Critical Circle Map (Silver Mean)
chaos · 36 views
chaos
What It Is
Sine circle map at K=1 critical coupling tuned to the silver-mean winding number sqrt(2)-1 (continued fraction [0;2,2,2,...]) --- a renormalization fixed point distinct from the golden mean's, with its own universal critical-scaling constants
Interpretation
Standard analysis sees: anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; low-dimensional; stationary. The atlas finds no named structure, but the source is distinctively extreme on Laplacian:laplacian_evolutionary_index (-3.0z) — beyond what the standard bank predicts for it. It sits beside Critical Circle Map in the atlas (standard-bank rank 31) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.38
asymmetry0.61
occupancy0.61
short-range corr0.09
long-range memory0.08
spectral colour0.89
periodicity0.80
complexity0.19
time-irreversibility0.10
volatility clustering0.00
multifractality0.04
dimensionality0.14
nonstationarity0.07
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Laplacian:laplacian_evolutionary_index | -3.0z | bank-miss 1.8σ |
Laplacian:curvature_autocorrelation | -2.6z | bank-miss 1.2σ |
p-Variation:volatility_clustering | -2.4z | bank-miss 1.7σ |
Laplacian:cross_scale_curvature_coherence | -2.4z | bank-miss 1.4σ |
Higher-Order Statistics:bicoherence_max | +2.2z | bank-miss 1.1σ |
Composition
dtypefloat64
range[0.005706, 0.9963]
unique values16384 / 16384
mean ± std0.499 ± 0.229
Render Gallery
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Native > phase_plotraw
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Native > timeseriesraw
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Native > triple_latticeraw
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Boltzmann > lag_map
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Boltzmann > spin_grid
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Cayley > default
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Chladni > plate_bottom
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Chladni > plate_top
_(centered)/signed_log_z/Critical_Circle_Map_(Silver_Mean).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Critical_Circle_Map_(Silver_Mean).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Critical_Circle_Map_(Silver_Mean).png)
Inflation (Substitution) > barcode
/d_curve/Critical_Circle_Map_(Silver_Mean).png)
Inflation (Substitution) > d_curve
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Information Theory > default
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Julia Set > escape_histogram
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Klein Bottle > default
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Laplacian > default
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Logarithmic Spiral > default
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Möbius-S³ > mobius_cylinder
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Möbius-S³ > s2_base_map
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Ordinal Partition > default
/phi_spectrum/Critical_Circle_Map_(Silver_Mean).png)
Penrose (Quasicrystal) > phi_spectrum
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Persistent Homology > barcode
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Persistent Homology > h1_diagram
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Predictability > default
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Recurrence Quantification > default
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Spectral Analysis > default
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Spectral Graph > default
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Spirograph > mode_spectrum
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Spirograph > phasor_trajectory
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Symplectic > flux_grid
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Symplectic > phase_space
/default/Critical_Circle_Map_(Silver_Mean).png)
Ulam Spiral (Sacks) > default
/default/Critical_Circle_Map_(Silver_Mean).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Critical Circle Map | 3.39 | |
| Critical Circle Map (Bronze Mean) | 3.44 | |
| Phyllotaxis | 4.07 | cross-domain |
Which Geometries Light Up
Boltzmann › Boltzmann:dominant_lag | rank 3/298 | 991.0000 |
Hodge–Laplacian › Hodge–Laplacian:curl_div_ratio | rank 294/298 | -0.7526 |
Laplacian › Laplacian:laplacian_evolutionary_index | rank 294/298 | -0.2637 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 298/298 | -0.6081 |
Modular Residue › Modular Residue:cycle_fraction | rank 5/298 | 0.9818 |
Navier-Stokes › Navier-Stokes:intermittency_autocorr | rank 298/298 | -0.5441 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 294/298 | 0.3170 |
Projective ℙ² › Projective ℙ²:cross_ratio_std | rank 5/298 | 40.0291 |
p-Variation › p-Variation:volatility_clustering | rank 294/298 | -0.5992 |
← / → within domain · ⇧← / ⇧→ alphabetical · ⇧← / ⇧→ inside an open render = same view across sources