Spectral Form Factor
random-matrix-quantum · 37 views
random-matrix-quantumspectral-RMT
What It Is
Ensemble-averaged spectral form factor log K(t) for a GUE Hamiltonian --- dip-ramp-plateau curve over log-spaced times; captures long-range eigenvalue correlations not seen by local spacings
Interpretation
Standard analysis sees: smooth / autocorrelated; long-range memory (persistent); strongly periodic; volatility-clustering (bursty); low-dimensional; nonstationary / drifting. The atlas finds no named structure, but the source is distinctively extreme on Ulam Spiral (Square):polynomial_concentration (+8.1z) — beyond what the standard bank predicts for it. It sits beside Regime Switching in the atlas (standard-bank rank 22) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.48
asymmetry0.79
occupancy0.50
short-range corr0.92
long-range memory0.92
spectral colour0.35
periodicity0.97
complexity0.27
time-irreversibility0.22
volatility clustering0.93
multifractality0.65
dimensionality0.06
nonstationarity0.97
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Ulam Spiral (Square):polynomial_concentration | +8.1z | bank-miss 1.1σ |
Nonstationarity:ac1_trend | -6.6z | bank-miss 2.0σ |
Nonstationarity:regime_persistence | +6.4z | bank-miss 2.1σ |
Julia Set:escape_entropy | -3.3z | bank-miss 1.2σ |
Laplacian:laplacian_evolutionary_index | +3.0z | bank-miss 3.2σ |
S² × ℝ (Thurston):height_drift | -2.8z | bank-miss 2.3σ |
H² × ℝ (Thurston):depth_height_corr | +2.8z | bank-miss 1.4σ |
Boltzmann:coupling_temporal_variance | +2.5z | bank-miss 3.0σ |
Composition
dtypefloat64
range[-11.08, -0.02565]
unique values16384 / 16384
mean ± std-5.27 ± 2.79
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
_(centered)/signed_log_z/Spectral_Form_Factor.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Spectral_Form_Factor.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Spectral_Form_Factor.png)
Inflation (Substitution) > barcode
/d_curve/Spectral_Form_Factor.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/Spectral_Form_Factor.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/Spectral_Form_Factor.png)
Ulam Spiral (Sacks) > default
/default/Spectral_Form_Factor.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Potomac River Flow | 4.04 | cross-origin |
| Regime Switching | 4.09 | cross-origin |
| ETH/BTC Ratio | 4.29 | cross-origin |
Which Geometries Light Up
Boltzmann › Boltzmann:coupling_temporal_variance | rank 3/306 | 1.6675 |
Catch24 › Catch24:SB_BinaryStats_diff_longstretch0 | rank 1/306 | 4014.5000 |
Catch24 › Catch24:CO_f1ecac | rank 3/306 | 2362.3976 |
Catch24 › Catch24:DN_OutlierInclude_p_001_mdrmd | rank 304/306 | -0.7830 |
Catch24 › Catch24:SP_Summaries_welch_rect_centroid | rank 304/306 | 0.0004 |
Catch24 › Catch24:FC_LocalSimple_mean1_tauresrat | rank 305/306 | 0.0003 |
Chladni › Chladni:nodal_clustering | rank 1/306 | 8.7156 |
Chladni › Chladni:nodal_gap_ratio | rank 304/306 | 0.0719 |
Dodecagonal (Stampfli) › Dodecagonal (Stampfli):pisot_triplet_coherence | rank 5/306 | 0.4974 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):escape_time_variance | rank 1/306 | 925.5001 |
Hodge–Laplacian › Hodge–Laplacian:poisson_recovery_error | rank 302/306 | 0.0227 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):temporal_variance | rank 303/306 | 0.0076 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):radial_temporal_memory | rank 2/306 | 4.9732 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):boundary_dynamics | rank 3/306 | 0.1332 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):depth_height_corr | rank 3/306 | 0.6722 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_step_dispersion | rank 304/306 | 0.0306 |
Julia Set › Julia Set:escape_entropy | rank 306/306 | 2.5153 |
Klein Bottle › Klein Bottle:linear_complexity | rank 305/306 | 0.0039 |
Laplacian › Laplacian:laplacian_evolutionary_index | rank 4/306 | 0.4843 |
Moiré › Moiré:moire_dilation_entropy | rank 3/306 | 4.1843 |
Moiré › Moiré:moire_integer_excess | rank 305/306 | -0.4318 |
Moiré › Moiré:moire_peak_alpha | rank 306/306 | 0.5000 |
Mostow Rigidity › Mostow Rigidity:margulis_ratio | rank 302/306 | 0.0003 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_slope | rank 302/306 | -0.5730 |
Möbius-S³ › Möbius-S³:phi_return_cv | rank 1/306 | 10.7464 |
Nonstationarity › Nonstationarity:regime_persistence | rank 1/306 | 1.0000 |
Nonstationarity › Nonstationarity:ac1_trend | rank 306/306 | -0.4697 |
Projective ℙ² › Projective ℙ²:distance_std | rank 305/306 | 0.0000 |
Projective ℙ² › Projective ℙ²:mean_distance | rank 305/306 | 0.0000 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_burstiness | rank 3/306 | 0.6941 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):trace_autocorrelation | rank 3/306 | 0.9993 |
Spirograph › Spirograph:angular_order | rank 2/306 | 15.0000 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):height_drift | rank 302/306 | -0.5165 |
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentration | rank 1/306 | 0.2121 |
Ulam Spiral (Square) › Ulam Spiral (Square):radial_dispersion | rank 2/306 | 0.7619 |
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