Fibonacci Tight-Binding
quantum · 36 views
quantum
What It Is
1D tight-binding chain with on-site potential modulated by Fibonacci substitution rule A->AB, B->A --- Cantor-set spectrum, multifractal eigenstates (deterministic multifractality from substitution-aperiodic mechanism)
Interpretation
Standard analysis sees: heavy-tailed; right-skewed; low-dimensional; nonstationary / drifting. The atlas additionally detects quasicrystalline order.
What standard analysis sees
tail heaviness0.95
asymmetry0.97
occupancy0.19
short-range corr0.50
long-range memory0.70
spectral colour0.44
periodicity0.67
complexity0.44
time-irreversibility0.48
volatility clustering0.52
multifractality0.84
dimensionality0.07
nonstationarity0.98
What the atlas adds
quasicrystalline order+2.4z
aperiodic long-range order at incommensurate (golden-ratio / Fibonacci) spacing — structure repeats at irrational ratios, not a fixed period
Atlas-extreme metrics the standard bank can’t predict for this source
Ordinal Partition:memory_order | +5.0z | bank-miss 1.6σ |
Attractor Reconstruction:lyap_entropy | +4.8z | bank-miss 1.2σ |
Laplacian:poisson_recovery_error | -2.4z | bank-miss 2.3σ |
Klein Bottle:rank_deficit | +2.4z | bank-miss 1.7σ |
Klein Bottle:rank_deficit_max | +2.1z | bank-miss 1.7σ |
Composition
dtypefloat64
range[3.191e-15, 0.001487]
unique values16384 / 16384
mean ± std6.1e-05 ± 0.000162
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Fibonacci_Tight-Binding.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Fibonacci_Tight-Binding.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Fibonacci_Tight-Binding.png)
Inflation (Substitution) > barcode
/d_curve/Fibonacci_Tight-Binding.png)
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
/phi_spectrum/Fibonacci_Tight-Binding.png)
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
/default/Fibonacci_Tight-Binding.png)
Ulam Spiral (Sacks) > default
/default/Fibonacci_Tight-Binding.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Minkowski Question Mark | 4.64 | cross-domain |
| Aubry-André Critical | 4.67 | |
| Multiplicative Cascade | 5.12 | cross-domain |
Which Geometries Light Up
Attractor Reconstruction › Attractor Reconstruction:lyap_entropy | rank 1/298 | 5.0000 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 298/298 | 0.1200 |
Chladni › Chladni:temporal_burstiness | rank 4/298 | 0.6536 |
Continued Fraction › Continued Fraction:gauss_kuzmin_distance | rank 4/298 | 0.7630 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):spatio_temporal_corr | rank 295/298 | -0.2945 |
Klein Bottle › Klein Bottle:rank_deficit | rank 1/298 | 1.0000 |
Klein Bottle › Klein Bottle:rank_deficit_max | rank 1/298 | 1.0000 |
Klein Bottle › Klein Bottle:linear_complexity | rank 298/298 | 0.0000 |
Ordinal Partition › Ordinal Partition:memory_order | rank 3/298 | 0.9870 |
Ordinal Partition › Ordinal Partition:statistical_complexity | rank 4/298 | 0.1034 |
Ordinal Partition › Ordinal Partition:time_irreversibility | rank 297/298 | 0.0007 |
Projective ℙ² › Projective ℙ²:distance_std | rank 294/298 | 0.0001 |
Projective ℙ² › Projective ℙ²:mean_distance | rank 294/298 | 0.0001 |
Spirograph › Spirograph:gear_rationality | rank 296/298 | 0.4557 |
Ulam Spiral (Square) › Ulam Spiral (Square):radial_periodicity | rank 2/298 | 0.8417 |
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentration | rank 297/298 | 0.0047 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):entropy_nonstationarity | rank 1/298 | 0.7319 |
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