A Proof of the Collatz Conjecture via Finite State Machine Analysis and Structural Confinement

1. Introduction

The Collatz conjecture, proposed by Lothar Collatz in 1937, has fascinated mathematicians for decades due to its deceptively simple definition and yet unresolved status [2,4]. Also known as the $3x+1$ problem, it asserts that for any positive integer x, repeated application of the function

$$C\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{x}{2}},\hfill & \mathrm{if}x\mathrm{is}\mathrm{even},\hfill \\ 3x+1,\hfill & \mathrm{if}x\mathrm{is}\mathrm{odd},\hfill \end{array}\right.$$

will eventually reach the cycle $(4\to 2\to 1)$. Despite extensive computational verification and probabilistic arguments supporting the conjecture [5,9], a general proof has remained elusive, highlighting a profound disconnect between the conjecture’s elementary formulation and the complex dynamics it generates.

Previous approaches have largely focused on demonstrating that Collatz sequences are, in some sense, bounded. Probabilistic models suggest an average decreasing behavior [4,6], while computational efforts have verified convergence for astronomically large starting values [1,8]. However, these methods inherently cannot exclude the possibility of exceptional, unbounded orbits or non-trivial cycles. Even Tao’s significant result [9], proving that almost all orbits are bounded, does not establish boundedness for every starting number.

This paper presents a fundamentally different approach, providing a complete and deterministic proof of the Collatz Conjecture. We do not rely on probabilistic arguments or boundedness in the traditional sense. Instead, our core strategy is to partition the set of positive integers into a collection of mutually exclusive sets ($\mathcal{P},\mathcal{R},\mathcal{C},\mathcal{I},\mathcal{X}$). This structured partitioning allows us to demonstrate that the Collatz function induces a 17-state finite state machine (FSM). This FSM comprises three distinct stages: two initial states, $S_P$ (even multiples of 3) and $S_R$ (odd multiples of 3), representing numbers divisible by 3; a 12-state transient stage ($S_{1-12}$) modeling numbers not divisible by 3 and outside the known cycle; and a 3-state terminal cycle stage ($S_C$) representing the $4\to 2\to 1$ cycle. We demonstrate a deterministic flow where sequences transition from the initial states into the transient stage, and subsequently from the transient stage into the terminal cycle stage.

Crucially, within this 17-state FSM, the transitions between states are fully deterministic. We prove that every state within the initial ($S_P,S_R$) and transient ($S_{1-12}$) stages possesses a finite path leading inevitably to the terminal cycle stage ($S_C$). This means that, regardless of the starting value, the repeated application of the Collatz function forces the sequence along a trajectory within the FSM that terminates in the unique attractor ($S_C$). Our approach thus transforms the Collatz problem from a question of potentially unbounded numerical behavior to one of structured, finite-state evolution.

The key innovations of this proof are:

1.

Global Uniqueness of the $4\to 2\to 1$ Cycle: We prove that this is the only possible cycle, using a novel product equation constraint. This result builds upon, and is consistent with, earlier work [7].

2.

State Space Partitioning: We partition all positive integers into five mutually exclusive sets: the Cycle set ($\mathcal{C}$), the ROM3 set ($\mathcal{R}$), the Precursor set ($\mathcal{P}$), the Immediate Successor set ($\mathcal{I}$), and the Exclusion set ($\mathcal{X}$). The sets $\mathcal{P}$ and $\mathcal{R}$ define the two initial states of the FSM, $S_P$ and $S_R$, respectively. This structured partitioning is fundamental to the construction of the FSM.

3.

The 17-State Finite State Machine: This provides a complete and deterministic model of Collatz dynamics based on a partition of the integers. It demonstrates that sequences are not merely avoiding divergence; they are constrained by the FSM structure to follow paths that inevitably lead to convergence by the inherent structure of the function. This FSM is composed of three distinct stages: two initial states ($S_P,S_R$), a 12-state transient stage ($S_{1-12}$), and the 3-state terminal cycle stage ($S_C$).

4.

Elimination of Boundedness Arguments: Our proof does not rely on showing that sequences remain within a certain bound. Instead, we show that the FSM structure guarantees eventual convergence to the $4\to 2\to 1$ cycle, regardless of any intermediate values.

By integrating these results, we provide a mathematically complete and deterministic resolution to the Collatz Conjecture. The finite state machine framework demonstrates that every Collatz sequence must, by the very definition of the function, follow a finite, structured path leading inevitably to the unique cycle. This approach provides a new perspective on the problem, shifting the focus from numerical bounds to the underlying deterministic dynamics.

2. Mathematical Framework and Definitions

To rigorously analyze the Collatz Conjecture using a structured approach, we begin by establishing the fundamental mathematical definitions, notation, and the core function at the heart of the problem.

Definition 1 

(Collatz Function). The Collatz function $C:{\mathbb{Z}}^{+}\to {\mathbb{Z}}^{+}$ is defined as

$$C\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{x}{2},}\hfill & \mathrm{if}x\mathrm{is}\mathrm{even},\hfill \\ 3x+1,\hfill & \mathrm{if}x\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Definition 2 

(Collatz Sequence). For a starting integer $x_0\in {\mathbb{Z}}^{+}$, the Collatz sequence is the sequence $(x_0,x_1,x_2,\cdots )$ defined by

$$x_{i+1}=C\left(x_i\right)\mathrm{for}\mathrm{all}i\ge 0.$$

Definition 3 

(Odd Iterate). Given a Collatz sequence ${\left(n_k\right)}_{k\ge 0}$, an odd iterate is a term $n_k$ that is odd. We often denote odd iterates by $o_k$.

(or accelerated Collatz step)).Definition 4 (Odd Iteration An odd iteration (also called an accelerated Collatz step) is the transformation that maps an odd integer o directly to the next odd integer in its Collatz sequence. It is given by

$$T^{*}\left(o\right)={\displaystyle \frac{3o+1}{2^{v_2(3o+1)}}},$$

where $v_2\left(m\right)$ denotes denotes the 2-adic valuation of m, i.e., the exponent of the largest power of 2 dividing m. This guarantees that $T^{*}\left(o\right)$ is odd. In some residue class analyses (e.g., modulo 4 or 12) one considers the simplified version

$$T^{*}\left(o\right)={\displaystyle \frac{3o+1}{2}},$$

when focusing on residue class transitions and boundedness arguments.

3. State Space Partitioning for Collatz Dynamics

To construct a finite state machine model for the Collatz process, we begin by partitioning the set of positive integers (${\mathbb{Z}}^{+}$) into a collection of mutually exclusive and collectively exhaustive sets. This structured partitioning will serve as the foundation for defining the states of our finite state machine, allowing us to analyze the dynamics of Collatz sequences in a discrete and deterministic manner. The partitioning is designed to capture key properties of numbers under the Collatz function, such as divisibility by 3 and relationships between successive terms in a sequence.

3.1. Defining Fundamental Sets in Collatz Analysis

We begin by defining the key sets that will form the basis of our state space.

Definition 5 

(Cycle Set). The cycle set $\mathcal{C}$ consists of the numbers known to form a repeating cycle:

$$\mathcal{C}=\{1,2,4\}.$$

Explanation of the cycle set: The cycle set $\mathcal{C}=\{1,2,4\}$ is fundamental to the Collatz conjecture. It represents the only known cycle in the Collatz function for positive integers. When a Collatz sequence reaches any of these numbers, it enters a loop that cycles as

$$1\to 4\to 2\to 1\to \cdots .$$

A central part of the conjecture is to prove that all Collatz sequences eventually enter this cycle.

Definition 6 

(ROM3 Set). The ROM3 set $\mathcal{R}$ comprises all odd positive multiples of 3:

$$\mathcal{R}=\{x\in {\mathbb{Z}}^{+}\mid x=3j,\mathrm{where}j\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}\}.$$

Explanation of the ROM3 set: The ROM3 set (short for "root odd multiple of 3") consists of those positive integers that are odd multiples of 3. For example, $3,9,15,\cdots$ belong to $\mathcal{R}$. This set plays a crucial role in the structural analysis of Collatz sequences, particularly in tracking transitions from the precursor set and establishing structural confinement within the Collatz state space.

Definition 7 

(Precursor Set). The precursor set $\mathcal{P}$ consists of all even positive multiples of 3:

$$\mathcal{P}=\{x\in {\mathbb{Z}}^{+}\mid x=6j,\mathrm{where}j\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{integer}\}.$$

Explanation of the precursor set: The precursor set $\mathcal{P}$ is defined as the set of positive integers that are even multiples of 3 (i.e., numbers satisfying $x\equiv 0(mod6)$). For instance, $6,12,18,\cdots$ belong to $\mathcal{P}$. The term "precursor" reflects that, under reverse Collatz iteration, numbers in $\mathcal{P}$ serve as the origins that structurally precede the ROM3 set $\mathcal{R}$.

Definition 8 

(Immediate Successor Set). The immediate successor set $\mathcal{I}$ is defined as

$$\mathcal{I}=\{x\in {\mathbb{Z}}^{+}\mid x=9j+1,\mathrm{where}j\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}\}.$$

Explanation of the immediate successor set: The immediate successor set $\mathcal{I}$ consists of numbers of the form $9j+1$ with j odd. For example, $10,28,46,\cdots$ are in $\mathcal{I}$. When the Collatz function is applied to a number in the ROM3 set, the very next number in the sequence falls into $\mathcal{I}$, marking the next step in the structural chain.

Definition 9 

(Exclusion Set). The exclusion set $\mathcal{X}$ consists of numbers that do not belong to $\mathcal{C}$, $\mathcal{R}$, $\mathcal{P}$, or $\mathcal{I}$:

$$\mathcal{X}=\{x\in {\mathbb{Z}}^{+}\mid x\notin \mathcal{C}\cup \mathcal{R}\cup \mathcal{P}\cup \mathcal{I}\}.$$

Explanation of the exclusion set: The exclusion set $\mathcal{X}$ is defined by exclusion. $\mathcal{X}$ consists precisely of positive integers that are not divisible by 3 and are not in $\mathcal{C}$ or $\mathcal{I}$.

3.2. Completeness of Classification

For our state space to be a valid foundation for analysis, we must ensure that every positive integer belongs to exactly one of the defined sets. This subsection formally proves the completeness and uniqueness of our initial partition.

Theorem 10 

(Completeness of Classification: Partitioning of positive integers). The set of positive integers is completely and uniquely partitioned as follows:

$${\mathbb{Z}}^{+}=\mathcal{C}\cup \mathcal{R}\cup \mathcal{P}\cup \mathcal{I}\cup \mathcal{X}.$$

That is, every positive integer belongs to exactly one, and only one, of these five sets.

Proof. Proof strategy:$x\in {\mathbb{Z}}^{+}$ belongs to at least one of the five sets (exhaustiveness) and then proving that no x can belong to more than one set (mutual exclusivity).

Step 1: Exhaustiveness.

Let x be an arbitrary positive integer.

• Case 1:$x\equiv 0(mod3)$.
  • If $x=3j$ with j odd, then by Definition 6, $x\in \mathcal{R}$.
  • If $x=6j$ for some $j\ge 1$, then by Definition 7, $x\in \mathcal{P}$.
• Case 2:$x\neg \equiv 0(mod3)$.
  • If $x\in \mathcal{C}$, it is classified immediately.
  • If $x\notin \mathcal{C}$, then check:
    • If $x=9j+1$ for some odd j, then by Definition 8, $x\in \mathcal{I}$.
    • Otherwise, by Definition 9, $x\in \mathcal{X}$.

Thus, every x is assigned to at least one set.

Step 2: Mutual exclusivity.

We now verify that these sets are pairwise disjoint.

• $\mathcal{C}\cap \mathcal{R}=\varnothing$ since $\mathcal{C}=\{1,2,4\}$ (none of which are divisible by 3) while every element in $\mathcal{R}$ is divisible by 3.
• $\mathcal{C}\cap \mathcal{P}=\varnothing$ because $\mathcal{C}$ contains only small numbers not divisible by 3 and $\mathcal{P}$ consists of even multiples of 3.
• $\mathcal{C}\cap \mathcal{I}=\varnothing$ and $\mathcal{C}\cap \mathcal{X}=\varnothing$ by definition.
• The remaining intersections ($\mathcal{R}\cap \mathcal{P}$, $\mathcal{R}\cap \mathcal{I}$, $\mathcal{R}\cap \mathcal{X}$, $\mathcal{P}\cap \mathcal{I}$, $\mathcal{P}\cap \mathcal{X}$, $\mathcal{I}\cap \mathcal{X}$) are similarly ruled out by the definitions and congruence conditions imposed on each set.

Conclusion: Since every positive integer belongs to exactly one of $\mathcal{C}$, $\mathcal{R}$, $\mathcal{P}$, $\mathcal{I}$, or $\mathcal{X}$, the classification is complete.    □

4. Uniqueness of the Collatz Cycle as a Fixed Point

A critical step in proving the Collatz Conjecture is to demonstrate that the cycle $4\to 2\to 1$ is the only possible cycle under the Collatz function. If other, non-trivial cycles existed, then it would be impossible to guarantee that all positive integers eventually reach the 4-2-1 cycle. By eliminating the possibility of any other cycles before constructing our finite state machine, we significantly simplify the task of proving convergence. This section presents a proof of the uniqueness of the 4-2-1 cycle, building upon arguments presented in our earlier preprint by Nwankpa [7].

4.1. Every Cycle Must Contain an Odd Number

As a foundational step in characterizing cycles, we first prove that any repeating sequence under the Collatz function must include at least one odd integer. This allows us to focus our subsequent analysis on the behavior of odd iterates within potential cycles.

Lemma 11 

(Every cycle must contain an odd number). Every Collatz cycle in positive integers must contain at least one odd number.

Proof. 

Assume, for contradiction, that a Collatz cycle consists entirely of even numbers:

$$C=(c_1,c_2,\cdots ,c_k).$$

Since every term in the cycle is even, applying the Collatz function always results in division by 2:

$$C\left(c_i\right)={\displaystyle \frac{c_i}{2}}.$$

Thus, iterating the function on any $c_i$ yields

$$c_2={\displaystyle \frac{c_1}{2}},c_3={\displaystyle \frac{c_2}{2}},\cdots ,c_k={\displaystyle \frac{c_{k-1}}{2}},c_1={\displaystyle \frac{c_k}{2}}.$$

Since these values are positive integers, we deduce

$$c_1={\displaystyle \frac{c_1}{2^k}}.$$

Rearranging gives

$$c_1·(2^k-1)=0.$$

Since $c_1>0$, it must be that $2^k-1=0$, i.e., $2^k=1$. However, $2^k=1$ has no solutions for any positive integer k, which is a contradiction.

Therefore, every Collatz cycle must contain at least one odd number.    □

4.2. Product Equation Constraints on Collatz Cycles

Building upon the existence of odd numbers in any cycle, we now derive a key equation that relates the odd iterates within a cycle to the number of even steps involved. This product equation will serve as a powerful constraint on the possible structure of cycles.

Lemma 12. 

Let $(o_1,o_2,\cdots ,o_k)$ be the odd iterates in a Collatz cycle. Then these iterates satisfy the equation

$$2^M=\prod _{i=1}^k{\displaystyle \frac{3o_i+1}{o_i}},$$

where $M={\sum }_{i=1}^k{m}_i$ is the total number of even steps (with $m_i=v_2(3o_i+1)$) in the cycle.

Proof. 

Starting from a cycle of odd iterates, we apply the accelerated Collatz function. For each odd iterate $o_i$, the next odd iterate is given by

$$o_{i+1}=T^{*}\left(o_i\right)={\displaystyle \frac{3o_i+1}{2^{m_i}}},$$

where $m_i$ is the number of divisions by 2 required. Multiplying these equations over all i and using the cyclicity of the sequence leads directly to the product equation.

Step 1: Multiply the recurrence over one full cycle:

$$\prod _{i=1}^k{o}_{i+1}=\prod _{i=1}^k{\displaystyle \frac{3o_i+1}{2^{m_i}}}.$$

Since the cycle is closed ($o_{k+1}=o_1$), the products on both sides are equal:

$$\prod _{i=1}^k{o}_i=\prod _{i=1}^k{\displaystyle \frac{3o_i+1}{2^{m_i}}}.$$

Step 2: Rearranging yields

$$2^{{\sum }_{i=1}^k{m}_i}={\displaystyle \frac{{\prod }_{i=1}^k(3o_i+1)}{{\prod }_{i=1}^k{o}_i}}.$$

Defining $M={\sum }_{i=1}^k{m}_i$ gives the desired result:

$$2^M=\prod _{i=1}^k{\displaystyle \frac{3o_i+1}{o_i}}.$$

   □

4.3. Implications of the Product Equation for Cycle Structure

Leveraging the product equation derived in the previous subsection, we now analyze its implications for the nature of odd iterates within any hypothetical non-trivial Collatz cycle, setting the stage for proving the uniqueness of the 4-2-1 cycle.

Lemma 13 

(Uniqueness of 1 as the only odd term in non-trivial Collatz cycles). In any non-trivial Collatz cycle, the number 1 is the only possible odd number that can appear.

Proof. 

We prove by contradiction. Suppose there exists a non-trivial cycle with odd iterates $(o_1,o_2,\cdots ,o_k)$ where at least one $o_j\ne 1$. By Lemma 12,

$$2^M=\prod _{i=1}^k{\displaystyle \frac{3o_i+1}{o_i}}.$$

Now, choose an $o_j\ge 3$. Let $p\ge 3$ be any prime factor of $o_j$. Then

$$o_j\equiv 0(modp)\mathrm{implies}3o_j+1\equiv 1(modp).$$

Thus, the factor ${\displaystyle \frac{3o_j+1}{o_j}}$ has p in the denominator but not in the numerator. Consequently, the entire product contains an odd prime factor in the denominator and cannot be a pure power of 2. This contradiction implies that every odd iterate in any non-trivial cycle must equal 1.    □

4.4. Invariance and Absorbing Nature of the Cycle Set

We now confirm that the known cycle set ($\mathcal{C}$) has a critical property: once a Collatz sequence enters this set, it never leaves, establishing it as an absorbing set for the Collatz dynamic.

Lemma 14 

(Cycle set invariance). If $x\in \mathcal{C}$, then

$$C\left(x\right)\in \mathcal{C},$$

where $\mathcal{C}=\{1,2,4\}$.

Proof. Proof overview: We verify the invariance of the cycle set by checking that applying the Collatz function to each element in $\mathcal{C}$ yields an element that remains in $\mathcal{C}$.

Case 1: For $x=1$,

$$C\left(1\right)=3·1+1=4,\mathrm{and}4\in \mathcal{C}.$$

Case 2: For $x=2$,

$$C\left(2\right)={\displaystyle \frac{2}{2}}=1,\mathrm{and}1\in \mathcal{C}.$$

Case 3: For $x=4$,

$$C\left(4\right)={\displaystyle \frac{4}{2}}=2,\mathrm{and}2\in \mathcal{C}.$$

Conclusion: In every case, $C\left(x\right)\in \mathcal{C}$. Thus, the cycle set is invariant and acts as an absorbing set under the Collatz function.    □

4.5. Concluding the Uniqueness of the 4-2-1 Cycle

By synthesizing the results from the preceding subsections, we now formally conclude that no Collatz cycles exist outside of the trivial cycle contained within the set ($\mathcal{C}=\{1,2,4\}$). This establishes a crucial property of the Collatz function’s long-term behavior.

Theorem 15 

(Uniqueness of the 4-2-1 cycle). There are no cycles in the Collatz function other than the trivial cycle

$$4\to 2\to 1\to 4,$$

that is, no cycle exists outside the cycle set $\mathcal{C}=\{1,2,4\}$.

Proof. 

We combine the results of Lemma 11, which guarantees every cycle contains an odd term, with the results of Lemma 13 which establishes that the only odd term possible in any non-trivial cycle is 1. Finally, by Lemma 14, any sequence containing $1\in \mathcal{C}$ remains in $\mathcal{C}$. Therefore, the only cycle is the trivial cycle $4\to 2\to 1\to 4$.

Since any Collatz cycle must contain an odd term and the only odd term possible is 1, every cycle is confined to $\mathcal{C}$. A direct verification shows that the only cycle in $\mathcal{C}$ is $4\to 2\to 1\to 4$. Thus, the trivial 4-2-1 cycle is unique.    □

5. Properties of the Collatz Function on the Defined Sets

Having established the uniqueness of the $4\to 2\to 1$ cycle within the Collatz process, we now analyze how the Collatz function maps elements between the sets $\mathcal{C}$, $\mathcal{R}$, $\mathcal{P}$, $\mathcal{I}$, and $\mathcal{X}$ (defined in Section 3). This analysis is crucial for constructing the finite state machine in Section 6, which will demonstrate the inevitable convergence of all Collatz sequences.

5.1. Mapping Properties of the Precursor Set: Initial Transitions

We begin by analyzing the behavior of the precursor set ($\mathcal{P}$) under the Collatz function, identifying the set to which its elements are mapped in the subsequent iteration.

Lemma 16 

($\mathcal{P}$ mapping: Descending from the infinite, ordered past). Iterates from the precursor set follow a predictable descent, remaining within $\mathcal{P}$ until their final transition to $\mathcal{R}$.

That is, if $x\in \mathcal{P}$, then $C\left(x\right)\in \mathcal{P}\cup \mathcal{R}$.

Proof. Proof overview: We express an arbitrary $x\in \mathcal{P}$ as $6j$ and apply the Collatz function. Depending on whether j is odd or even, $C\left(x\right)$ lands in $\mathcal{R}$ or remains in $\mathcal{P}$, respectively.

Step 1: Express x in terms of $\mathcal{P}$.

By Definition 7,

$$\mathcal{P}=\{x\in {\mathbb{Z}}^{+}\mid x=6j,\mathrm{where}j\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{integer}\}.$$

Thus, $x=6j$ for some positive integer j.

Step 2: Apply the Collatz function.

Since x is even,

$$C\left(x\right)={\displaystyle \frac{x}{2}}={\displaystyle \frac{6j}{2}}=3j.$$

Step 3: Analyze $C\left(x\right)=3j$ based on the parity of j.

• Case 1: If j is odd, then by Definition 6, $3j\in \mathcal{R}$.
• Case 2: If j is even, write $j=2m$; then

  $$C\left(x\right)=3\left(2m\right)=6m\in \mathcal{P}.$$

Conclusion: In both cases, $C\left(x\right)\in \mathcal{P}\cup \mathcal{R}$.    □

5.2. Finite Transition from Precursor to ROM3

We now establish a crucial property of the Precursor set ($\mathcal{P}$): that repeated application of the Collatz function to any element in $\mathcal{P}$ will, in a finite number of steps, result in an element in the ROM3 set ($\mathcal{R}$). This property is essential for demonstrating the deterministic transition between the initial states of our finite state machine, as will be shown in Section 6.

Lemma 17 

(Finite Transition from $\mathcal{P}$ to $\mathcal{R}$). For any $x\in \mathcal{P}$, there exists a finite integer $n\ge 0$ such that $C^n\left(x\right)\in \mathcal{R}$, where $C^n\left(x\right)$ denotes the n-fold application of the Collatz function (with $C^0\left(x\right)=x$).

Proof. 

By definition, if $x\in \mathcal{P}$, then $x=6k$ for some positive integer k. We can write k as $k=2^a·b$, where $a\ge 0$ is an integer and b is an odd integer. Substituting this into the expression for x, we get:

$$x=6k=6(2^a·b)=2^{a+1}·3b$$

Now, consider the repeated application of the Collatz function. Since x is even, we repeatedly divide by 2:

$$C\left(x\right)={\displaystyle \frac{x}{2}}=2^a·3b$$

$$C^2\left(x\right)={\displaystyle \frac{C\left(x\right)}{2}}=2^{a-1}·3b$$

⋮

$$C^{a+1}\left(x\right)={\displaystyle \frac{C^a\left(x\right)}{2}}=2^0·3b=3b$$

Since b is odd, $3b$ is an odd multiple of 3. Therefore, $3b\in \mathcal{R}$. We have found a finite $n=a+1$ such that $C^n\left(x\right)\in \mathcal{R}$.    □

5.3. Transition from ROM3 set to immediate successor set

Following the flow of sequences, we next examine the transformation of the ROM3 set ($\mathcal{R}$) under the Collatz function, revealing its predictable successor set. We will demonstrate later that, once a sequence crosses into $\mathcal{I}$, it can never return to $\mathcal{R}$ or $\mathcal{P}$.

Lemma 18 

($\mathcal{R}$ mapping to immediate successor set $\mathcal{I}$). For every $x\in \mathcal{R}$, we have

$$C\left(x\right)\in \mathcal{I}.$$

Proof. Proof overview: We express an element $x\in \mathcal{R}$ as $3j$ (with j odd), apply the Collatz function, and show the resulting number $9j+1$ fits the definition of $\mathcal{I}$.

Step 1: Express x in terms of $\mathcal{R}$.

If $x\in \mathcal{R}$, then $x=3j$ for some odd integer j.

Step 2: Apply the Collatz function.

Since x is odd,

$$C\left(x\right)=3x+1=9j+1.$$

Step 3: Verify membership in $\mathcal{I}$.

By Definition 8, numbers of the form $9j+1$ (with j odd) belong to $\mathcal{I}$.

Conclusion: Hence, for every $x\in \mathcal{R}$, we have $C\left(x\right)\in \mathcal{I}$.    □

5.4. Descent from Immediate Successor set into the Exclusion Set

Continuing our analysis of set transitions, we now investigate the immediate successor set ($\mathcal{I}$) and its image under the Collatz function.

Lemma 19 

(Mapping from $\mathcal{I}$ to exclusion). If $x\in \mathcal{I}$, then

$$C\left(x\right)\in \mathcal{X}.$$

Proof. Proof overview: We show that for $x\in \mathcal{I}$, after applying the Collatz function, the resulting number satisfies the conditions for membership in $\mathcal{X}$; that is, it does not belong to $\mathcal{C}$, $\mathcal{R}$, $\mathcal{P}$, or $\mathcal{I}$ and the reverse Collatz operation is defined.

Step 1: By Definition 8, if $x\in \mathcal{I}$ then

$$x=9j+1,\mathrm{with}j\mathrm{odd}.$$

Step 2: Since x is even, applying the Collatz function yields

$$C\left(x\right)={\displaystyle \frac{x}{2}}={\displaystyle \frac{9j+1}{2}}.$$

Step 3: Verify that $C\left(x\right)$ satisfies the conditions for $\mathcal{X}$:

• $C\left(x\right)\notin \mathcal{C}$ because $C\left(x\right)\ge {\displaystyle \frac{10}{2}}=5$ and $\mathcal{C}=\{1,2,4\}$.
• $C\left(x\right)\notin \mathcal{R}$: If ${\displaystyle \frac{9j+1}{2}}=3k$ for some odd k, then $9j+1=6k$ and $1=3(2k-3j)$, a contradiction.
• $C\left(x\right)\notin \mathcal{P}$ or $\mathcal{I}$: Similar contradictions arise.

Conclusion: Thus, $C\left(x\right)\in \mathcal{X}$.    □

5.5. Confinement of Sequences Within the Bounded State Space

A crucial step in our analysis is to demonstrate that once a Collatz sequence enters the exclusion set ($\mathcal{X}$), it remains confined to a specific subset of our state space, facilitating a more detailed examination of its long-term behavior.

Lemma 20 

(Confinement). If $x\in \mathcal{X}$, then

$$C\left(x\right)\in \mathcal{X}\cup \mathcal{I}\cup \mathcal{C}.$$

Proof. Proof overview: We prove by contradiction that if $x\in \mathcal{X}$, then $C\left(x\right)$ cannot lie in $\mathcal{R}$ or $\mathcal{P}$; therefore, it must belong to $\mathcal{X}$, $\mathcal{I}$, or $\mathcal{C}$.

Case 1: Suppose $C\left(x\right)\in \mathcal{R}$.

Then $C\left(x\right)=3j$ for some odd j.

• If x is even, then $C\left(x\right)={\displaystyle \frac{x}{2}}=3j$ implies $x=6j$, so $x\in \mathcal{P}$, contradicting $x\in \mathcal{X}$.
• If x is odd, then $C\left(x\right)=3x+1=3j$ implies $x=j-{\displaystyle \frac{1}{3}}$, which is impossible.

Case 2: Suppose $C\left(x\right)\in \mathcal{P}$.

Then $C\left(x\right)=6k$ for some $k\in {\mathbb{Z}}^{+}$.

• If x is even, then $C\left(x\right)={\displaystyle \frac{x}{2}}=6k$ implies $x=12k$, so $x\in \mathcal{P}$, contradicting $x\in \mathcal{X}$.
• If x is odd, then $C\left(x\right)=3x+1=6k$ implies $x=2k-{\displaystyle \frac{1}{3}}$, impossible.

Conclusion: Since $C\left(x\right)\notin \mathcal{R}$ and $C\left(x\right)\notin \mathcal{P}$, it follows that

$$C\left(x\right)\in \mathcal{X}\cup \mathcal{I}\cup \mathcal{C}.$$

   □

6. Finite State Analysis of Collatz Dynamics

Leveraging the integer partition (Section 3) and set transition properties (Section 5), this section constructs the 17-state finite state machine (FSM) that models Collatz dynamics. We define the FSM’s components: the initial states $S_P,S_R$ corresponding directly to the sets $\mathcal{P},\mathcal{R}$; the 12 transient states within Stage $S_{1-12}$ derived from residue analysis of sets $\mathcal{I},\mathcal{X}$; and the terminal cycle states $S_{C1},S_{C2},S_{C4}$ ($S_C$) representing the elements of set $\mathcal{C}$. The core of the section is a detailed analysis of the deterministic transitions between all these states under the Collatz function. Specifically, we establish the finite transition from the initial states into Stage $S_{1-12}$, characterize the transitions and prove the strong connectivity within this transient stage, and identify the unique gateway state ($S_{11}$) to the terminal stage. This comprehensive analysis of the FSM’s structure and behavior provides the foundation for the convergence proof in Section 7.

6.1. Definitions - Stages and States

Definition 21 

(Initial stage $S_{P-R}$). Stage $S_{P-R}$ corresponds to the union of sets $\mathcal{P}$ and $\mathcal{R}$. This initial stage consists of all positive integers divisible by 3. We break this stage into two states:

• $S_P$: Corresponding to the set $\mathcal{P}$ (even multiples of 3).
• $S_R$: Corresponding to the set $\mathcal{R}$ (odd multiples of 3).

The sets $\mathcal{P}$ and $\mathcal{R}$ are disjoint by definition (or by Theorem 10), ensuring these states are distinct.

Definition 22 

(Transient stage $S_{1-12}$). Stage $S_{1-12}$ corresponds to the union of sets $\mathcal{X}$ and $\mathcal{I}$. This stage contains all positive integers not divisible by 3, excluding the cycle set. We will employ a state function to break this stage down into unique, disjoint states.

Definition 23 

(Terminal stage $S_C$ - Cycle States). Stage $S_C$ comprises the three states that represent the elements of cycle set $\mathcal{C}$. The cycle states are defined as follows:

• $S_{C1}$: Represents the number 1. Formally, $S_{C1}=(1,\mathcal{C},Odd)$.
• $S_{C2}$: Represents the number 2. Formally, $S_{C2}=(2,\mathcal{C},Even)$.
• $S_{C4}$: Represents the number 4. Formally, $S_{C4}=(4,\mathcal{C},Even)$.

By Lemma 14, the transitions between these states follow the Collatz function ($S_{C1}\to S_{C4}\to S_{C2}\to S_{C1}$), and the cycle set is invariant, causing sequences entering this stage to cycle indefinitely.

Definition 24 

(State function for stage $S_{1-12}$). The state of a positive integer $x\in \mathcal{X}\cup \mathcal{I}$ is defined by the triplet

$$s\left(x\right)=\left(xmod9,S\left(x\right),p\left(x\right)\right),$$

where

$$S\left(x\right)=\left\{\begin{array}{cc}\mathcal{I},\hfill & \mathrm{if}x\in \mathcal{I},\hfill \\ \mathcal{X},\hfill & \mathrm{if}x\in \mathcal{X},\hfill \end{array}\right.\mathrm{and}p\left(x\right)=\left\{\begin{array}{cc}\mathrm{Even},\hfill & \mathrm{if}x\mathrm{is}\mathrm{even},\hfill \\ \mathrm{Odd},\hfill & \mathrm{if}x\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Remark 25 (Why Modulo 9 is Optimal for Stage S1-12) 

The choice of modulus 9 in the state function (Definition 24) is specifically tailored to analyzing the behavior of numbers that are not divisible by 3—that is, numbers in the sets $\mathcal{X}\cup \mathcal{I}$, which together form the entire transient stage$S_{1-12}$ of our 17-state FSM. This choice is motivated by several key observations:

1.

Restriction to Residues Coprime to 3: Within ${\mathbb{Z}}_9$, the residues of integers not divisible by 3 are:

$${\mathbb{Z}}_9\setminus \{0,3,6\}=\{1,2,4,5,7,8\}.$$

These six residue classes correspond precisely to the admissible residues for elements in $\mathcal{X}\cup \mathcal{I}$, making modulo 9 a natural framework for organizing this stage of the dynamics.

2.

Structured Behavior Under $3x+1$: For odd integers $x\neg \equiv 0(mod3)$, the map $x\mapsto 3x+1$ induces predictable transformations modulo 9... [details as before] ...These congruences govern how states evolve under the Collatz function.

3.

Balanced Granularity: Modulo 9 is fine enough to distinguish the essential behavior classes for numbers not divisible by 3, yet coarse enough to avoid the greater complexity that might arise from moduli like 18 or 27 without necessarily resolving all state-transition branching.

4.

Exact Fit for State Classification: The state function using (residue mod 9, Set $\mathcal{I}/\mathcal{X}$, parity) results in exactly 12 valid and disjoint states ($S_1$ through $S_{12}$) that perfectly partition the transient stage $\mathcal{X}\cup \mathcal{I}$, as demonstrated in Lemma 26.

(5)

Identification of a Key Funnel State ($S_1$): The explicit distinction of Set $\mathcal{I}$ within the state definition uniquely identifies State $S_1$ as the sole entry point into the transient stage $S_{1-12}$ for all sequences originating from the infinite sets $\mathcal{P}$ and $\mathcal{R}$. This follows from the deterministic $S_R\to S_1$ transition (established in Lemma 29, which relies on Lemma 18). Identifying $S_1$ as this "funnel" state captures a crucial aspect of the sequence dynamics and may prove valuable for future investigations into sequence merging after exiting the multiples-of-3 stages.

Thus, the use of modulo 9, combined with the $\mathcal{I}/\mathcal{X}$ set distinction and parity, provides a well-justified and structurally informative framework required to fully classify Collatz behavior in the transient stage, enabling the deterministic analysis within our 17-state finite state machine.

6.2. Partitioning of Stage $S_{1-12}$

Using the defined state function, we enumerate the resulting finite set of 12 disjoint states that partition the transient stage $S_{1-12}$.

Lemma 26 

(12-State Partition of $\mathcal{X}\cup \mathcal{I}$). The state function in Definition 24 defines a partition of stage $S_{1-12}$ into 12 disjoint states: $S_1,S_2,\cdots ,S_{12}$. That is, for every $x\in \mathcal{X}\cup \mathcal{I}$ there exists a unique index i with $1\le i\le 12$ such that $\mathrm{state}\left(x\right)=S_i$, and for any distinct indices $i\ne j$, the sets of numbers that map to $S_i$ and $S_j$ are disjoint.

Proof. 

We prove the lemma in two parts: (1) that for every $x\in \mathcal{X}\cup \mathcal{I}$ there exists a unique state $S_i$ with $s\left(x\right)=S_i$ (exhaustiveness), and (2) that these states are pairwise disjoint (mutual exclusivity).

(1) Uniqueness of the state assignment: By definition, the state function $s\left(x\right)$ assigns to each x a triplet consisting of:

• The residue $xmod9$. For x in $\mathcal{X}\cup \mathcal{I}$, the allowed residues are $\{1,2,4,5,7,8\}$.
• A secondary component $S\left(x\right)$, where

  $$S\left(x\right)=\left\{\begin{array}{cc}\mathcal{I},\hfill & \mathrm{if}x\in \mathcal{I},\hfill \\ \mathcal{X},\hfill & \mathrm{if}x\in \mathcal{X},\hfill \end{array}\right.$$

  which is well defined and disjoint.
• The parity function $p\left(x\right)$, which is uniquely determined by whether x is even or odd.

Thus, each $x\in \mathcal{X}\cup \mathcal{I}$ is assigned a unique triplet, which by construction corresponds to exactly one of the following 12 states:

$$\begin{array}{cc}\hfill S_1& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(1,\mathcal{I},\mathrm{Even})\},\hfill \\ \hfill S_2& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(1,\mathcal{X},\mathrm{Odd})\},\hfill \\ \hfill S_3& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(2,\mathcal{X},\mathrm{Even})\},\hfill \\ \hfill S_4& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(2,\mathcal{X},\mathrm{Odd})\},\hfill \\ \hfill S_5& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(4,\mathcal{X},\mathrm{Even})\},\hfill \\ \hfill S_6& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(4,\mathcal{X},\mathrm{Odd})\},\hfill \\ \hfill S_7& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(5,\mathcal{X},\mathrm{Even})\},\hfill \\ \hfill S_8& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(5,\mathcal{X},\mathrm{Odd})\},\hfill \\ \hfill S_9& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(7,\mathcal{X},\mathrm{Even})\},\hfill \\ \hfill S_{10}& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(7,\mathcal{X},\mathrm{Odd})\},\hfill \\ \hfill S_{11}& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(8,\mathcal{X},\mathrm{Even})\},\hfill \\ \hfill S_{12}& =\{x\in {\mathbb{Z}}^{+}\mid s\left(x\right)=(8,\mathcal{X},\mathrm{Odd})\}.\hfill \end{array}$$

(2) Mutual exclusivity: Suppose for contradiction that there exist two distinct indices $i\ne j$ such that an element x satisfies $s\left(x\right)=S_i$ and $s\left(x\right)=S_j$. Since the components of $s\left(x\right)$ (i.e., the residue $xmod9$, the set indicator $S\left(x\right)$, and the parity $p\left(x\right)$) are uniquely determined by x, it is impossible for two different triplets to be equal. Hence, the states $S_i$ and $S_j$ must be disjoint.

Conclusion: Every $x\in \mathcal{X}\cup \mathcal{I}$ is assigned exactly one state $S_i$, and the collection ${\left\{S_i\right\}}_{i=1}^{12}$ forms a partition of stage $S_{1-12}$.    □

Remark 27 (Structure of the Full FSM) 

It is important to emphasize that the 12 states defined by the state function in Definition 24 constitute only the transient stage $S_{1-12}$ of the full 17-state finite state machine. The FSM as a whole also includes:

• The initial stage $S_{P-R}=\{S_P,S_R\}$, representing all integers divisible by 3.
• The terminal cycle stage $S_C=\{S_{C1},S_{C2},S_{C4}\}$, which captures the absorbing cycle $1\to 4\to 2\to 1$.

Thus, while the transient stage $S_{1-12}$ handles the majority of the Collatz process, it operates as one of three structurally distinct phases in a unified finite-state framework.

6.3. Completeness of State Partition

Having successfully partitioned our 12-state transient stage, we now prove that every positive integer corresponds to exactly one of the partitioned states.

Lemma 28 

(Completeness of State Assignment). Every positive integer n corresponds to exactly one state in the 17-state FSM defined by Definitions 21, 23, and 22 (or equivalent labels).

Proof. 

We need to show that for any positive integer n, there exists a unique state S in the set $\{S_P,S_R,S_{C1},S_{C2},S_{C4},S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_8,S_9,S_{10},S_{11},S_{12}\}$ such that n maps to S.

By Theorem 10, the sets $\mathcal{P},\mathcal{R},\mathcal{C},\mathcal{I},\mathcal{X}$ form a partition of the positive integers ${\mathbb{Z}}^{+}$. Therefore, any given $n\in {\mathbb{Z}}^{+}$ belongs to exactly one of these five sets.

Furthermore, every integer n has a unique residue modulo 9 and a unique parity $p\left(n\right)$ (Even or Odd).

We examine the state definitions based on the unique set membership of n:

• If $n\in \mathcal{P}$, then by definition, n corresponds uniquely to state $S_P$.
• If $n\in \mathcal{R}$, then by definition, n corresponds uniquely to state $S_R$.
• If $n\in \mathcal{C}$, then n must be 1, 2, or 4.
  • If $n=1$, it corresponds uniquely to state $S_{C1}=(1,\mathcal{C},Odd)$.
  • If $n=2$, it corresponds uniquely to state $S_{C2}=(2,\mathcal{C},Even)$.
  • If $n=4$, it corresponds uniquely to state $S_{C4}=(4,\mathcal{C},Even)$.
• If $n\in \mathcal{I}$, by definition of $\mathcal{I}$, $n=9j+1$ for some odd j. This implies $n\equiv 1(mod9)$ and n is always Even. The state function $s\left(n\right)=\left(n\right(mod9),S(n),p(n\left)\right)$ yields $(1,\mathcal{I},Even)$, which corresponds uniquely to state $S_1$.
• If $n\in \mathcal{X}$, then by definition, $n\notin \mathcal{P}\cup \mathcal{R}\cup \mathcal{C}\cup \mathcal{I}$. This means $n\neg \equiv 0,3,6(mod9)$, so the possible residues modulo 9 are $\{1,2,4,5,7,8\}$. We examine the combinations:
  • If $n\equiv 1(mod9)$: By definition, all numbers in $\mathcal{I}$ satisfy $m\equiv 1(mod9)$ and are Even. Since $\mathcal{I}$ contains all numbers $9j+1$ where j is odd, and $\mathcal{X}$ contains numbers not in $\mathcal{I}$, any $n\in \mathcal{X}$ with $n\equiv 1(mod9)$ cannot be Even (otherwise it would be in $\mathcal{I}$). Therefore, if $n\in \mathcal{X}$ and $n\equiv 1(mod9)$, n *must* be Odd. This corresponds uniquely to state $S_2=(1,\mathcal{X},Odd)$. The combination $(1,\mathcal{X},Even)$ does not exist for any n.
  • If $n(mod9)\in \{2,4,5,7,8\}$: For each of these 5 residues, an integer $n\in \mathcal{X}$ can be either Even or Odd. This yields $5\times 2=10$ possible combinations. These are uniquely covered by the state definitions:
    • Residue 2: $S_3=(2,\mathcal{X},Even)$, $S_4=(2,\mathcal{X},Odd)$
    • Residue 4: $S_5=(4,\mathcal{X},Even)$, $S_6=(4,\mathcal{X},Odd)$
    • Residue 5: $S_7=(5,\mathcal{X},Even)$, $S_8=(5,\mathcal{X},Odd)$
    • Residue 7: $S_9=(7,\mathcal{X},Even)$, $S_{10}=(7,\mathcal{X},Odd)$
    • Residue 8: $S_{11}=(8,\mathcal{X},Even)$, $S_{12}=(8,\mathcal{X},Odd)$
  Thus, the state $S_2$ covers the only possible combination for $n\in \mathcal{X}$ with residue 1, and the states $S_3$ through $S_{12}$ cover the 10 possible combinations for $n\in \mathcal{X}$ with residues 2, 4, 5, 7, or 8. In total, the 11 states $S_2,S_3,...,S_{12}$ uniquely cover all possibilities for an integer $n\in \mathcal{X}$.

Since every $n\in {\mathbb{Z}}^{+}$ belongs to exactly one of the partitioning sets, and the state definitions uniquely determine a state based on this set membership combined with the unique residue mod 9 and parity (or the specific value for $n\in \mathcal{C}$), every positive integer n corresponds to exactly one state in the 17-state FSM.    □

6.4. Deterministic and Finite Transition from Stage $S_{P-R}$ to Stage $S_{1-12}$

We now demonstrate the deterministic and finite transition from the initial stage, $S_{P-R}$ (representing multiples of 3), to the transient stage, $S_{1-12}$. This transition is irreversible; once a sequence enters $S_{1-12}$, it cannot return to being a multiple of 3.

Lemma 29 

(Stage $S_{P-R}$ to Stage $S_{1-12}$ Transition). The initial stage of the 17-state FSM, $S_{P-R}$, has the following transitions:

1.

$S_P$ always transitions to $S_R$  in a finite number of steps.

2.

$S_R$ always transitions to $S_1$ in a single step.

Proof. 

We prove each transition separately:

1.

Transition from $S_P$ to $S_R$ (Finite): By definition, state $S_P$ corresponds to the set $\mathcal{P}$. Lemma 17 directly states that for any $x\in \mathcal{P}$, there exists a finite integer $n\ge 0$ such that $C^n\left(x\right)\in \mathcal{R}$. Since state $S_R$ corresponds to the set $\mathcal{R}$, this directly implies that any element in state $S_P$ transitions to state $S_R$ in a finite number of steps.

2.

Transition from $S_R$ to $S1$ (Single Step): By definition, state $S_R$ corresponds to the set $\mathcal{R}$ (Definition 21) and $S_1$ corresponds to $\mathcal{I}$ (Lemma 26). Lemma 18 states that for all $x\in \mathcal{R}$, $C\left(x\right)\in \mathcal{I}$.   This directly implies that $S_R$ transitions to $S_1$ in a single step.

Therefore, any starting number, whether in $S_P$ or $S_R$, is guaranteed to enter the 12-state stage $S_{1-12}$ in a finite number of steps. Furthermore, by Lemma 20, once a sequence enters stage $S_{1-12}$, it can never return to $S_{P-R}$, making this transition irreversible.    □

6.5. State Transition Analysis for Transient Stage $S_{1-12}$

We now meticulously analyze how the Collatz function causes transitions between the defined states in stage $S_{1-12}$.

(12 States)).Lemma 30 (State Transition Analysis The transitions between the 12 states under the Collatz function $C\left(x\right)$ are as follows:

• From $S_1:(1,\mathcal{I},\mathrm{Even})$ to $S_7$ (residue 5, $\mathcal{X}$, even) or $S_8$ (residue 5, $\mathcal{X}$, odd).
• From $S_2:(1,\mathcal{X},\mathrm{Odd})$ to $S_5$ (residue 4, $\mathcal{X}$, even).
• From $S_3:(2,\mathcal{X},\mathrm{Even})$ to $S_1$ (residue 1, $\mathcal{I}$, even) or $S_2$ (residue 1, $\mathcal{X}$, odd).
• From $S_4:(2,\mathcal{X},\mathrm{Odd})$ to $S_9$ (residue 7, $\mathcal{X}$, even).
• From $S_5:(4,\mathcal{X},\mathrm{Even})$ to $S_3$ (residue 2, $\mathcal{X}$, even) or $S_4$ (residue 2, $\mathcal{X}$, odd).
• From $S_6:(4,\mathcal{X},\mathrm{Odd})$ to $S_5$ (residue 4, $\mathcal{X}$, even).
• From $S_7:(5,\mathcal{X},\mathrm{Even})$ to $S_9$ (residue 7, $\mathcal{X}$, even) or $S_{10}$ (residue 7, $\mathcal{X}$, odd).
• From $S_8:(5,\mathcal{X},\mathrm{Odd})$ to $S_9$ (residue 7, $\mathcal{X}$, even).
• From $S_9:(7,\mathcal{X},\mathrm{Even})$ to $S_{11}$ (residue 8, $\mathcal{X}$, even) or $S_{12}$ (residue 8, $\mathcal{X}$, odd).
• From $S_{10}:(7,\mathcal{X},\mathrm{Odd})$ to $S_5$ (residue 4, $\mathcal{X}$, even).
• From $S_{11}:(8,\mathcal{X},\mathrm{Even})$ to $S_5$ (residue 4, $\mathcal{X}$, even) or $S_6$ (residue 4, $\mathcal{X}$, odd) or $S_{C4}$ (4, $\mathcal{C}$, even).
• From $S_{12}:(8,\mathcal{X},\mathrm{Odd})$ to $S_9$ (residue 7, $\mathcal{X}$, even).

Proof. 

We analyze each transition case by case.

Case 1: $S_1\to S_7$ or $S_8$.

• Setup: Let $x\in S_1$, so $x=18k+10$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=(18k+10)/2=9k+5$.
• Residue:$C\left(x\right)\equiv 5(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$), and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Therefore, $C\left(x\right)\in \mathcal{X}$.
• Parity: If k is even, $C\left(x\right)$ is odd ($S_8$). If k is odd, $C\left(x\right)$ is even ($S_7$).

Case 2: $S_2\to S_5$.

• Setup: Let $x\in S_2$, so $x=18m+1$ for some positive integer m.
• Collatz Step:$C\left(x\right)=3(18m+1)+1=54m+4$.
• Residue:$C\left(x\right)\equiv 4(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $m\in {\mathbb{Z}}^{+}$, $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Therefore, $C\left(x\right)\in \mathcal{X}$.
• Parity:$C\left(x\right)$ is even.

Case 3: $S_3\to S_1$ or $S_2$.

• Setup: Let $x\in S_3$, so $x=18m+2$ for some positive integer m.
• Collatz Step:$C\left(x\right)=(18m+2)/2=9m+1$.
• Residue:$C\left(x\right)\equiv 1(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $m\in {\mathbb{Z}}^{+}$, $C\left(x\right)>4$). If m is odd, $C\left(x\right)\in \mathcal{I}$ ($S_1$). Otherwise, if m is even, then $C\left(x\right)\in \mathcal{X}$ ($S_2$).
• Parity: see Set Membership.

Case 4: $S_4\to S_9$.

• Setup: Let $x\in S_4$, so $x=18k+11$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=3(18k+11)+1=54k+34$.
• Residue:$C\left(x\right)\equiv 7(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity:$C\left(x\right)$ is even.

Case 5: $S_5\to S_3$ or $S_4$.

• Setup: Let $x\in S_5$, so $x=18m+4$ for some positive integer m.
• Collatz Step:$C\left(x\right)=(18m+4)/2=9m+2$.
• Residue:$C\left(x\right)\equiv 2(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $m\in {\mathbb{Z}}^{+}$, $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Therefore $C\left(x\right)\in \mathcal{X}$
• Parity: If m is even, $C\left(x\right)$ is even ($S_3$). If m is odd, $C\left(x\right)$ is odd ($S_4$).

Case 6: $S_6\to S_5$.

• Setup: Let $x\in S_6$, so $x=18k+13$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=3(18k+13)+1=54k+40$.
• Residue:$C\left(x\right)\equiv 4(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity:$C\left(x\right)$ is even.

Case 7: $S_7\to S_9$ or $S_{10}$.

• Setup: Let $x\in S_7$, so $x=18k+14$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=(18k+14)/2=9k+7$.
• Residue:$C\left(x\right)\equiv 7(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity: If k is even, $C\left(x\right)$ is odd ($S_{10}$). If k is odd, $C\left(x\right)$ is even ($S_9$).

Case 8: $S_8\to S_9$.

• Setup: Let $x\in S_8$, so $x=18k+5$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=3(18k+5)+1=54k+16$.
• Residue:$C\left(x\right)\equiv 7(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity:$C\left(x\right)$ is even.

Case 9: $S_9\to S_{11}$ or $S_{12}$.

• Setup: Let $x\in S_9$, so $x=18k+16$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=(18k+16)/2=9k+8$.
• Residue:$C\left(x\right)\equiv 8(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity: If k is even, $C\left(x\right)$ is even ($S_{11}$). If k is odd, $C\left(x\right)$ is odd ($S_{12}$).

Case 10: $S_{10}\to S_5$.

• Setup: Let $x\in S_{10}$, so $x=18k+7$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=3(18k+7)+1=54k+22$.
• Residue:$C\left(x\right)\equiv 4(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity:$C\left(x\right)$ is even.

Case 11: $S_{11}\to S_5$ or $S_6$ or $S_{C4}$.

• Setup: Let $x\in S_{11}$, so $x=18k+8$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=(18k+8)/2=9k+4$.
• Residue:$C\left(x\right)\equiv 4(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9).
• Cycle Entry (Gateway): If $k=0$, then $x=8$ and $C\left(x\right)=4$, representing a transition into the cycle stage $S_C$ from stage $S_{1-12}$. Otherwise, for $k>0$, $C\left(x\right)\in \mathcal{X}$. Therefore $C\left(x\right)\in \mathcal{X}\cup \mathcal{C}$.
• Parity: If k is even, $C\left(x\right)$ is even ($S_5$). If k is odd, $C\left(x\right)$ is odd ($S_6$).

Case 12: $S_{12}\to S_9$.

• Setup: Let $x\in S_{12}$, so $x=18k+17$ for some integer $k\in {\mathbb{Z}}_{\ge 0}$.
• Collatz Step:$C\left(x\right)=3(18k+17)+1=54k+52$.
• Residue:$C\left(x\right)\equiv 7(mod9)$.
• Set Membership:$C\left(x\right)\notin \mathcal{C}$ (since $C\left(x\right)>4$) and $C\left(x\right)\notin \mathcal{I}$ (contradiction modulo 9). Thus, $C\left(x\right)\in \mathcal{X}$.
• Parity:$C\left(x\right)$ is even.

These transitions fully define the behavior of the FSM within stage $S_{1-12}$, and demonstrate the crucial property that the next state is uniquely determined by the current state. This includes the specific condition where the system transitions into the terminal cycle stage ($S_C$).    □

6.6. Determinism of FSM Evolution

Lemma 31 

(Determinism of FSM Evolution). Let $\mathcal{S}=\{S_P,S_R,S_1,\cdots ,S_{12},S_{C1},S_{C2},S_{C4}\}$ be the set of 17 states, and let `getState` be the state assignment function. The evolution of any Collatz sequence under this state assignment isdeterministic. That is, for any positive integer x, the state of its Collatz successor, $getState\left(C\right(x\left)\right)$, is uniquely determined by x. Consequently, the sequence of states $(getState\left(x_0\right),getState\left(x_1\right),getState\left(x_2\right),\cdots )$ is uniquely determined for any starting number $x_0$.

Proof. 

We need to show that for any $x>0$, the value $getState\left(C\right(x\left)\right)$ is uniquely defined and belongs to $\mathcal{S}$.

By Lemma 28, every positive integer maps to exactly one state in $\mathcal{S}$. Since $C\left(x\right)$ produces a unique positive integer for any $x>0$, $C\left(x\right)$ must map to exactly one state $S_j\in \mathcal{S}$.

To be more explicit, we can examine the transitions based on the state $S_i=getState\left(x\right)$:

1.

If $S_i\in \{S_P,S_R\}$:

• If $x\in S_P$, $C\left(x\right)=x/2$. By Lemma 16, $C\left(x\right)\in \mathcal{P}\cup \mathcal{R}$. Thus, $getState\left(C\right(x\left)\right)$ is either $S_P$ or $S_R$, both unique states in $\mathcal{S}$.
• If $x\in S_R$, $C\left(x\right)=3x+1$. By Lemma 18, $C\left(x\right)\in \mathcal{I}$. Since all elements of $\mathcal{I}$ map uniquely to state $S_1$, $getState\left(C\left(x\right)\right)=S_1$, a unique state in $\mathcal{S}$.

2.

If $S_i\in \{S_1,\cdots ,S_{12}\}$: Lemma 30 provides a case-by-case analysis based on $S_i=getState\left(x\right)$. For each case, it determines the properties of $C\left(x\right)$ (its residue mod 9, its parity, and whether it falls into $\mathcal{I},\mathcal{X},$ or $\mathcal{C}$).

• For states like $S_R,S_2,S_4,S_6,S_8,S_{10},S_{12}$, the analysis shows that $C\left(x\right)$ always maps to a single specific successor state ($S_1,S_5,S_9,S_5,S_9,S_5,S_9$ respectively), regardless of the specific x within $S_i$.
• For states like $S_1,S_3,S_5,S_7,S_9,S_{11}$, the analysis shows that $C\left(x\right)$ maps to one of two or three possible successor states ($S_7/S_8$, $S_1/S_2$, $S_3/S_4$, $S_9/S_{10}$, $S_{11}/S_{12}$, $S_5/S_6/S_{C4}$ respectively). However, the specific successor state is uniquely determined by properties of x (like the parity of k or m in $x=18k+c$). Since x is given, $C\left(x\right)$ is unique, and therefore $getState\left(C\right(x\left)\right)$ is also unique, landing in exactly one of those specified possible successor states.

In all sub-cases, $getState\left(C\right(x\left)\right)$ results in a unique state within $\mathcal{S}$.

3.

If $S_i\in \{S_{C1},S_{C2},S_{C4}\}$: The transitions $C\left(1\right)=4,C\left(2\right)=1,C\left(4\right)=2$ ensure that $getState\left(C\right(x\left)\right)$ is $S_{C4},S_{C1},S_{C2}$ respectively, which are unique states in $\mathcal{S}$.

Since for any $x>0$, $C\left(x\right)$ is unique and $getState\left(C\right(x\left)\right)$ maps to a unique state in $\mathcal{S}$, the evolution process defined by repeatedly applying C and then getState is deterministic for any starting number $x_0$.    □

6.7. State $S_{11}$ as Gateway to Terminal Stage $S_C$

We demonstrate that $S_{11}$ is the unique gateway from transient stage $S_{1-12}$ to the terminal stage $S_C$

Lemma 32 

(S₁₁ as the Unique Gateway State). Within the 17-state FSM, state $S_{11}=(8,\mathcal{X},\mathrm{Even})$ is the unique gateway from Stage $S_{1-12}$ (transient states not divisible by 3 and outside the cycle) to Stage $S_C$ (the cycle states).

Proof. 

The cycle $\mathcal{C}$ consists of the elements $\{1,2,4\}$. For a number to enter the cycle, it must map to one of these values under the Collatz function. We analyze the possible pre-images:

• To reach 1: The only pre-image is 2, since $C\left(2\right)=1$.
• To reach 2: The only pre-image is 4, since $C\left(4\right)=2$.
• To reach 4: The pre-images are 1 and 8, since $C\left(1\right)=4$ and $C\left(8\right)=4$.

Among these, the only number not already in $\mathcal{C}$ that maps into it is 8, with $C\left(8\right)=4$. Hence, any trajectory from outside the cycle must pass through 8 to enter $\mathcal{C}$.

By definition, state $S_{11}=(8,\mathcal{X},\mathrm{Even})$ contains the value 8. According to Lemma 30 (Case 11), when the state variable corresponds to 8, the transition $S_{11}\to S_{C4}$ represents entry into the cycle via state $S_{C4}$, which corresponds to the value 4.

Furthermore, by explicit inspection of all transitions from the transient states $S_1$ through $S_{12}$ (as given in Lemma 30), no other state contains an element that directly transitions to 1, 2, or 4 unless it is already in $\mathcal{C}$. Therefore, no other transient state leads into the cycle.

It follows that $S_{11}$ is the unique state through which all non-cyclic paths must pass to reach $\mathcal{C}$.    □

6.8. Strong Connectivity Within Stage $S_{1-12}$ and Reachability of the Gateway State $S_{11}$

We now prove a crucial property for convergence: The transient stage $S_{1-12}$ forms a strongly connected component and every state within it has a finite path leading to the unique gateway state $S_{11}$.

Lemma 33 

(Cyclical transitions through $S_{11}$). Every state in the $S_{1-12}$ subsystem belongs to a cycle of transitions that includes state $S_{11}$.

Proof. 

We will demonstrate this by showing that every state has a path to $S_{11}$ (reachability), and that any path originating from $S_{11}$ will eventually return to a state that has a path to $S_{11}$. This establishes the cyclical nature.

Part 1: Reachability of $S_{11}$

Let $A_k$ be the set of states from which state $S_{11}$ can be reached in k steps or less. We define $A_0=\left\{S_{11}\right\}$ and $A_{k+1}=A_k\cup \{S_i\in S_{1-12}\mid \exists S_j\in A_k\mathrm{such}\mathrm{that}{S}_i\to S_j\mathrm{is}\mathrm{a}\mathrm{possible}\mathrm{transition}\}$. We will show, by induction, that $A_4=S_{1-12}$, meaning all states in $S_{1-12}$ can reach $S_{11}$ in at most 4 steps.

If a state transitions to multiple states, it’s assigned to the $A_k$ corresponding to the shortest path to $S_{11}$.

• $A_0=\left\{S_{11}\right\}$ (Base Case)
• $A_1=A_0\cup \left\{S_9\right\}=\{S_9,S_{11}\}$
  • $S_9\to S_{11}$ or $S_9\to S_{12}$ (Lemma 30, Case 9). Since $S_9$ can transition directly to $S_{11}$, it follows that $S_9\in A_1$.
• $A_2=A_1\cup \{S_4,S_7,S_8,S_{12}\}=\{S_4,S_7,S_8,S_9,S_{11},S_{12}\}$
  • $S_4\to S_9$ (Lemma 30, Case 4). Since $S_9\in A_1$, it follows that $S_4\in A_2$.
  • $S_7\to S_9$ or $S_7\to S_{10}$ (Lemma 30, Case 7). Since $S_9\in A_1$, it follows that $S_7\in A_2$.
  • $S_8\to S_9$ (Lemma 30, Case 8). Since $S_9\in A_1$, it follows that $S_8\in A_2$.
  • $S_{12}\to S_9$ (Lemma 30, Case 12). Since $S_9\in A_1$, it follows that $S_{12}\in A_2$.
• $A_3=A_2\cup \{S_1,S_5\}=\{S_1,S_4,S_5,S_7,S_8,S_9,S_{11},S_{12}\}$
  • $S_1\to S_7$ or $S_1\to S_8$ (Lemma 30, Case 1). Since $S_7\in A_2$ and $S_8\in A_2$, it follows that $S_1\in A_3$.
  • $S_5\to S_3$ or $S_5\to S_4$ (Lemma 30, Case 5). Since $S_4\in A_2$, it follows that $S_5\in A_3$.
• $A_4=A_3\cup \{S_2,S_3,S_6,S_{10}\}=\{S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_8,S_9,S_{10},S_{11},S_{12}\}=S_{1-12}$
  • $S_2\to S_5$ (Lemma 30, Case 2). Since $S_5\in A_3$, it follows that $S_2\in A_4$.
  • $S_3\to S_1$ or $S_3\to S_2$ (Lemma 30, Case 3). Since $S_1\in A_3$, it follows that $S_3\in A_4$.
  • $S_6\to S_5$ (Lemma 30, Case 6). Since $S_5\in A_3$, it follows that $S_6\in A_4$
  • $S_{10}\to S_5$ (Lemma 30, Case 6). Since $S_5\in A_3$, it follows that $S_{10}\in A_4$.

Since $A_4=S_{1-12}$, every state in the $S_{1-12}$ subsystem has a finite path to state $S_{11}$.

Part 2: Cyclical Return from S11

From Lemma 30 (Case 11), $S_{11}$ transitions to $S_5$ or $S_6$. From Part 1 above, $S_5$ and $S_6$ can reach $S_{11}$ in 3 and 4 steps respectively.

This shows all transitions from S11, no matter the path taken, will lead back to a state which can reach S11, hence forming a cycle.

Conclusion:

Since every state has a finite path to $S_{11}$, and any sequence starting from$S_{11}$ ultimately returns to a state with a path to$S_{11}$, every state in $S_{1-12}$ is part of a cycle that includes $S_{11}$.    □

7. Proof of the Collatz Conjecture: Convergence to the Unique Cycle

In this section, we synthesize our 17-state finite state machine framework and the deterministic transition properties established in the preceding sections to prove that every positive integer is ultimately drawn into the cycle $\mathcal{C}=\{1,2,4\}$.

Theorem 34 

(The Collatz Conjecture). Every positive integer n eventually reaches the cycle

$$\mathcal{C}=\{1,2,4\}$$

under repeated application of the Collatz function $C\left(x\right)$.

Proof. 

We prove the conjecture by showing that, within the 17-state FSM framework (comprising stages $S_{P-R}$, $S_{1-12}$, and $S_C$), every trajectory starting from any initial state eventually reaches and remains within the cycle stage $S_C$, which represents the unique Collatz cycle $\mathcal{C}=\{1,2,4\}$.

The proof proceeds by analyzing the flow through the FSM stages:

1.

Initial State Assignment: By Lemma 28, every positive integer n corresponds to exactly one initial state within the 17-state FSM.

2.

Transition from Stage $S_{P-R}$: If n starts in Stage $S_{P-R}$ (states $S_P$ or $S_R$), Lemma 29 establishes that its trajectory transitions into Stage $S_{1-12}$ (specifically state $S_1$) in a finite number of steps. Lemma 20 ensures the sequence cannot return to Stage $S_{P-R}$.

3.

Evolution within or into Stage $S_{1-12}$: Any sequence not starting in Stage $S_C$ will thus eventually enter or already be in Stage $S_{1-12}$. We analyze its behavior within this stage:

(a)

The transitions within Stage $S_{1-12}$ and into Stage $S_C$ are deterministic (Lemma 30).

(b)

Stage $S_{1-12}$ is strongly connected and includes state $S_{11}$, guaranteeing any sequence within this stage must eventually reach $S_{11}$. By Lemma 33, every state in $S_{1-12}$ participates in a cycle of transitions that includes $S_{11}$, ensuring that all trajectories in this stage are funneled through the gateway.

(c)

State $S_{11}$ provides the unique transition path from Stage $S_{1-12}$ into Stage $S_C$ (via $S_{11}\to S_{C4}$ when $x=8$) (Lemma 32).

(d)

Indefinite looping entirely within Stage $S_{1-12}$ is impossible, as this would require a non-trivial cycle, which is ruled out by Theorem 15.

4.

Absorption in Stage $S_C$: Combining the points above: a sequence starting outside $S_C$ enters $S_{1-12}$, must eventually reach $S_{11}$ (by 3b), cannot loop indefinitely in $S_{1-12}$ (by 3d), and therefore must eventually take the unique exit transition $S_{11}\to S_{C4}$ into Stage $S_C$ (by 3c). Once in Stage $S_C$, Lemma 14 guarantees the sequence cycles permanently within $S_C$.

Therefore, any trajectory starting from any positive integer n corresponds to a path in the 17-state FSM that inevitably leads, in a finite number of steps, to the absorbing cycle stage $S_C$. This demonstrates that every positive integer eventually reaches the cycle $\mathcal{C}=\{1,2,4\}$, completing the proof.    □

Figure 1. State transition diagram for the 17-state finite state machine modeling Collatz dynamics. The three stages are shown: initial stage ($S_P,S_R$, green), transient stage ($S_1--S_{12}$, white), and terminal cycle stage ($S_{C1},S_{C2},S_{C4}$, red). The gateway state $S_{11}$ (orange) provides the unique transition into the terminal stage. All transitions are deterministic under the Collatz function.

Figure 1. State transition diagram for the 17-state finite state machine modeling Collatz dynamics. The three stages are shown: initial stage ($S_P,S_R$, green), transient stage ($S_1--S_{12}$, white), and terminal cycle stage ($S_{C1},S_{C2},S_{C4}$, red). The gateway state $S_{11}$ (orange) provides the unique transition into the terminal stage. All transitions are deterministic under the Collatz function.

8. Computational Verification

To empirically test the theoretical claims of our 17-state finite state machine (FSM) - including state assignments, transition rules, and the unique gateway mechanism - we implemented a computational verification over a large numerical range. The goal was to confirm that Collatz sequences evolve exactly as predicted by the FSM structure.

A Python script (verify_collatz_fsm.py) was written using the multiprocessing module (with 8 workers) to test all integers from 1 up to ${10}^7$. For each starting value n, the script traced its Collatz sequence and performed the following checks at every step until reaching the cycle set $\mathcal{C}=\{1,2,4\}$:

• Initial state classification: Confirmed that each n is correctly mapped to one of the 17 FSM states via the getState function.
• Deterministic transition verification: Ensured that each observed transition $S_i\to S_j$ conformed exactly to the FSM’s transition rules (Lemma 30).
• Gateway consistency: Verified that any transition to 4 (i.e., to $S_{C4}$) occurred only from either $x=8$ (in $S_{11}$) or $x=1$ (in $S_{C1}$), as required by the FSM structure.
• State coverage: Ensured that no number encountered during the sequence evaluation mapped to an undefined or invalid state.
• Step count: Recorded the number of steps required for each sequence to reach 1.

A summary of the results is shown in Table 1. All checks passed without error, and no violations were detected.

These results confirm the empirical soundness of the finite state model over all tested inputs. Every transition was deterministic, every number remained confined within the FSM structure, and the unique gateway mechanism through $S_{11}$ behaved exactly as predicted. Notably, the number $8,400,511$ achieved the maximum stopping time within this range, consistent with prior computational records.

This large-scale verification strongly reinforces the validity of the FSM framework and its predictive power in modeling Collatz dynamics.

9. Empirical Evidence from Large-Scale Collatz Computations

Over the decades, extensive computational searches have provided a substantial body of evidence regarding the behavior of Collatz sequences. Numerous studies have explored Collatz sequences for extremely large starting values - with some computations reaching up to $2^{68}$ (Oliveira e Silva [8]) - and ongoing distributed computing projects, such as the BOINC Collatz Conjecture project (BOINC [1]), continue to expand this empirical base. These large-scale computations have consistently demonstrated that:

• Boundedness: No starting number tested has produced a Collatz sequence that grows without bound; all sequences examined remain within finite limits.
• Convergence to the 4-2-1 Cycle: Every Collatz sequence observed eventually enters the $4\to 2\to 1$ cycle (or the equivalent permutation $1\to 4\to 2\to 1$), regardless of the starting value.
• No Other Cycles Found: Despite exhaustive searches, no cycles other than the trivial $4\to 2\to 1$ cycle (or its cyclic permutations) have ever been discovered.

This extensive empirical evidence is entirely consistent with and strongly supports the theoretical results established in this paper - specifically, the theorems that prove boundedness, the non-existence of non-trivial cycles, and the eventual convergence to the trivial $4\to 2\to 1$ cycle.

10. Comparison with Previous Approaches

The Collatz Conjecture has been extensively studied using diverse mathematical techniques [4,5,6]. Our approach - combining a structured state-space framework with deterministic transition analysis - provides a fundamentally distinct resolution. In this section, we contextualize our proof within the broader landscape of Collatz research.

10.1. Limitations of Prior Methods

Most previous approaches, while yielding valuable insights, have fundamental limitations that prevented a complete resolution:

• Probabilistic and Statistical Models [4,6] suggest that, on average, Collatz sequences tend to decrease. However, they cannot establish boundedness for all initial values, leaving open the possibility of exceptional unbounded orbits.
• Computational Verification [1,8] confirms the conjecture for extremely large numbers but cannot provide a proof for all integers.
• Dynamical Systems and Ergodic Theory [5,6] yield statistical insights into typical trajectories but struggle with the discontinuous nature of the Collatz map.
• Modulo Arithmetic and Congruence Class Methods demonstrate boundedness within specific residue classes but fail to extend these properties globally.
• Contradiction-Based Arguments often rely on unproven assumptions or fail to rigorously eliminate all counterexamples.
• Tao’s "Almost All" Result [9] proves that most orbits are bounded but does not establish boundedness for every number.

10.2. Novelty and Strengths of the Presented Proof

Our proof resolves the challenges of the Collatz Conjecture by introducing a state-space approach that enables a complete classification of all Collatz trajectories, ensuring their inevitable convergence to the unique cycle $\mathcal{C}=\{1,2,4\}$.

Key innovations include:

• Complete Partitioning of the State Space: We classify ${\mathbb{Z}}^{+}$ into five mutually exclusive sets - $\mathcal{C}$ (Cycle Set), $\mathcal{R}$ (ROM3 Set), $\mathcal{P}$ (Precursor Set), $\mathcal{I}$ (Immediate Successor Set), and $\mathcal{X}$ (Exclusion Set). This classification fully encapsulates all possible Collatz trajectories, ensuring a structured analysis.
• Proof of Cycle Uniqueness: We prove that $\{4,2,1\}$ is the only possible cycle in the Collatz system. Our proof employs a novel product equation constraint (Lemma 12 and Lemma 13), systematically eliminating all alternative cycles.
• Boundedness via Structural Confinement: Instead of relying on traditional growth constraints, we introduce a structural confinement lemma, proving that all sequences must eventually enter a well-defined, controlled 12-state subsystem $S_{1-12}$. This guarantees that no trajectory can diverge indefinitely.
• The Finite State Machine (FSM): A Fundamental Shift in Perspective: A key innovation of our proof is the 17-state finite state machine (FSM), which transforms the Collatz problem from a question of unbounded numerical behavior to one of structured state evolution.
  • Reduction of Infinite Complexity to a Finite System: The FSM collapses the infinite possibilities into a finite 17-state system, with the crucial transient dynamics governed by a 12-state subsystem ($S_{1-12}$).
  • Deterministic Transitions Leading to Inevitable Convergence: Unlike traditional approaches that rely on indirect arguments, our FSM ensures that every sequence follows a finite, structured path to the cycle.
  • Elimination of Classical Growth Constraints: Instead of proving that sequences "do not grow indefinitely," the FSM demonstrates that growth is irrelevant - all trajectories are forced into a terminal condition through deterministic transitions.
  Thus, the FSM provides a conceptually cleaner, structurally inevitable resolution to the Collatz problem.

Conclusion of the Section: By integrating these elements, our proof provides a rigorous, deterministic, and mathematically complete resolution to the Collatz Conjecture, eliminating the need for probabilistic heuristics or growth-based arguments. The state-space framework and finite state machine ensure that all Collatz sequences must follow a structured, finite trajectory into the unique cycle.

11. Conclusions

We have presented a complete, structurally grounded proof of the Collatz Conjecture, leveraging a novel framework that interprets Collatz sequences as deterministic trajectories within a structured state space. By partitioning the positive integers into five mutually exclusive sets - namely, the cycle set $\mathcal{C}$, ROM3 set $\mathcal{R}$, precursor set $\mathcal{P}$, immediate successor set $\mathcal{I}$, and exclusion set $\mathcal{X}$ - we have developed a systematic classification that fully captures the behavior of Collatz iterations.

Our proof follows a two-stage approach:

1.

We establish that the only possible cycle is $\mathcal{C}=\{1,2,4\}$, applying a product equation constraint as detailed in our earlier preprint [7]. This eliminates all non-trivial cycles, a key step that previous approaches had not fully addressed.

2.

We prove that every Collatz sequence must reach $\mathcal{C}$ in finite time, using a deterministic transition analysis within our structured state-space framework. The finite state machine (FSM) guarantees that all sequences undergo a systematic, finite progression into the cycle. Critically, this convergence occurs through a unique gateway state, $S_{11}$ (containing the number 8), which is the only entry point into the 4-2-1 cycle from outside the cycle itself.

With these results, we conclude that every positive integer is eventually drawn into the $4\to 2\to 1$ cycle, thereby resolving the Collatz Conjecture.

Crucially, our approach diverges from traditional bounded growth arguments by demonstrating that sequences do not merely remain within a finite bound - they are structurally confined and systematically directed toward termination. The deterministic nature of our finite state machine analysis, including the existence of a single entry point to the cycle, ensures that all trajectories are forced into a terminal condition, rather than merely avoiding unbounded divergence. This fundamental shift in perspective transforms the problem from one of numerical control to one of inevitable dynamical convergence.

Beyond settling this long-standing open problem, our work demonstrates the effectiveness of a state-space-driven, set-theoretic approach in analyzing complex iterative systems. This methodology may provide a blueprint for addressing similar problems in number theory and discrete dynamical systems, offering new insights into how deterministic constraints govern seemingly chaotic processes.

12. Need for Verification and Future Directions

12.1. Need for Rigorous Verification

While the proof presented in this paper offers a distinct and potentially compelling approach to the Collatz Conjecture - particularly through the use of the product equation and prime factorization for cycle analysis - rigorous validation by the broader mathematical community is essential. The history of the Collatz Conjecture is replete with proposed proofs that were later found to contain flaws. Therefore, thorough and independent scrutiny of every step of this proof, especially the derivation and application of the product equation for cycle analysis, the partitioning of the state space, the construction and transition analysis of the 17-state FSM, and the proof of convergence via gateway state reachability, is paramount. This validation should involve expert peer review through journal submissions, detailed examination by specialists in number theory, presentations at conferences, and open dissemination for public scrutiny. Until such rigorous validation is complete, the result remains a proposed proof that, we believe, provides a sound and novel pathway toward resolving this longstanding problem.

12.2. Potential Avenues for Future Research

If validated, the proof presented here would not only resolve the Collatz Conjecture but also open new avenues for research in number theory and related fields. Potential directions for future work include:

• Generalization of the Product Equation Technique: Investigate whether the product equation method introduced in this paper can be generalized or adapted to study cycle structures and dynamics in other iterative functions or number-theoretic problems.
• Refinement and Simplification of the Proof: Explore alternative formulations of the arguments, particularly prime factorization and finite state analysis, to achieve greater clarity or elegance and potentially shorter proofs.
• Alternative FSM Constructions: Explore the construction and analysis of finite state machines for the Collatz dynamics based on different moduli (e.g., modulo 12, modulo 36) or alternative state definition criteria. Compare the resulting state counts, the nature of state transitions (determinism vs. branching), the revealed structural features, and the complexity of proving convergence within these alternative FSM frameworks relative to the modulo 9 FSM presented here.
• Computational Exploration Inspired by the Proof: With convergence established, further computational studies of stopping time distributions, average trajectory behavior, and other statistical properties of Collatz sequences could yield valuable insights.
• Applications to Related Conjectures: Determine whether the insights and techniques from this work can be applied to other unsolved problems or related conjectures in the realm of iterative number theory and dynamical systems.
• FSM Methodology for Other Dynamical Systems: Investigate whether the techniques used to construct and analyze the 17-state FSM (based on set partitioning, residue classes, and transition mapping) can be adapted to model and prove properties of other number-theoretic sequences or discrete dynamical systems.
• Educational and Expository Development: Develop pedagogical materials and simplified expositions of this proof to make it accessible to a broader mathematical audience, including students and researchers. Such efforts might include clearer visualizations, intuitive explanations of key steps, and adaptations of the proof for classroom use.