How constrained the data's 8-byte structure is relative to the densest sphere packing in 8 dimensions.
Takes each group of 8 consecutive bytes, normalizes them, and finds the closest of the 240 root vectors of E8. These roots come in two families: 112 "integer" roots (pairs of coordinate axes) and 128 "half-integer" roots (all coordinates ±1/2 with even sign parity). The geometry also snaps each 8-byte window to the nearest actual E8 lattice point via parity correction, measuring the residual distance. Data with algebraic constraints concentrates on a few roots; unconstrained data spreads evenly across all 240.
Average distance from each 8-byte window to its nearest E8 lattice point. Critical circle map scores 0.84 (farthest from the lattice — its dynamics explore 8D space in a way that consistently avoids lattice alignment). Logistic edge-of-chaos (0.79) is close behind. Morse code scores near 0 (its binary structure snaps almost exactly onto lattice points). High lattice distance means the data lives in the interstices of E8, not on it.
Variation in how well windows align with their nearest root direction. Morse code (1.41) has the highest spread: some windows align perfectly (long dashes), others are orthogonal (transitions). ECG signals cluster near the top (1.19, 1.16) — the heartbeat's mix of P-waves, QRS complexes, and baselines creates windows with wildly different alignment quality. Constants and periodic orbits score 0.0 (every window looks the same).
Fraction of windows closest to half-integer roots vs. integer roots. Logistic period-2, L-System Dragon, and phyllotaxis all score 1.0 (every window aligns with a half-integer root). Square wave scores 0.012 (nearly all integer-root alignment). This separates data whose internal 8-byte structure has even vs. odd parity character.
Fraction of windows whose nearest E8 lattice point lies in the half-integer coset (as opposed to the integer coset). Phyllotaxis and circle map quasiperiodic both score 1.0 — their irrational-rotation structure places every 8-byte window closer to a half-integer lattice point. Fibonacci word scores 0.0 (always snaps to the integer coset). This metric and coset_balance measure different things: coset_balance is about root-direction preference; snap_coset_fraction is about actual lattice proximity.
| Source | Domain | Value |
|---|---|---|
| Morse Code | waveform | 1.4125 |
| ECG Fusion | medical | 1.1937 |
| ECG Supraventr. | medical | 1.1559 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| L-System (Dragon Curve) | exotic | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Square Wave | waveform | 0.0122 |
| Source | Domain | Value |
|---|---|---|
| Codon Usage | bio | 6.6754 |
| DNA Chimp | bio | 6.6721 |
| DNA Phage Lambda | bio | 6.6585 |
| ··· | ||
| Logistic r=3.5 (Period-4) | chaos | -0.0000 |
| Logistic Edge-of-Chaos | chaos | -0.0000 |
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Source | Domain | Value |
|---|---|---|
| Critical Circle Map | chaos | 0.8386 |
| Logistic Edge-of-Chaos | chaos | 0.7941 |
| Langton's Ant | exotic | 0.7704 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Morse Code | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Phyllotaxis | bio | 1.0000 |
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Triangle Wave | waveform | 0.9469 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| DNA Chimp | bio | 119.0000 |
| Earthquake Depths | geophysics | 119.0000 |
| Codon Usage | bio | 119.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 1.0000 |
E8 Lattice is the highest-dimensional root system geometry in the framework (window size 8), complementing G2 (2), H3 (3), D4 and H4 (4). Its lattice_distance metric provides a unique signal: data that fills 8D space without aligning to E8's exceptionally dense packing is structurally different from data that happens to land on lattice points. The coset metrics distinguish two flavors of algebraic constraint — integer vs. half-integer parity — that lower-dimensional root systems cannot see.