | 1.0000 |
| Noisy Period-2 | iterated-maps | 1.0000 |
| Aliquot Orbit Lengths | number-theory | 0.9995 |
| ··· | ||
| Sawtooth Wave | signal-synthesis | 0.0000 |
| Period-Doubled Wave | signal-synthesis | 0.0000 |
| Sine Wave | signal-synthesis | 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. lattice_closure and edge_walk_fraction shift focus from static root statistics to dynamics in root space — how the signal's trajectory through the 600-cell evolves over time.