Collatz Cycle Word
What It Is
Lower-Christoffel / Sturmian word of slope log2(3): the balanced binary sequence s_i = floor((i+1)d) - floor(i*d) with d = 1/(1+log2(3)) ~ 0.3868 (the 1-density). Like the Fibonacci Word it has minimal complexity p(n) = n+1 (the fewest distinct length-n blocks an aperiodic sequence can have), but its irrational slope is log2(3) --- the growth constant of the Collatz / 3n+1 map --- rather than the golden ratio, making it the symbolic cycle/parity word of that dynamics. Stationary and non-monotone. Included as a contrast to the golden-slope Fibonacci Word: is the log2(3) slope geometrically special, or just another minimal-complexity Sturmian?
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 H3 Icosahedral:nn_enrichment (+4.1z) — beyond what the standard bank predicts for it.
H3 Icosahedral:nn_enrichment | +4.0z | bank-miss 2.0σ |
Hyperbolic (Poincaré):temporal_variance | +3.9z | bank-miss 1.1σ |
H3 Icosahedral:temporal_coherence | +3.7z | bank-miss 2.3σ |
Wavelet Cascade:scale_gini_dispersion | +3.4z | bank-miss 1.3σ |
D4 Triality:triplet_temporal | -2.8z | bank-miss 1.3σ |
Nonstationarity:metric_volatility | +2.6z | bank-miss 1.0σ |
Isochronicity:frequency_shear | +2.4z | bank-miss 1.0σ |
Modular Residue:occupancy_entropy | -2.3z | bank-miss 1.1σ |
Composition
Binary sequence — two distinct symbols.
Render Gallery
_(centered)/signed_log_z/Collatz_Cycle_Word.png)
_(centered)/xy_path/Collatz_Cycle_Word.png)
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/d_curve/Collatz_Cycle_Word.png)
/phi_spectrum/Collatz_Cycle_Word.png)
/default/Collatz_Cycle_Word.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Penrose Substitution | 2.64 | cross-origin |
| Fibonacci Word | 2.85 | cross-origin |
| Periodic Word | 3.20 | cross-origin |
Which Geometries Light Up
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 4/306 | 4.2018 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 5/306 | 0.3664 |
Laplacian › Laplacian:poisson_recovery_error | rank 4/306 | 15.4462 |
Projective ℙ² › Projective ℙ²:mean_distance | rank 2/306 | 0.3852 |
Projective ℙ² › Projective ℙ²:distance_std | rank 3/306 | 0.2586 |
Projective ℙ² › Projective ℙ²:collinearity | rank 302/306 | 0.9354 |
Symplectic › Symplectic:windowed_area_cv | rank 303/306 | 0.0254 |
Time Reversibility › Time Reversibility:increment_skewness_max | rank 304/306 | 0.0003 |
Time Reversibility › Time Reversibility:ordinal_reversal_distance | rank 305/306 | 0.0001 |
Visibility Graph › Visibility Graph:assortativity | rank 302/306 | -0.3730 |
Wavelet Cascade › Wavelet Cascade:scale_gini_dispersion | rank 5/306 | 0.2577 |