Stern-Brocot Walk
number_theory · 36 views
number_theory
What It Is
Stern diatomic sequence --- every positive rational visited once, fractal self-similar structure
Interpretation
Standard analysis sees: left-skewed. The atlas additionally detects combinatorially flat (normal-sequence). It sits beside Aliquot Orbit Lengths in the atlas (standard-bank rank 181) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.79
asymmetry0.02
occupancy0.44
short-range corr0.48
long-range memory0.69
spectral colour0.29
periodicity0.77
complexity0.35
time-irreversibility0.54
volatility clustering0.17
multifractality0.35
dimensionality0.62
nonstationarity0.56
What the atlas adds
combinatorially flat (normal-sequence)+2.9z
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
Ordinal Partition:memory_order | +5.5z | bank-miss 1.4σ |
D4 Triality:triplet_temporal | -4.8z | bank-miss 1.6σ |
Nonstationarity:change_quantiles_low | +4.0z | bank-miss 2.7σ |
AutoRegressive:ar_coef_8 | +2.7z | bank-miss 1.5σ |
Moiré:moire_peak_alpha | +2.4z | bank-miss 2.2σ |
AutoRegressive:ar_coef_9 | -2.2z | bank-miss 2.3σ |
H² × ℝ (Thurston):depth_height_corr | +2.1z | bank-miss 1.1σ |
Spectral Analysis:spectral_bandwidth | +2.1z | bank-miss 2.2σ |
Composition
dtypefloat64
range[1.099, 7.857]
unique values1817 / 16384
mean ± std6.33 ± 0.749
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/Stern-Brocot_Walk.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Stern-Brocot_Walk.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Stern-Brocot_Walk.png)
Inflation (Substitution) > barcode
/d_curve/Stern-Brocot_Walk.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/Stern-Brocot_Walk.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/Stern-Brocot_Walk.png)
Ulam Spiral (Sacks) > default
/default/Stern-Brocot_Walk.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Goldbach r(2n) | 5.26 | |
| Collatz Flights | 5.40 | |
| Aliquot Orbit Lengths | 5.53 |
Which Geometries Light Up
Catch24 › Catch24:DN_Mean | rank 3/298 | 0.7606 |
D4 Triality › D4 Triality:triplet_temporal | rank 298/298 | -1.4754 |
Hodge–Laplacian › Hodge–Laplacian:solenoidal_fraction | rank 295/298 | 0.2617 |
Moiré › Moiré:moire_peak_alpha | rank 5/298 | 3.2500 |
Nonstationarity › Nonstationarity:change_quantiles_low | rank 3/298 | 0.1412 |
Ordinal Partition › Ordinal Partition:memory_order | rank 2/298 | 1.0766 |
Ordinal Partition › Ordinal Partition:time_irreversibility | rank 298/298 | 0.0006 |
Piecewise-Linear › Piecewise-Linear:envelope_area | rank 4/298 | 11401.2515 |
Predictability › Predictability:entropy_decay_rate | rank 295/298 | -0.1814 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 2/298 | 0.2093 |
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