Gaussian Collatz Orbit
number_theory · 36 views
number_theory
What It Is
Collatz map over Gaussian integers Z[i] with divisor pi=1+i and multiplier 3. Signal is split into 4 segments: 3 seeded from the proven period-16 cycle starts (2-8i, -48+4i, -43-35i) and 1 from random init showing the divergent regime. Captures both the proven cycle dynamics and the surrounding chaotic divergence
Interpretation
Standard analysis sees: anti-persistent; nonstationary / drifting. The atlas additionally detects combinatorially flat (normal-sequence). It sits beside Gut Motility in the atlas (standard-bank rank 144) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.78
asymmetry0.60
occupancy0.35
short-range corr0.52
long-range memory0.10
spectral colour0.53
periodicity0.53
complexity0.46
time-irreversibility0.42
volatility clustering0.67
multifractality0.52
dimensionality0.34
nonstationarity0.97
What the atlas adds
combinatorially flat (normal-sequence)+2.3z
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:vol_of_vol | +4.5z | bank-miss 1.8σ |
Zipf–Mandelbrot (8-bit):entropy_nonstationarity | +4.5z | bank-miss 1.3σ |
Nonstationarity:ac1_trend | -3.5z | bank-miss 1.2σ |
Möbius-S³:spinorial_asymmetry | +3.2z | bank-miss 1.3σ |
D4 Triality:triplet_temporal | -2.7z | bank-miss 3.8σ |
Spirograph:angular_order | +2.6z | bank-miss 2.6σ |
Hyperbolic (Poincaré):spatio_temporal_corr | +2.1z | bank-miss 2.3σ |
Composition
dtypefloat64
range[-16.04, 16.01]
unique values3090 / 16384
mean ± std-0.134 ± 5.24
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/Gaussian_Collatz_Orbit.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Gaussian_Collatz_Orbit.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Gaussian_Collatz_Orbit.png)
Inflation (Substitution) > barcode
/d_curve/Gaussian_Collatz_Orbit.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/Gaussian_Collatz_Orbit.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/Gaussian_Collatz_Orbit.png)
Ulam Spiral (Sacks) > default
/default/Gaussian_Collatz_Orbit.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Accel Walk | 5.45 | cross-domain |
| Speech "Zero" | 5.56 | cross-domain |
| Speech "Nine" | 5.59 | cross-domain |
Which Geometries Light Up
Catch24 › Catch24:DN_OutlierInclude_n_001_mdrmd | rank 1/298 | 0.7272 |
Catch24 › Catch24:DN_OutlierInclude_p_001_mdrmd | rank 2/298 | 0.7280 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 2/298 | 0.8460 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):spatio_temporal_corr | rank 3/298 | 0.8467 |
Hölder Regularity › Hölder Regularity:holder_mean | rank 1/298 | 1.3452 |
Laplacian › Laplacian:gradient_curvature_anticorrelation | rank 294/298 | -0.0793 |
Möbius-S³ › Möbius-S³:spinorial_asymmetry | rank 3/298 | 0.1883 |
Nonstationarity › Nonstationarity:regime_persistence | rank 2/298 | 1.0000 |
Nonstationarity › Nonstationarity:variance_trend | rank 2/298 | 0.7917 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 2/298 | 2.9432 |
Nonstationarity › Nonstationarity:metric_volatility | rank 295/298 | 0.2886 |
Nonstationarity › Nonstationarity:ac1_trend | rank 296/298 | -0.2601 |
Spirograph › Spirograph:petal_symmetry | rank 1/298 | 0.7404 |
Spirograph › Spirograph:angular_order | rank 5/298 | 8.0000 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):entropy_nonstationarity | rank 2/298 | 0.5980 |
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