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.
Fisher kurtosis of raw alignment values (Euclidean distance to nearest root). G2's bimodal root lengths (1 vs sqrt(3)) make raw alignments bimodal — negative kurtosis.
kurtosis_angular minus kurtosis_raw. Large for data where G2's specific root-length ratio matters; near zero when it doesn't. This is the key D1 metric: Thomson control (equal-length roots) has no bimodality, so kurtosis_raw is unimodal and the differential collapses. Evolved via ShinkaEvolve atlas v1.
Fraction of the 12 G2 roots actually used. High diversity means the signal's byte pairs explore all hexagonal directions; low means they are confined to a subset.
| Source | Domain | Value |
|---|---|---|
| Fibonacci Word | exotic | 1.3953 |
| Logistic r=3.5 (Period-4) | chaos | 1.3851 |
| Random Telegraph | exotic | 1.3775 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.3067 |
| Forest Fire | exotic | 0.5149 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 1.3721 |
| Forest Fire | exotic | 1.2745 |
| Speech "Five" | speech | 1.2155 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 611.0234 |
| Accel Sit | motion | 294.9696 |
| OpenBSD ELF x86-64 | binary | 111.6549 |
| ··· | ||
| Logistic r=3.5 (Period-4) | chaos | -2.0000 |
| Morse Code | waveform | -1.9994 |
| Square Wave | waveform | -1.9962 |
| Source | Domain | Value |
|---|---|---|
| Accel Sit | motion | 4.6407 |
| Rainfall (ORD Hourly) | climate | 2.7264 |
| Gaussian Collatz Orbit | number_theory | 1.7508 |
| ··· | ||
| Pomeau-Manneville | chaos | -5.7322 |
| Geomagnetic ap Index | geophysics | -2.8922 |
| Ambient Microseism | geophysics | -1.0210 |
| Source | Domain | Value |
|---|---|---|
| Dice Rolls | exotic | 0.8786 |
| AES Encrypted | binary | 0.8777 |
| Gzip (level 9) | binary | 0.8777 |
| ··· | ||
| Constant 0xFF | noise | -0.0000 |
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Morse Code | waveform | 0.1116 |
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 1.0000 |
| Morse Code | waveform | 1.0000 |
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| ··· | ||
| Baker Map | chaos | 0.2956 |
| Tent Map | chaos | 0.2991 |
| Bernoulli Shift | chaos | 0.3039 |
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.