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.
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.
Triality-folded polarity of edge orthogonality: measures whether the three triality images of each root prefer to land on the same D4 edge or on orthogonal ones. Rudin-Shapiro (0.755), Sawtooth Wave (0.741), and Triangle Wave (0.740) saturate near the top; Morse Code and constants collapse to 0. Renamed from diversity_ratio (which was a degenerate alphabet probe); the polarity reading exploits D4's order-3 automorphism specifically rather than counting used roots. Renamed 2026-05-25.
Consistency of the Gram matrix of consecutive 4-byte windows under D4's triality action — how much the inner-product structure between consecutive windows is preserved when each is replaced by its triality image. Ramanujan Tau (2.52), ChaCha20 (2.50), and Golden Ratio Digits (2.50) lead — their algebraic / high-entropy structure is invariant under the triality permutation; Morse Code (0.0) collapses. F-stat 10.07, top quartile of atlas discriminators. Renamed from structural_coherence; the Gram-matrix reading is mechanism-explicit. Renamed 2026-05-25.
| Source | Origin | Value |
|---|---|---|
| Rudin-Shapiro | symbolic-dynamics | 0.7570 |
| Sawtooth Wave | signal-synthesis | 0.7409 |
| Triangle Wave | signal-synthesis | 0.7398 |
| ··· | ||
| Morse Code | signal-synthesis | 0.0000 |
| Square Wave | signal-synthesis | 0.0074 |
| Mian-Chowla | number-theory | 0.0112 |
| Source | Origin | Value |
|---|---|---|
| Ramanujan Tau | number-theory | 2.5203 |
| ChaCha20 | algorithmic-bytes | 2.5026 |
| Golden Ratio Digits | number-theory | 2.4969 |
| ··· | ||
| Morse Code | signal-synthesis | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 2.0000 |
| Champernowne | number-theory | 1.0513 |
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 0.8100 |
| ··· | ||
| Stern-Brocot Walk | number-theory | -1.4754 |
| Penrose Substitution | symbolic-dynamics | -0.8326 |
| Fibonacci Word | symbolic-dynamics | -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. The two renamed metrics — edge_ortho_polarity and gram_consistency — make explicit that what D4 actually probes is the triality action on consecutive windows, not generic root-usage statistics. D4's 4-byte window size complements G2 (pairs) and E8 (octets), catching intermediate-scale correlations with crystallographic structure.