Aubry-André Critical

quantum · 36 views

quantum

What It Is

Almost-Mathieu operator at critical coupling λ=2 with golden-ratio incommensurate potential --- multifractal eigenstate intensity (deterministic multifractality, canonical exemplar from quasiperiodic origin)

Interpretation

Standard analysis sees: no strongly notable standard properties. The atlas finds no named structure, but the source is distinctively extreme on G2 Root System:kurtosis_differential (-14.5z) — beyond what the standard bank predicts for it. It sits beside Solar Flares Daily Peak in the atlas (standard-bank rank 40) — a neighbor conventional features miss.

What the atlas adds

Atlas-extreme metrics the standard bank can’t predict for this source

`G2 Root System:kurtosis_differential`	-14.5z	bank-miss 1.2σ
`Symplectic:windowed_area_cv`	+6.8z	bank-miss 1.4σ
`Mostow Rigidity:spectral_rigidity`	-6.4z	bank-miss 1.1σ
`Higher-Order Statistics:skew_mean`	+5.6z	bank-miss 1.5σ
`2-adic:mean_distance`	-4.3z	bank-miss 1.1σ
`Higher-Order Statistics:kurt_max`	+4.1z	bank-miss 1.3σ
`Zariski:nonsep_fraction`	+4.1z	bank-miss 2.3σ
`Higher-Order Statistics:c3_energy`	+3.6z	bank-miss 2.3σ

Composition

dtypefloat64

range[1.677e-14, 0.02408]

unique values16384 / 16384

mean ± std6.1e-05 ± 0.000385

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

Nearest neighbor	Distance
Solar Flares Daily Peak	4.23	cross-domain
Anderson 1D Localized	4.66
Fibonacci Tight-Binding	4.67

Open in Atlas →

Which Geometries Light Up

2-adic › `2-adic:distance_entropy`	rank 294/298	0.7532
2-adic › `2-adic:mean_distance`	rank 295/298	0.1079
Attractor Reconstruction › `Attractor Reconstruction:lyap_sum`	rank 2/298	3.5017
Attractor Reconstruction › `Attractor Reconstruction:lyap_entropy`	rank 3/298	3.5017
Catch24 › `Catch24:DN_HistogramMode_10`	rank 1/298	3.6296
Catch24 › `Catch24:DN_HistogramMode_5`	rank 1/298	7.3919
Catch24 › `Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff`	rank 295/298	0.0142
Catch24 › `Catch24:DN_Mean`	rank 298/298	0.0018
Catch24 › `Catch24:DN_Spread_Std`	rank 298/298	0.0133
Chladni › `Chladni:temporal_burstiness`	rank 1/298	0.7671
Continued Fraction › `Continued Fraction:gauss_kuzmin_distance`	rank 1/298	0.8724
G2 Root System › `G2 Root System:kurtosis_differential`	rank 298/298	-37.1001
Higher-Order Statistics › `Higher-Order Statistics:kurt_max`	rank 1/298	7.7395
Higher-Order Statistics › `Higher-Order Statistics:skew_mean`	rank 1/298	3.2447
Hodge–Laplacian › `Hodge–Laplacian:source_fraction`	rank 4/298	0.6662
Klein Bottle › `Klein Bottle:rank_deficit`	rank 3/298	0.9919
Klein Bottle › `Klein Bottle:rank_deficit_max`	rank 4/298	0.9996
Mostow Rigidity › `Mostow Rigidity:distance_rigidity`	rank 296/298	0.0361
Mostow Rigidity › `Mostow Rigidity:spectral_rigidity`	rank 297/298	0.5348
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_fine`	rank 294/298	0.0011
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_mean`	rank 295/298	0.0011
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_std`	rank 295/298	0.0002
Piecewise-Linear › `Piecewise-Linear:envelope_area`	rank 298/298	29.0448
Predictability › `Predictability:cond_entropy_k1`	rank 294/298	0.0135
SL(2,ℝ) (Thurston) › `SL(2,ℝ) (Thurston):lyapunov_exponent`	rank 1/298	1.9883
Sol (Thurston) › `Sol (Thurston):z_variance`	rank 297/298	0.0176
Spherical S² › `Spherical S²:concentration`	rank 2/298	0.9992
Spherical S² › `Spherical S²:hemisphere_balance`	rank 294/298	0.0003
Spherical S² › `Spherical S²:angular_spread`	rank 296/298	0.0453
Spirograph › `Spirograph:gear_rationality`	rank 294/298	0.5052
Symplectic › `Symplectic:phase_reflection_symmetry`	rank 1/298	0.9999
Symplectic › `Symplectic:recurrence_rate`	rank 1/298	0.9831
Symplectic › `Symplectic:windowed_area_cv`	rank 2/298	9.2084
S² × ℝ (Thurston) › `S² × ℝ (Thurston):sphere_concentration`	rank 2/298	0.9994
S² × ℝ (Thurston) › `S² × ℝ (Thurston):great_circle_dispersion`	rank 295/298	0.0071
Ulam Spiral (Square) › `Ulam Spiral (Square):polynomial_concentration`	rank 294/298	0.0204
Visibility Graph › `Visibility Graph:degree_r_squared`	rank 2/298	0.9778
Wasserstein › `Wasserstein:dist_from_uniform`	rank 1/298	0.4837
Wasserstein › `Wasserstein:concentration`	rank 2/298	31.7109
Wasserstein › `Wasserstein:entropy`	rank 297/298	0.0925
Wavelet Cascade › `Wavelet Cascade:cascade_coupling`	rank 2/298	0.9350
Zariski › `Zariski:nonsep_fraction`	rank 2/298	0.9767
Zipf–Mandelbrot (16-bit) › `Zipf–Mandelbrot (16-bit):gini_coefficient`	rank 2/298	0.9744
Zipf–Mandelbrot (8-bit) › `Zipf–Mandelbrot (8-bit):gini_coefficient`	rank 3/298	0.9629
Zipf–Mandelbrot (8-bit) › `Zipf–Mandelbrot (8-bit):hapax_ratio`	rank 3/298	0.3175

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