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. Rainfall (1.37) scores highest: the heavy-tailed distribution creates some pairs that align perfectly (near-zero values) and others that scatter wildly (storm spikes). Forest fire (1.27) is similar — bursty dynamics with long quiet intervals. Logistic period-2 scores 0.0 (every pair is identical in normalized coordinates, so alignment is constant).
Fraction of windows assigned to the 6 short roots (out of all 12 roots). Morse code, logistic period-2, and square wave all score 1.0 (exclusively short-root alignment — their binary structure restricts byte pairs to directions that happen to coincide with the short roots). Baker map scores 0.29 (strong long-root preference). This distinguishes data whose pair correlations are axial (short roots = coordinate-aligned) from data with diagonal pair structure (long roots = 30-degree offset).
Average alignment quality across all byte pairs. Fibonacci word (1.40) and logistic period-4 (1.39) score highest: their constrained dynamics produce byte pairs that consistently point toward root directions. Rainfall (0.30) is the lowest non-degenerate score — the exponential distribution creates pairs that scatter across 2D without directional preference.
| Source | Domain | Value |
|---|---|---|
| Fibonacci Word | exotic | 1.3953 |
| Logistic r=3.5 (Period-4) | chaos | 1.3851 |
| Logistic r=3.83 (Period-3 Window) | chaos | 1.3768 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.3008 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 1.3727 |
| Forest Fire | exotic | 1.2745 |
| Speech "Five" | speech | 1.2155 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Sunspot Number | astro | 0.8333 |
| Accel Jog | motion | 0.8333 |
| Speech "Zero" | speech | 0.8333 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0833 |
| Source | Domain | Value |
|---|---|---|
| Dice Rolls | exotic | 0.8786 |
| Pi Digits | number_theory | 0.8781 |
| Bzip2 (level 1) | binary | 0.8778 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Morse Code | waveform | 1.0000 |
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Square Wave | waveform | 0.9979 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Baker Map | chaos | 0.2945 |
G2 is the complement to E8: where E8 examines long-range 8-byte windows, G2 probes the shortest possible temporal structure (adjacent pairs). The short_long_ratio metric is unique to G2 — no other geometry separates axial from diagonal pair correlations. In the symmetry view, G2 alignment_std contributes to separating bursty environmental signals (rainfall, forest fire) from periodic and quasiperiodic sources, a distinction the higher-dimensional root systems blur because they average over longer windows.