Whether 4-byte structural patterns respect the Spin(8) triality symmetry.
Takes each group of 4 consecutive values and projects them onto the 24 roots of D4 — the vectors ±eᵢ ± eⱼ in 4D space. D4 is the only Lie algebra with triality: an order-3 automorphism that cyclically permutes three fundamentally different representations of Spin(8). The geometry asks: does the data's root usage look the same after applying this triality rotation? If so, the data has a structural symmetry that only D4 can detect.
How symmetric is the root distribution under the triality automorphism? 1.0 means the distribution is unchanged by the permutation. Critical circle map scores 0.63 (highest in the atlas): its dynamics at the golden-mean rotation number create 4-byte patterns with near-perfect triality. Circle map quasiperiodic (0.62) and phyllotaxis (0.62) are close behind — both driven by irrational rotations. White noise scores 0.49 (close to the theoretical expectation for uniform random root assignment). Logistic period-2 scores exactly 0.0: alternating between two values creates a degenerate root pattern with no triality.
What fraction of the 24 roots are actually used? Solar wind, speech, and PRNG outputs reach 0.50 (12 of 24 roots). Logistic period-4 uses just 1 root (0.04). High diversity means the data explores the full D4 geometry; low diversity means the dynamics are confined to a low-dimensional subspace of the root system.
Shannon entropy of the root usage distribution, normalized by the maximum. Nikkei returns (0.64) and DNA (0.64) have the most uniform root distributions — their 4-byte structure genuinely samples the full D4 polytope. Periodic orbits score 0.0 (all weight on one root).
How well do the 4-byte windows align with the nearest root? Langton's Ant (1.90) and circle map (1.89) produce windows that sit almost exactly on a root direction. Constants score 0.0 — degenerate windows that don't align with any root. This measures how "crystalline" the data's 4-byte structure is in D4 coordinates.
| Source | Domain | Value |
|---|---|---|
| Langton's Ant | exotic | 1.9013 |
| Circle Map Quasiperiodic | chaos | 1.8930 |
| Phyllotaxis | bio | 1.8930 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Morse Code | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Solar Wind IMF | astro | 0.5000 |
| Solar Wind Speed | astro | 0.5000 |
| Speech "Five" | speech | 0.5000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0417 |
| Source | Domain | Value |
|---|---|---|
| Nikkei Returns | financial | 0.6386 |
| DNA Thermus | bio | 0.6375 |
| DNA Phage Lambda | bio | 0.6261 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Critical Circle Map | chaos | 0.6339 |
| Circle Map Quasiperiodic | chaos | 0.6179 |
| Phyllotaxis | bio | 0.6178 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
D4 triality is encoding-invariant: the root projection depends on the relative order of the 4 bytes, not their absolute values. Its distinctive contribution to the ordinal view is detecting 4-byte rotational symmetry that the other ordinal geometries (which operate on pairs or 5-grams) miss. The critical circle map / quasiperiodic cluster at the top of the triality ranking connects to the golden ratio: irrational rotations produce 4-byte patterns that are maximally balanced across the triality orbits. This is a structural echo of the fact that D4 triality and quasicrystalline order are both connected to the exceptional algebra structure of Spin(8).