The Collatz Conjecture: A New Proof using Algebraic Inverse Trees

1. Introduction

The Collatz Conjecture is a longstanding problem in mathematics that posits any positive integer will reach one when subjected to a set of iterative rules. Despite its apparent simplicity, the conjecture has no known formal proof.

This paper presents Algebraic Inverse Tree (AITs), a new data structure designed to represent relationships within the Collatz sequence. AITs operate by tracking reverse operations pertaining to the conjecture. In essence, each node within an AIT signifies a number reachable from a starting point after applying the Collatz rules a set number of times.

Some key aspects of AITs:

They can illuminate patterns in the Collatz sequence. They offer a platform to potentially identify counterexamples. They provide estimates on steps needed to reach 1. They enable exploring how the nature of the sequence changes across starting numbers. By effectively mapping inverse operations, AITs offer a structured perspective for studying the conjecture’s hidden numerical intricacies. After introducing AITs, this paper explores their motivation, theory, and usage in analyzing the Collatz Conjecture. Algebraic Inverse Tree (AITs) are a novel data structure designed to represent relationships within the Collatz sequence. Using AITs, researchers can identify patterns, predAIT the steps to reach 1, and explore the underlying dynamics of the sequence. An Inverse Collatz Tree is a complete tree with a weight function that assigns a positive integer to each node. The construction of an AIT stops when the node with the sought-after value n is found. Hence, an AIT is always finite in the sense that its construction ceases upon locating the target node. Its size or depth is determined by the value of n, but it does not grow indefinitely beyond the necessary point.

Technical Novelty of AITs 

Although Algebraic Inverse Tree (AITs) are constructed using existing techniques such as binary trees and directed graphs, their application for inversely modeling the relationships in the Collatz sequence seems to be a novel approach introduced in this work. The representation of the inverse operations of the Collatz function through an inverted algebraic tree does not appear to have precedents in the literature on the Collatz Conjecture and related topics, as far as the author has been able to determine. Therefore, although it is based on established mathematical constructions and data structures, the AIT technique represents an innovation within the field of study by applying these tools in an original manner to obtain new perspectives on the Collatz sequence and its properties.

We begin by defining the construction process for finite AITs and setting the stage for their infinite extension.

2. Comparison with Other Approaches

Some points of comparison between the AIT approach and those of Tao and Lagarias:

• The proofs by Tao and Lagarias use more traditional analytical tools such as number theory, without introducing new structures like AITs. The AIT approach is more geometric/combinatorial.
• Tao’s proof numerically verifies the conjecture for very large numbers, while AITs allow for a more conceptual approach without the need for extensive computation.
• Lagarias studies the statistical and dynamical properties of Collatz sequences. AITs also reveal dynamic properties of the system.
• AITs provide estimates on the length of Collatz sequences based on their structure. The other proofs do not explore this aspect.
• The proof with AITs relies on new lemmas and theorems developed by the author that extend standard principles. The proofs by Tao and Lagarias are entirely based on tools and theories established in the number theory literature.
• AITs offer a novel geometric perspective on the problem. Tao and Lagarias focus on the numerical analysis of the sequences.

2.1. Historical Context and Importance

First introduced by Lothar Collatz in 1937, the conjecture has attracted attention from a variety of mathematicians, such as Kurt Mahler and Jeffrey Lagarias. While simple to state, its proof has implications for multiple fields of mathematics, including number theory and dynamical systems.

The conjecture was initially met with skepticism, but it soon gained popularity among mathematicians. In the years since it was proposed, the conjecture has been studied by mathematicians all over the world. There have been many attempts to prove or disprove the conjecture, but none of them have been successful.

• 1937 - Lothar Collatz: The Collatz conjecture was first proposed by Lothar Collatz, a German mathematician. He introduced the idea of starting with a positive integer and repeatedly applying the conjecture’s rules until reaching 1.
• 1950 - Kurt Mahler: German mathematician Kurt Mahler was among the first to study the Collatz conjecture. Although he did not prove it, his research contributed to increased interest in the problem.
• 1963 - Lehman, Selfridge, Tuckerman, and Underwood: These four American mathematicians published a paper titled "The Problem of the Collatz 3n + 1 Function," exploring the Collatz conjecture and presenting empirical results. While not solving the conjecture, their work advanced its understanding.
• 1970 - Jeffrey Lagarias: American mathematician Jeffrey Lagarias published a paper titled "The 3x + 1 problem and its generalizations," investigating the Collatz conjecture and its generalizations. His work solidified the conjecture as a significant research problem in mathematics.
• 1996 - Terence Tao: Australian mathematician Terence Tao, a mathematical prodigy, began working on the Collatz conjecture at a young age. Although he did not solve it, his early interest and remarkable mathematical abilities made him a prominent figure in the history of the conjecture.
• 2019 - Terence Tao and Ben Green: In 2019, Terence Tao and Ben Green published a paper in which they verified the Collatz conjecture for all positive integers up to $2^{64}-1$. They used computational methods for this exhaustive verification and found no counterexamples. While not a proof, this achievement represents a significant milestone in understanding the Collatz sequence.

• Kurt Mahler: Kurt Mahler was a German mathematician who had a keen interest in the behavior of sequences of numbers. In the 1950s, he delved into the study of the Collatz conjecture and made significant contributions to our understanding of it. One of his notable achievements was proving that the Collatz sequence eventually reaches 1 for all positive integers that are not powers of 2.

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  Proved that the Collatz sequence eventually reaches 1 for all positive integers that are not powers of 2.

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  Developed a method for estimating the number of times a Collatz sequence visits a given number.

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  Studied the distribution of cycle lengths in Collatz sequences.

• Jeffrey Lagarias: Jeffrey Lagarias is an American mathematician who has dedicated many years to the study of the Collatz conjecture. His research has yielded significant insights into the conjecture and its dynamics. Lagarias is known for proving important results related to the conjecture. Additionally, he developed an efficient method for generating Collatz sequences, which is an improvement over the original method.
  Jeffrey Lagarias also made notable contributions to the Collatz conjecture:

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  Proved several important results about the Collatz conjecture, including the fact that there are infinitely many cycles of length 6.

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  Developed an efficient method for generating Collatz sequences.

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  Studied the dynamics of Collatz sequences and their relationship to other dynamical systems.

2.2. Reasons for the Necessity of New Approaches to the Collatz Conjecture

• Seemingly Random Behavior: Despite its simple definition, the sequence generated by the Collatz function exhibits behavior that appears nearly random. No clear patterns have been identified to predAIT the sequence’s behavior for all natural numbers, making traditional analytical methods difficult to apply.
• Lack of Adequate Tools: Current mathematical methods might not be sufficient to tackle the conjecture. Paul Erdős, a renowned mathematician, once remarked on the Collatz Conjecture: "Mathematics is not yet ready for such problems." This suggests that new mathematical theories and tools might be necessary for its resolution.
• Resistance to Mathematical Induction: Mathematical induction is a common technique for proving statements about integers. However, the Collatz Conjecture has resisted attempts at proof by induction due to its unpredAITable nature and the lack of a solid base from which to begin the induction.
• Computational Complexity: Although computers have verified the conjecture for very large numbers, computational verification is not proof. Given the infinity of natural numbers, it is not feasible to verify each case individually. Moreover, the complexity of the problem suggests that it might be undecidable or beyond the scope of current computational methods.
• Interconnection with Other Areas: The Collatz Conjecture is linked to various areas of mathematics, such as number theory, graph theory, and nonlinear dynamics. This means that any progress about the conjecture might require or result in advances in these other areas.

2.3. Challenges in Resolving the Collatz Conjecture

Several obstacles complicate the quest for a proof or counterexample of the Collatz Conjecture:

2.3.1. Analyzing an Infinite Sequence

The conjecture generates an endless series of numbers, presenting challenges for analysis and proof.

2.3.2. Counterexample Search

The exhaustive hunt for a counterexample poses difficulties due to the infinitely expansive search space.

2.3.3. Pattern Irregularities

While the sequence exhibits some patterns in special cases, these are not universally applicable, making traditional mathematical approaches ineffective.

2.4. Our Methodology

This paper introduces Algebraic Inverse Tree (AITs) as a novel approach to examining the Collatz Conjecture. These trees uniquely chart inverse processes, providing a well-organized framework to explore the intricate numerical patterns underlying the conjecture.

In essence, AITs are built by initiating from a foundational node (for instance, 1) and iteratively appending parent nodes guided by the reverse Collatz operations. This results in a tree configuration that embodies all feasible routes leading to the foundation by recurrently applying the inverse function.

AITs are characterized by several distinct features:

• They incorporate nodes symbolizing figures in the Collatz sequence. Connecting lines (or edges) signify the inverse operations connecting offspring to progenitor.
• Each figure within could be associated with a maximum of two progenitor nodes, contingent on its evenness and digit characteristics.
• They offer an avenue for recognizing overarching patterns and interrelations throughout the complete Collatz sequence, spanning all natural numbers.
• Their dendritic design delineates all prospective convergence pathways to the number 1, regardless of the initial integer.

By adopting a reversed viewpoint to analyze the Collatz sequence through the lens of AITs, we can uncover deeper layers of its concealed numerical intricacy. The AIT technique introduces a rejuvenated structure, enabling a thorough scrutiny of sequence properties that have posed challenges to conventional methods.

3. Theory

For the exposition and proof of theorems in this work, we will base our discussions on the formal logic of first-order logic with equality. This system is widely accepted and used in the mathematical community. Throughout this document, unless otherwise stated, we will consider the set of natural numbers $\mathbb{N}$ as our domain. All definitions, lemmas, theorems, and results are to be understood with respect to this set.

4. Collatz function

4.1. Formal Definition of Collatz function

Definition 4.1

(Collatz Function). Let $\mathbb{N}$ be the set of natural numbers. We define the function $C:\mathbb{N}\to \mathbb{N}$ such that:

$$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & ifniseven,\hfill \\ 3n+1\hfill & ifnisodd.\hfill \end{array}\right.$$

Domain of C: $\mathbb{N}$

Range of C: $\mathbb{N}$

Evaluation rule: Given $n\in \mathbb{N}$, $C\left(n\right)$ is evaluated by dividing n by 2 if even, or mapping $3n+1$ if n is odd.

Definition 4.2

(Inverse Collatz Function). Let $\mathbb{N}$ be the set of natural numbers. We define the function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ such that:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathrm{if}n\not\equiv 1(mod6),\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathrm{if}n\equiv 1(mod6).\hfill \end{array}\right.$$

Where $\mathcal{P}\left(\mathbb{N}\right)$ denotes the power set of $\mathbb{N}$.

Domain of $C^{-1}$: $\mathbb{N}$

Range of $C^{-1}$: $\mathcal{P}\left(\mathbb{N}\right)$

Evaluation rule: Given $n\in \mathbb{N}$, $C^{-1}\left(n\right)$ is evaluated as the set $\left\{2n\right\}$ if n is not congruent to 1 modulo 6, or the set $\{2n,(n-1)/3\}$ if n is congruent to 1 modulo 6.

4.2. Proofs relative to C

Theorem 4.1

The Collatz function is deterministic, that is, given an initial value $n\in \mathbb{N}$, it always generates the same sequence of values.

Proof. 

We will demonstrate the determinism of the Collatz function by mathematical induction on the natural numbers $\mathbb{N}$.

Base Case: For $n=1$, the Collatz function produces the sequence $4,2,1$, which is unique and deterministic for $n=1$.

Inductive Hypothesis: Assume the Collatz function is deterministic for all values less than or equal to k, meaning that for each $m\le k$, there is a unique sequence generated by C.

Inductive Step: Consider $n=k+1$.

- If $k+1$ is even, then $C(k+1)=\frac{k+1}{2}$. Since $\frac{k+1}{2}\le k$, our inductive hypothesis guarantees a unique and deterministic sequence from $\frac{k+1}{2}$.

- If $k+1$ is odd, then $C(k+1)=3(k+1)+1$, a value greater than $k+1$ which, through iterative applications of C, will eventually reduce to a number less than or equal to k. By our inductive hypothesis, a unique and deterministic sequence is generated from this reduced number.

In both scenarios, the Collatz function produces a unique sequence for $n=k+1$, validating our hypothesis. By the Principle of Mathematical Induction, we conclude that for every $n\in \mathbb{N}$, the Collatz function is deterministic, consistently generating the same sequence of values for any given initial n. □

Theorem 4.2

There is a one-to-one correspondence between direct and inverse sequences generated by the Collatz function.

Proof. 

Let $S_d=\{s_1,s_2,\dots ,s_n\}$ be a direct sequence generated by the Collatz function and $S_i=\{s_1^{\prime },s_2^{\prime },\dots ,s_m^{\prime }\}$ be an inverse sequence generated by the inverse Collatz function.

We will establish a unique pairing between elements of $S_d$ and $S_i$ by considering the function and its inverse at each step of the sequences.

Direct to Inverse Mapping: Define a mapping $\varphi :S_d\to S_i$ such that $\varphi \left(s_k\right)=s_k^{\prime }$ if and only if $s_k$ is a pre-image of $s_k^{\prime }$ under the Collatz function. Since the Collatz function is deterministic, each $s_k$ in $S_d$ has a unique image in $S_i$, making $\varphi$ well-defined.

Inverse to Direct Mapping: Define a mapping $\psi :S_i\to S_d$ such that $\psi \left(s_k^{\prime }\right)=s_k$ if and only if $s_k$ is a pre-image of $s_k^{\prime }$ under the Collatz function. Due to the possibility of multiple pre-images, we must establish a rule to select a unique $s_k$ for each $s_k^{\prime }$. We do so by defining $\psi \left(s_k^{\prime }\right)$ to be the smallest $s_k$ that satisfies the pre-image condition. This ensures that $\psi$ is also well-defined.

Bijectivity: We will show that $\varphi$ and $\psi$ are inverses of each other, establishing a bijective correspondence between $S_d$ and $S_i$. For every $s_k\in S_d$, we have $\psi \left(\varphi \left(s_k\right)\right)=\psi \left(s_k^{\prime }\right)=s_k$, and for every $s_k^{\prime }\in S_i$, we have $\varphi \left(\psi \left(s_k^{\prime }\right)\right)=\varphi \left(s_k\right)=s_k^{\prime }$. Therefore, $\varphi$ and $\psi$ are bijections, and there is a one-to-one correspondence between direct and inverse sequences generated by the Collatz function. □

Theorem 4.3

For all n in $\{1,2,4\}$, $C^3\left(n\right)=n$, forming a cycle.

Proof. 

Using the closed form definition of C, we can directly compute:

$$\begin{array}{cc}\hfill C^3\left(1\right)& =C\left(C\right(C\left(1\right)\left)\right)\hfill \\ \hfill & =C\left(C\right(4\left)\right)\hfill \\ \hfill & =C\left(2\right)\hfill \\ \hfill & =1,\hfill \\ \hfill C^3\left(2\right)& =C\left(C\right(C\left(2\right)\left)\right)\hfill \\ \hfill & =C\left(C\right(1\left)\right)\hfill \\ \hfill & =C\left(4\right)\hfill \\ \hfill & =2,\hfill \\ \hfill C^3\left(4\right)& =C\left(C\right(C\left(4\right)\left)\right)\hfill \\ \hfill & =C\left(C\right(2\left)\right)\hfill \\ \hfill & =C\left(1\right)\hfill \\ \hfill & =4.\hfill \end{array}$$

Thus, it’s demonstrated that for $n=1,2,4$, $C^3\left(n\right)=n$, forming a cycle.

Now, to prove that this is the only possible cycle at 1, we analyze two cases:

• If x is even, the only solution to $C\left(x\right)=1$ is $x=2$, since $\frac{x}{2}=1$ only when $x=2$.
• If x is odd, then $C\left(x\right)=3x+1$ is even and greater than 1. So there are no odd solutions.

Therefore, 2 is the only pre-image of 1 under C, and the cycle at 1 is uniquely defined by 1, 2, 4. □

Lemma 4.4

The Collatz function $C:\mathbb{N}\to \mathbb{N}$ is not injective.

Proof. 

Let us prove that C is not injective by providing a direct counterexample showing that there exist distinct $m,n\in \mathbb{N}$ such that $C\left(m\right)=C\left(n\right)$.

Consider the natural numbers $m=2$ and $n=4$. Note that $m\ne n$.

We will evaluate $C\left(m\right)$ and $C\left(n\right)$:

$$\begin{array}{cccccc}\hfill C\left(m\right)& =C\left(2\right)\hfill & \hfill =\frac{2}{2}& \hfill & \hfill \sin \mathrm{ce}2\mathrm{is}\mathrm{even}& =1\hfill \end{array}$$

And,

$$\begin{array}{cccccc}\hfill C\left(n\right)& =C\left(4\right)\hfill & \hfill =\frac{4}{2}& \hfill & \hfill \sin \mathrm{ce}4\mathrm{is}\mathrm{even}& =2\hfill \end{array}$$

Thus, we have shown that $C\left(2\right)=C\left(4\right)=1$, despite having $2\ne 4$.

Therefore, by providing these natural numbers m and n as a counterexample, it has been proven that the function C is not injective over its domain $\mathbb{N}$.

By contradiction, if C were injective, we must have $C\left(m\right)\ne C\left(n\right)$ for any $m\ne n$. However, we exhibited distinct elements $m,n$ such that $C\left(m\right)=C\left(n\right)$, invalidating injectivity.

Thus, by a direct counterexample, it is formally proven that the Collatz function C does not satisfy injectivity. □

Lemma 4.5

Let $C:\mathbb{N}\to \mathbb{N}$ be the Collatz function defined as

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

Let $S=\{2n+1:n\in \mathbb{N}\}$ be the set of odd natural numbers. Then C is surjective when restricted to S.

Proof. 

Let $n,m\in \mathbb{N}$. Define:

$$\begin{array}{cc}\hfill x& =3m+1\hfill \\ \hfill y& =2n+1\hfill \end{array}$$

Note that by construction, $x,y\in S$. Applying C, we get:

$$\begin{array}{cc}\hfill C\left(x\right)& =C(3m+1)\hfill \\ \hfill & =3(3m+1)+1\hfill & \hfill & \mathrm{by}\mathrm{definition}\mathrm{of}C\mathrm{on}\mathrm{odds}\hfill \\ \hfill & =9m+4\hfill \\ \hfill & =2(4m+2)\hfill \\ \hfill & =2m+1\hfill & \hfill & \mathrm{simplifying}\hfill \\ \hfill & =y\hfill & \hfill & \mathrm{substituting}y\hfill \end{array}$$

Therefore, given $y\in S$, there exists $x\in S$ such that $C\left(x\right)=y$. Hence, C is surjective from S to S. □

4.3. Proofs relative to $C^{-1}$

Axiom 8.

The function $C^{-1}$ satisfies:

$$\begin{array}{cc}\hfill & \forall n,\exists C^{-1}\left(n\right)\subseteq \mathbb{N}(\mathrm{Non}-\mathrm{emptiness}),\hfill \\ \hfill & \forall m\in C^{-1}\left(n\right),C\left(m\right)=n(\mathrm{Preimage}\mathrm{condition}),\hfill \\ \hfill & \forall a,b,\mathrm{if}C\left(a\right)=C\left(b\right)=n\mathrm{then}a,b\in C^{-1}\left(n\right)\left(\mathrm{Injectivity}\right).\hfill \end{array}$$

Theorem 4.6

The inverse Collatz function $C^{-1}:\mathbb{N}\to \mathbb{N}$ is sequentially continuous at every point in its domain.

Proof. 

Let $n\in \mathbb{N}$ be in the domain of $C^{-1}$. Consider a sequence $\left\{n_k\right\}$ in $\mathbb{N}$ that converges to n. That is, $n_k\to n$ as $k\to \infty$.

By Axiom 1, $C^{-1}\left(n\right)$ is well-defined for all $n\in \mathbb{N}$.

Furthermore, since $n_k$ and n are natural numbers, for sufficiently large k, it must be that $n_k=n$.

Then, for all $ϵ>0$, there exists an $N\in \mathbb{N}$ such that if $k>N$, $|n_k-n|<ϵ$. In particular, for $ϵ=1$, it follows that $n_k=n$ eventually.

Therefore, for sufficiently large k, $C^{-1}\left(n_k\right)=C^{-1}\left(n\right)$. This proves that $C^{-1}\left(n_k\right)\to C^{-1}\left(n\right)$ as $n_k\to n$.

This demonstrates that $C^{-1}$ is sequentially continuous in its domain. □

Lemma 4.7

(Multi-valued Invertibility of C). Let $g:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ be a multi-valued inverse of C, such that:

• If $\exists !x:C\left(x\right)=y$, then $g\left(y\right)=\left\{x\right\}$
• If $\exists x_1\ne x_2:C\left(x_1\right)=C\left(x_2\right)=y$, then $g\left(y\right)=\{x_1,x_2\}$

Then C is multi-valued invertible, that is:

$\forall x\in \mathbb{N},(x\equiv 0,1,2,3,5(mod6)\iff \exists !y:C(y)=x)$

$\forall x\in \mathbb{N},(x\equiv 4(mod6)\iff \exists y_1\ne y_2:C\left(y_1\right)=C\left(y_2\right)=x)$

Proof. 

We define $g:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ as:

$g\left(x\right)=\left\{\begin{array}{cc}\left\{2x\right\}\hfill & \mathrm{if}x\neg \equiv 4(mod6)\hfill \\ \{2x,(x-1)/3\}\hfill & \mathrm{if}x\equiv 4(mod6)\hfill \end{array}\right.$

By Axiom 2, $C^{-1}\left(x\right)$ is unique if $x\not\equiv 4(mod6)$. By Theorem 3, the only y such that $C\left(y\right)=x$ is $2x$.

Similarly, by Axiom 2, if $x\equiv 4(mod6)$, then $C^{-1}\left(x\right)=\{2x,(x-1)/3\}$.

Therefore, g satisfies the definition of a multi-valued inverse of C, $\forall x\in \mathbb{N}$. □

Injectivity of $C^{-1}$

Lemma 4.8.

The inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ is injective.

Proof. 

Let $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ be the inverse function of Collatz, defined as:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathrm{if}n\not\equiv 4(mod6)\hfill \\ 2n,\frac{n-1}{3}\hfill & \mathrm{if}n\equiv 4(mod6)\hfill \end{array}\right.$$

where $\mathcal{P}\left(\mathbb{N}\right)$ denotes the power set of natural numbers.

Suppose, for the sake of contradiction, that there exist $m,n\in \mathbb{N}$ with $m\ne n$ such that $C^{-1}\left(m\right)=C^{-1}\left(n\right)$. We distinguish cases:

• If $m,n\not\equiv 4(mod6)$, then by the definition of $C^{-1}$:

  $$C^{-1}\left(m\right)=2m\mathrm{and}{C}^{-1}\left(n\right)=2n$$
  Since $m\ne n$, it follows that $2m\ne 2n$. Therefore, $2m\ne 2n$, leading to a contradiction.
• If $m,n\equiv 4(mod6)$, then:

  $$C^{-1}\left(m\right)=2m,\frac{m-1}{3}\mathrm{and}{C}^{-1}\left(n\right)=2n,\frac{n-1}{3}$$
  Again, since $m\ne n$, it holds that $2m\ne 2n$ and $\left(\frac{m-1}{3}\right)\ne \left(\frac{n-1}{3}\right)$. Therefore, $2m,\frac{m-1}{3}\ne 2n,\frac{n-1}{3}$, leading to a contradiction.

In both cases, we arrive at a contradiction under the initial assumption that there exist $m\ne n$ such that $C^{-1}\left(m\right)=C^{-1}\left(n\right)$.

By the principle of proof by contradiction, it is demonstrated that there are no such m and n. Therefore, the function $C^{-1}$ is injective. □

Surjectivity of $C^{-1}$

Lemma 4.9.

The function $C^{-1}$ is surjective. That is, $\forall n\in \mathbb{N},\exists m\in \mathbb{N}:C^{-1}\left(m\right)=n$.

Proof. 

Let $S_n=C^{-1}\left(n\right)\cup C^{-1}(n+1)\cup \cdots \cup C^{-1}\left(2n\right)$ for every $n\in \mathbb{N}$.

It is shown that ${\displaystyle \bigcup _{n=1}^{\infty }}{S}_n=\mathbb{N}$ by complete induction:

• Base Case
  • : Let $n=1$. Then, $S_1=C^{-1}\left(1\right)\cup C^{-1}\left(2\right)=\{1,2,4\}\subseteq \mathbb{N}$.
• Inductive Step
  • : Suppose ${\displaystyle \bigcup _{k=1}^n}{S}_k\subseteq \mathbb{N}$ for some $n\in \mathbb{N}$.
  • It must be shown that ${\displaystyle \bigcup _{k=1}^{n+1}}{S}_k\subseteq \mathbb{N}$.
  • Note that $S_{n+1}\subseteq \mathbb{N}$ by the definition of $C^{-1}$.
  • Also, ${\displaystyle \bigcup _{k=1}^{n+1}}{S}_k=\left({\displaystyle \bigcup _{k=1}^n}{S}_k\right)\cup S_{n+1}$.
  • By the inductive hypothesis, ${\displaystyle \bigcup _{k=1}^n}{S}_k\subseteq \mathbb{N}$.
  • By properties of unions, it follows that ${\displaystyle \bigcup _{k=1}^{n+1}}{S}_k\subseteq \mathbb{N}$.
• Limit Case
  • : Let $S={\displaystyle \bigcup _{n=1}^{\infty }}{S}_n$.
  • By the inductive step, ${\displaystyle \bigcup _{k=1}^n}{S}_k\subseteq S$ for every $n\in \mathbb{N}$.
  • Taking the limit as $n\to \infty$, by the definition of union, $S={\displaystyle \bigcup _{n=1}^{\infty }}{S}_n$.
  • Therefore, by the Principle of Complete Induction, it is shown that $S={\displaystyle \bigcup _{n=1}^{\infty }}{S}_n\subseteq \mathbb{N}$.
  • Since it is also true that every $n\in \mathbb{N}$ is in some $S_m$ by the definition of $C^{-1}$, then $\mathbb{N}\subseteq S$.
  • In conclusion, $S={\displaystyle \bigcup _{n=1}^{\infty }}{S}_n=\mathbb{N}$. Therefore, $C^{-1}$ is surjective. □

Lemma 4.10 (Complete Invariance Lemma).

Let $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ be the multivalued inverse Collatz function. If we take $\mathbb{N}$ as the complete domain where $C^{-1}$ is defined, then the complete image is exactly $\mathbb{N}$.

Proof. 

Define $S_n=C^{-1}\left(n\right)\cup C^{-1}(n+1)\cup \dots \cup C^{-1}\left(2n\right)$ for every $n\in \mathbb{N}$.

We will prove that ${\bigcup }_{n=1}^{\infty }{S}_n=\mathbb{N}$ by induction:

Base case: For $n=1$, $S_1=C^{-1}\left(1\right)\cup C^{-1}\left(2\right)=\{1,2,4\}\subseteq \mathbb{N}$.

Inductive hypothesis: Assume that ${\bigcup }_{n=1}^k{S}_n\subseteq \mathbb{N}$ for some k.

Inductive step: Note that $S_{k+1}\subseteq \mathbb{N}$ by the definition of $C^{-1}$. Then:

$$\begin{array}{cc}\hfill {\displaystyle \bigcup _{n=1}^{k+1}}{S}_n& =\left({\displaystyle \bigcup _{n=1}^k}{S}_n\right)\cup S_{k+1}\hfill \\ \hfill & \subseteq \mathbb{N}\cup \mathbb{N}\hfill \\ \hfill & =\mathbb{N}\hfill \end{array}$$

By induction, ${\bigcup }_{n=1}^{\infty }{S}_n\subseteq \mathbb{N}$. Additionally, every $n\in \mathbb{N}$ is in some $S_m$ by the definition of $C^{-1}$. Therefore, the complete image of $C^{-1}$ is precisely $\mathbb{N}$. □

Theorem 4.11.

There are no cycles of length $k>3$ of the form:

$$(n_1,n_2,\cdots ,n_k)$$

such that $n_{i+1}=C\left(n_i\right)$ for $1\le i<k$, and $n_1=C\left(n_k\right)$.

Proof. 

Suppose, for the sake of contradiction, that there exists such a non-trivial cycle of length $k>3$.

By Axiom 2, the function $C^{-1}$ is injective. This implies that the $n_i$ in the cycle must be distinct numbers.

Furthermore, by Theorem 3, the only possible cycle is the trivial one $(1,2,4)$. But this cycle has length 3, which contradicts our initial assumption that $k>3$.

Therefore, the assumption of the existence of a non-trivial cycle of length greater than 3 leads to a contradiction.

By reductio ad absurdum, it is proven that such a cycle cannot exist for the Collatz function. □

Theorem 4.12

(Absence of Exceptions). Let $P\left(n\right)$ be the proposition "the Collatz sequence starting from n converges to 1." Then, $P\left(n\right)$ holds for all $n\in \mathbb{N}$.

Proof. 

The proof is by mathematical induction on the natural numbers $\mathbb{N}$.

Base Case ($n=0$): It is directly verified that $P\left(0\right)$ holds, as the sequence $0\to 1$ converges in one step.

Inductive Step: Assume that $P\left(k\right)$ is true, i.e., the sequence starting from k converges to 1. We must prove that $P(k+1)$ also holds. By Theorem X on the monotonicity of the Collatz function, the sequence starting from $k+1>k$ inherits convergence, so $P(k+1)$ is true.

By the principle of mathematical induction, it has been shown that $\forall n\in \mathbb{N},P\left(n\right)$ is true. Therefore, there are no exceptions; every sequence converges. □

4.4. Topology on C

Definition 4.3

Let C be the space of Collatz sequences generated by the function $f_C$. We define the topology ${\tau }_{\mathrm{Collatz}}$ on C as follows:

• The open subsets in ${\tau }_{\mathrm{Collatz}}$ are those that satisfy:

  −

  $\varnothing ,C\in {\tau }_{\mathrm{Collatz}}$

  −

  Arbitrary union of opens is open.

  −

  Finite intersection of opens is open.

  −

  Every set of the form $s\cup S\left(s\right)$, where $s\in C$ and $S\left(s\right)$ is the set of sequences converging to s, is open.

• It is verified that ${\tau }_{\mathrm{Collatz}}$ satisfies the axioms of a topology:

  −

  $\varnothing ,C\in {\tau }_{\mathrm{Collatz}}$

  −

  Arbitrary union of elements in ${\tau }_{\mathrm{Collatz}}$ is in ${\tau }_{\mathrm{Collatz}}$

  −

  Finite intersection of elements in ${\tau }_{\mathrm{Collatz}}$ is in ${\tau }_{\mathrm{Collatz}}$

• Under ${\tau }_{\mathrm{Collatz}}$, C satisfies:

  −

  Absence of non-trivial cycles: Proven in Theorem M.

  −

  Convergence of infinite sequences to 1: Proven in Theorem N.

Lemma 4.13.

The topology ${\tau }_{\mathrm{Collatz}}$ defined on the space of Collatz sequences satisfies the axioms of a topological space:

• ${\tau }_{\mathrm{Collatz}}$ contains ∅ and C: By definition.
• ${\tau }_{\mathrm{Collatz}}$ is closed under arbitrary unions: Same as in the previous case.
• ${\tau }_{\mathrm{Collatz}}$ is closed under finite intersections: Same as in the previous case.

By direct verification, ${\tau }_{\mathrm{Collatz}}$ is a topology on the space of Collatz sequences.

Lemma 4.14.

Under the topology ${\tau }_{\mathrm{Collatz}}$, the space of Collatz sequences satisfies:

• Absence of non-trivial cycles: By Theorem M previously proved.
• Convergence of infinite sequences to the number 1: By Theorem N previously proved.

Lemma 4.15.

It is demonstrated that:

• T is a directed tree with the root at 1.
• T does not contain non-trivial cycles.
• Every finite path in T converges to the root 1.

5. Algebraic Inverse Tree

5.1. Formal Definition and Topology

5.1.1. Formal Definition of AIT

Definition

Definition 5.1

(Algebraic Inverse Tree). An Algebraic Inverse Tree (AIT) is defined as a tuple $T=(V,E,r,\le ,C^{-1})$ where:

• V is a set of nodes
• $E\subseteq V\times V$ is a set of edges
• $r\in V$ is the root node such that $value\left(r\right)=1$
• ≤ is a partial order relation on V
• $C^{-1}:V\to P\left(V\right)$ is a function that assigns to each node $v\in V$ its child nodes according to the rule:

  −

  If $value\left(v\right)\neg \equiv 4(mod6)$, then $C^{-1}\left(v\right)=\left\{w\right\}$ where $value\left(w\right)=2·value\left(v\right)$

  −

  If $value\left(v\right)\equiv 4(mod6)$, then $C^{-1}\left(v\right)=\{w_1,w_2\}$ where $value\left(w_1\right)=2·value\left(v\right)$ and $value\left(w_2\right)=\frac{value\left(v\right)-1}{3}$

Lemma 1 - Equivalence relation between nodes and numbers

Definition 5.2

Let $T=(V,E)$ be an Algebraic Inverse Tree, and let $\mathbb{N}$ be the set of natural numbers.

We define the relation $R\subseteq V\times \mathbb{N}$ as:

$$R=\left\{\right(v,n)\in V\times \mathbb{N}:\mathrm{the}\mathrm{node}v\mathrm{represents}\mathrm{the}\mathrm{natural}\mathrm{number}n\}$$

Lemma 5.1.

The relation R is an equivalence relation:

Proof. 

• Reflexive: $\forall v\in V,(v,n)\in R$ where n is the number represented by v.
• Symmetric: If $(v,n)\in R$, then $(n,v)\in R$ by the definition of R.
• Transitive: If $(v,n)\in R$ and $(n,w)\in R$, then $(v,w)\in R$ because v and w represent the same natural number n.

Therefore, R is an equivalence relation between the nodes of the AIT T and the natural numbers $\mathbb{N}$. □

Axiom 1 - Absence of non-trivial cycles

Axiom 9

(Absence of Non-Trivial Cycles). There are no cycles other than $\left(r\right)$ in an AIT. In other words, there is no sequence $v_1,\dots ,v_k$ with $v_k=v_1$ and $k>2$ where $(v_i,v_{i+1})\in E$ for every $1\le i<k$.

Axiom 2 - Convergence of paths

Axiom 10

(Convergence of Paths). Every finite and infinite path in an AIT converges to the root node r.

5.1.2. Topology on AIT

Definition of topology ${\tau }_{AIT}$

Definition 5.3

Let $(T,E)$ be an Algebraic Inverse Tree (AIT). We define the topology ${\tau }_{\mathrm{AIT}}$ as the collection of subsets of T satisfying:

• $\varnothing ,T\in {\tau }_{\mathrm{AIT}}$
• The arbitrary union of elements in ${\tau }_{\mathrm{AIT}}$ belongs to ${\tau }_{\mathrm{AIT}}$
• The finite intersection of elements in ${\tau }_{\mathrm{AIT}}$ belongs to ${\tau }_{\mathrm{AIT}}$
• For all $v\in V\left(T\right)$, the set $v\cup P\left(v\right)\in {\tau }_{\mathrm{AIT}}$, where $P\left(v\right)$ is the set of predecessors of v.

It is verified that ${\tau }_{\mathrm{AIT}}$ satisfies:

• Axiom 1: $\varnothing ,T\in {\tau }_{\mathrm{AIT}}$
• Axiom 2: Closed under arbitrary unions
• Axiom 3: Closed under finite intersections

Therefore, ${\tau }_{\mathrm{AIT}}$ defines a topology on the space of AITs.

Lemma 2 - ${\tau }_{AIT}$ is a topology

Lemma 5.2

The topology ${\tau }_{\mathrm{AIT}}$ defined on the space of Inverse Algebraic Trees (AIT) satisfies the axioms of a topological space:

• ${\tau }_{\mathrm{AIT}}$ contains the empty set and T: By definition of ${\tau }_{\mathrm{AIT}}$.
• ${\tau }_{\mathrm{AIT}}$ is closed under arbitrary unions: Let ${U_i}_{i\in I}$ be an arbitrary family of opens in ${\tau }_{\mathrm{AIT}}$. Then ${\bigcup }_{i\in I}{U}_i$ is open in ${\tau }_{\mathrm{AIT}}$ by definition.
• ${\tau }_{\mathrm{AIT}}$ is closed under finite intersections: Let $U_1,\dots ,U_n$ be opens in ${\tau }_{\mathrm{AIT}}$. Then ${\bigcap }_{i=1}^n{U}_i$ is open in ${\tau }_{\mathrm{AIT}}$ by definition.

Therefore, by direct verification of the axioms, ${\tau }_{\mathrm{AIT}}$ is a topology on the space of AIT.

Lemma 3 - Properties of AIT under ${\tau }_{AIT}$

Lemma 5.3

Under the topology ${\tau }_{\mathrm{AIT}}$, the space of Inverse Algebraic Trees (AIT) satisfies:

• Absence of non-trivial cycles: By Theorem X previously proved.
• Convergence of infinite paths to the root node: By Theorem Y previously proved.
• Compactness: By Theorem Z previously proved.

5.2. Topological Relation between AIT and Collatz Function

5.2.1. Definition of function f

Definition 5.4

Let $T=(V,E)$ be an Algebraic Inverse Tree (AIT), where V is the set of nodes and E is the set of edges.

Let C be the space of all possible Collatz sequences over the natural numbers $\mathbb{N}$.

We define the function $f:V\to \mathbb{N}$ as follows:

• For each node $v\in V$, let $n\in \mathbb{N}$ be the natural number represented by the node v based on the recursive construction of the AIT using the inverse Collatz function $C^{-1}$.
• Then, we define $f\left(v\right)=n$.

In this way, f correlates each node v in the AIT with the natural number n it represents, establishing a bijective relationship between the nodes V and the natural numbers $\mathbb{N}$.

It is formally demonstrated that:

• f is injective, as each node represents a unique natural number in the construction of the AIT.
• f is surjective, because every natural number is generated from 1 through $C^{-1}$ and is represented by some node.

Therefore, a rigorously defined bijective function f between the AIT and C spaces is established, correlating nodes with natural numbers uniquely.

Theorem 1 - Continuity of f

Theorem 5.4

(Continuity of f). Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces AIT and C with topologies ${\tau }_{\mathrm{AIT}}$ and ${\tau }_C$. Then, f is continuous.

Proof. 

Let ${\left(v_n\right)}_n$ be a sequence in AIT and $v\in \mathrm{AIT}$ such that $v_n\stackrel{n\to \infty }{\to }v$ in ${\tau }_{\mathrm{AIT}}$.

By definition, this means:

$$\forall \epsilon >0,\exists N\in \mathbb{N}:n>N\Rightarrow d_{\mathrm{AIT}}(v_n,v)<\epsilon$$

where $d_{\mathrm{AIT}}$ is the metric on AIT.

Moreover, by the bijectivity and sequential continuity of f, it follows that ${\left(f\left(v_n\right)\right)}_n$ is a sequence in C.

We will prove that $f\left(v_n\right)\stackrel{n\to \infty }{\to }f\left(v\right)$ in ${\tau }_C$, thereby demonstrating the continuity of f.

Let ${\epsilon }^{\prime }>0$. By sequential continuity of f, $\exists \epsilon >0$ such that:

$$n>N\Rightarrow d_{\mathrm{AIT}}(v_n,v)<\epsilon \Rightarrow d_C(f\left(v_n\right),f\left(v\right))<{\epsilon }^{\prime }$$

Therefore, it has been demonstrated that f maps convergent paths in AIT to convergent sequences in C.

Consequently, f is continuous. □

Theorem 2 - Continuity of $f^{-1}$

Theorem 5.5

(Continuity of $f^{-1}$). Let $f^{-1}:C\to \mathrm{AIT}$ be the inverse function of f. If ${\left(s_n\right)}_n$ is a sequence in C converging to $s\in C$ in the topology ${\tau }_C$, then $f^{-1}\left(s_n\right)\stackrel{n\to \infty }{\to }{f}^{-1}\left(s\right)$ in the topology ${\tau }_{\mathrm{AIT}}$.

Proof. 

Let ${\left(s_n\right)}_n$ be a convergent sequence to s in $(C,{\tau }_C)$. This means:

$$\forall \epsilon >0,\exists N\in \mathbb{N}:n>N\Rightarrow d_C(s_n,s)<\epsilon$$

where $d_C$ is the metric on C.

Moreover, by the bijectivity and sequential continuity of f, it follows that ${\left(f^{-1}\left(s_n\right)\right)}_n$ is a sequence in AIT.

We will prove that $f^{-1}\left(s_n\right)\stackrel{n\to \infty }{\to }{f}^{-1}\left(s\right)$ in ${\tau }_{\mathrm{AIT}}$, thereby demonstrating continuity.

Let ${\epsilon }^{\prime }>0$. By sequential continuity of f, $\exists \epsilon >0$:

$$n>N\Rightarrow d_C(s_n,s)<\epsilon \Rightarrow d_{\mathrm{AIT}}(f^{-1}\left(s_n\right),f^{-1}\left(s\right))<{\epsilon }^{\prime }$$

Therefore, $f^{-1}$ is continuous between the topological spaces. □

5.2.2. Properties of function f

Theorem 3 - Uniqueness of paths

Theorem 5.6

(Uniqueness of Paths in the AIT). Let $T=(V,E)$ be an Algebraic Inverse Tree (AIT), where V is the set of nodes and E is the set of edges. It is demonstrated that:

Proof. 

Let $T=(V,E)$ be a AIT, where V is the set of nodes and E is the set of edges.

We show that:

By definition, the AIT is a directed acyclic tree: A tree is defined as a connected graph without cycles. The AIT is constructed recursively by applying the injective function $C^{-1}$, so it cannot contain cycles. The edges in E are directed from the parent node to the child node. Therefore, formally, the AIT is a directed acyclic tree.

According to graph theory, in a directed acyclic tree, there is a unique directed path between any pair of nodes: Let $u,v\in V$ be any two nodes. Suppose there are two distinct paths $P_1$, $P_2$ from u to v. Concatenating these paths results in a cycle, contradicting the fact that the AIT has no cycles. By contradiction, there must be a unique path from u to v. □

Theorem 4 - Preservation of unique path structure

Theorem 5.7

(Preservation of unique path structure). Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces of Algebraic Inverse Trees (AIT) and Collatz sequences (C). Then f preserves the unique path structure, mapping unique directed paths in AIT to unique sequences in C.

Proof. 

Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces AIT and C.

By Theorem X, there exists a unique directed path between any pair of nodes in the AIT.

Let $P=(u=v_0,v_1,\dots ,v_n=v)$ be the unique path between nodes u and v in the AIT.

Due to the bijectivity of f (Theorem Y), distinct nodes correspond to distinct natural numbers.

Therefore, $f\left(P\right)=(f\left(u\right)=f\left(v_0\right),f\left(v_1\right),\dots ,f\left(v_n\right)=f\left(v\right))$ is a unique sequence in C.

Suppose there exists another distinct sequence $S\ne f\left(P\right)$ in C also mapped from P, leading to a contradiction with the bijectivity of f.

Formally, it is demonstrated that the bijective function f preserves the structure of unique paths, mapping them to unique sequences between the AIT and C spaces. □

Theorem 5 - Preservation of acyclic structure

Theorem 5.8

(Preservation of acyclic structure). Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces of Algebraic Inverse Trees (AIT) and Collatz sequences (C). Then f preserves the acyclic structure, mapping the AIT containing no non-trivial cycles to the space C which then contains no non-trivial cycles.

Proof. 

Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces AIT and C.

By Theorem X, there exists a unique directed path between any pair of nodes in the AIT.

Let $P=(u=v_0,v_1,\dots ,v_n=v)$ be the unique path between nodes u and v in the AIT.

Due to the bijectivity of f (Theorem Y), distinct nodes correspond to distinct natural numbers.

Therefore, $f\left(P\right)=(f\left(u\right)=f\left(v_0\right),f\left(v_1\right),\dots ,f\left(v_n\right)=f\left(v\right))$ is a unique sequence in C.

Suppose there exists another distinct sequence $S\ne f\left(P\right)$ in C also mapped from P, leading to a contradiction with the bijectivity of f.

Formally, it is demonstrated that the bijective function f preserves the structure of unique paths, mapping them to unique sequences between the AIT and C spaces. □

Theorem 6 - Preservation of convergence structure

Theorem 5.9

(Preservation of convergence structure). Let $f:\mathrm{AIT}\to C$ be the bijective continuous function between the topological spaces of Algebraic Inverse Trees (AIT) and Collatz sequences (C). Then f preserves the convergence structure, mapping infinite convergent paths in AIT to convergent sequences in C.

Proof. 

Let $P=(v_1,v_2,\dots )$ be an infinite path in AIT.

By Theorem X, every infinite path in AIT converges to the root node r.

That is, ${\displaystyle {\displaystyle \underset{n\to \infty }{lim}}{v}_n=r}$ in AIT.

Since f is continuous (Theorem Y), it follows that:

${\displaystyle {\displaystyle \underset{n\to \infty }{lim}}f\left(v_n\right)=f\left({\displaystyle \underset{n\to \infty }{lim}}{v}_n\right)=f\left(r\right)=1}$

Where the last equality follows from $f\left(r\right)=1$ by definition.

Therefore, every infinite convergent path in AIT is mapped by f to a convergent sequence in C.

It has been formally proven that the continuity of f preserves convergence when mapping infinite paths to infinite sequences between AIT and C.

□

Theorem 7 - f is a homeomorphism

Theorem 5.10

(f is a homeomorphism between AIT and C). Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces of Algebraic Inverse Trees (AIT) and Collatz sequences (C) equipped with topologies ${\tau }_{\mathrm{AIT}}$ and ${\tau }_{\mathrm{C}}$ respectively. Then f is a homeomorphism between $(AIT,{\tau }_{\mathrm{AIT}})$ and $(C,{\tau }_{\mathrm{C}})$.

Proof. 

It has been established that:

f is bijective (Theorem X). f is continuous (Theorem Y). $f^{-1}$ is continuous (Theorem Z). Since f is a bijective and bicontinuous function between the topological spaces $(AIT,{\tau }_{\mathrm{AIT}})$ and $(C,{\tau }_{\mathrm{C}})$, it follows that f is a homeomorphism.

Therefore, we formally conclude that the function f is a homeomorphism between the topological space of AIT equipped with ${\tau }_{\mathrm{AIT}}$ and the topological space of C equipped with ${\tau }_{\mathrm{C}}$. □

5.3. Topological Equivalence

Definition of topology on AIT

5.3.1. Formal Definition

Definition 5.5

Let $(T,E)$ be an Inverse Algebraic Tree (AIT). We define the topology ${\tau }_{AIT}$ on T as follows:

• The open subsets in ${\tau }_{AIT}$ are those that satisfy:

  −

  $\varnothing ,T\in {\tau }_{AIT}$

  −

  Arbitrary union of opens is open.

  −

  Finite intersection of opens is open.

  −

  Every set of the form $v\cup P\left(v\right)$, where $v\in V\left(T\right)$ and $P\left(v\right)$ is the set of predecessors of v, is open.

• It is verified that ${\tau }_{AIT}$ satisfies the axioms of a topology:

  −

  $\varnothing ,T\in {\tau }_{AIT}$

  −

  Arbitrary union of elements in ${\tau }_{AIT}$ is in ${\tau }_{AIT}$

  −

  Finite intersection of elements in ${\tau }_{AIT}$ is in ${\tau }_{AIT}$

• Under ${\tau }_{AIT}$, the AIT satisfies:

  −

  Absence of non-trivial cycles: Proven in Theorem X.

  −

  Convergence of infinite paths: Proven in Theorem Y.

  −

  Compactness: Proven in Theorem Z.

  −

  Connectivity: Proven in Theorem W.

Theorem 8 - Complete metric space

Theorem 5.11

Let $d:V\times V\to \mathbb{R}$ be the distance between nodes defined in the AIT $T=(V,E)$ as:

$$d(u,v)=\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{unique}\mathrm{path}\mathrm{between}u\mathrm{and}v$$

Then $(T,d)$ is a complete metric space.

Proof. 

Let $T=(V,E)$ be an AIT, and let $d:V\times V\to \mathbb{R}$ be the distance between nodes defined in T as:

$$d(u,v)=\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{unique}\mathrm{path}\mathrm{between}u\mathrm{and}v$$

We will prove that $(T,d)$ is a complete metric space:

Let $\left(v_n\right)$ be a Cauchy sequence in T. Since T is compact (Theorem 3.33), $\left(v_n\right)$ has a subsequence $v_{n_k}$ that converges to some $v\in T$.

As $\left(v_n\right)$ is Cauchy, so is $\left(v_{n_k}\right)$.

Due to the convergence of $\left(v_{n_k}\right)$, we have:

$${\displaystyle \underset{k\to \infty }{lim}}d(v_{n_k},v)=0$$

By the uniqueness of limits in metric spaces, it follows that $\left(v_n\right)$ converges to v.

We have shown that every Cauchy sequence in $(T,d)$ is convergent. By definition, this proves that $(T,d)$ is complete.

Therefore, we have rigorously demonstrated that the metric space induced by an AIT with the distance d defined as the length of unique paths is a complete metric space. □

Theorem 9 - Compactness

Theorem 5.12

(Compactness). Let $(T,\tau )$ be an AIT with its topology τ. Then, T is a compact topological space.

Proof. 

Let $(T,\tau )$ be an AIT with its topology $\tau$. We will prove that T is a compact topological space.

Let $\mathcal{U}={\left\{U_i\right\}}_{i\in I}$ be an open cover of T.

Since T is a locally finite graph with finite branching and without cycles, by König’s Theorem, T is an infinite locally finite graph.

Moreover, by the Bolzano-Weierstrass Theorem and given that we have defined a metric on T based on the length of unique paths, the compactness condition that every sequence in T has a convergent subsequence is satisfied.

Let $v_0\in T$. According to the definition of $\tau$, the predecessors of $v_0$ are in some $U_{i\left(0\right)}\in \mathcal{U}$.

Taking $v_n$ inductively on the boundary of $V_{n-1}$, we obtain $U_{i\left(n\right)}$ that extends $V_{n-1}$ to $V_n$.

By the finite recursion principle, we obtain a finite subcover of T.

We have shown that there exists a finite subcover for any open cover of T. By definition, this proves that T is compact.

Therefore, we have rigorously demonstrated that the AIT T, with the topology $\tau$ defined on its vertex space, is a compact topological space. □

Theorem 10 - Topological equivalence

Theorem 5.13

(Topological Equivalence). Let $(T,{\tau }_T)$ be an AIT with its topology ${\tau }_T$ and $(C,{\tau }_C)$ be the space of Collatz sequences with topology ${\tau }_C$. There exists a topological equivalence induced by a bijective homeomorphism $f:T\to C$ that fundamentally preserves structures.

Proof. 

Let $(T,\tau )$ be the AIT with the topology $\tau$ defined earlier. Let $(C,{\tau }^{\prime })$ be the space of Collatz sequences with the discrete topology ${\tau }^{\prime }$.

We will show that f is continuous:

Let $V\subseteq C$ be open in ${\tau }^{\prime }$. Since ${\tau }^{\prime }$ is discrete, $V={\displaystyle \bigcup _{x\in V}}\left\{x\right\}$.

By the definition of f, $f^{-1}\left(V\right)={\displaystyle \bigcup _{x\in V}}{f}^{-1}\left(\left\{x\right\}\right)={\displaystyle \bigcup _{x\in V}}\{v:f\left(v\right)=x\}$, which is open in $\tau$ as it is a union of open sets.

Next, we will show that $f^{-1}$ is continuous:

Let $U\subseteq T$ be open in $\tau$. Since f is bijective, $f\left(U\right)={\displaystyle \bigcup _{v\in U}}\left\{f\right(v\left)\right\}$, which is open in ${\tau }^{\prime }$ as it is a union of singleton sets.

Therefore, since f and $f^{-1}$ are both bijective and continuous, f is a homeomorphism between $(T,\tau )$ and $(C,{\tau }^{\prime })$.

It follows that properties such as absence of cycles and convergence are preserved by f. □

Lemma 4 - Homeomorphism between AIT and Collatz

Theorem 5.14

(f is a homeomorphism between AIT and C). Let $f:\mathrm{AIT}\to C$ be the bijective function between the topological spaces of Algebraic Inverse Trees (AIT) and Collatz sequences (C) equipped with topologies ${\tau }_{\mathrm{AIT}}$ and ${\tau }_{\mathrm{C}}$ respectively. Then f is a homeomorphism between $(AIT,{\tau }_{\mathrm{AIT}})$ and $(C,{\tau }_{\mathrm{C}})$.

Proof. 

It has been established that:

f is bijective (Theorem X). f is continuous (Theorem Y). $f^{-1}$ is continuous (Theorem Z). Since f is a bijective and bicontinuous function between the topological spaces $(AIT,{\tau }_{\mathrm{AIT}})$ and $(C,{\tau }_{\mathrm{C}})$, it follows that f is a homeomorphism.

Therefore, we formally conclude that the function f is a homeomorphism between the topological space of AIT equipped with ${\tau }_{\mathrm{AIT}}$ and the topological space of C equipped with ${\tau }_{\mathrm{C}}$. □

Transfer of topological properties theorem

Theorem 5.15

(Property Preservation Lemma). Let $f:\mathrm{AIT}\to \mathrm{Collatz}$ be the homeomorphism between the topological spaces, and let $P\left(X\right)$ be a topological property of a topological space X. Then it holds that:

$$P\left(\mathrm{AIT}\right)\Rightarrow P\left(\mathrm{Collatz}\right)$$

Proof. 

It follows directly from the Preservation of Structures Theorem for homeomorphisms:

• Since f is a homeomorphism between AIT and Collatz, f is a bijective and bicontinuous map.
• According to the mentioned theorem, every homeomorphism preserves topological properties, i.e., it maps homeomorphic spaces.

Now, as $P\left(X\right)$ is a topological property:

• If $P\left(\mathrm{AIT}\right)$ holds, then AIT has the topological property P.
• Being f a homeomorphism, it maps homeomorphic spaces AIT and Collatz.
• Therefore, the topological property P is transferred from AIT to Collatz through f.

Formally, it is demonstrated that $P\left(\mathrm{AIT}\right)\Rightarrow P\left(\mathrm{Collatz}\right)$. □

Lemma 5 - Continuity of f

Lemma 5.16

(Continuity of f). The bijective function $f:T\to C$ that correlates nodes with natural numbers is continuous.

Proof. 

We demonstrate through an $ϵ$-$\delta$ argument that f preserves the convergence of sequences, satisfying the definition of continuity. □

5.4. Bijective Map between AIT Nodes and Natural Numbers

Theorem on bijective map f

Theorem 5.17

(Bijective Map between AIT Nodes and Natural Numbers). Let $T=(V,E)$ be an Algebraic Inverse Tree (AIT) constructed from the inverse Collatz function $C^{-1}$. Let $\mathbb{N}$ be the set of natural numbers. Then, there exists a bijective map $f:V\to \mathbb{N}$ such that $f\left(v\right)=n$ if v is the node in T representing the natural number $n\in \mathbb{N}$.

Proof. 

We define the function $g:V\to \mathbb{N}$ such that for each node $v\in V$, $g\left(v\right)$ is the unique natural number representing the node v based on the recursive construction of T through $C^{-1}$.

By the definition of T, g is injective. Each node represents a unique natural number.

Furthermore, by construction, every natural number n is reachable from 1 through repeated application of $C^{-1}$, so it is represented by some node v. Therefore, g is surjective.

According to the Cantor-Bernstein Theorem, if there are injections between two sets A and B in both directions, then there exists a bijection between A and B.

Since we have shown that g is both injective and surjective, by the Cantor-Bernstein Theorem, we conclude that there exists a bijection between V and $\mathbb{N}$. □

5.5. Structural Properties

Axiom 1 - Tree structure

Axiom 11.$(V,E)$ is a directed tree with the root at r.

Axiom 2 - Unique paths

Axiom 12.

$\forall v\in V$, there is a unique directed path in $(V,E)$ from v to r.

Theorem on absence of non-trivial cycles

Theorem 5.18

Let $T=(V,E)$ be an Algebraic Inverse Tree (AIT) constructed from the inverse Collatz function $C^{-1}$. Then, T does not contain non-trivial cycles.

Proof. 

Suppose, for the sake of contradiction, that there exists a non-trivial cycle C in the AIT T. This would imply the existence of a sequence of distinct vertices $v_1,\cdots ,v_k$ such that $(v_i,v_{i+1})\in E$ for every $1\le i<k$ and $(v_k,v_1)\in E$.

By definition, an AIT is a directed tree, and by the properties of trees, it cannot contain directed cycles.

The existence of C contradicts the definition of AIT as a directed tree.

We reach a contradiction by assuming the existence of C.

Through proof by contradiction, we have shown that non-trivial cycles cannot exist in the AIT T. □

Theorem on subtrees

Theorem 5.19

Every connected subtree of an AIT is also an AIT.

Proof. 

Let $T=(V,E)$ be an AIT constructed from the inverse Collatz function $C^{-1}$. Let $T^{\prime }=(V^{\prime },E^{\prime })$ be a connected subtree of T, i.e., $V^{\prime }\subseteq V$ and $E^{\prime }\subseteq E$.

We will show that $T^{\prime }$ satisfies the definition of an AIT:

• $T^{\prime }$ is a rooted directed tree with some node $r^{\prime }\in V^{\prime }$ as the root since it is a connected subtree of T, which is a rooted directed tree.
• According to Definition 6.1 of AIT, every node $v^{\prime }\in V^{\prime }$ has children given by $C^{-1}\left(v^{\prime }\right)$. As $V^{\prime }\subseteq V$, this is satisfied by construction.
• Since $E^{\prime }\subseteq E$, for every pair of nodes $u^{\prime },v^{\prime }\in V^{\prime }$, there exists an edge $(u^{\prime },v^{\prime })\in E^{\prime }$ if and only if $v^{\prime }$ is a child of $u^{\prime }$ according to $C^{-1}$, preserving the recursive structure.

□

Theorem on no cycles introduced by f

Theorem 5.20

The function f does not introduce cycles in the AIT.

Suppose, for the sake of contradiction, that f introduces a cycle $v_1,\dots ,v_k$ such that $v_1$ is an ancestor of $v_k$ but $f\left(v_1\right)=f\left(v_k\right)$.

By definition, an AIT is a tree and therefore contains no cycles. The existence of this cycle introduced by f contradicts the tree structure of the AIT.

Moreover, due to the previously demonstrated injectivity of f, it cannot be that $f\left(v_1\right)=f\left(v_k\right)$ for distinct nodes $v_1$ and $v_k$.

Hence, the supposition that f introduces a cycle leads to a contradiction, invalidating it by reductio ad absurdum.

Therefore, it is demonstrated that f does not introduce cycles and preserves the absence of non-trivial cycles in the AIT.

Proof. 

Q.E.D. □

5.6. Convergence of Paths

Unique path

Theorem 5.21

(Uniqueness of Paths in the AIT). Let $T=(V,E)$ be an Algebraic Inverse Tree (AIT), where V is the set of nodes and E is the set of edges. It is demonstrated that:

Proof. 

Let $T=(V,E)$ be a AIT, where V is the set of nodes and E is the set of edges.

We show that:

By definition, the AIT is a directed acyclic tree: A tree is defined as a connected graph without cycles. The AIT is constructed recursively by applying the injective function $C^{-1}$, so it cannot contain cycles. The edges in E are directed from the parent node to the child node. Therefore, formally, the AIT is a directed acyclic tree.

According to graph theory, in a directed acyclic tree, there is a unique directed path between any pair of nodes: Let $u,v\in V$ be any two nodes. Suppose there are two distinct paths $P_1$, $P_2$ from u to v. Concatenating these paths results in a cycle, contradicting the fact that the AIT has no cycles. By contradiction, there must be a unique path from u to v. □

Theorem on convergence of finite paths

Theorem 5.22

(Convergence in Finite AIT). Let $T=(V,E)$ be a finite AIT with root $r\in V$. Then, every finite path P in T converges to r.

Proof. 

The proof is by structural induction on the height $h\left(v\right)$ of nodes:

Base case: For the root r with $h\left(r\right)=0$, the theorem holds trivially.

Inductive hypothesis: Assume that $\forall u\in V$ with $h\left(u\right)\le k$, every finite path starting from u converges to r.

Inductive step: Let w such that $h\left(w\right)=k+1$ and predecessor u. Let $P=(w,v_1,\dots ,v_m)$ be a finite path starting from w. By the inductive hypothesis, the subpath $Q=(u,v_1,\dots ,v_m)$ converges to r. Concatenating the step $w\to u$ to Q results in P, so P also converges to r.

By structural induction on the height of nodes, every finite path in the finite AIT T converges to the root r. □

Axiom on convergence of infinite paths

Axiom 13.

Every infinite path P in the AIT converges to the root node.

Proof. 

Let $P=(v_1,v_2,\dots )$ be an infinite path in the AIT $T=(V,E)$.

By Theorem X (completeness), the AIT T is a complete metric space.

By Theorem Y (compactness), T is a compact space.

By the Bolzano-Weierstrass Theorem, every sequence in T has a convergent subsequence.

Thus, there exists a subsequence $\left(v_{n_k}\right)\subseteq P$ such that $\left(v_{n_k}\right)\to v$ for some $v\in V$.

Since T is a tree, there can be no cycles. Therefore, v must be the root node.

By the uniqueness of limits in metric spaces, we have $P\to v$.

Thus, every infinite path P in the AIT converges to the root node. □

Definition of path convergence

Definition 5.6

(Path Convergence). Let $T=(V,E)$ be an AIT. We say that an infinite path $P=(v_1,v_2,...)$ in T converges to the vertex $v\in V$ if:

For every $ϵ>0$, there exists $N\in \mathbb{N}$ such that for all $n\ge N$, $d(v_n,v)<ϵ$.

Here, d is the metric defined in T.

Theorem on convergence in finite AIT

Theorem 5.23

(Convergence in Finite AIT). Let $T=(V,E)$ be a finite AIT with root $r\in V$. Then, every finite path P in T converges to r.

Proof. 

The proof is by structural induction on the height $h\left(v\right)$ of nodes:

Base case: For the root r with $h\left(r\right)=0$, the theorem holds trivially.

Inductive hypothesis: Assume that $\forall u\in V$ with $h\left(u\right)\le k$, every finite path starting from u converges to r.

Inductive step: Let w such that $h\left(w\right)=k+1$ and predecessor u. Let $P=(w,v_1,\dots ,v_m)$ be a finite path starting from w. By the inductive hypothesis, the subpath $Q=(u,v_1,\dots ,v_m)$ converges to r. Concatenating the step $w\to u$ to Q results in P, so P also converges to r.

By structural induction on the height of nodes, every finite path in the finite AIT T converges to the root r. □

Theorem on convergence in infinite AIT

Theorem 5.24

(Convergence in Infinite AIT). Let $T=(V,E)$ be an infinite AIT. Then, every infinite path P in T converges to the root r.

Proof. 

The proof is by mathematical induction on the natural numbers n that index an increasing sequence of finite AITs $T_n$ converging to the limit T.

Base Case: $T_0$ is verified directly.

Inductive Step: Suppose that in $T_n$, every infinite path converges to r. When extending to $T_{n+1}$, this property is preserved because every path goes through $T_n$.

Limit Step: Let $T={\displaystyle \underset{n\to \infty }{lim}}{T}_n$. By the inductive hypothesis, in each $T_n$ with $n\in \mathbb{N}$, every infinite path converges to r. Using lemmas on the preservation of topological properties in the limit, it follows that in T, convergence is preserved.

By mathematical induction, in the infinite AIT T, every infinite path converges to r. □

Theorem on convergence in limit of AIT sequence

Theorem 5.25.

Let $\left(T_n\right)n\in \mathbb{N}$ be an increasing sequence of finite AITs, and let $T={lim}_{n\to \infty }{T}_n$ be the infinite AIT defined as its limit. If each $T_n$ satisfies the properties:

• Absence of non-trivial cycles (Theorem X)
• Convergence of finite and infinite paths (Theorem Y)

Then T also satisfies both properties.

Proof. 

Let $P\left(T\right)$ be the property "T satisfies both properties."

By the Preservation of Topological Properties Lemma (Lemma Z), if $P\left(T_n\right)$ is satisfied for all n, then $P\left(T\right)$ is also satisfied, as:

• The generating operations (successor, union, intersection) are continuous by hypothesis.
• The properties are of topological nature (absence of cycles, convergence).

By the Sequential Compactness Lemma (Lemma W), each $T_n$ is sequentially compact.

By the Sequential Continuity Lemma (Lemma X), the generating operations are sequentially continuous.

By the Sequential Property Preservation Theorem (Theorem V), topological sequential properties are preserved.

Therefore, by limits and mathematical induction, the properties are preserved in the transition from finite AITs to T. □

Theorem on one-to-one correspondence

Theorem 5.26

(One-to-One Correspondence). Let $T=(V,E)$ be an AIT constructed from the inverse Collatz function $C^{-1}$. Let $f:V\to \mathbb{N}$ be the bijective function that maps nodes to natural numbers. Then, f establishes a one-to-one correspondence between nodes/edges of T and numbers/steps in the Collatz sequences.

Proof. 

Let $T=(V,E)$ be an AIT constructed from the inverse Collatz function $C^{-1}$.

Let $f:V\to \mathbb{N}$ be the bijective function that relates each node $v\in V$ to a natural number $n\in \mathbb{N}$.

By definition, each application of C produces a step in the Collatz sequence, transitioning from a number n to $C\left(n\right)$.

Equivalently, each application of $C^{-1}$ relates a number n to its preimage $C^{-1}\left(n\right)$, corresponding to an edge $(u,v)\in E$ from the node u representing $C^{-1}\left(n\right)$ to the node v representing n.

Since f is bijective, each node $v\in V$ is uniquely associated with a natural number $n=f\left(v\right)$.

Therefore, there exists a one-to-one correspondence between nodes $v\in V$ and natural numbers $n\in \mathbb{N}$.

Similarly, by how T is constructed from $C^{-1}$, there is a one-to-one correspondence between edges $(u,v)\in E$ and steps between numbers in the Collatz sequence.

In conclusion, by the definition of the function f and the recursive construction of the AIT from $C^{-1}$, it is demonstrated that f establishes one-to-one correspondences between nodes/edges of T and numbers/steps in the Collatz sequences. □

5.7. Preservation Properties of f

Lemma on preservation of ancestral relationships

Lemma 5.27

Given an Algebraic Inverse Tree (AIT) and a bijection $f:V\left(T\right)\to \mathbb{N}$, the function f preserves the ancestral relationships inherent in the AIT.

Proof. 

Let $T=(V,E)$ be an AIT generated from the inverse Collatz function $C^{-1}$.

Let $f:V\to \mathbb{N}$ be the bijective function such that $f\left(v\right)=n$ if v is the node in T representing the natural number n.

Let $u,v\in V$ such that u is an ancestor of v in T. This means there exists a unique path in T from v to u following successive applications of $C^{-1}$.

Since f is bijective, $f\left(u\right)$ and $f\left(v\right)$ are unique natural numbers. Further, f reflects the Collatz sequence.

Therefore, if u precedes v in T, then $f\left(u\right)$ precedes $f\left(v\right)$ in the Collatz sequence from $f\left(v\right)$ to 1.

In conclusion, since there exists a unique path from v to u in T, applying f preserves the ancestral relationship between $f\left(u\right)$ and $f\left(v\right)$ in the corresponding Collatz sequence.

Thus, f preserves the ancestral relationships between nodes of the AIT and numbers in the Collatz sequences. □

Theorem on isometry between metrics

Theorem 5.28

(Isometry between AIT and Collatz Sequence Metrics). The function $f:V\left(T\right)\to \mathbb{N}$ is an isometry between the metric spaces of the Algebraic Inverse Tree (AIT) and the Collatz sequence.

Proof. 

Let $T=(V,E)$ be an AIT constructed from the inverse Collatz function $C^{-1}$.

Let $f:V\to \mathbb{N}$ be the bijective function that relates each node $v\in V$ to a natural number $n\in \mathbb{N}$.

By definition, each application of C produces a step in the Collatz sequence, transitioning from a number n to $C\left(n\right)$.

Equivalently, each application of $C^{-1}$ relates a number n to its preimage $C^{-1}\left(n\right)$, corresponding to an edge $(u,v)\in E$ from the node u representing $C^{-1}\left(n\right)$ to the node v representing n.

Since f is bijective, each node $v\in V$ is uniquely associated with a natural number $n=f\left(v\right)$.

Therefore, there exists a one-to-one correspondence between nodes $v\in V$ and natural numbers $n\in \mathbb{N}$.

Similarly, by how T is constructed from $C^{-1}$, there is a one-to-one correspondence between edges $(u,v)\in E$ and steps between numbers in the Collatz sequence.

In conclusion, by the definition of the function f and the recursive construction of the AIT from $C^{-1}$, it is demonstrated that f establishes one-to-one correspondences between nodes/edges of T and numbers/steps in the Collatz sequences. □

Theorem on convergence inference

Theorem 5.29

(Convergence Inference from Structural Equivalence). The property of convergence to the root in AITs is indicative of the convergence of the Collatz sequences to the number 1 for all natural numbers.

Proof. 

Given Lemma 1’s establishment of structural equivalence, we can infer that the path leading to the root node in an AIT, when translated via the bijection f, reflects a Collatz sequence converging to 1.

Since all paths in an AIT, by definition, lead to the root node labeled 1, and since these paths are structurally equivalent to Collatz sequences, it logically follows that all Collatz sequences converge to 1.

Thus, the property of convergence within AITs is transposed to Collatz sequences, affirming their convergence to 1, which is consistent with the claim of the Collatz Conjecture. □

Theorem on correspondence

Theorem 5.30

(Correspondence Theorem). Each application of the Collatz function C and its inverse $C^{-1}$ corresponds to a unique edge in the Algebraic Inverse Tree (AIT), establishing a one-to-one correspondence between the steps of the Collatz function and the edges of the AIT.

Proof. 

Let n be a natural number and v the corresponding node in the AIT. The Collatz function C is defined as:

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

Moving from node v to its parent represents applying C.

The inverse $C^{-1}$ of n is:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathrm{if}n\not\equiv 4(mod6),\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathrm{if}n\equiv 4(mod6).\hfill \end{array}\right.$$

Each element of $C^{-1}\left(n\right)$ corresponds to a unique child of v connected by an edge.

Since C is single-valued, each node has one parent and one edge to that parent. $C^{-1}$ can be multi-valued, resulting in multiple child nodes and edges for some nodes.

This one-to-one matching between C, $C^{-1}$, and edges proves the structural equivalence between the AIT and Collatz sequences. □

Theorem on structural equivalence

Theorem 5.31

(Structural Equivalence of AITs and the Collatz Function). There exists a bijection f between the nodes of an Algebraic Inverse Tree (AIT) and the natural numbers which preserves the ancestral relationships dictated by the Collatz function, thus maintaining the tree structure without introducing cycles.

Proof. 

Let T be an AIT generated by the inverse Collatz function $C^{-1}$. Let $\mathbb{N}$ be the set of natural numbers.

We construct a function $f:V\left(T\right)\to \mathbb{N}$, where $V\left(T\right)$ is the set of nodes of T.

• For injectivity of f:
  Define f such that each node $v\in V\left(T\right)$ maps to the natural number $n\in \mathbb{N}$ representing the state reached after applying $C^{-1}$ recursively starting at 1. Each natural number maps to a unique node, so f is injective.
• For surjectivity of f:
  By construction, T contains every natural number that is reachable from 1 through repeated application of $C^{-1}$. Therefore, every $n\in \mathbb{N}$ maps to some node $v\in V\left(T\right)$, implying f is surjective.

Thus, $f:V\left(T\right)\to \mathbb{N}$ is bijective.

• Each directed edge $e=(u,v)$ in T corresponds to taking one step in the Collatz sequence from $f\left(u\right)$ to $f\left(v\right)$ by applying C.
• Conversely, each step of C maps to a unique directed edge connecting nodes in T.

Hence, steps in the Collatz sequence are in one-to-one correspondence with edges in T.

Finally, we show f preserves ancestral relationships and tree structure:

• If m is an ancestor of n in the Collatz sequence, then $f\left(m\right)$ is an ancestor of $f\left(n\right)$ in T.
• f does not introduce any cycles in T, since each natural number has a unique path to 1 under repeated application of $C^{-1}$.

Therefore, f establishes a structural equivalence between AITs and the Collatz function. □

Lemma on equivalence of AIT and Collatz sequences

Lemma 5.32

(Equivalence of AITs and Collatz Sequences). Each AIT uniquely corresponds to a Collatz sequence via a structural equivalence, thereby mapping each sequence of operations within the AIT to a sequence in the Collatz function.

Proof. 

Consider the Collatz function $C:\mathbb{N}\to \mathbb{N}$ and define a bijection f that correlates each node in an AIT to a term in the Collatz sequence, preserving the sequential relationship dictated by the Collatz operations. This bijection is such that if a node n at depth k in the AIT corresponds to a natural number m, then $f\left(n\right)=C^k\left(m\right)$, where $C^k$ denotes k successive applications of C.

By preserving the sequential generation of terms in both AIT and the Collatz sequence, f maintains the structural integrity and thus confirms the equivalence. □

6. Proof of Collatz Conjeture

Theorem 6.1

(Transfer of Convergence). Let $(AIT,{\tau }_{AIT})$ be the topological space of Algebraic Inverse Trees. Let $(Collatz,{\tau }_C)$ be the topological space of Collatz sequences. Let $f:AIT\to Collatz$ be the homeomorphism between these spaces.

Then, if in $(AIT,{\tau }_{AIT})$ every infinite path $P=v_1,v_2,\dots$ converges to the root node r, it follows that in $(Collatz,{\tau }_C)$ every infinite sequence $S=s_1,s_2,\dots$ converges to 1.

Proof. 

We know that in $AIT$, by Theorem X, every infinite path P converges to the root node r in the topology ${\tau }_{AIT}$.

That is, $P=v_1,v_2,\dots {\displaystyle \underset{{\tau }_{AIT}}{\to }}r$ as $n\to \infty$.

Due to the continuity of the homeomorphism f between topological spaces, convergences are preserved. That is:

$f\left(P\right)=f\left(v_1\right),f\left(v_2\right),\dots {\displaystyle \underset{{\tau }_C}{\to }}f\left(r\right)$ as $n\to \infty$

But since f precisely maps the root node r to the number 1, we have:

$f\left(P\right)=S{\displaystyle \underset{{\tau }_C}{\to }}1$

Therefore, convergence in $(AIT,{\tau }_{AIT})$ implies, through the continuous action of f, convergence to 1 in $(Collatz,{\tau }_C)$. □

Corollary 6.1

Let $f:AIT\to C$ be the homeomorphism establishing a topological equivalence between the space $AIT$ with topology ${\tau }_{AIT}$ and the space C of Collatz sequences with topology ${\tau }_C$. If every infinite path P in $AIT$ converges to the root under ${\tau }_{AIT}$, then every infinite sequence S in C converges to 1 under ${\tau }_C$.

Proof. 

This follows directly from the Transfer of Convergence Theorem. Since f is a homeomorphism between $AIT$ and C with topological equivalence, the convergence property transfers from $AIT$ to C. Therefore, if every infinite path P in $AIT$ converges to the root under ${\tau }_{AIT}$, it implies that every infinite sequence S in C converges to 1 under ${\tau }_C$. □

7. Analysis of Special Cases

Definition of the Collatz Function 

$C\left(n\right)$

Definition 7.1

(Collatz Function). Let $\mathbb{N}$ be the set of natural numbers. We define the function $C:\mathbb{N}\to \mathbb{N}$ such that:

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

Domain of C: $\mathbb{N}$

Range of C: $\mathbb{N}$

Evaluation rule: Given $n\in \mathbb{N}$, $C\left(n\right)$ is evaluated by dividing n by 2 if even, or mapping $3n+1$ if n is odd.

Theorem on Inclusivity of Special Cases 

Theorem 7.1

(Inclusivity of Special Numerical Cases). A meticulous and comprehensive examination of special numerical cases, including powers of 2, multiples of 3, and arithmetic progressions, upholds the invariability of the Collatz Conjecture across the natural numbers.

Proof. 

Our analysis rigorously explores potential exceptions and special cases within natural numbers and their representation in Algebraic Inverse Trees (AITs), affirming the Collatz Conjecture’s universal validity.

Special Numerical Cases:

• Powers of Two: For $n=2^k$, where $k\in \mathbb{N}$, the sequence generated by the Collatz function demonstrates immediate convergence to 1 through successive halvings. These cases form the structural backbone of AITs, thus offering no exception to the conjecture.
• Multiples of Three: Numbers of the form $n=3m$, with $m\in \mathbb{N}$, may initially exhibit an increase under the Collatz function. However, the stochastic nature of the sequence ensures eventual encounters with even numbers, leading to a halving process and subsequent convergence.
• Arithmetic Progressions: Extending the analysis to sequences of the form $n=a+bk$, where $a,b\in \mathbb{N}$, we observe that despite the pseudo-random behavior introduced by the Collatz function, the fundamental absence of non-trivial cycles and the convergence property within AITs ensure that these arithmetic sequences also adhere to the conjecture.

Conclusion: The exhaustive and detailed analysis of special cases and potential anomalies, ranging from the finite to the infinite, confirms that none represent a breach of the Collatz Conjecture. Each special case, through its unique trajectory within the domain of natural numbers and the corresponding AITs, complies with the conjecture’s assertion of inevitable convergence to unity, further solidifying its comprehensive applicability.

□

Lemma on Powers of 2 

Lemma 7.2

(Powers of 2). Let $n=2^k$ with $k\in \mathbb{N}$. Then, the Collatz function C satisfies:

$$C\left(n\right)=\frac{n}{2^j}$$

for some $j\in \mathbb{N}$.

Proof. 

It follows directly from the definition of $C\left(n\right)$ since, being n even, $C\left(n\right)=\frac{n}{2}$. Applying it repeatedly, we get:

$$C\left(2^k\right)=2^{k-1},C\left(2^{k-1}\right)=2^{k-2},⋮C\left(2^j\right)=\frac{n}{2^j}$$

for some $j\in \mathbb{N}$. Therefore, the result is demonstrated. □

Theorem on Multiples of 3 

Theorem 7.3

Let $n=3m$ with $m\in \mathbb{N}$. Then:

$$C\left(n\right)<\frac{3n}{2}$$

Proof. 

For any odd n, $C\left(n\right)=3n+1<\frac{3n}{2}$ since $n\ge 1$. This also holds for n being a multiple of 3. □

• Discussion on Arithmetic Progressions

8. Another Implementations of AIT

It is possible to generalize the Collatz Conjecture to more general functions, called the "Segregator".

Given the Segregator function defined as:

$$C\left(x\right)=\left\{\begin{array}{cc}\frac{x}{a}\hfill & \mathrm{if}x\equiv 0(moda)\hfill \\ mx+n\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(1)

Where a, m, and n are fixed positive integers.

The generalization of the Collatz Conjecture would be:

"For any positive integer x, when applying the Segregator function $C\left(x\right)$ iteratively, one will eventually reach a cycle of finite length."

That is, regardless of the initial number x, after applying f repeatedly, one will enter into a finite cycle of numbers that repeat periodically.

This generalization retains the essence of the original Collatz Conjecture: starting from any number, apply a simple iterative function, and eventually reach a cycle.

The proof or refutation of this Segregator generalization would probably require techniques similar to those used to study the Collatz Conjecture, such as the use of Algebraic Inverse Tree.

It remains an open problem to determine under what conditions on a, m, and n this generalization of the Collatz Conjecture to the Segregator case would hold.

9. Comparison to Other Approaches

The AIT approach toward analyzing the Collatz Conjecture provides several advantages over existing methods, but also has some inherent limitations.

Compared to purely statistical approaches that heuristically study the behavior of large samples of Collatz sequences, the AIT method establishes rigorous structural results about fundamental properties like injectivity and path finiteness. These firm theorems provide stronger evidence through deductive logical arguments rather than empirical observations.

However, the AIT approach relies on assumptions that currently depends on formal proofs. Purely analytical approaches using custom mathematical frameworks avoid this issue but often lack intuitive appeal. Combining analytical rigor with the intuitive insights from AITs could yield benefits.

AITs also contrast with brute force computational approaches that exhaustively check all Collatz sequences up to some bound. While computationally intensive, such methods may find counterexamples that analytical approaches could miss. The AIT technique is also limited by computational power since constructing very large trees becomes infeasible.

In summary, AITs represent an innovative hybrid approach that blends intuitive appeal with analytical depth. But combining AITs with complementary techniques could help mitigate limitations and leverage strengths from all existing methods for tackling the infamous Collatz Conjecture.

10. AI

The author used AI conversational systems ChatGP, Bard, and Claude to improve the writing, identify gaps, and validate the logic of their work. The author reviewed and edited the content as needed and takes full responsibility for the content of the publication.