Critical Circle Map (Silver Mean)
iterated-maps · 37 views
iterated-mapsquasiperiodic
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 additionally detects Time Reversibility:ordinal_reversal_distance. 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
asymmetry0.60
occupancy0.64
short-range corr0.08
long-range memory0.09
spectral colour0.89
periodicity0.80
complexity0.20
time-irreversibility0.10
volatility clustering0.00
multifractality0.04
dimensionality0.15
nonstationarity0.07
What the atlas adds
Time Reversibility:ordinal_reversal_distance+3.1z
time-reversal symmetry breaking via Bandt-Pompe ordinal patterns
Atlas-extreme metrics the standard bank can’t predict for this source
Boltzmann:dominant_lag | +3.9z | bank-miss 1.2σ |
Spirograph:petal_symmetry | +3.0z | bank-miss 1.2σ |
Laplacian:laplacian_evolutionary_index | -3.0z | bank-miss 1.9σ |
Laplacian:curvature_autocorrelation | -2.6z | bank-miss 1.6σ |
Hölder Regularity:holder_mean | -2.5z | bank-miss 1.3σ |
p-Variation:volatility_clustering | -2.5z | bank-miss 2.2σ |
Laplacian:cross_scale_curvature_coherence | -2.4z | bank-miss 1.4σ |
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 > free_plate_nodal
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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 (Bronze Mean) | 3.33 | |
| Critical Circle Map | 3.34 | |
| Standard Map K=0.5 (Mixed) | 4.12 |
Which Geometries Light Up
Boltzmann › Boltzmann:dominant_lag | rank 3/306 | 991.0000 |
Hodge–Laplacian › Hodge–Laplacian:curl_div_ratio | rank 302/306 | -0.7636 |
Laplacian › Laplacian:laplacian_evolutionary_index | rank 302/306 | -0.2637 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 306/306 | -0.6081 |
Modular Residue › Modular Residue:cycle_fraction | rank 5/306 | 0.9828 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 302/306 | 0.3188 |
Projective ℙ² › Projective ℙ²:cross_ratio_std | rank 4/306 | 41.0035 |
p-Variation › p-Variation:volatility_clustering | rank 303/306 | -0.5992 |
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