Goldbach r(2n)
number_theory · 36 views
number_theory
What It Is
Goldbach r(2n) --- number of unordered representations of 2n as p+q with both p,q prime. Highly erratic (Hardy-Littlewood: mean ~2*C_2*S(n)*2n/log^2(2n)) with strong primorial bias (r(2n) jumps when 2n is divisible by many small primes). Distinct mechanism from Divisor Count d(n): additive prime structure vs multiplicative divisor structure
Interpretation
Standard analysis sees: anti-correlated (alternating); blue spectrum (high-frequency power); strongly periodic; homoskedastic; monofractal. The atlas additionally detects combinatorially flat (normal-sequence). It sits beside Stern-Brocot Walk in the atlas (standard-bank rank 117) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.67
asymmetry0.84
occupancy0.49
short-range corr0.10
long-range memory0.39
spectral colour0.90
periodicity0.88
complexity0.55
time-irreversibility0.24
volatility clustering0.06
multifractality0.12
dimensionality0.40
nonstationarity0.62
What the atlas adds
combinatorially flat (normal-sequence)+2.5z
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
Nonstationarity:variance_trend | +8.2z | bank-miss 1.3σ |
Nonstationarity:adf_pvalue | +4.8z | bank-miss 2.9σ |
Composition
dtypefloat64
range[498, 2613]
unique values1498 / 16384
mean ± std923 ± 366
Render Gallery
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Native > phase_plotraw
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Native > timeseriesraw
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Native > triple_latticeraw
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Boltzmann > lag_map
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Boltzmann > spin_grid
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Cayley > default
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Chladni > plate_bottom
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Chladni > plate_top
_(centered)/signed_log_z/Goldbach_r(2n).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Goldbach_r(2n).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Goldbach_r(2n).png)
Inflation (Substitution) > barcode
/d_curve/Goldbach_r(2n).png)
Inflation (Substitution) > d_curve
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Information Theory > default
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Julia Set > escape_histogram
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Klein Bottle > default
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Laplacian > default
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Logarithmic Spiral > default
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Möbius-S³ > mobius_cylinder
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Möbius-S³ > s2_base_map
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Ordinal Partition > default
/phi_spectrum/Goldbach_r(2n).png)
Penrose (Quasicrystal) > phi_spectrum
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Persistent Homology > barcode
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Persistent Homology > h1_diagram
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Predictability > default
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Recurrence Quantification > default
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Spectral Analysis > default
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Spectral Graph > default
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Spirograph > mode_spectrum
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Spirograph > phasor_trajectory
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Symplectic > flux_grid
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Symplectic > phase_space
/default/Goldbach_r(2n).png)
Ulam Spiral (Sacks) > default
/default/Goldbach_r(2n).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Aliquot Orbit Lengths | 5.17 | |
| Stern-Brocot Walk | 5.26 | |
| Champernowne | 5.48 |
Which Geometries Light Up
Catch24 › Catch24:DN_OutlierInclude_p_001_mdrmd | rank 4/298 | 0.5231 |
Catch24 › Catch24:DN_OutlierInclude_n_001_mdrmd | rank 295/298 | -0.3687 |
Fisher Information › Fisher Information:effective_dimension | rank 294/298 | 2.7883 |
Hölder Regularity › Hölder Regularity:holder_std | rank 1/298 | 1.9736 |
Nonstationarity › Nonstationarity:variance_trend | rank 1/298 | 0.9387 |
Nonstationarity › Nonstationarity:adf_pvalue | rank 4/298 | 0.7946 |
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