Riemann-Hardy-Littlewood
exotic · 36 views
exotic
What It Is
Riemann's 1859 'monster' R(t) = sum_{n>=1} sin(pi n^2 t)/(pi n^2). Continuous everywhere, differentiable only at rationals (2p+1)/(2q+1) where R' = -1/2 (Gerver 1970), nowhere else (Hardy 1916). Quadratic frequencies n^2 with 1/n^2 amplitude decay --- the steepest known cosine fractal: ~92% of energy in the fundamental, non-differentiability emerging from the long thin Hausdorff-dimensional tail.
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 Multi-Scale Wasserstein:w_max_ratio (+8.7z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.17
asymmetry0.41
occupancy0.82
short-range corr0.99
long-range memory0.99
spectral colour0.30
periodicity0.99
complexity0.48
time-irreversibility0.85
volatility clustering0.97
multifractality0.57
dimensionality0.11
nonstationarity0.90
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Multi-Scale Wasserstein:w_max_ratio | +8.7z | bank-miss 1.3σ |
Septagonal (Danzer):cubic_coherence | +2.9z | bank-miss 1.0σ |
Septagonal (Danzer):z_primary | +2.5z | bank-miss 1.2σ |
Navier-Stokes:sl_fit_quality | +2.0z | bank-miss 2.3σ |
Composition
dtypefloat64
range[-0.4, 0.4]
unique values16384 / 16384
mean ± std-1.81e-05 ± 0.234
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/Riemann-Hardy-Littlewood.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Riemann-Hardy-Littlewood.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Riemann-Hardy-Littlewood.png)
Inflation (Substitution) > barcode
/d_curve/Riemann-Hardy-Littlewood.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/Riemann-Hardy-Littlewood.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/Riemann-Hardy-Littlewood.png)
Ulam Spiral (Sacks) > default
/default/Riemann-Hardy-Littlewood.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| fBm (Persistent) | 3.44 | cross-domain |
| Perlin Noise | 4.05 | cross-domain |
| ETH/BTC Ratio | 4.12 | cross-domain |
Which Geometries Light Up
Ammann-Beenker (Octagonal) › Ammann-Beenker (Octagonal):convergent_profile | rank 295/298 | -0.1715 |
Catch24 › Catch24:CO_HistogramAMI_even_2_5 | rank 1/298 | 1.5592 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 1/298 | 0.8704 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 3/298 | 0.9999 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 296/298 | 0.0089 |
Chladni › Chladni:plate_low_mode_fraction | rank 1/298 | 0.9956 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughness | rank 295/298 | 0.0053 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):z_rate_spectral_entropy | rank 5/298 | 0.8774 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):radial_temporal_memory | rank 4/298 | 4.9664 |
Inflation (Substitution) › Inflation (Substitution):return_concentration | rank 295/298 | 0.0822 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_max_ratio | rank 1/298 | 3027.3606 |
Navier-Stokes › Navier-Stokes:sl_fit_quality | rank 1/298 | 0.9581 |
Navier-Stokes › Navier-Stokes:ess_quality | rank 294/298 | 0.0001 |
Predictability › Predictability:excess_predictability | rank 2/298 | 2.9203 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):trace_autocorrelation | rank 4/298 | 0.9992 |
Septagonal (Danzer) › Septagonal (Danzer):cubic_coherence | rank 3/298 | 0.2286 |
Sol (Thurston) › Sol (Thurston):z_variance | rank 2/298 | 23.0570 |
Sol (Thurston) › Sol (Thurston):sol_step_persistence | rank 4/298 | 0.8878 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 295/298 | 0.0009 |
Spherical S² › Spherical S²:hemisphere_balance | rank 4/298 | 0.9999 |
Spherical S² › Spherical S²:spectral_gini | rank 5/298 | 0.9991 |
p-Variation › p-Variation:var_p2 | rank 295/298 | 0.0000 |
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