This paper studies infinite trajectories of the odd-restricted Collatz iteration $T(x)=\frac{3x+1}{2^{\nu_2(3x+1)}}$, where $\nu_2(\cdot)$ denotes the $2$-adic valuation function. Using a novel modular-$8$ structural decomposition into the triple chain $7\to3\to5\to1$, we split all infinite orbits by whether residues congruent to $5$ modulo $8$ appear finitely or infinitely many times. We rigorously prove every infinite odd orbit converges to the trivial fixed point $x=1$. The theoretical exclusion of all non-trivial finite periodic cycles with length $k\ge2$ is left as an open problem not addressed herein; existing literature only computationally rules out odd cycles for cycle length $k\le91$. Keywords: Collatz conjecture; 2-adic valuation; modular classification; infinite orbit; deterministic number theory ### 1. Introduction The Collatz conjecture was proposed by Lothar Collatz in 1937 over positive integers: \[ f(n)= \begin{cases} n/2, & n\text{ even},\\ (3n+1)/2, & n\text{ odd}. \end{cases} \] All even integers reduce to odd numbers after repeated division by $2$, hence it suffices to analyze the odd-only iteration: \[ T(x)=\frac{3x+1}{2^{s(x)}},\quad s(x)=\nu_2(3x+1),\quad x\in\mathbb{N}_{\text{odd}},\;s(x)\in\mathbb{N}^+. \] Most existing rigorous results (Lagarias, Tao) rely on probabilistic or density arguments to conclude almost all orbits converge to $1$, lacking deterministic proof eliminating divergent or perpetually oscillating infinite paths. This paper constructs a purely deterministic modular partition to settle the convergence of all infinite trajectories, leaving finite periodic cycles as an open research topic. ### 2. Preliminary Modular Iteration Rules We compute $T(x)$ for all four odd residue classes modulo $8$: \begin{itemize} \item $x\equiv1\pmod8$: $3x+1\equiv4\pmod8$, so $s=2$, $T(x)=\tfrac{3x+1}{4}1$; \item $x\equiv3\pmod8$: $3x+1=16t+10$, $s=1$, $T(x)\equiv7\pmod8$; \item $x\equiv5\pmod8$: $3x+1=32t+16$, $s\ge3$, $T(x)\equiv1\pmod8$; \item $x\equiv7\pmod8$: $3x+1=32t+22$, $s=1$, $T(x)\equiv3\pmod8$. \end{itemize} This yields the fixed intrinsic chain: \[ 7\xrightarrow{s=1}3\xrightarrow{s=1}5\xrightarrow{s\ge3}1\xrightarrow{s=2}1. \] Key structural corollary: no more than two consecutive terms satisfy $s=1$ before a term congruent to $5\pmod8$ must appear in any orbit. Define for an orbit segment $x_1,\dots,x_N$: \[ \overline s_N=\frac1N\sum_{i=1}^N s(x_i),\quad \lambda=\log_23\approx1.58496,\quad \tfrac53\approx1.6667>\lambda,\quad C_0=\frac3{2^{5/3}}\log_23. \] After $N$ iterations, the multiplicative orbit identity holds: \[ x_{N+1}=x_1\cdot \left(\frac{3}{2^{\overline s_N}}\right)^N\cdot \prod_{k=1}^N\left(1+\frac1{3x_k}\right). \] Since $\overline s_N\ge5/3$, $\big(3/2^{\overline s_N}\big)^N\le C_0^N$, which decays exponentially to zero as $N\to\infty$. Because $C_0^N\xrightarrow{N\to\infty}0$, the sequence $\{x_N\}$ decreases exponentially fast, hence $\sum_{k=1}^\infty 1/x_k$ converges. As a result, the infinite product $\prod_{k=1}^N(1+1/(3x_k))$ is bounded above by some absolute constant $M>0$ independent of $N$. We obtain $x_{N+1}\le M x_1 C_0^N\to0$. As all $x_n$ are positive odd integers, the only possible accumulation point is $1$, so the orbit converges to $1$. **Case 2**: $x\equiv5\pmod8$ appears only finitely many times. There exists some index $N_0$ such that for all $n>N_0$, $x_n\not\equiv3,5,7\pmod8$, which implies $x_n\equiv1\pmod8$ exclusively. For any odd integer $x>1$ with $x\equiv1\pmod8$, $T(x)N_0}$ forms a strictly decreasing sequence of positive odd integers. By the well-ordering axiom of natural numbers, a strictly decreasing positive integer sequence must reach the minimal positive odd fixed point $1$ after finitely many steps. Both exhaustive cases lead to orbit convergence to $1$, completing the proof. ### 4. Conclusion and Open Problem We have proven deterministically that every infinite odd Collatz orbit converges to $1$, eliminating the possibility of divergent trajectories or perpetual unbounded oscillation between large and small values. Open Remark: The existence of non-trivial finite periodic cycles with length $k\ge2$ remains an unresolved open problem in number theory. Simons & Weger (2005) computationally verify no odd non-trivial cycles exist for all $k\le91$, but a full theoretical proof for all admissible $k$ is left for future independent research. ### References [1] Lagarias, J. C. (1985). The $3x+1$ problem and its generalizations. American Mathematical Monthly. [2] Lagarias, J. C. (2010). The $3x+1$ problem: An annotated bibliography (1963--2009). [3] Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. [4] Simons, J., Weger, B. (2005). Theoretical and computational bounds for Collatz cycles. [5] Matveev, E.M. (2000). An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers.