Free Group F₂ Walk
What It Is
Non-backtracking random walk on the Cayley graph of the free group F₂ = ⟨a, b⟩ --- the 4-regular tree, canonical δ-hyperbolic (δ=0) space. At each step the next generator is drawn uniformly from {a, a⁻¹, b, b⁻¹} excluding the inverse of the previous step, so the walk strictly extends a reduced word. Emits step labels as a 4-level uint8 sequence {0, 85, 170, 255}. Word length grows linearly in step count (tree metric) while ball volume grows like 4·3^{n-1} (exponential) --- the textbook Gromov-hyperbolic positive control for Cayley:delta_hyp_norm, growth_exponent, saturation_radius, spectral_gap.
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; blue spectrum (high-frequency power). The atlas finds no named structure, but the source is distinctively extreme on Zariski:algebraic_residual (+2.5z) — beyond what the standard bank predicts for it. It sits beside DNA SARS-CoV-2 in the atlas (standard-bank rank 43) — a neighbor conventional features miss.
Zariski:algebraic_residual | +2.5z | bank-miss 2.1σ |
Composition
Fixed alphabet — only 4 distinct symbols across 16384 samples.
Render Gallery
_(centered)/signed_log_z/Free_Group_F₂_Walk.png)
_(centered)/xy_path/Free_Group_F₂_Walk.png)
/barcode/Free_Group_F₂_Walk.png)
/d_curve/Free_Group_F₂_Walk.png)
/phi_spectrum/Free_Group_F₂_Walk.png)
/default/Free_Group_F₂_Walk.png)
/default/Free_Group_F₂_Walk.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Codon Usage | 3.80 | cross-domain |
| Categorical Sensor | 4.00 | |
| DNA SARS-CoV-2 | 4.25 | cross-domain |
Which Geometries Light Up
Moiré › Moiré:moire_max_coherence | rank 297/298 | 0.0225 |
Zariski › Zariski:algebraic_residual | rank 5/298 | 0.0604 |