Champernowne
number-theory · 37 views
number-theoryhigh-entropy
What It Is
Provably normal number in base 256 --- uniform digit distribution but deterministic concatenation
Interpretation
Standard analysis sees: anti-correlated (alternating); time-irreversible (slow rise, sharp collapse); homoskedastic. The atlas finds no named structure, but the source is distinctively extreme on Predictability:transition_entropy_variance (+7.1z) — beyond what the standard bank predicts for it. It sits beside Chirikov Standard Map in the atlas (standard-bank rank 42) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.52
asymmetry0.83
occupancy0.35
short-range corr0.12
long-range memory0.65
spectral colour0.52
periodicity0.79
complexity0.27
time-irreversibility0.12
volatility clustering0.07
multifractality0.57
dimensionality0.24
nonstationarity0.56
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Predictability:transition_entropy_variance | +7.1z | bank-miss 1.4σ |
Gottwald-Melbourne:k_variance | +5.1z | bank-miss 1.3σ |
Predictability:entropy_decay_rate | -3.7z | bank-miss 1.8σ |
D4 Triality:triplet_temporal | +3.2z | bank-miss 3.0σ |
WaveformAsymmetry:waveform_asymmetry | +3.2z | bank-miss 1.1σ |
Zipf–Mandelbrot (16-bit):zipf_r_squared | -2.4z | bank-miss 2.2σ |
Zipf–Mandelbrot (16-bit):hapax_ratio | +2.2z | bank-miss 2.8σ |
Laplacian:curvature_autocorrelation | -2.0z | bank-miss 1.5σ |
Composition
dtypeuint8
range[0, 255]
unique values256 / 16384
mean ± std69.6 ± 88.3
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/Champernowne.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Champernowne.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Champernowne.png)
Inflation (Substitution) > barcode
/d_curve/Champernowne.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/Champernowne.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/Champernowne.png)
Ulam Spiral (Sacks) > default
/default/Champernowne.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Copeland-Erdős | 4.82 | |
| Goldbach r(2n) | 5.24 | |
| Chirikov Standard Map | 5.27 | cross-origin |
Which Geometries Light Up
Attractor Reconstruction › Attractor Reconstruction:d2_saturation | rank 4/306 | 0.9967 |
D4 Triality › D4 Triality:triplet_temporal | rank 2/306 | 1.0513 |
Gottwald-Melbourne › Gottwald-Melbourne:k_variance | rank 2/306 | 0.8397 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):radial_temporal_memory | rank 1/306 | 4.9771 |
Laplacian › Laplacian:poisson_recovery_error | rank 5/306 | 15.1811 |
Predictability › Predictability:transition_entropy_variance | rank 1/306 | 0.1319 |
Predictability › Predictability:entropy_decay_rate | rank 304/306 | -0.2172 |
WaveformAsymmetry › WaveformAsymmetry:waveform_asymmetry | rank 3/306 | 0.9714 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):hapax_ratio | rank 1/306 | 0.9461 |
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