Collatz Trajectory
number-theory · 37 views
number-theoryarithmetic
What It Is
Collatz orbit values log1p(n) along the trajectory, restarting from consecutive integers to avoid 4-2-1 cycle collapse. Preserves the multiplicative walk structure (3n+1 then halve) on a compressed scale.
Interpretation
Standard analysis sees: time-irreversible (sharp rises, slow decay). The atlas additionally detects Time Reversibility:ordinal_reversal_distance.
What standard analysis sees
tail heaviness0.36
asymmetry0.66
occupancy0.75
short-range corr0.69
long-range memory0.70
spectral colour0.36
periodicity0.45
complexity0.26
time-irreversibility0.99
volatility clustering0.69
multifractality0.60
dimensionality0.36
nonstationarity0.60
What the atlas adds
Time Reversibility:ordinal_reversal_distance+3.1z
time-reversal symmetry breaking via Bandt-Pompe ordinal patterns
Atlas-extreme metrics the standard bank can’t predict for this source
Navier-Stokes:cascade_asymmetry | +3.4z | bank-miss 1.2σ |
Möbius-S³:spinorial_asymmetry | +3.1z | bank-miss 2.5σ |
Composition
dtypefloat64
range[1.792, 19.85]
unique values2838 / 16384
mean ± std9.31 ± 4.13
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/Collatz_Trajectory.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Collatz_Trajectory.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Collatz_Trajectory.png)
Inflation (Substitution) > barcode
/d_curve/Collatz_Trajectory.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/Collatz_Trajectory.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/Collatz_Trajectory.png)
Ulam Spiral (Sacks) > default
/default/Collatz_Trajectory.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Hawkes Process | 4.06 | cross-origin |
| Ocean Wind (Buoy) | 4.46 | cross-origin |
| Solar Wind Speed | 4.49 | cross-origin |
Which Geometries Light Up
Fisher Information › Fisher Information:effective_dimension | rank 305/306 | 2.3978 |
Möbius-S³ › Möbius-S³:spinorial_asymmetry | rank 4/306 | 0.1802 |
Navier-Stokes › Navier-Stokes:cascade_asymmetry | rank 5/306 | 0.8212 |
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