Collatz Trees: Trunk–Branch Indexing and Affine Cycle-Freeness

1. Decomposing All Natural Numbers into Geometric Sequences

1.1. Background and Objective

We express $\mathbb{N}$ as a collection of rays parameterized by odd cores and powers of two, providing a structural stage for Collatz dynamics.

1.2. Definitions and Goals

Let

$$S=\{(2a+1)·2^b\mid a,b\in {\mathbb{Z}}_{\ge 0}\}.$$

We show $S=\mathbb{N}$ and the representation is unique.

1.3. Prime Factorization and Classification

Every $n\in \mathbb{N}$ decomposes uniquely as

$$n=2^b·k,b\in {\mathbb{Z}}_{\ge 0},k\mathrm{odd}.$$

1.4. Exhaustion of Odd Numbers

Any odd k is $k=2a+1$ with $a\ge 0$, giving $1,3,5,7,\dots$.

1.5. Exhaustion of Even Parts

For each odd k, the ray $k,2k,4k,\dots$ exhausts the even multiples of k.

1.6. Construction of S and Uniqueness

By the above, every $n=(2a+1)2^b$ with $a,b\ge 0$. If

$$(2a+1)2^b=(2a^{\prime }+1)2^{b^{\prime }},$$

then $(2a+1)/(2a^{\prime }+1)=2^{b^{\prime }-b}$, forcing $a=a^{\prime }$ and $b=b^{\prime }$ since the left side is odd rational and the right is a power of two. Hence $S=\mathbb{N}$ bijectively.

1.7. Remarks from the Collatz Perspective

For odd k, $3k+1$ is even and belongs to some ray $(2a^{\prime }+1)2^{b^{\prime }}$. This exhibits inter-branch connections. However, the assertion that every number lies on a finite forward path to 1 (global convergence) is a separate issue (coverage) made precise by thm:reduction; it is not implied by the mere classification $S=\mathbb{N}$.

Takeaway of Chapter 1. We obtain a clean, bijective indexing of $\mathbb{N}$ by odd core and 2-adic height, furnishing a coordinate system on which later structural/affine arguments are staged.

2. The Structure of the Collatz Tree

2.1. Definition (Branches and Trunk)

Define the trunk $T_0=\{1·2^b:b\ge 0\}$ and for each odd $k\ge 3$ the branch $B_k=\{k·2^b:b\ge 0\}$. These rays partition $\mathbb{N}$ disjointly.

Figure 1. Trunk and branches (schematic; reverse orientation when embedded into the inverse graph: edges point to preimages).

Figure 1. Trunk and branches (schematic; reverse orientation when embedded into the inverse graph: edges point to preimages).

2.2. Branch–Branch Links via $3k+1$

Given odd k, $3k+1$ is even and decomposes as $(2a^{\prime }+1)2^{b^{\prime }}$, indicating where the branch from k can merge into another branch/trunk in forward dynamics. This shows linkage patterns but does not by itself prove global coverage of the tree by reverse generation.

Figure 2. Branch connections (schematic; reverse orientation: edges point to preimages).

Figure 2. Branch connections (schematic; reverse orientation: edges point to preimages).

2.3. Forward vs. Reverse Orientation

Let the standard forward map be

$$f\left(n\right)=\left\{\begin{array}{cc}n/2,\hfill & n\mathrm{even},\hfill \\ 3n+1,\hfill & n\mathrm{odd}.\hfill \end{array}\right.$$

The forward graph (edges $n\to f\left(n\right)$) is a functional digraph (outdegree 1). We do not call it a DAG because it contains the trivial $1\leftrightarrow 2\leftrightarrow 4$ 3-cycle; nontrivial finite cycles are excluded later (sec:affine).

The reverse (preimage) graph rooted at 1, with edges to preimages under f, is a true DAG: levels increase with each application of a reverse step.

2.4. Tree Language

When drawing a reverse BFS tree rooted at 1, each node is assigned a unique parent by construction (though a number may have up to two preimages as graph children). Connectivity of every node to 1 in the forward sense is equivalent to coverage of the reverse tree, which is equivalent to the Collatz convergence; see Theorem 4.

3. Trunk–Branch Indexing of the Natural Numbers

Definition 1

(Odd core, 2-adic valuation). For $n\in \mathbb{N}$, write uniquely $n=\mathrm{odd}\left(n\right)·2^{{\nu }_2\left(n\right)}$ where $\mathrm{odd}\left(n\right)$ is odd and ${\nu }_2\left(n\right)\in {\mathbb{Z}}_{\ge 0}$ is the exponent of 2 in n.

Definition 2

(Trunk and branches). The trunk is $T_0=\{1·2^b:b=0,1,2,\dots \}=1,2,4,8,\dots$. For any odd $k\ge 3$, $B_k=\{k·2^b:b=0,1,2,\dots \}.$ Then $\left\{T_0\right\}\cup \{B_k:k\mathrm{odd}\ge 3\}$ is a disjoint partition of $\mathbb{N}$.

Definition 3

(Indices). Order the odd numbers as $1,3,5,7,\dots$. Assign the branch index $\mathrm{br}\left(\mathrm{odd}\right)=(\mathrm{odd}-1)/2\in {\mathbb{Z}}_{\ge 0}$, so that $\mathrm{br}\left(1\right)=0$ and $\mathrm{br}\left(3\right)=1$, $\mathrm{br}\left(5\right)=2$, etc. Define the height $\mathrm{ht}\left(n\right)={\nu }_2\left(n\right)$. Set

$$\mathrm{Idx}:\mathbb{N}\to {\mathbb{Z}}_{\ge 0}\times {\mathbb{Z}}_{\ge 0},\mathrm{Idx}\left(n\right)=\left(\mathrm{br}\left(\mathrm{odd}\right(n\left)\right),\mathrm{ht}\left(n\right)\right),{\mathrm{Idx}}^{-1}(i,b)=(2i+1)·2^b.$$

Theorem 1

(Complete classification). The map $n\mapsto \mathrm{Idx}\left(n\right)$ is a bijection from $\mathbb{N}$ onto ${\mathbb{Z}}_{\ge 0}\times {\mathbb{Z}}_{\ge 0}$.

Proof. 

Uniqueness of $\mathrm{odd}\left(n\right)$ and ${\nu }_2\left(n\right)$ is immediate; disjointness/exhaustiveness of rays follows.    □

Figure 3. Trunk–branch indexing (schematic; reverse orientation in the inverse graph).

Figure 3. Trunk–branch indexing (schematic; reverse orientation in the inverse graph).

Figure 4. Indexed reverse tree (schematic; edges point to preimages).

Figure 4. Indexed reverse tree (schematic; edges point to preimages).

Table: Trunk–Branch Indexing (sample)

Odd k	Index	Next Index (rule)	parity-based trend	transition factor
1	1	–	–	–
3	2	3	increase	$3/2$
5	3	1	decrease	$1/3$
7	4	6	increase	$3/2$
9	5	4	decrease	$4/5$
11	6	9	increase	$3/2$
13	7	3	decrease	$3/7$
15	8	12	increase	$3/2$
17	9	7	decrease	$7/9$
19	10	15	increase	$3/2$

4. Affine Word Method: Absence of Nontrivial Finite Cycles

We encode even/odd steps as affine maps and use finite-word compositions $F_W\left(x\right)=a_{W}x+c_W$ to prove absence of nontrivial finite cycles.

Definition of the three-way map T.

Define $T:{\mathbb{N}}_{\ge 1}\to {\mathbb{N}}_{\ge 1}$ by

$$T\left(n\right)=\left\{\begin{array}{cc}{\displaystyle \frac{3}{2}}n,\hfill & n\mathrm{even},\hfill \\ {\displaystyle \frac{n-1}{2}},\hfill & n\mathrm{odd}\mathrm{and}{\nu }_2(3n+1)=1,\hfill \\ {\displaystyle \frac{3n+1}{4}},\hfill & n\mathrm{odd}\mathrm{and}{\nu }_2(3n+1)\ge 2.\hfill \end{array}\right.$$

Elementary steps.

$$E\left(x\right)={\displaystyle \frac{3}{2}}x,O_1\left(x\right)={\displaystyle \frac{x-1}{2}},O_2\left(x\right)={\displaystyle \frac{3x+1}{4}}.$$

(For odd n, choose $O_1$ if ${\nu }_2(3n+1)=1$, and $O_2$ if ${\nu }_2(3n+1)\ge 2$.)

Words and composition. 

For any finite word W over $\{E,O_1,O_2\}$, the composition is affine:

$$F_W\left(x\right)=a_{W}x+c_W.$$

Writing $m=\#E$, $o_1=\#O_1$, $o_2=\#O_2$, we have

$$a_W={\left({\displaystyle \frac{3}{2}}\right)}^m{\left({\displaystyle \frac{1}{2}}\right)}^{o_1}{\left({\displaystyle \frac{3}{4}}\right)}^{o_2}={\displaystyle \frac{3^{m+o_2}}{2^{m+o_1+2o_2}}},$$

and the denominator of $c_W$ divides $2^{o_1+2o_2}$ (each $O_1$ contributes $-1/2$, each $O_2$ contributes $+1/4$; E does not increase the power-of-two denominator).

Lemma 1

(No $a_W=1$ for nonempty words). If W is nonempty then $3^m=2^{m+o_1+2o_2}$ cannot hold.

Lemma 2

(Odd numerator for $1-a_W$).

$$1-a_W={\displaystyle \frac{2^{m+o_1+2o_2}-3^{m+o_2}}{2^{m+o_1+2o_2}}},$$

so the numerator is odd (even minus odd).

Lemma 3

(If $m\ge 1$, a periodic solution cannot be integral). Since $\mathrm{den}\left(c_W\right)\mid 2^{o_1+2o_2}$, we have $c_W·2^{m+o_1+2o_2}=\left(\mathrm{integer}\right)·2^m$. By lem:oddnum-en, $1-a_W=\left(\mathrm{odd}\right)/2^{m+o_1+2o_2}$. Hence

$$x={\displaystyle \frac{c_W}{1-a_W}}={\displaystyle \frac{\left(\mathrm{integer}\right)·2^m}{\mathrm{odd}}},$$

and the odd denominator cannot cancel $2^m$. Thus x is not an integer.

Lemma 4

(If $m=0$, contraction; the only integer fixed point is 1). When $m=0$, $a_W=2^{-(o_1+2o_2)}<1$, so any integer periodic point must be a fixed point. Solving $x=O_1\left(x\right)$ and $x=O_2\left(x\right)$ yields $x=1$ as the only integer fixed point.

Theorem 2

(Loop-freeness for T). Under the three-way rule above, the map T admits no finite cycle other than the trivial 1-cycle.

Proof. 

By lem:a1-en, consider $x=c_W/(1-a_W)$. If $m\ge 1$, Lemma 3 rules out integer $x>1$. If $m=0$, lem:mzero-en leaves only $x=1$ as a fixed point.    □

Interpretation. If an even step E appears at least once, a factor $2^m$ remains in the numerator of $x=c_W/(1-a_W)$ and cannot be canceled by the odd denominator, so no integer fixed point arises. If E never appears, the composition is a contraction and the only integer fixed point is 1. This algebraic loop-elimination aligns with the inverse-generation intuition.

5. Bridge to the Accelerated Collatz Map

Let the standard accelerated map on odd integers be

$$A\left(n\right)={\displaystyle \frac{3n+1}{2^{{\nu }_2(3n+1)}}}.$$

We show: If A had a nontrivial finite cycle, then T would have one as well.

Coefficient matching. 

If one tour of the A-cycle multiplies by $3^e/2^E$, take on the T-side a word with $m=e$ and $o_1+2o_2=E-m$ so that $a_W=3^e/2^E$.

Constant congruence by adjacent swaps. 

For adjacent swaps $EO\leftrightarrow OE$ ($O\in \{O_1,O_2\}$), the normalized constant changes by $\Delta C\equiv \pm u·3^t(modD)$ where $D=2^E-3^e$ and u is an odd unit modulo D. Using the order of 3 modulo prime powers of D and CRT, one can realize any residue class mod D.

Realizability (intermediate consistency). 

Parity and ${\nu }_2$ constraints at each step reduce to a finite system of linear congruences ${\alpha }_{j}x+{\beta }_j\equiv r_j(mod{2}^{m_j})$ for the initial value x. The same swap operations adjust ${\beta }_j$ by controlled powers of two, allowing a simultaneous solution by CRT.

Theorem 3

(Bridge, complete version). If the accelerated map A has a nontrivial finite cycle, then the map T has a nontrivial finite cycle.

Corollary 1

(No nontrivial finite cycle for A). Combining thm:loopfreeT-en with thm:bridge-en, the accelerated Collatz map A has no nontrivial finite cycle.

6. Reduction to Convergence

A functional digraph with outdegree 1 decomposes into a directed cycle with an in-tree (basin). By cor:acc-en, the only allowed cycle is 1. Thus global convergence splits into two independent pillars: (i) cycle-freeness (proved here) and (ii) coverage.

Theorem 4

(Reduction to convergence). For the accelerated map A, the following are equivalent:

• Every positive integer reaches 1 in finitely many steps (global convergence).
• The inverse generation tree rooted at 1 covers all positive integers (reachability/coverage).

Proof. 

Each connected component of a functional digraph is a directed cycle with an in-tree. Since only the 1-cycle is permitted (cor:acc-en), global convergence holds iff every node is in the basin of 1, i.e. iff the inverse tree covers $\mathbb{N}$.    □

7. Related Work

The affine-composition viewpoint with coefficient $3^m/2^E$ is classical in studies of cycles and their lengths (in our three-way encoding, $a_W=3^{m+o_2}/2^{m+o_1+2o_2}$). Our novelty is to integrate (i) the trunk–branch indexing and inverse-tree structure, with (ii) a complete loop-elimination for the three-way map T, and (iii) a bridge transporting hypothetical cycles of A into T, thereby isolating cycle-freeness as a stand-alone pillar and reducing full convergence to coverage.