Hexagonal symmetry in consecutive byte pairs.
Projects each pair of adjacent bytes onto the 12 roots of G2 — 6 short roots at 60-degree intervals, and 6 long roots at 30-degree offsets with length proportional to the square root of 3. This 2D root system captures the simplest non-trivial Lie algebra structure, operating at the smallest possible window size. Because it works on pairs, it acts as a fast correlation probe: any directional preference in the (byte_t, byte_{t+1}) scatterplot shows up as root concentration.
Fraction of windows assigned to the 6 short roots (out of all 12). Binary-valued signals score 1.0 (exclusively short-root, coordinate-aligned); baker map prefers long roots (30-degree offset). Separates axial from diagonal pair correlations.
Fisher kurtosis of the |cos| alignment between each UNIT-normalized pair vector and its nearest of the 12 root directions. Pure angles: every pair contributes equally regardless of magnitude, so heavy-tailed marginals cannot move it. High when pair angles pile onto specific root directions (slow coherent oscillation, quasiperiodic pair structure — Fibonacci Tight-Binding fires at shuffle-null z=+12.6 with ac1≈0); negative when angles spread more uniformly than an i.i.d. re-pairing of the same values (Logistic Chaos, z=−6.9). Replaces kurtosis_differential (evolved, ShinkaEvolve v1), REFUTED 2026-06-09: both of its components carried the un-normalized pair magnitude linearly, making every fire a marginal-kurtosis tail-pairing lottery — original values sat inside their own 40-permutation shuffle null (z ≤ 1.2) on all top sources. The earlier two-length "bimodality" story died with it; this metric uses only the 12 directions. Same disease killed kurtosis_angular in the 2026-05-21 prune.
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 326.5138 |
| Square Wave | signal-synthesis | 286.3648 |
| Mian-Chowla | number-theory | 102.1322 |
| ··· | ||
| Logistic r=3.5 (Period-4) | iterated-maps | -2.0000 |
| Liouville Function | number-theory | -1.9996 |
| Rule 30 | symbolic-dynamics | -1.9987 |
| Source | Origin | Value |
|---|---|---|
| Morse Code | signal-synthesis | 1.0000 |
| Square Wave | signal-synthesis | 0.9979 |
| Mian-Chowla | number-theory | 0.9953 |
| ··· | ||
| Van der Pol Oscillator | continuous-flows | 0.0662 |
| Pomeau-Manneville | iterated-maps | 0.2062 |
| Baker Map | iterated-maps | 0.2948 |
G2 is the complement to E8: where E8 examines 8-byte windows, G2 probes the shortest temporal structure (adjacent pairs). Both surviving metrics read the direction structure of the (byte_t, byte_{t+1}) scatterplot against G2's 30-degree-spaced root fan: short_long_ratio asks which sectors pairs fall in (axial vs offset — no other geometry separates these), pair_angle_kurtosis asks how concentrated they are on root directions at all. Scramble-faithfulness 2026-06-09: pair_angle_kurtosis 0.49/0.65, short_long_ratio 0.38/0.39 — both faithful to the actual G2 arrangement vs random 12-direction fans.