Collatz Parity
What It Is
Collatz parity sequence --- binary trace (even/odd) of the full 3n+1 orbit. Each odd step (3n+1) always lands on an even number, halved ~2 times on average (geometric, mean 2), so it behaves like a biased coin with P(odd) ≈ 1/3 --- one odd per ~2 halvings. 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 (+4.2z) — beyond what the standard bank predicts for it. It sits beside Symbolic Henon in the atlas (standard-bank rank 31) — a neighbor conventional features miss.
Hyperbolic (Poincaré):temporal_variance | +4.2z | bank-miss 1.4σ |
Composition
Binary sequence — two distinct symbols.
Render Gallery
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/phi_spectrum/Collatz_Parity.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Golden-Mean β-Shift | 2.53 | cross-origin |
| Gap-Word β-Shift | 2.96 | cross-origin |
| Symbolic Henon | 3.40 | cross-origin |
Which Geometries Light Up
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):temporal_variance | rank 2/306 | 18.9112 |
Moiré › Moiré:moire_invariance_breadth | rank 303/306 | 0.0031 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 304/306 | 0.0002 |
Visibility Graph › Visibility Graph:degree_exponent_gamma | rank 4/306 | 2.7659 |
Zariski › Zariski:residual_convexity | rank 4/306 | 14.4790 |