Collatz Gap Lengths

What It Is

BSL gap vector from Collatz orbits --- number of halvings between consecutive odd steps (v_i in the Böhm-Sontacchi-Lagarias equation). Distribution approximately geometric(1/2) with correlations encoding the multiplicative walk structure of 3n+1

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; few distinct values; aperiodic / broadband. The atlas finds no named structure, but the source is distinctively extreme on Attractor Reconstruction:lyap_sum (-3.9z) — beyond what the standard bank predicts for it. It sits beside Categorical Sensor in the atlas (standard-bank rank 53) — a neighbor conventional features miss.

Composition

Fixed alphabet — only 14 distinct symbols across 16384 samples.

Render Gallery

Native > phase_plotraw

Native > timeseriesraw

Native > triple_latticeraw

Boltzmann > lag_map

Boltzmann > spin_grid

Cayley > default

Chladni > plate_bottom

Chladni > plate_top

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

Inflation (Substitution) > d_curve

Information Theory > default

Julia Set > escape_histogram

Klein Bottle > default

Laplacian > default

Logarithmic Spiral > default

Möbius-S³ > mobius_cylinder

Möbius-S³ > s2_base_map

Ordinal Partition > default

Penrose (Quasicrystal) > phi_spectrum

Persistent Homology > barcode

Persistent Homology > h1_diagram

Predictability > default

Recurrence Quantification > default

Spectral Analysis > default

Spectral Graph > default

Spirograph > mode_spectrum

Spirograph > phasor_trajectory

Symplectic > flux_grid

Symplectic > phase_space

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

Open in Atlas →

Which Geometries Light Up