Gaussian Collatz Orbit
number-theory · 37 views
number-theoryarithmetic
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 finds no named structure, but the source is distinctively extreme on Nonstationarity:vol_of_vol (+5.6z) — beyond what the standard bank predicts for it. It sits beside Speech "Zero" in the atlas (standard-bank rank 69) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.78
asymmetry0.59
occupancy0.35
short-range corr0.52
long-range memory0.10
spectral colour0.54
periodicity0.52
complexity0.46
time-irreversibility0.42
volatility clustering0.67
multifractality0.52
dimensionality0.34
nonstationarity0.97
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Nonstationarity:vol_of_vol | +5.6z | bank-miss 2.2σ |
Zipf–Mandelbrot (8-bit):entropy_nonstationarity | +4.8z | bank-miss 1.3σ |
D4 Triality:triplet_temporal | -2.7z | bank-miss 3.8σ |
Predictability:entropy_decay_rate | -2.3z | bank-miss 1.6σ |
Hyperbolic (Poincaré):spatio_temporal_corr | +2.2z | bank-miss 1.8σ |
Composition
dtypefloat64
range[-16.04, 16.01]
unique values3098 / 16384
mean ± std-0.131 ± 5.26
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/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 | |
|---|---|---|
| Linux ELF x86-64 | 5.16 | cross-origin |
| Speech "Zero" | 5.19 | cross-origin |
| Speech "Nine" | 5.29 | cross-origin |
Which Geometries Light Up
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 3/307 | 0.8020 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):spatio_temporal_corr | rank 3/307 | 0.8514 |
Hölder Regularity › Hölder Regularity:holder_mean | rank 1/307 | 1.3449 |
Möbius-S³ › Möbius-S³:spinorial_asymmetry | rank 3/307 | 0.1954 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 2/307 | 3.4781 |
Nonstationarity › Nonstationarity:regime_persistence | rank 3/307 | 0.9700 |
Nonstationarity › Nonstationarity:metric_volatility | rank 304/307 | 0.2888 |
Spirograph › Spirograph:petal_symmetry | rank 2/307 | 0.6780 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):entropy_nonstationarity | rank 1/307 | 0.5895 |
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