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 discrete Levy area z_{n+1} = z_n + x_n·Δy_{n+1} (accumulated x-position times the y-increment) itself. Each step fires exactly one generator, so x and y are NOT independent: exactly one coordinate moves by ±1 while the other holds (Δx,Δy ∈ {-1,0,+1} with probabilities 1/4, 1/2, 1/4), so each marginally is a ±1 walk idle ~half the steps. This coupling is the defining anisotropy: the Levy area grows like Var(z_n) ~ n²/8 in the number n of (x,y) steps (the vertical z scales as n while the horizontal x,y scale as √n) — the Carnot–Carathéodory homogeneity of nilpotent geometry.
Interpretation
Standard analysis sees: left-skewed; anti-correlated (alternating); homoskedastic; nonstationary / drifting. The atlas finds no named structure, but the source is distinctively extreme on Ordinal Partition:memory_order (+7.8z) — beyond what the standard bank predicts for it.
Ordinal Partition:memory_order | +7.8z | bank-miss 1.8σ |
Nonstationarity:ac1_trend | -2.7z | bank-miss 1.2σ |
Wasserstein:recurrence_distance | +2.7z | bank-miss 1.0σ |
Spectral Analysis:spectral_bandwidth | +2.6z | bank-miss 1.0σ |
Wasserstein:distributional_stationarity | -2.5z | bank-miss 3.0σ |
Projective ℙ²:distance_std | +2.5z | bank-miss 2.8σ |
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 | 5.05 | cross-origin |
| Windows PE x86-64 | 5.57 | cross-origin |
| Champernowne | 5.59 | cross-origin |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_2 | rank 2/306 | 0.6900 |
Boltzmann › Boltzmann:spectral_gap_J | rank 5/306 | 0.9927 |
Boltzmann › Boltzmann:nn_dominance | rank 303/306 | 0.0090 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):boundary_dynamics | rank 5/306 | 0.1003 |
Hölder Regularity › Hölder Regularity:alpha_autocorrelation | rank 3/306 | 0.3738 |
Hölder Regularity › Hölder Regularity:holder_mean | rank 303/306 | -1.8842 |
Isochronicity › Isochronicity:amplitude_exploration | rank 2/306 | 2.0540 |
Laplacian › Laplacian:biharmonic_ratio | rank 2/306 | 15.9087 |
Lorentzian › Lorentzian:causal_persistence | rank 1/306 | 0.9856 |
Möbius-S³ › Möbius-S³:hopf_fiber_coherence | rank 305/306 | 0.0106 |
Nonstationarity › Nonstationarity:ac1_trend | rank 304/306 | -0.1898 |
Ordinal Partition › Ordinal Partition:memory_order | rank 1/306 | 1.4040 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 1/306 | 0.2394 |
Spirograph › Spirograph:rational_grid_proximity | rank 305/306 | 0.4454 |