Phi-Squared β-Shift

What It Is

Greedy β-expansion digits of a random x in [0,1) at base β = φ² = (3+√5)/2 ≈ 2.618. The expansion of 1 is 2·1^∞ (eventually periodic but infinite), so this β-shift is SOFIC BUT NOT a shift of finite type --- the strictly-sofic middle of the Parry trichotomy (SFT ⊂ sofic ⊂ general subshift). Ternary alphabet, entropy log φ² ≈ 0.962. A higher-entropy sofic anchor for the soficity probe, alongside the Golden-Mean β-Shift.

Interpretation

Standard analysis sees: few distinct values; anti-correlated (alternating); blue spectrum (high-frequency power); homoskedastic; monofractal; high-dimensional / space-filling. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_high (+4.2z) — beyond what the standard bank predicts for it.

Composition

Fixed alphabet — only 3 distinct symbols across 16384 samples.

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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