Fibonacci Tight-Binding

random-matrix-quantum · 37 views

random-matrix-quantumquasiperiodic

What It Is

1D tight-binding chain with hopping (off-diagonal) amplitudes t_n in {t_A, t_B} (t_A/t_B=2) modulated by Fibonacci substitution rule A->AB, B->A --- Cantor-set spectrum, multifractal critical eigenstate at the chiral-protected band center E=0 (Kohmoto-Kadanoff-Tang 1983; deterministic multifractality from substitution-aperiodic mechanism)

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; low-dimensional; nonstationary / drifting. The atlas additionally detects discrete-map sensitive dependence, Attractor Reconstruction:lyap_entropy, Higher-Order Statistics:c3_energy.

What standard analysis sees

tail heaviness0.95

asymmetry0.97

occupancy0.20

short-range corr0.49

long-range memory0.70

spectral colour0.44

periodicity0.67

complexity0.43

time-irreversibility0.48

volatility clustering0.51

multifractality0.84

dimensionality0.07

nonstationarity0.98

What the atlas adds

discrete-map sensitive dependence+9.3z

deterministic chaos (positive λ_max, sensitive dependence)

names a discrete-map-scoped estimate, NOT chaos in general — continuous-flow chaos (Lorenz, Rössler) reads weak/neutral here despite being genuinely chaotic; spiky arithmetic sources can false-positive on the finite-time estimate

Attractor Reconstruction:lyap_entropy+4.5z

Kolmogorov-Sinai entropy production rate (sum of positive λ)

Higher-Order Statistics:c3_energy+2.8z

third-cumulant burst energy (SOC irreducibility signature)

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

`Ulam Spiral (Square):polynomial_concentration`	+5.8z	bank-miss 2.6σ
`H3 Icosahedral:temporal_coherence`	+4.5z	bank-miss 1.7σ
`Ulam Spiral (Square):radial_periodicity`	+3.6z	bank-miss 2.1σ
`Hodge–Laplacian:source_fraction`	+3.3z	bank-miss 1.6σ
`H3 Icosahedral:nn_enrichment`	+3.2z	bank-miss 1.9σ
`Nonstationarity:variance_trend`	-3.1z	bank-miss 2.0σ
`AutoRegressive:ar_coef_2`	+2.9z	bank-miss 1.3σ
`Wavelet Cascade:intermittency_slope`	-2.6z	bank-miss 1.2σ

Composition

dtypefloat64

range[8.793e-15, 0.02254]

unique values16384 / 16384

mean ± std6.1e-05 ± 0.000413

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

Nearest neighbor	Distance
Aubry-André Critical	4.50
Solar Flares Daily Peak	5.12	cross-origin
Anderson 1D Localized	5.16

Open in Atlas →

Which Geometries Light Up

Attractor Reconstruction › `Attractor Reconstruction:lyap_entropy`	rank 1/307	5.0000
Attractor Reconstruction › `Attractor Reconstruction:lyapunov_max`	rank 1/307	0.8698
AutoRegressive › `AutoRegressive:ar_coef_2`	rank 1/307	1.5382
Cantor Set › `Cantor Set:bit_plane_autocorrelation`	rank 305/307	0.0007
Catch24 › `Catch24:DN_HistogramMode_10`	rank 2/307	2.5806
Catch24 › `Catch24:DN_HistogramMode_5`	rank 2/307	5.3090
Catch24 › `Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff`	rank 304/307	0.0094
Catch24 › `Catch24:DN_Mean`	rank 305/307	0.0027
Catch24 › `Catch24:DN_Spread_Std`	rank 305/307	0.0183
Chladni › `Chladni:temporal_burstiness`	rank 4/307	0.6034
Continued Fraction › `Continued Fraction:gauss_kuzmin_distance`	rank 2/307	0.8713
H3 Icosahedral › `H3 Icosahedral:temporal_coherence`	rank 2/307	0.4179
Higher-Order Statistics › `Higher-Order Statistics:skew_mean`	rank 2/307	3.0869
Higher-Order Statistics › `Higher-Order Statistics:kurt_max`	rank 4/307	7.1664
Hodge–Laplacian › `Hodge–Laplacian:source_fraction`	rank 1/307	0.7595
Klein Bottle › `Klein Bottle:rank_deficit`	rank 5/307	0.9521
Laplacian › `Laplacian:poisson_recovery_error`	rank 2/307	15.6161
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_std`	rank 303/307	0.0003
Nonstationarity › `Nonstationarity:variance_trend`	rank 305/307	-0.3267
SL(2,ℝ) (Thurston) › `SL(2,ℝ) (Thurston):cocycle_growth_rate`	rank 4/307	1.9804
Spectral Analysis › `Spectral Analysis:spectral_bandwidth`	rank 3/307	0.2088
Spherical S² › `Spherical S²:concentration`	rank 4/307	0.9986
Spherical S² › `Spherical S²:angular_spread`	rank 303/307	0.0646
Symplectic › `Symplectic:phase_reflection_symmetry`	rank 2/307	0.9994
Symplectic › `Symplectic:windowed_area_cv`	rank 3/307	9.0044
Symplectic › `Symplectic:recurrence_rate`	rank 4/307	0.9751
S² × ℝ (Thurston) › `S² × ℝ (Thurston):sphere_concentration`	rank 3/307	0.9989
Ulam Spiral (Square) › `Ulam Spiral (Square):polynomial_concentration`	rank 3/307	0.1706
Ulam Spiral (Square) › `Ulam Spiral (Square):radial_dispersion`	rank 3/307	0.7148
Visibility Graph › `Visibility Graph:degree_r_squared`	rank 3/307	0.9745
Wasserstein › `Wasserstein:dist_from_uniform`	rank 2/307	0.4831
Wasserstein › `Wasserstein:concentration`	rank 4/307	31.4375
Wasserstein › `Wasserstein:entropy`	rank 305/307	0.1580
Wavelet Cascade › `Wavelet Cascade:cascade_coupling`	rank 4/307	0.8097
Zariski › `Zariski:nonsep_fraction`	rank 4/307	0.9527

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