Parity vectors and paradoxical sequences in the accelerated Collatz map
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2026
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| _version_ | 1866916032676888576 |
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| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $Ω_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. As for the numerical piece, among the seven $(j, q)$ pairs that show up in the Rozier-Terracol enumeration with first term $n \le 10^9$, every paradoxical reduced ratio $q/j$ turns out to be a left convergent, a left semiconvergent, or a Stern-Brocot mediant of adjacent convergents/semiconvergents of $\log_3 2$. The three theorems are unconditional. The fourth observation is verified for $n \le 10^7$ and conjectured for all $n$. We make no claim toward the Collatz conjecture or Terras's coefficient-stopping-time conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13886 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parity vectors and paradoxical sequences in the accelerated Collatz map Niu, Tong Number Theory 11B83, 11A07, 37P10, 11A55 This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $Ω_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. As for the numerical piece, among the seven $(j, q)$ pairs that show up in the Rozier-Terracol enumeration with first term $n \le 10^9$, every paradoxical reduced ratio $q/j$ turns out to be a left convergent, a left semiconvergent, or a Stern-Brocot mediant of adjacent convergents/semiconvergents of $\log_3 2$. The three theorems are unconditional. The fourth observation is verified for $n \le 10^7$ and conjectured for all $n$. We make no claim toward the Collatz conjecture or Terras's coefficient-stopping-time conjecture. |
| title | Parity vectors and paradoxical sequences in the accelerated Collatz map |
| topic | Number Theory 11B83, 11A07, 37P10, 11A55 |
| url | https://arxiv.org/abs/2605.13886 |