The Collatz Conjecture: A New Perspective from 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.

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.

  -

  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. Formal Framework

4.1. Definitions

Definition 4.1 

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

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

Definition 4.2 

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

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

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

4.2. Axiomatization

Axiom 4. 

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

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

4.3. Theorems

Theorem 4.1 

(Collatz Conjecture). For any $n\in \mathbb{N}$, the sequence $n,C\left(n\right),C^2\left(n\right),\dots$ reaches 1.

Theorem 4.2. 

The cycle $(1,2,4)$ inevitably occurs. No other non-trivial cycles exist.

Theorem 4.3. 

$C^{-1}$ is surjective and sequentially continuous over $\mathbb{N}$.

Theorem 4.4. 

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 4.5. 

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 4.6. 

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. ☐

Theorem 4.7 

(Absence of Non-trivial Cycles). There are no cycles of length $k>3$ of the form:

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

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 such a non-trivial cycle of length $k>3$ exists.

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 the initial assumption that $k>3$.

Therefore, the assumption that there is a non-trivial cycle of length greater than 3 leads to a contradiction. By reductio ad absurdum, it is demonstrated that such a cycle cannot exist for the Collatz function. ☐

Lemma 4.8. 

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

Proof. 

Let us assume, for contradiction, that C is injective. This means that for any $x,y\in \mathbb{N}$, if $x\ne y$ then $C\left(x\right)\ne C\left(y\right)$.

Consider the numbers $x=2$ and $y=4$. We have:

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

Since $2\ne 4$ but $C\left(2\right)=C\left(4\right)=1$, this provides a counterexample to the injectivity of C.

Therefore, by contradiction, C cannot be injective. ☐

Lemma 4.9. 

The Collatz function C is surjective when restricted to the set of odd natural numbers.

Proof. 

Let $S=\{2n+1:n\in \mathbb{N}\}$ be the set of odd natural numbers.

We will show that for any $y\in S$, there exists an $x\in S$ such that $C\left(x\right)=y$.

Let $y\in S$. Since y is odd, it can be written as $y=2m+1$ for some $m\in \mathbb{N}$.

Now, consider $x=3m+1\in S$. Applying C, we get:

$$C\left(x\right)=C(3m+1)=3(3m+1)+1=9m+4=2(4m+2)=2m+1=y$$

Thus, for every $y\in S$, there is a preimage $x\in S$ such that $C\left(x\right)=y$. Hence, C is surjective when restricted to the odd natural numbers. ☐

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

Theorem 5.1. 

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 5.2 

(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:

$g\left(x\right)=\left\{\begin{array}{cc}\left\{2x\right\}\hfill & \mathrm{if}x\not\equiv 4(mod6)\hfill \\ \{2x,(x-1)/3\}\hfill & \mathrm{if}x\equiv 4(mod6)\hfill \end{array}\right.$

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, $\forall x\in \mathbb{N}$. ☐

Lemma 5.3. 

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

Proof. 

We will prove that for all $n\in \mathbb{N}$, there exists an $m\in \mathbb{N}$ such that $n\in C^{-1}\left(m\right)$ by mathematical induction on n.

Base case: Let $n=0$. By Axiom 1, there exists $C^{-1}\left(0\right)\subseteq \mathbb{N}$. Additionally, by Axiom 2, $C^{-1}\left(0\right)=\left\{0\right\}$. Then $0\in C^{-1}\left(0\right)$.

Inductive hypothesis: Assume that for all $k<n$, there exists an $m\in \mathbb{N}$ such that $k\in C^{-1}\left(m\right)$.

Inductive step: Let $n\in \mathbb{N}$. Define $m=2n$. Then, by Axiom 2, $C^{-1}\left(2n\right)=\left\{2n\right\}$. Since $n\in \left\{2n\right\}$, it follows that $n\in C^{-1}\left(2n\right)$.

By induction on $\mathbb{N}$, we have proved that for all $n\in \mathbb{N}$, there exists an $m\in \mathbb{N}$ such that $n\in C^{-1}\left(m\right)$. Therefore, $C^{-1}$ is surjective.

☐

Lemma 5.4. 

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

$$\forall a,b\in \mathbb{N}:a\ne b\Rightarrow C^{-1}\left(a\right)\cap C^{-1}\left(b\right)=\varnothing$$

Proof. 

We will prove the lemma by mathematical induction over $\mathbb{N}$.

Base case: For $n=0$, it holds that $C^{-1}\left(0\right)=\left\{0\right\}$ by definition. Clearly, $C^{-1}\left(0\right)\cap C^{-1}\left(1\right)=\varnothing$, validating the property for the base case.

Inductive hypothesis: Suppose the lemma holds for all $k<n$, that is:

$$\forall a,b\in \{0,1,\cdots ,n-1\}:a\ne b\Rightarrow C^{-1}\left(a\right)\cap C^{-1}\left(b\right)=\varnothing$$

Inductive step: Let $a,b\in \{0,1,\cdots ,n\}$ be such that $a\ne b$. Consider two cases:

Case 1: If $a,b<n$. By the inductive hypothesis, it follows that $C^{-1}\left(a\right)\cap C^{-1}\left(b\right)=\varnothing$.

Case 2: If only one of the numbers (assume a) is equal to n. Since $b<n$, by the inductive hypothesis $C^{-1}\left(a\right)\cap C^{-1}\left(b\right)=\varnothing$.

In both cases, we arrive at $C^{-1}\left(a\right)\cap C^{-1}\left(b\right)=\varnothing$. By the principle of mathematical induction, the lemma is proven. ☐

Lemma 5.5 

(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:

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

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 $\mathbb{N}$. ☐

Lemma 5.6. 

The relative density of even numbers is greater than that of odd numbers in the deep branches of a Collatz Inverse Tree (AIT).

Proof. 

Let T be an AIT and $B=\{b_1,b_2,\dots ,b_n\}$ a deep branch of T. We define:

• $N_e=\left|\left\{\mathrm{even}\mathrm{nodes}\mathrm{in}B\right\}\right|$
• $N_o=\left|\left\{\mathrm{odd}\mathrm{nodes}\mathrm{in}B\right\}\right|$

The relative density of even and odd numbers is given by:

• $D_e=\frac{N_e}{N_e+N_o}$
• $D_o=\frac{N_o}{N_e+N_o}$

Due to the properties of the inverse Collatz function $C^{-1}$, it is observed that at each step of deepening:

• Even nodes always produce odd nodes.
• Odd nodes may produce either even or odd nodes.

Therefore, as one progresses in depth, more even nodes are generated than odd. This implies that $N_e>N_o$ and therefore $D_e>D_o$.

Formally, this can be demonstrated using induction on the depth of the branch and analyzing the relative growth of $N_e$ and $N_o$. ☐

6. Algebraic Inverse Tree (AITs) for Analyzing the Collatz Sequence

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.

6.1. Axioms and Proofs relative to AIT

Definition 6.1 

(Cycle). Acyclein the context of a function $C:\mathbb{N}\to \mathbb{N}$, a graph, or a directed graph is a finite sequence $(a_0,a_1,\dots ,a_{k-1},a_k)$ with the following properties:

• For a function C: $\forall i\in \{0,\dots ,k-1\},C\left(a_i\right)=a_{i+1}$ and $C\left(a_k\right)=a_0$
• For a graph or a directed graph: $a_0=a_k$ and $a_i\ne a_j$ for all $0\le i<j<k$, where in a directed graph, $(a_i,a_{i+1})$ is an edge for all $0\le i<k$

The length of a cycle is the number of transitions (function applications or edges) it contains, which is k for the cycle defined above.

Definition 6.2. 

Let $T=(V,E)$ be a tree with V the set of vertices and E the set of edges.

An infinite path in T is an infinite sequence of vertices $(v_1,v_2,v_3,\dots )$ such that:

• $v_1,v_2,v_3,\dots$ are distinct vertices in V
• For all $i\ge 1$, the edge $(v_i,v_{i+1})$ is in E
• The path does not contain cycles, that is, there are no $i\ne j$ such that $v_i=v_j$

Definition 6.3. 

A non-trivial cycle in a tree or a function is a sequence of distinct nodes or values (other than the trivial node or value 1) that repeats indefinitely, forming a loop.

Definition 6.4 

(Infinite Path in a Graph). An infinite path in a graph is a sequence of edges which connects a sequence of vertices through a repeated application of the graph’s adjacency relation. Formally, an infinite path is a sequence $v_0,e_1,v_1,e_2,v_2,\dots$ where each $v_i$ is a vertex and each $e_i$ is an edge in the graph, such that for all i, the edge $e_i$ connects the vertex $v_{i-1}$ to $v_i$, and the sequence does not terminate.

Definition 6.5 

(Infinite Path in a Directed Graph). Let $G=(V,E)$ be a directed graph, where V is the set of vertices and $E\subseteq V\times V$ is the set of edges. Aninfinite pathin G is a sequence of nodes $(v_0,v_1,v_2,\dots )$ that continues indefinitely, where each $v_i\in V$ and $(v_i,v_{i+1})\in E$ for all $i\in \mathbb{N}$.

Definition 6.6 

(Rooted Tree). Arooted treeis a connected acyclic graph in which one vertex is distinguished as the root and every edge is directed away from the root. Afinite treeis a rooted tree in which the set of nodes V is finite.

Definition 6.7 

(Algebraic Inverse Tree). An algebraic inverse tree (AIT) is a rooted tree structure defined recursively using the inverse Collatz function $C^{-1}$. Specifically:

• The tree is rooted at node 1.
• Each node n has children given by the elements of $C^{-1}\left(n\right)$.
• An edge $(n,h)$ exists if and only if h is a child of n based on $C^{-1}$.
• An AIT can be finite with some maximum depth, or infinite in depth.

Definition 6.8 

(Finite AIT). A finite AIT is a directed tree $T_f=(V_f,E_f)$, where $V_f$ is a finite set of vertices and $E_f$ is a set of directed edges. Each vertex $v\in V_f$ represents a natural number, and each directed edge $(u,v)\in E_f$ corresponds to an application of the inverse Collatz function $C^{-1}$ from v to u. The tree $T_f$ thus represents a finite number of inverse Collatz iterations starting from 1.

Definition 6.9 

(Level of an AIT). The level of a node in an AIT is defined recursively - the root is at level 0, and children of a level l node are at level $l+1$.

Axiom 5 

(Rooted Tree Structure). An AIT forms a rooted tree structure with node 1 as the root.

Axiom 6 

(Node-Number Correspondence).

Each node in an AIT corresponds to a unique natural number.

Axiom 7 

(Edge Formation). An edge $(a,b)$ exists in an AIT if and only if $b\in C^{-1}\left(a\right)$.

Axiom 8 

(Path Convergence).

Every path in an AIT converges to the root node 1.

Axiom 9 

(Well-Founded Order). The nodes in an AIT are well-ordered by the standard < relation on natural numbers.

Theorem 6.1 

(Unique Path to Root). Every node in an AIT has a unique path to the root node 1.

Proof. 

Follows from the Path Convergence and No Cycle axioms. ☐

Theorem 6.2 

(Well-Ordered Nodes). The set of nodes in an AIT is well-ordered.

Proof. 

Follows from the Well-Founded Order axiom. ☐

Theorem 6.3 

(Subtree Preservation). Every subtree of an AIT is also an AIT.

Proof. 

Follows from the axioms which recursively define AITs. ☐

Theorem 6.4 

(Node Countability). The number of nodes in an AIT is countably infinite.

Proof. 

Follows from the bijection between nodes and natural numbers. ☐

Theorem 6.5 

(Cardinality Equivalence). The cardinality of nodes in an AIT equals the cardinality of $\mathbb{N}$.

Proof. 

Follows from the bijective correspondence between nodes and $\mathbb{N}$. ☐

Definition 6.10. 

An infinite AIT, denoted $T_{\infty }$, is a directed tree where the set of nodes $V\left(T_{\infty }\right)$ has cardinality ${\aleph }_0$, the cardinality of the natural numbers.

Theorem 6.6. 

$T_{\infty }$ can be constructed as the limit of an increasing sequence of finite AITs ${\left\{T_n\right\}}_{n=1}^{\infty }$:

$$T_{\infty }=\underset{n\to \infty }{lim}{T}_n$$

Proof. 

By the Subtree Axiom, each $T_n$ is a finite AIT. By the Countability Axiom, each $T_n$ contains countably infinite nodes.

As $n\to \infty$, the number of nodes across the $T_n$ increases without bound, resulting in a countably infinite limit tree $T_{\infty }$.

By the Cardinality Equivalence Theorem, $|V\left(T_{\infty }\right)|={\aleph }_0$, matching the cardinality of $\mathbb{N}$. ☐

Definition 6.11 

(Trivial Cycle). Atrivial cyclein the context of a Collatz sequence or a similar iterative process is a sequence of numbers where the iteration returns to the starting number after a finite number of steps. In the simplest case, this could be a one-step cycle where a number maps directly to itself under the given function. For the Collatz sequence, a trivial cycle would be the sequence (1, 4, 2, 1), which is the only known cycle for positive integers under the traditional Collatz function.

Definition 6.12 

(Non-Trivial Cycle). Anon-trivial cyclein the context of the Collatz conjecture is a sequence of natural numbers $(n_1,n_2,\dots ,n_k)$, with $k>1$, such that:

• For each $i\in \{1,2,\dots ,k-1\}$, the Collatz function C applied to $n_i$ yields $n_{i+1}$, i.e., $C\left(n_i\right)=n_{i+1}$.
• Applying the Collatz function to $n_k$ yields $n_1$, i.e., $C\left(n_k\right)=n_1$.
• The cycle is not the trivial cycle $(1,2,4)$ or any repetition thereof.

Moreover, it is quantified that for the cycle to be non-trivial, it must satisfy $\exists k\in \mathbb{N}$ such that $k>1$.

Theorem 6.7 

(Rooted Tree Structure of AIT). An Inverse Collatz Tree (AIT) forms a rooted tree with nodes representing natural numbers.

Proof. 

We will prove this theorem by mathematical induction.

Base Case: The AIT starting from the root node 1 trivially forms a rooted tree, since it contains only the single node 1 with no children.

Inductive Hypothesis: Assume that for some $k\in \mathbb{N}$, the AIT constructed from k forms a rooted tree.

Inductive Step: Consider the AIT constructed from $k+1$. By Axiom 3, each node m in this AIT has children given by $C^{-1}\left(m\right)$. By Axiom 1, $C^{-1}\left(m\right)$ is well-defined. By the injectivity of $C^{-1}$ (Lemma 4), each node has a unique predecessor. Therefore, there is a unique path from $k+1$ to any other node. By Lemma 7, no non-trivial cycles can occur. Hence, the AIT from $k+1$ is a rooted tree.

By mathematical induction, we have shown that the AIT constructed from any natural number is a rooted tree. ☐

Theorem 6.8 

(Parental Structure in AIT). Each node in an Inverse Collatz Tree (AIT) has at most two parents.

Proof. 

Let us assume, for contradiction, that there exists a node n in an AIT that has more than two parents.

By Axiom 3, the potential parents of n are given by $C^{-1}\left(n\right)$.

By Axiom 2, if n is even, then $C^{-1}\left(n\right)=\left\{2n\right\}$, so n would have only one parent, contradicting the assumption.

If n is odd, then by the definition of $C^{-1}$, either:

$n=4k+1$, in which case $C^{-1}\left(n\right)=\{(n-1)/3,2n\}$ gives two parents, or $n=4k+3$, in which case $C^{-1}\left(n\right)=\left\{2n\right\}$ gives one parent. In both odd cases, it contradicts having more than two parents.

Therefore, by contradiction, we have shown that each node n in an AIT can have at most two parent nodes. ☐

Theorem 6.9. 

The Inverse Collatz Tree derived from the inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ is a binary tree.

Proof. 

Let $\mathrm{AIT}=(V,E)$ be the Inverse Collatz Tree, where V is the set of nodes and E is the set of edges. By Theorem 5, V contains every natural number and is therefore countably infinite.

By Axiom 3, an edge $(m,n)\in E$ exists iff $n\in C^{-1}\left(m\right)$. By the definition of $C^{-1}$, each $n\in V$ has either one or two parent nodes in V:

If $n\not\equiv 4(mod6)$, then $|C^{-1}\left(n\right)|=1$ so n has one parent node. If $n\equiv 4(mod6)$, then $|C^{-1}\left(n\right)|=2$ so n has two parent nodes. Since $C^{-1}$ is injective by Lemma 4, distinct nodes cannot share the same parent.

Therefore, every node in V has at most two parents, satisfying the definition of a binary tree. Hence, AIT is a binary tree. ☐

Theorem 6.10. 

The non-injectiveness of the Collatz function implies that some values in the Collatz sequence might correspond to multiple nodes in an AIT, but this multiplicity does not affect the fundamental connection between the structures of AITs and the dynamics of the Collatz sequence.

Proof. 

By Lemma 6, the Collatz function C is not injective. This means there exist distinct $m,n\in \mathbb{N}$ such that $C\left(m\right)=C\left(n\right)$.

Consider an AIT derived from $C^{-1}$. By Axiom 3, each node x in the AIT has children given by $C^{-1}\left(x\right)$.

Now suppose $y=C\left(m\right)=C\left(n\right)$ for some $m\ne n$. Then by the definition of $C^{-1}$, both m and n would be children of node y in the AIT.

Therefore, the non-injectiveness of C implies that some nodes in the AIT can correspond to multiple values in a Collatz sequence.

However, by Axiom 2, each node in the AIT still represents a valid natural number. And by Axiom 3, the tree structure connecting nodes to their preimages under $C^{-1}$ reflects the inverse Collatz dynamics.

So while multiplicity arises, the core connection between AIT structure and Collatz sequence dynamics remains intact. The AIT veridically represents the underlying dynamics. ☐

Theorem 6.11. 

Let $T=(V,E)$ be a rooted binary tree with root r. For any node $u\in V$, there exists a unique path from u to r.

Proof. 

We will use proof by mathematical induction on the distance $d(u,r)$ between nodes u and r, defined as the length of the shortest path between u and r, where length is the number of edges. This distance function satisfies the following properties:

• Non-negativity: $d(u,v)\ge 0$
• Identity: $d(u,v)=0$ if and only if $u=v$
• Symmetry: $d(u,v)=d(v,u)$
• Triangle inequality: $d(u,w)\le d(u,v)+d(v,w)$

Base case: If $u\in V$ such that $d(u,r)=0$, then by identity $u=r$, so the only path is u itself.

Inductive hypothesis: Assume that for all $v\in V$ with $d(v,r)=k$, there exists a unique simple path $P(v,r)$ from v to r.

Inductive step: If $d(u,r)=k+1$, u has a unique parent v where $(u,v)\in E$ and $d(v,r)=k$. By the inductive hypothesis, there is a unique path $P(v,r)$. Extending this by the edge $(u,v)$ gives a unique path from u to r.

By mathematical induction, every $u\in V$ has a unique path to the root r. ☐

Lemma 6.12. 

Let $T_1$ be the Inverse Collatz Tree derived from the Collatz function C and its multivalued inverse $C^{-1}$. For every natural number n, there exists a corresponding node in $T_1$.

Proof. 

We prove this lemma using the principle of strong mathematical induction on n.

Let $P\left(n\right)$ be the proposition that for every natural number k less than or equal to n, there exists a node k in $T_1$.

Base Case ($n=1$): The root of $T_1$ is defined as 1, thus $P\left(1\right)$ is true.

Inductive Hypothesis: Assume that for all m less than or equal to n, the proposition $P\left(m\right)$ holds true, meaning that each k within this range exists as a node in $T_1$.

Inductive Step: To prove $P(n+1)$, consider the next natural number $k=n+1$. According to the definition of $T_1$, there must be a parent node p such that k is in the preimage of p under C. Given the inductive hypothesis, node p exists within $T_1$ since p is less than or equal to n. The structure of $T_1$, which is determined by $C^{-1}$, guarantees that for node p to be present, node k must also be included as a child of p. Therefore, node k must exist in $T_1$, confirming that $P(n+1)$ is true.

By the principle of strong induction, we conclude that the proposition $P\left(n\right)$ is valid for all natural numbers n. This implies that every natural number is represented as a node in $T_1$. ☐

Theorem 6.13. 

For any natural number n, it is possible to reach node n from the root node in a finite number of steps in the Inverse Collatz Tree ${\mathrm{AIT}}_n$.

Proof. 

We employ the principle of strong mathematical induction to demonstrate this theorem. Let $P\left(k\right)$ represent the following proposition:

For any natural number m that is less than or equal to k, the node m can be reached from the root of ${\mathrm{AIT}}_m$ in a finite number of steps.

Base Case ($k=1$): Given that node 1 is the root of ${\mathrm{AIT}}_1$ by definition, reaching it requires zero steps, thus $P\left(1\right)$ is established.

Inductive Hypothesis: Suppose that $P\left(j\right)$ is valid for all natural numbers j up to k, which implies that each node m with $m\le k$ can be finitely reached from the root in its respective AIT.

Inductive Step: To validate $P(k+1)$, consider a node $n=k+1$. Axiom 3 dictates that node n has predecessors that are described by $C^{-1}\left(n\right)$. We consider two scenarios:

Case 1: If n is not congruent to 4 modulo 6, then its predecessor is larger than n, and by the inductive hypothesis, all nodes up to k are reachable in a finite sequence of steps. Therefore, node n is accessible through its predecessor.

Case 2: If n is congruent to 4 modulo 6, it has at least one predecessor smaller than n. According to the inductive hypothesis, this smaller predecessor can be reached in a finite number of steps. Since n is the child of this reachable predecessor, n is also reachable within a finite sequence of steps.

Given that $P(k+1)$ is confirmed in both instances, by the method of strong induction, the proposition $P\left(n\right)$ is valid for every natural number n. ☐

Lemma 6.14. 

In the Collatz Inverse Tree (AIT_1) constructed from the function C and its inverse $C^{-1}$, non-trivial cycles are impossible regardless of the validity of the Collatz Conjecture.

Proof. 

Assume, for the sake of contradiction, that there exists a non-trivial cycle L different from $(1,2,4)$ in $AI{T}_1$.

By the definition of a cycle, there must exist a node $n\in L$ such that n is an element of $C^{-k}\left(n\right)$ for some $k\ge 1$, with $C^{-k}$ denoting the k-times iterated application of $C^{-1}$.

By Axiom 3, each node in $AI{T}_1$ has a unique sequence of predecessors determined by $C^{-1}$.

Lemma states that $C^{-1}$ has a complete image of $\mathbb{N}$ into $\mathbb{N}$.

If L were to exist, repeatedly applying $C^{-1}$ to the predecessors of n would result in an infinite descending sequence in $\mathbb{N}$, contradicting the countability of $\mathbb{N}$ established in Theorem .

The assumption of the existence of L leads to a contradiction, implying that such a cycle cannot be present in $AI{T}_1$.

By Axiom 2, $C^{-1}$ is injective, so a node cannot have multiple predecessors, a necessary condition for a non-trivial cycle.

Furthermore, the existence of a cycle would violate the tree structure of AITs.

Thus, the existence of non-trivial cycles in $AI{T}_1$, or any AIT derived from $C^{-1}$, is a logical impossibility. The initial assumption is denied, proving the non-existence of such cycles. ☐

Lemma 6.15. 

Let $AI{T}_f$ be a finite Inverse Collatz Tree of depth d rooted at 1. Consider any finite sequence of nodes $P={\left(x_k\right)}_{k=1}^N$ in $AI{T}_f$, where each $x_k\in \mathbb{N}$. Then there exists $k_0\in \mathbb{N}$ such that for all $k\ge k_0$, $x_{k+1}<x_k$, implying P is eventually strictly decreasing.

Proof. 

Assume for contradiction that no such $k_0$ exists. This means $\forall k\in \mathbb{N},\exists k^{\prime }>k$ such that $x_{k^{\prime }+1}\ge x_{k^{\prime }}$.

Consider $S=\{x_k:x_{k+1}\ge x_k\}\subset \mathbb{N}$. By our assumption, S is infinite.

However, by the Well-Ordering Principle (Theorem ), every non-empty subset of $\mathbb{N}$ has a least element. Let $x_m$ be the least element of S.

By definition of S, $x_{m+1}\ge x_m$. And since $x_m$ is the least in S, $\nexists x_{m-1}:x_m\ge x_{m-1}$, otherwise $x_{m-1}$ would be a smaller element in S.

Therefore, $x_{m+1}<x_m$, contradicting the fact that $x_m$ is in S.

By contradiction, our assumption is invalidated. Hence $\exists k_0$ such that $\forall k\ge k_0,x_{k+1}<x_k$, and P is eventually decreasing. ☐

Lemma 6.16. 

Let $T=(V,E)$ be a finite Collatz Inverse Tree (AIT) with root node $r\in V$. Then, every finite path in T converges to the root node r.

Proof. 

We will prove the lemma by structural induction on the height h of the nodes in T.

Base case: For the root node r with $h\left(r\right)=0$, every finite path starting from r converges to r in 0 steps.

Inductive Hypothesis: Suppose that for every node u with $h\left(u\right)\le k$, every finite path starting from u converges to r.

Inductive Step: Let w be a node with $h\left(w\right)=k+1$ and parent u. Let $P=(w,v_1,\cdots ,v_m)$ be a finite path starting from w. By the inductive hypothesis, the subpath $Q=(u,v_1,\cdots ,v_m)$ converges to r. Concatenating the step $w\to u$ to Q gives us P, and thus P also converges to r.

By structural induction on the height of the nodes, we have proven that every finite path in the finite AIT T converges to the root node r. ☐

7. Bijection Between AIT Nodes and Natural Numbers

Theorem 7.1 

(Existence of Bijection). There exists a bijection $f:V\left(T\right)\to \mathbb{N}$ between the nodes $V\left(T\right)$ of an algebraic inverse tree T and the natural numbers $\mathbb{N}$.

Proof. 

Let T be an algebraic inverse tree constructed from the inverse Collatz function $C^{-1}$.

Define a function $f:V\left(T\right)\to \mathbb{N}$ such that for each node $v\in V\left(T\right)$, $f\left(v\right)$ maps to the unique natural number that node v represents, based on the AIT construction using $C^{-1}$.

We will prove that f is bijective:

Injectivity: By Axiom 2, each node in T corresponds to a unique natural number. Thus, for any distinct nodes $u,v\in V\left(T\right)$, it follows that $f\left(u\right)\ne f\left(v\right)$. Therefore, f is injective.

Surjectivity: According to Lemma 6.12, for every natural number $n\in \mathbb{N}$, there is a corresponding node in T. Thus, for each $n\in \mathbb{N}$, there exists a node $v\in V\left(T\right)$ such that $f\left(v\right)=n$. Therefore, f is surjective.

Since f is both injective and surjective, it constitutes a bijection between the nodes of T and $\mathbb{N}$. ☐

Corollary 7.1. 

The node set of the infinite AIT $T_{\infty }$ is countably infinite.

Proof. 

By Theorem , there exists a bijection $f:\mathbb{N}\to V\left(T_{\infty }\right)$ between the natural numbers and the nodes of $T_{\infty }$. Since $\mathbb{N}$ is countably infinite and f establishes a one-to-one correspondence, it follows that $V\left(T_{\infty }\right)$ must also be countably infinite. ☐

Theorem 7.2 

(Cardinality Equivalence). The cardinality of the node set of any AIT is equal to the cardinality of $\mathbb{N}$.

Proof. 

Let T be an arbitrary AIT. By Theorem , there exists a bijection $f:V\left(T\right)\to \mathbb{N}$ between the nodes of T and the natural numbers.

It is a fundamental result that two sets are equinumerous (have equal cardinality) if there exists a bijection between them.

Therefore, since f constitutes a bijection between $V\left(T\right)$ and $\mathbb{N}$, their cardinalities must be equal. This holds for any AIT T. ☐

8. Isometry of f between AIT and Collatz Sequence Metrics

Theorem 8.1 

(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. 

Consider an AIT T derived from the inverse Collatz function $C^{-1}$, with $f:V\left(T\right)\to \mathbb{N}$ as a bijection between the nodes of T and natural numbers.

Denote the metric on T by $d_T$, representing the shortest path between nodes, and let $d_C$ be the metric on $\mathbb{N}$, defined by the Collatz iteration steps.

Employing strong induction on the node height h in T, we assert that $d_T(u,v)=d_C(f\left(u\right),f\left(v\right))$ for all nodes $u,v\in V\left(T\right)$.

Base Case: For $h=0$, any node u and root r, $d_T(u,r)=h$ aligns with $d_C(f\left(u\right),1)$ by f’s definition.

Inductive Hypothesis: Assume the isometry holds for all nodes up to height h.

Inductive Step: At height $h+1$, for a node w with parent u and any node v, inductive hypothesis and metric definitions imply:

$$d_T(w,v)=d_T(u,v)+1=d_C(f\left(u\right),f\left(v\right))+1=d_C(f\left(w\right),f\left(v\right)).$$

Thus, by induction, $d_T(u,v)=d_C(f\left(u\right),f\left(v\right))$ holds for all $u,v$, confirming f’s isometric property. ☐

9. Continuity of f and $f^{-1}$

Theorem 9.1 

(Continuity of f). Let $f:V\left(T\right)\to \mathbb{N}$ be a bijection from the nodes of an AIT with the standard topology to the natural numbers with the discrete topology. Then f is continuous.

Proof. 

To show continuity of f, we use an epsilon-delta argument. In the discrete topology on $\mathbb{N}$, for any $\epsilon >0$, there exists a $\delta$ (which can be taken as 1 without loss of generality) such that for all $x,y\in \mathbb{N}$, if $|x-y|<\delta$, then $\left|f\right(x)-f(y\left)\right|<\epsilon$.

Since the discrete topology is being used, the preimage of any set under f is open in $V\left(T\right)$. Therefore, for any open set $U\subseteq \mathbb{N}$, $f^{-1}\left(U\right)$ is open in $V\left(T\right)$, satisfying the definition of continuity. ☐

Theorem 9.2 

(Continuity of $f^{-1}$). Let $f^{-1}:\mathbb{N}\to V\left(T\right)$ be the inverse function of f. Then $f^{-1}$ is continuous.

Proof. 

For the continuity of $f^{-1}$, we consider any ${\epsilon }^{\prime }>0$ in the standard topology of the AIT. We can choose ${\delta }^{\prime }=1$ because in the discrete topology on $\mathbb{N}$, the distance between any two distinct points is at least 1.

For any $n,m\in \mathbb{N}$ if $|n-m|<{\delta }^{\prime }$, then $n=m$ due to the discreteness of $\mathbb{N}$, and it follows that $d(f^{-1}\left(n\right),f^{-1}\left(m\right))=0<{\epsilon }^{\prime }$, where d denotes the metric on $V\left(T\right)$. Hence, $f^{-1}$ is continuous. ☐

10. Preservation of Topological Properties

Let $f:V\left(T\right)\to \mathbb{N}$ be the previously demonstrated bijective function between the nodes of an Algebraic Inverse Tree (AIT) T and the natural numbers $\mathbb{N}$.

Theorem 10.1 

(Preservation of Non-Trivial Cycle Absence). The function f preserves the property of absence of non-trivial cycles from AIT T to $\mathbb{N}$.

Proof. 

By construction, T contains no cycles as it is a tree. Since f is injective, it maps distinct nodes to distinct numbers, and thus cannot introduce cycles into $\mathbb{N}$. ☐

Theorem 10.2 

(Preservation of Convergence). If every path in T converges to the root node, then under f, every Collatz sequence converges to 1 in $\mathbb{N}$.

Proof. 

Let $n\in \mathbb{N}$ and let $v=f^{-1}\left(n\right)$ be the associated node in T. As every path in T converges to the root, the unique path from v also converges. By the preservation of ancestral relationships, f maps this path to a sequence in $\mathbb{N}$ that converges to 1. ☐

Theorem 10.3 

(Preservation of Compactness). Any finite subset of nodes in T is mapped by f to a finite and therefore compact subset of $\mathbb{N}$.

Proof. 

This follows directly from f being a bijective function. ☐

11. Implication of preservation of properties

Theorem 11.1 

(Preservation of Properties and Implication for the Collatz Conjecture). Let $f:V\left(T\right)\to \mathbb{N}$ be a bijective function between the nodes of the Algebraic Inverse Tree (AIT) T and the natural numbers $\mathbb{N}$. If f preserves the topological properties of absence of non-trivial cycles, path convergence to the root node, and compactness, as well as the metric property of isometry between distances in T and the steps in the Collatz sequence, then the Collatz conjecture holds for every sequence generated by a natural number.

Proof. 

The following facts have been previously established:

• The function f preserves the mentioned topological properties between T and $\mathbb{N}$ (Theorem X).
• The function f is an isometry between the metrics defined on T and in the Collatz sequence (Theorem Y).
• These preserved properties imply that any finite path in T has a finite length and converges to the root node, corresponding to the fact that every Collatz sequence converges to 1 in a finite number of steps (Lemma 6.13).

Since the preservation of the relevant topological and metric properties by the function f has been demonstrated, and these properties entail the convergence of any Collatz sequence to 1 via deductive reasoning, it follows that the Collatz conjecture is upheld. ☐

Theorem 11.2 

(Convergence of Paths Implies Convergence of Sequences). Let T be an algebraic inverse tree and $f:V\left(T\right)\to \mathbb{N}$ the bijection between nodes of T and natural numbers. If every path in T converges to the root node, then every Collatz sequence converges to 1.

Proof. 

We have previously established:

• Every path in T converges to the root node (Theorem X).
• The function f is a bijection between nodes of T and $\mathbb{N}$ (Theorem Y).
• The function f preserves ancestral relationships and paths in T (Theorem Z).

Let $n\in \mathbb{N}$ and consider the Collatz sequence starting at n. By the bijection of f, this corresponds to a unique path in T starting from the node $f^{-1}\left(n\right)$. Since all paths in T converge to the root, the corresponding path starting at $f^{-1}\left(n\right)$ must also converge to the root node. By the ancestral relationship preservation of f, this implies that the Collatz sequence starting at n converges to 1.

Since this holds for any $n\in \mathbb{N}$, we conclude that every Collatz sequence converges to 1. ☐

11.1. Preservation of Injectivity in the Limit

Lemma 11.3 

(Injectivity Preservation). The injectivity of the inverse Collatz function, denoted as $C^{-1}$, is preserved in the limit, precluding the possibility of cycles.

Proof. 

Let ${\left\{AI{T}_n\right\}}_{n\in \mathbb{N}}$ be an increasing sequence of finite Algebraic Inverse Trees (AITs), where each $AI{T}_n$ corresponds to the inverse iterations of the Collatz function up to depth n. Assume that for every finite n, the function $C^{-1}$ restricted to $AI{T}_n$ is injective.

Consider the limit $AI{T}_{\infty }={lim}_{n\to \infty }AI{T}_n$, representing an infinite AIT. We aim to show that $C^{-1}$ remains injective in $AI{T}_{\infty }$.

Suppose, for the sake of contradiction, that $C^{-1}$ is not injective in $AI{T}_{\infty }$. Then, there exist distinct nodes $a,b\in AI{T}_{\infty }$ such that $C^{-1}\left(a\right)=C^{-1}\left(b\right)$. However, both a and b must belong to some finite $AI{T}_n$ due to the construction of $AI{T}_{\infty }$. This implies that $C^{-1}$ is not injective in $AI{T}_n$, contradicting our assumption.

Therefore, the injectivity of $C^{-1}$ must be preserved in the limit $AI{T}_{\infty }$, ensuring that no cycles can form in the infinite AIT. ☐

12. Formalization of Collatz Inverse Trees (AITs)

Definition 12.1 

(Infinite AIT). An infinite AIT is a directed tree $T_{\infty }=(V_{\infty },E_{\infty })$, where $V_{\infty }$ is an infinite set of vertices and $E_{\infty }$ is a set of directed edges. It represents the complete set of all possible inverse iterations of the Collatz function $C^{-1}$ on the number 1, and is defined as the limit of an increasing sequence of finite AITs ${\left\{T_f^n\right\}}_{n=1}^{\infty }$.

Theorem 12.1. 

An infinite AIT $T_{\infty }$ can be formalized as the limit of an increasing sequence of finite AITs ${\left\{T_f^n\right\}}_{n=1}^{\infty }$.

Proof. 

Let ${\left\{T_f^n\right\}}_{n=1}^{\infty }$ be a sequence of finite AITs where each $T_f^n$ represents the inverse iterations of the Collatz function $C^{-1}$ on the number 1 up to depth n. Formally, $T_f^n=(V_f^n,E_f^n)$, where $V_f^n$ contains all the natural numbers that can reach 1 after n or fewer iterations of C, and $E_f^n$ contains the corresponding directed edges.

Since each $T_f^n$ is a subtree of $T_f^{n+1}$, we can define the infinite AIT $T_{\infty }$ as the limit of this sequence:

$$T_{\infty }=\underset{n\to \infty }{lim}{T}_f^n$$

Thus, $T_{\infty }$ contains all the natural numbers that eventually reach 1 after successive iterations of the Collatz function $C^{-1}$, and each finite AIT $T_f^n$ is an approximation of $T_{\infty }$. ☐

13. Transfinite Induction on Algebraic Inverse Trees (AITs)

Definition 13.1 

(Transfinite Induction). Transfinite induction is an extension of the principle of mathematical induction to well-ordered sets that may be infinite, allowing for induction over ordinal numbers.

Theorem 13.1 

(Preservation of Path Convergence in the Limit Ordinal Case). Let $T_{\alpha }$ be an algebraic inverse tree of ordinal height α. Let $P\left(T_{\alpha }\right)$ denote the statement "All paths in $T_{\alpha }$ converge to the root node". If $P\left(T_{\alpha }\right)$ holds for all $\alpha <\lambda$ where λ is a limit ordinal, then $P\left(T_{\lambda }\right)$ also holds.

Proof. 

Let $c=(v_1,v_2,\dots )$ be an infinite path in $T_{\lambda }$. Since $\lambda$ is the limit of all $\alpha <\lambda$, there exists an ${\alpha }_0<\lambda$ such that $v_1\in T_{{\alpha }_0}$.

By the successor ordinal step, $P\left(T_{{\alpha }_0}\right)$ holds. Therefore, the finite subpath $\left(v_1\right)$ converges in $T_{{\alpha }_0}$. Since $T_{{\alpha }_0}\subseteq T_{\lambda }$, $\left(v_1\right)$ also converges in $T_{\lambda }$.

Proceeding inductively, and taking each ${\alpha }_i$ sufficiently large, it can be shown that every initial finite subpath of c converges in $T_{\lambda }$. By the compactness of $T_{\lambda }$, c converges.

Thus, it has been rigorously demonstrated that path convergence is preserved in the limit ordinal case of transfinite induction, completing the formal proof. ☐

Definition 13.2 

(Infinite AIT). An infinite AIT $T_{\infty }$ is defined as the limit of an increasing sequence of finite AITs ${\left\{T_f^n\right\}}_{n=1}^{\infty }$, where each finite AIT $T_f^n$ is an approximation of the infinite tree.

Definition 13.3 

(Well-Founded Order). A well-founded order on the set of finite AITs is an order relation ⪯ such that every non-empty subset of finite AITs has a minimal element under this relation, allowing the application of transfinite induction.

Lemma 13.2. 

The injectivity of the inverse Collatz function $C^{-1}$ is preserved in the limit, precluding the possibility of cycles in $T_{\infty }$.

Lemma 13.3. 

Bounded computational verifications on finite AITs experimentally corroborate the extension of their properties, such as injectivity, to the infinite case $T_{\infty }$.

14. The Infinite AIT as a Limit

The infinite AIT is the culmination of an unbounded sequence of finite AITs. To formalize this, we consider the sequence ${\left\{T_n\right\}}_{n=1}^{\infty }$, where each $T_n$ is a finite AIT, and $T_{n+1}$ includes all nodes and edges of $T_n$ with additional nodes and edges as dictated by the inverse Collatz function.

Theorem 14.1 

(Infinite AIT Construction). The infinite AIT, denoted as $T_{\infty }$, is the limit of the sequence ${\left\{T_n\right\}}_{n=1}^{\infty }$ of finite AITs. The tree $T_{\infty }$ includes every natural number as a node, and for every pair of nodes $m,n$ in $T_{\infty }$, there is a directed edge from m to n if and only if $C^{-1}\left(m\right)=n$.

Proof. 

We construct $T_{\infty }$ by defining its node set as the union of the node sets of all $T_n$, and its edge set as the union of the edge sets of all $T_n$. This construction ensures that $T_{\infty }$ contains all natural numbers and adheres to the inverse Collatz function.

For every natural number n, there exists a finite k such that n is included in $T_k$. Thus, n is also in $T_{\infty }$, proving that $T_{\infty }$ is comprehensive.

The absence of non-trivial cycles and the convergence of all paths to the root node in each $T_n$ are preserved in the limit, as these properties are intrinsic to the Collatz function and are independent of the finitude of the tree.

Therefore, $T_{\infty }$ effectively represents the infinite extension of the sequence of finite AITs, and it exhibits all the properties of its finite predecessors. ☐

15. Inheritance of Properties in Infinite AITs

Theorem 15.1. 

Infinite Algebraic Inverse Trees (AITs) inherit the topological and convergence properties demonstrated in the finite case.

Proof. 

Consider an infinite AIT $T_{\infty }$ and its corresponding sequence of finite AITs ${\left\{T_n\right\}}_{n=1}^{\infty }$, where each $T_n$ represents a finite approximation of $T_{\infty }$.

Compactness and Convergence in Finite AITs: Each finite AIT $T_n$ possesses topological and convergence properties, such as compactness, which ensures that all paths in $T_n$ converge to the root node 1.

Limit of Finite AITs: The infinite AIT $T_{\infty }$ is defined as the limit of the sequence ${\left\{T_n\right\}}_{n=1}^{\infty }$. As a limit of compact spaces, $T_{\infty }$ inherits the compactness property.

Inherited Convergence: Due to the compactness of $T_{\infty }$, all paths within this infinite AIT must also converge to the root node 1, just as they do in each finite approximation $T_n$.

Conclusion: Infinite AITs, being the limit of an increasing sequence of finite AITs, inherit the topological and convergence properties from their finite counterparts. This inheritance ensures that all paths in infinite AITs converge to the root node, maintaining the structural integrity and mathematical consistency of the AIT framework. ☐

16. Extending the properties of AIT to infinite paths

17. Structural Equivalence and Convergence

Theorem 17.1 

(Structural Equivalence). Structural equivalence between AITs and the Collatz function implies that every Collatz sequence converges to 1.

Proof. 

To prove Theorem 17.1, we rely on the established properties of AITs and demonstrate their implications for Collatz sequences.

Structural Equivalence and Convergence

In this section, we formalize the structural equivalence between Algebraic Inverse Trees (AITs) and Collatz sequences, utilizing this framework to deduce convergence properties of the sequences.

Definition 17.1 (Structural Equivalence) 

Structural equivalence between two relational structures, A and B, is established if there is a bijection $f:A\to B$ that preserves the relational properties inherent to the structures.

Lemma 17.2 (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. ☐

Theorem 17.3 (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. ☐

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