Heisenberg Walk
What It Is
Random walk on the discrete Heisenberg group H(ℤ) with generators drawn uniformly from {(±1,0,0), (0,±1,0)} (the standard symmetric Carnot generating set). Emits the (x,y) coordinates interleaved as a byte stream (out[2i]=x_i mod 256, out[2i+1]=y_i mod 256), which is exactly the encoding the framework's Heisenberg geometry consumes — its embed() builds the path from consecutive byte pairs and accumulates the Levy area z_{n+1} = z_n + x_n·y_{n+1} itself. Pure Heisenberg-walk reference: x,y are independent ±1 random walks, so the Levy area grows like Var(z_n) ~ n²/8 (faster than the n^{3/2} that an independent (x,y) byte pair would give) — the defining Carnot–Carathéodory anisotropy of nilpotent geometry.
Interpretation
Standard analysis sees: bounded / light-tailed; anti-correlated (alternating); low-dimensional. The atlas finds no named structure, but the source is distinctively extreme on Ordinal Partition:memory_order (+7.2z) — beyond what the standard bank predicts for it. It sits beside Copeland-Erdős in the atlas (standard-bank rank 76) — a neighbor conventional features miss.
Ordinal Partition:memory_order | +7.2z | bank-miss 2.0σ |
Isochronicity:amplitude_exploration | +5.3z | bank-miss 1.5σ |
H² × ℝ (Thurston):boundary_dynamics | +3.8z | bank-miss 1.6σ |
Möbius-S³:hopf_fiber_coherence | -3.5z | bank-miss 1.1σ |
Projective ℙ²:distance_std | +2.9z | bank-miss 2.4σ |
Wasserstein:self_similarity | -2.7z | bank-miss 1.3σ |
Spectral Analysis:spectral_bandwidth | +2.5z | bank-miss 2.0σ |
Composition
Render Gallery
_(centered)/signed_log_z/Heisenberg_Walk.png)
_(centered)/xy_path/Heisenberg_Walk.png)
/barcode/Heisenberg_Walk.png)
/d_curve/Heisenberg_Walk.png)
/phi_spectrum/Heisenberg_Walk.png)
/default/Heisenberg_Walk.png)
/default/Heisenberg_Walk.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Copeland-Erdős | 5.34 | cross-domain |
| Windows PE x86-64 | 5.79 | cross-domain |
| Gut Motility | 5.86 | cross-domain |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_2 | rank 2/298 | 0.6983 |
Boltzmann › Boltzmann:nn_dominance | rank 295/298 | 0.0198 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):boundary_dynamics | rank 5/298 | 0.1029 |
Hölder Regularity › Hölder Regularity:alpha_autocorrelation | rank 2/298 | 0.3748 |
Hölder Regularity › Hölder Regularity:holder_mean | rank 296/298 | -1.8990 |
Inflation (Substitution) › Inflation (Substitution):return_concentration | rank 297/298 | 0.0763 |
Isochronicity › Isochronicity:amplitude_exploration | rank 2/298 | 2.0540 |
Laplacian › Laplacian:biharmonic_ratio | rank 2/298 | 15.9184 |
Lorentzian › Lorentzian:causal_persistence | rank 1/298 | 0.9867 |
Möbius-S³ › Möbius-S³:hopf_fiber_coherence | rank 297/298 | 0.0111 |
Ordinal Partition › Ordinal Partition:memory_order | rank 1/298 | 1.3699 |
Projective ℙ² › Projective ℙ²:distance_std | rank 4/298 | 0.2350 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 1/298 | 0.2292 |
Spirograph › Spirograph:gear_rationality | rank 297/298 | 0.4554 |
Wasserstein › Wasserstein:self_similarity | rank 294/298 | 0.8385 |
p-Variation › p-Variation:increment_persistence | rank 297/298 | -0.9885 |