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.
Variation in how tightly byte pairs align with the nearest G2 root. Bursty signals (rainfall, forest fire) score highest — some pairs align perfectly, others scatter wildly. Periodic orbits score 0.0 (every pair identical).
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.
Average alignment quality across all byte pairs. Constrained dynamics (Fibonacci word, logistic period-4) score highest; heavy-tailed distributions (rainfall) score lowest.
Uniformity of root usage across all 12 G2 roots.
Fisher kurtosis of angular alignments (projections onto root directions). G2's 12 roots at uniform 30-degree spacing produce unimodal angular distributions for most data.
kurtosis_angular minus the Fisher kurtosis of raw alignment values (Euclidean distance to nearest root; bimodal for G2 because root lengths come in 1:sqrt(3)). Large for data where G2's specific root-length ratio matters; near zero when it doesn't. Thomson control (equal-length roots) has no bimodality, so the raw kurtosis is unimodal and the differential collapses. The standalone raw-kurtosis and 12-root diversity-ratio components are atlas-excluded — diversity_ratio buckets almost every 2D-filling source to exactly 10/12 due to floating-point argmax ties on two cardinal-axis antipodal pairs, making it a degenerate 1-bit classifier redundant with normalized_entropy. Evolved via ShinkaEvolve atlas v1.
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 5.6056 |
| Accel Sit | motion | 4.6407 |
| Intermittency Type-III | chaos | 2.7376 |
| ··· | ||
| Aubry-André Critical | quantum | -37.1001 |
| Solar Flares Daily Peak | astro | -21.0532 |
| S&P 500 Returns | financial | -4.0114 |
| Source | Domain | Value |
|---|---|---|
| Morse Code | waveform | 1.0000 |
| Square Wave | waveform | 0.9979 |
| Mian-Chowla | number_theory | 0.9953 |
| ··· | ||
| Van der Pol Oscillator | exotic | 0.0662 |
| Pomeau-Manneville | chaos | 0.2062 |
| Baker Map | chaos | 0.2954 |
G2 is the complement to E8: where E8 examines 8-byte windows, G2 probes the shortest temporal structure (adjacent pairs). The kurtosis_differential metric exploits G2's unique two-length root structure (short:long = 1:sqrt(3)) — the only simple Lie algebra where root lengths differ. Thomson-6 control has equal-length directions, so the bimodality signal vanishes. The short_long_ratio metric remains unique to G2: no other geometry separates axial from diagonal pair correlations.