Champernowne
number_theory · 36 views
number_theory
What It Is
Provably normal number in base 256 --- uniform digit distribution but deterministic concatenation
Interpretation
Standard analysis sees: right-skewed; anti-correlated (alternating); time-irreversible (slow rise, sharp collapse); homoskedastic. The atlas additionally detects combinatorially flat (normal-sequence).
What standard analysis sees
tail heaviness0.74
asymmetry0.86
occupancy0.32
short-range corr0.12
long-range memory0.65
spectral colour0.57
periodicity0.79
complexity0.27
time-irreversibility0.06
volatility clustering0.05
multifractality0.61
dimensionality0.27
nonstationarity0.58
What the atlas adds
combinatorially flat (normal-sequence)+3.5z
every fixed-length block is near-equiprobable and longer context yields no extra predictability — the De Bruijn / normal-number signature (distinct from random noise, which sits mid-scale)
names deterministic flatness — IID noise sits neutral, NOT at this poleAtlas-extreme metrics the standard bank can’t predict for this source
D4 Triality:triplet_temporal | +3.3z | bank-miss 1.8σ |
Zipf–Mandelbrot (16-bit):zipf_r_squared | -2.5z | bank-miss 2.2σ |
Laplacian:laplacian_evolutionary_index | -2.2z | bank-miss 1.3σ |
Zipf–Mandelbrot (16-bit):hapax_ratio | +2.1z | bank-miss 2.8σ |
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 > 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 | 4.89 | |
| Goldbach r(2n) | 5.48 | |
| Chirikov Standard Map | 5.55 | cross-domain |
Which Geometries Light Up
Attractor Reconstruction › Attractor Reconstruction:d2_saturation | rank 4/298 | 0.9967 |
Boltzmann › Boltzmann:spectral_gap_J | rank 5/298 | 0.9908 |
D4 Triality › D4 Triality:triplet_temporal | rank 2/298 | 1.0513 |
Gottwald-Melbourne › Gottwald-Melbourne:k_variance | rank 2/298 | 0.8397 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):radial_temporal_memory | rank 1/298 | 4.9771 |
Laplacian › Laplacian:poisson_recovery_error | rank 3/298 | 15.1811 |
Predictability › Predictability:transition_entropy_variance | rank 1/298 | 0.1319 |
Predictability › Predictability:entropy_decay_rate | rank 296/298 | -0.2172 |
WaveformAsymmetry › WaveformAsymmetry:waveform_asymmetry | rank 3/298 | 0.9714 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):hapax_ratio | rank 1/298 | 0.9461 |
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