Collatz Parity
number_theory · 36 views
number_theory
What It Is
Collatz parity sequence --- binary trace (even/odd) of the 3n+1 orbit, conjectured to behave like a biased coin flip with P(odd) ≈ log₂(3)/(1+log₂(3)). Restarts from consecutive integers on cycle collapse
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); blue spectrum (high-frequency power); homoskedastic. The atlas finds no named structure, but the source is distinctively extreme on Hyperbolic (Poincaré):temporal_variance (+4.7z) — beyond what the standard bank predicts for it. It sits beside Symbolic Henon in the atlas (standard-bank rank 29) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.12
asymmetry0.80
occupancy0.02
short-range corr0.07
long-range memory0.23
spectral colour0.92
periodicity0.33
complexity0.30
time-irreversibility0.46
volatility clustering0.03
multifractality0.42
dimensionality0.51
nonstationarity0.23
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Hyperbolic (Poincaré):temporal_variance | +4.7z | bank-miss 1.2σ |
Zariski:heyting_gap | +4.1z | bank-miss 1.1σ |
Projective ℙ²:collinearity | -2.7z | bank-miss 1.5σ |
Fractal (Mandelbrot):escape_entropy | -2.2z | bank-miss 1.5σ |
Composition
dtypefloat64
range[0, 1]
unique values2 / 16384
mean ± std0.345 ± 0.475
Binary sequence — two distinct symbols.
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/Collatz_Parity.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Collatz_Parity.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Collatz_Parity.png)
Inflation (Substitution) > barcode
/d_curve/Collatz_Parity.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_Parity.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_Parity.png)
Ulam Spiral (Sacks) > default
/default/Collatz_Parity.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Symbolic Henon | 3.83 | cross-domain |
| Liouville Function | 4.32 | |
| Penrose Substitution | 4.47 | cross-domain |
Which Geometries Light Up
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):temporal_variance | rank 1/298 | 18.9112 |
Moiré › Moiré:moire_invariance_breadth | rank 296/298 | 0.0031 |
Projective ℙ² › Projective ℙ²:mean_distance | rank 5/298 | 0.3552 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 297/298 | 0.0002 |
Visibility Graph › Visibility Graph:degree_exponent_gamma | rank 4/298 | 2.7095 |
Zariski › Zariski:residual_convexity | rank 2/298 | 14.4790 |
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