Parity vectors and paradoxical sequences in the accelerated Collatz map

TL;DR

This paper investigates parity vectors and paradoxical sequences in the accelerated Collatz map, providing new theorems on their density, enumeration, and distribution, supported by numerical evidence up to large bounds.

Contribution

It proves three new theorems on parity-vector density, paradoxical sequence counts, and their density-zero property, extending prior work and adding explicit formulas and constants.

Findings

Sharp finitary form of Terras's parity-vector density

Closed-form count of paradoxical sequences of fixed length

Density-zero theorem for bounded-length paradoxical sequences

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 $\Omega_k(n)$ 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.