Periodic Word
What It Is
Fixed binary pattern of period 7 (…1010010…) tiled with a random phase per trial. Trivially SOFIC --- finitely many follower sets, bounded factor complexity --- and zero topological entropy. The sofic, zero-entropy pole contrasted against the Sturmian words (Fibonacci / Collatz Cycle / Pell), which are NON-sofic at zero entropy (complexity n+1, infinitely many follower sets). Sanity anchor for the soficity probe: clean separation here confirms the atlas sees the easy end of the soficity gradient.
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; homoskedastic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_mid (+6.6z) — beyond what the standard bank predicts for it.
Nonstationarity:change_quantiles_mid | +6.6z | bank-miss 3.7σ |
Julia Set:potential_variance | +4.5z | bank-miss 1.4σ |
Julia Set:potential_smoothness | +4.5z | bank-miss 1.4σ |
Hyperbolic (Poincaré):temporal_variance | +4.4z | bank-miss 1.7σ |
Klein Bottle:linear_complexity | -3.8z | bank-miss 1.3σ |
Chladni:modal_nodal_cascade | +3.4z | bank-miss 1.2σ |
Wavelet Cascade:scale_gini_dispersion | +3.2z | bank-miss 1.3σ |
H3 Icosahedral:temporal_coherence | +2.8z | bank-miss 3.0σ |
Composition
Binary sequence — two distinct symbols.
Render Gallery
_(centered)/signed_log_z/Periodic_Word.png)
_(centered)/xy_path/Periodic_Word.png)
/barcode/Periodic_Word.png)
/d_curve/Periodic_Word.png)
/phi_spectrum/Periodic_Word.png)
/default/Periodic_Word.png)
/default/Periodic_Word.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Collatz Cycle Word | 3.20 | cross-origin |
| Penrose Substitution | 3.70 | |
| Tribonacci Word | 3.91 |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_residual_frac | rank 306/306 | 0.0000 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 302/306 | 0.1260 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 302/306 | 0.0000 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 305/306 | 0.1200 |
Chladni › Chladni:modal_nodal_cascade | rank 3/306 | 2.0618 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):temporal_variance | rank 1/306 | 19.7630 |
Information Theory › Information Theory:compression_ratio | rank 304/306 | 0.0024 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 302/306 | -0.5477 |
Möbius-S³ › Möbius-S³:hopf_fiber_coherence | rank 303/306 | 0.1885 |
Nonstationarity › Nonstationarity:change_quantiles_mid | rank 2/306 | 0.8572 |
Nonstationarity › Nonstationarity:dynamic_coupling | rank 5/306 | 8.3234 |
Projective ℙ² › Projective ℙ²:distance_std | rank 2/306 | 0.2604 |
Projective ℙ² › Projective ℙ²:mean_distance | rank 3/306 | 0.3848 |
Sol (Thurston) › Sol (Thurston):integrated_walk_smoothness | rank 306/306 | -0.2712 |
Symplectic › Symplectic:windowed_area_cv | rank 306/306 | 0.0000 |
Time Reversibility › Time Reversibility:increment_skewness_max | rank 306/306 | 0.0002 |
Visibility Graph › Visibility Graph:assortativity | rank 303/306 | -0.4009 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 302/306 | -0.3333 |