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.
Temporal coherence of root assignment across overlapping 4-byte windows. Measures whether the root trajectory has predictable structure rather than random jumps. Higher for signals with smooth dynamics.
Asymmetry between the 16 edge-neighbors per root in the D4 graph. Measures whether transitions favor certain neighbor directions over others. High for signals with directional bias in 4D root space.
Combined structural regularity of root usage patterns. Captures how "crystalline" the data's 4-byte structure is in D4 coordinates.
Projects root-to-root transition dynamics onto D4's edge-adjacency eigenspaces {-8, 0, 16}, returns weighted (E₁₆ - E₋₈) / E_total. Thomson control points have no edges, so this eigenspace decomposition doesn't exist — giving strong D1 drop. Evolved via ShinkaEvolve atlas v1.
Fraction of the 24 roots actually used. High diversity means the data explores the full D4 geometry; low diversity means it's confined to a subspace.
Shannon entropy of root usage, normalized by maximum. Uniform root distributions score high; periodic orbits score 0.
| Source | Domain | Value |
|---|---|---|
| Langton's Ant | exotic | 0.7583 |
| Rudin-Shapiro | number_theory | 0.7570 |
| Sawtooth Wave | waveform | 0.7409 |
| ··· | ||
| Morse Code | waveform | 0.0000 |
| Square Wave | waveform | 0.0074 |
| Mian-Chowla | number_theory | 0.0112 |
| Source | Domain | Value |
|---|---|---|
| Ramanujan Tau | number_theory | 2.5205 |
| ChaCha20 | binary | 2.5026 |
| Golden Ratio Digits | number_theory | 2.4969 |
| ··· | ||
| Morse Code | waveform | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Earthquake Magnitudes | geophysics | 0.2483 |
| Poisson Spacings | quantum | 0.2480 |
| Ramanujan Tau | number_theory | 0.2480 |
| ··· | ||
| Quartic Map (Feigenbaum) | chaos | -0.0417 |
| Logistic Edge-of-Chaos | chaos | -0.0417 |
| Logistic r=3.5 (Period-4) | chaos | -0.0417 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 2.0000 |
| Champernowne | number_theory | 1.0513 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.8100 |
| ··· | ||
| Stern-Brocot Walk | number_theory | -1.4754 |
| Penrose Substitution | exotic | -0.8326 |
| Fibonacci Word | exotic | -0.8320 |
D4 triality is the only Lie algebra with an order-3 automorphism cyclically permuting three representations of Spin(8). The spectral_transition metric exploits D4's unique edge-adjacency spectrum: the eigenvalues {-8, 0, 16} are algebraic properties of the 24-root graph that no generic point arrangement shares. D4's 4-byte window size complements G2 (pairs) and E8 (octets), catching intermediate-scale correlations with crystallographic structure.