Golden-Mean β-Shift

What It Is

Greedy β-expansion digits of a random x in [0,1) at base β = φ ≈ 1.618 (golden ratio). The φ-β-shift is the golden-mean shift of finite type --- the binary sequences with no two consecutive 1s --- which is SOFIC (recognized by a 2-state labeled graph). Topological entropy log φ ≈ 0.481. The sofic pole of a matched-entropy soficity contrast: paired with the Gap-Word β-Shift (non-sofic, β ≈ 1.55) it holds entropy roughly fixed while soficity varies, testing whether the atlas separates sofic from non-sofic beyond entropy / complexity.

Interpretation

Standard analysis sees: few distinct values; anti-correlated (alternating); blue spectrum (high-frequency power); homoskedastic; low-dimensional. The atlas finds no named structure, but the source is distinctively extreme on Hyperbolic (Poincaré):temporal_variance (+4.0z) — beyond what the standard bank predicts for it. It sits beside Symbolic Henon in the atlas (standard-bank rank 28) — a neighbor conventional features miss.

Composition

Binary sequence — two distinct symbols.

Render Gallery

Native > phase_plotraw

Native > timeseriesraw

Native > triple_latticeraw

Boltzmann > lag_map

Boltzmann > spin_grid

Cayley > default

Chladni > free_plate_nodal

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 →

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