Parity vectors and paradoxical sequences in the accelerated Collatz map

Abstract

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.