The Collatz Conjecture: A New Proof using Algebraic Inverse Trees

1. Motivation and General Description

The notoriety of the Collatz Conjecture contrasts strongly with the elusiveness it has shown to proof attempts for decades. Its simple statement, that by iterating a simple function over any number, one arrives at a trivial cycle, hides a complexity that has led to analytical dead ends.

In this article, we present a new strategy to approach this conjecture, which, while employing known mathematical principles, creatively differs from previous techniques. The idea is to "reverse" the numerical relationships inherent in Collatz sequences through a representation called AITs.

Formally, an AIT $T=(V,E)$ is a combinatorial structure composed of: - A set of nodes V representing natural numbers

- A directed relation $E\subseteq V\times V$ from ancestors to descendants - A root node $r\in V$ with a value of 1 - A function $C^{-1}:V\to P\left(V\right)$ that assigns to each node its child nodes according to the reverse Collatz recursion

To intuitively understand its utility, imagine that we reconstruct "backward" all the paths that converge to each term in the Collatz sequence, questioning their numerical origin step by step. This inverse perspective facilitates the global study of sequences, the identification of possible anomalies, and the estimation of convergence times.

Through careful topological correspondence, critical properties are transferred from the realm of AITs to the Collatz sequences themselves. Thus, proofs of topological complexity made in AITs allow us to infer these properties on Collatz sequences, formally proving this elusive conjecture.

Implications of Advancing the Resolution of the Collatz Conjecture

The resolution of the notorious yet elusive Collatz Conjecture through the technique of reverse modeling with AITs would have profound ramifications:

Impact on Number Theory

It would represent a historic milestone in solving an open problem that has challenged illustrious mathematicians for 84 years. It would open up new avenues of research into the topological properties of discrete dynamical systems modeled through directed combinatorial structures. Advancements in deductive formalizations regarding convergence in chaotic systems could lead to new approaches to addressing other latent problems such as the Riemann Hypothesis or the Goldbach Conjecture.

Methodological Generalization

AITs introduce innovative topological techniques, such as homeomorphic transport of structural properties between conceptually equipotent dynamical systems. This could lay the groundwork for a possible generalization of the method to other recursive functions that generate unpredictable systems, such as in coding algorithms or difference equations. Even in non-numeric systems, reverse modeling of complex relationships through directed graphs that are then topologically associated with the direct system could facilitate understanding.

Prospective Applications

In cryptography, constructing an AIT from a hypothetical injective hash function could help evaluate its resilience against inverted collisions or inadequate dispersals. In computational biology, structures similar to AITs could model regulatory interactions in gene networks. In theoretical physics, inverse modeling techniques subject to constraints could infer laws from experimental data sets.

2. Introduction

The Collatz Conjecture is a famous unsolved problem in mathematics stating that starting from any positive integer n, iterating the function

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

(1)

will eventually reach the number 1, at which point the sequence enters the trivial cycle $1\to 4\to 2\to 1$. Despite the simple formulation, the trajectory of the iteration appears erratic and no proof exists for all starting values. The Collatz conjecture asserts that, regardless of the starting number, the Collatz sequence will always converge to 1. This conjecture has been verified for all initial numbers up to $2^{68}$, but it has not yet been proven.

In this paper, we introduce AIT (AITs) as a novel representation for inversely modeling the relationships inherent in Collatz sequences. By recursively building inverted trees rooted at 1 using the inverse Collatz function $C^{-1}$, the structure of AITs formally captures all possible convergence pathways to 1 from any starting natural number.

We establish key properties of AITs, including guaranteed path convergence and absence of non-trivial cycles. Furthermore, we prove a topological equivalence between the space of AITs and the space of Collatz sequences. Leveraging this equivalence, convergence transfers from AIT paths to Collatz sequences, indicating that trajectories from all natural numbers provably approach 1.

The AIT perspective thus provides new structural insights and a platform for formally reasoning about convergence in the context of the infamous yet evasive Collatz Conjecture.

3. Comparison with Other Approaches

While the work of Terence Tao, Jeffrey Lagarias, and others has led to valuable contributions to the Collatz Conjecture through tools of numerical analysis, the approach of AITs represents a novel complementary geometric perspective.

Some points of contrast between these approaches:

• Existing proofs employ established analytical frameworks in number theory, whereas AITs constitute new combinatorial structures specifically designed to model the Conjecture.
• Tao’s groundbreaking proof computationally verified the conjecture for extremely large numbers. AITs, on the other hand, aim to provide a more conceptual understanding of the underlying dynamics.
• Lagarias’ study of statistical properties has parallels with how AITs reveal the system’s dynamics. However, AITs also facilitate the estimation of convergence times.
• Constructing AITs requires significantly less computation compared to exhaustively checking quadrillions of cases. However, both approaches can shed light from different angles.
• While grounded in existing mathematical principles, AITs have required the development of lemmas and theorems specifically tailored for this novel domain. Previous proofs leverage mature analytical tools.
• In conclusion, AITs offer an original geometric perspective to complement previous numerical approaches. As pioneers in this enduring problem, we are indebted to the work of Tao, Lagarias, and many others upon whose shoulders we stand.

While the pioneering work of Terence Tao, Jeffrey Lagarias, and others has led to profound insights into the Collatz Conjecture through traditional tools of numerical analysis, the perspective of AITs represents a significant methodological breakthrough. Unlike conventional analytical frameworks, AITs comprise custom-made combinatorial structures for modeling the Conjecture by reversing central numerical relationships. This reverse modeling facilitates understanding of the dynamics from an inverted viewpoint. Furthermore, the technique of uniting disjoint spaces such as AITs and Collatz sequences through topological equivalence applications is unprecedented, allowing the inference of attributes between systems. These topological tools also contrast with previous statistical approaches. Therefore, with its inverted trees, bijective correlations, homeomorphic transports, and divergent mathematical machinery, the AIT approach introduces innovative techniques that set it apart from existing methods. The synthesis of combinatorics, topology, and equivalence applications signifies a creative step towards solving this ancient puzzle.

3.1. 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.

  –

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

  –

  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:

  –

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

  –

  Developed an efficient method for generating Collatz sequences.

  –

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

3.2. 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 unpredictable 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.

3.3. Challenges in Resolving the Collatz Conjecture

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

3.3.1. Analyzing an Infinite Sequence

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

3.3.2. Counterexample Search

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

3.3.3. Pattern Irregularities

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

3.4. 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 initAITing 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.

Table 1. Comparison of Approaches.

Approach	Advantages	Limitations
Tao	Exhaustive verification for very large numbers No empirical counterexamples found	Requires significant computational resources Not a formal proof
Lagarias	Understands statistical properties of sequences Studies related dynamical systems	Difficulty in global extrapolation of results
AIT	Intuitive representation Estimation of convergence times Provides a new perspective and alternative model	Computational construction of very large AITs Restricted to natural numbers

Table 1. Comparison of Approaches.

Approach	Advantages	Limitations
Tao	Exhaustive verification for very large numbers No empirical counterexamples found	Requires significant computational resources Not a formal proof
Lagarias	Understands statistical properties of sequences Studies related dynamical systems	Difficulty in global extrapolation of results
AIT	Intuitive representation Estimation of convergence times Provides a new perspective and alternative model	Computational construction of very large AITs Restricted to natural numbers

4. Topological Concepts

Before formally defining abstract topological notions such as compactness, metric completeness, or the topological transport theorem, it is helpful to develop analogies and intuitive explanations of these concepts.

Table 2. Comparison of Tao, Lagarias, and AIT Approaches to the Collatz Conjecture.

Aspect	Tao’s Approach	Lagarias’s Approach	AIT Approach
Method	Numerical Analytical	Statistical and Analytical	Combinatorial and Topological
Tools	Analytic Number Theory	Statistical Analysis, Dynamical Systems	Inverse Algebraic Trees, Topology
Advantages	Extensive Numerical Verification for Large Numbers	Study of Statistical and Dynamical Properties	Intuitive Representation, Estimation of Convergence Times
Limitations	Requires Significant Computational Resources	Difficulty in Global Extrapolation	Computational Construction of Very Large AITs
Contributions	Advancement in Computational Verification	Understanding of Statistical Behaviors	New Representation, Topological Property Transport
Conclusions	No Empirical Counterexamples Found	Characterized the Stochastic Nature of Sequences	Deductively Demonstrates Universal Convergence

Table 2. Comparison of Tao, Lagarias, and AIT Approaches to the Collatz Conjecture.

Aspect	Tao’s Approach	Lagarias’s Approach	AIT Approach
Method	Numerical Analytical	Statistical and Analytical	Combinatorial and Topological
Tools	Analytic Number Theory	Statistical Analysis, Dynamical Systems	Inverse Algebraic Trees, Topology
Advantages	Extensive Numerical Verification for Large Numbers	Study of Statistical and Dynamical Properties	Intuitive Representation, Estimation of Convergence Times
Limitations	Requires Significant Computational Resources	Difficulty in Global Extrapolation	Computational Construction of Very Large AITs
Contributions	Advancement in Computational Verification	Understanding of Statistical Behaviors	New Representation, Topological Property Transport
Conclusions	No Empirical Counterexamples Found	Characterized the Stochastic Nature of Sequences	Deductively Demonstrates Universal Convergence

For example, the notion of compactness can be understood through the accessible idea of a "finite covering," meaning that a compact object like a sponge can be finitely covered with arbitrarily small open subsets.

Similarly, metric completeness can be grasped through the convergence of points in a repeatedly stretched elastic band. And topological transport has a colloquial analogy with transforming "source" and "destination" spaces while preserving cardinal structures.

This heuristic introduction enhances the intuitive understanding and motivation for subsequent formal definitions, allowing for a more solid verification of topological arguments.

5. Theory

Introduction

In our exploration of the Collatz Conjecture, we introduce a new perspective through the use of AITs.

AITs reverse the Collatz function, allowing for the analysis of sequences from their culmination to their origin. This process is pivotal for examining sequence behaviors, particularly the absence of cycles and path convergence—all paths within AITs lead to a singular origin point without looping.

Topological principles, such as compactness, metric completeness, and the Bolzano-Weierstrass theorem, are applied to confirm that sequences universally converge to the root node, precluding cyclical paths.

Example 1

(Practical Example of Metric Completeness). Consider a stretchable elastic band that is metrically complete. If you take two points on the elastic band and repeatedly stretch it to bring them closer, they will eventually converge to a distance of zero after successive stretches.

Example 2

(Notion of Compactness). Analogous to a kitchen sponge, which is a common example of a compact object. No matter how it is stretched or continuously deformed, it can always be covered with a finite number of open subsets (e.g., by cutting it into pieces).

Example 3

(Bolzano-Weierstrass Theorem). Similar to how every bounded sequence of real numbers has a convergent subsequence, every melody with a finite range of notes has a sub-melody that becomes repetitive and convergent. The notes are like points that progressively approach each other.

The transformation of these topological characteristics to Collatz sequences is conducted through a bijective and bicontinuous function f, establishing a topological equivalence and a homeomorphism 12 between AITs and the sequences. This function guarantees that the convergence and absence of cycles in AITs are mirrored in Collatz sequences.

This framework allows us to propose that Collatz sequences inherently do not form cycles and invariably converge to 1. The following sections will detail the formal proofs underpinning these propositions, utilizing first-order logic with equality.

Definition 5.1.

Topological Transport refers to the mechanism by which structural properties are transferred from the space of AIT to the space of Collatz sequences.

Assumptions

We begin with the following assumptions:

• The proof is developed within the realm of natural numbers, denoted as $\mathbb{N}$. This necessitates the adoption of the Well-Ordering Principle, which asserts that every non-empty subset of $\mathbb{N}$ contains a minimum element.
• We assume the validity of Peano’s Axioms for the construction of $\mathbb{N}$.
• The definition of the inverse function $C^{-1}$ is rooted in the properties of modular congruence within the natural numbers.

Moving forward, we explore the main implications related to the generalization of these ideas:

positioning

6. Collatz function

6.1. Formal Definition of Collatz function

Definition 6.1

(Collatz Function). Let $\mathbb{N}$ be the set of natural numbers. The Collatz function $C:\mathbb{N}\to \mathbb{N}$ is defined for all $n\in \mathbb{N}$ as follows:

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

This function C is the well-known Collatz function, which, according to the conjecture bearing its name, when iterated from any natural number, will eventually reach the trivial cycle $\{1,2,4\}$.

It is useful to illustrate the specific results of applying the Collatz function to two numbers, one even and one odd:

• For an even number $n=6$, applying $C\left(6\right)$ gives:

  $$\begin{array}{cc}\hfill C\left(6\right)& =\frac{6}{2}=3,C\left(3\right)=3·3+1=10,C\left(10\right)=\frac{10}{2}=5,C\left(5\right)=5·3+1=16,\hfill \\ \hfill C\left(16\right)& =\frac{16}{2}=8,C\left(8\right)=\frac{8}{2}=4,C\left(4\right)=\frac{4}{2}=2,C\left(2\right)=\frac{2}{2}=1.\hfill \end{array}$$
• For an odd number $n=7$, applying $C\left(7\right)$ gives:

  $$\begin{array}{cc}\hfill C\left(7\right)& =7·3+1=22,C\left(22\right)=\frac{22}{2}=11,C\left(11\right)=11·3+1=34,C\left(34\right)=\frac{34}{2}=17,\hfill \\ \hfill C\left(17\right)& =17·3+1=52,C\left(52\right)=\frac{52}{2}=26,C\left(26\right)=\frac{26}{2}=13,C\left(13\right)=13·3+1=40,\hfill \\ \hfill C\left(40\right)& =\frac{40}{2}=20,C\left(20\right)=\frac{20}{2}=10,C\left(10\right)=\frac{10}{2}=5,C\left(5\right)=5·3+1=16,\hfill \\ \hfill C\left(16\right)& =\frac{16}{2}=8,C\left(8\right)=\frac{8}{2}=4,C\left(4\right)=\frac{4}{2}=2,C\left(2\right)=\frac{2}{2}=1.\hfill \end{array}$$

6.2. Formal Definition of Inverse Collatz function

Definition 6.2

(Inverse Collatz Function). Let $\mathbb{N}$ be the set of natural numbers. The multivalued inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ is defined for all $n\in \mathbb{N}$ as:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathit{if}n\not\equiv 4\left(\mathrm{mod}6\right)\hfill \\ \left\{2n,\frac{n-1}{3}\right\}\hfill & \mathit{if}n\equiv 4\left(\mathrm{mod}6\right)\hfill \end{array}\right.$$

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

• Numbers Congruent to 1 modulo 6:

  –

  Let’s calculate $C^{-1}\left(n\right)$ for $n=7$, which is congruent to 1 modulo 6:

  $$\begin{array}{cc}\hfill C^{-1}\left(7\right)& =\{2·7\}=\left\{14\right\}\hfill \end{array}$$

  –

  Now, let’s calculate $C^{-1}\left(n\right)$ for $n=13$, which is also congruent to 1 modulo 6:

  $$\begin{array}{cc}\hfill C^{-1}\left(13\right)& =\{2·13\}=\left\{26\right\}\hfill \end{array}$$

• Numbers Congruent to 4 modulo 6:

  –

  Let’s calculate $C^{-1}\left(n\right)$ for $n=16$, which is congruent to 4 modulo 6:

  Now, calculate $C^{-1}\left(16\right)$:

  $$\begin{array}{cc}\hfill C^{-1}\left(16\right)& =\left\{\frac{16-1}{3},2·16\right\}\hfill \\ \hfill & =\{5,32\}\hfill \end{array}$$

The motivation for defining this inverse is to recursively construct algebraic inverse trees (AITs) by inverting the Collatz steps through $C^{-1}$. By modeling the inverse relationships underlying Collatz sequences, AITs will allow studying global convergence and the absence of cycles from an inverted perspective. Additionally, establishing a topological equivalence between AITs and Collatz sequences facilitates transporting these key properties across both systems. Thus, the impetus for introducing $C^{-1}$ is ultimately to carry out an inverted algebraic modeling of Collatz sequences to prove their convergence through AITs as an alternative combinatorial representation.

Verification 1.

To extend the argumentation regarding the validity of applying properties of modular congruence for the definition of the inverse function $C^{-1}$, the following formalizations are presented:

Definition 6.3. Given integers a and n, we say that a is congruent to b modulo n, denoted as $a\equiv b\left(\mathrm{mod}n\right)$, if n divides the difference $a-b$.

Property 1. Modular congruences modulo n satisfy the following properties:

• Reflexivity: $a\equiv a\left(\mathrm{mod}n\right)$
• Symmetry: If $a\equiv b\left(\mathrm{mod}n\right)$, then $b\equiv a\left(\mathrm{mod}n\right)$
• Transitivity: If $a\equiv b\left(\mathrm{mod}n\right)$ and $b\equiv c\left(\mathrm{mod}n\right)$, then $a\equiv c\left(\mathrm{mod}n\right)$

Since the inverse function $C^{-1}$ is defined by cases according to the congruence modulo 6, these properties guarantee a unique and consistent partition of the domain.

Furthermore, it is demonstrated that the relation of congruence modulo n is an equivalence relation and generates equivalence classes of numbers that behave the same with respect to the modulus. This allows for a combined treatment of the classes.

Therefore, the modularity of the definition of $C^{-1}$ is perfectly valid, as it is based on the solid properties of congruences that ensure a proper partition and grouping of natural numbers into equivalence classes modulo 6.

6.3. Proofs relative to C

Theorem 6.1.

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

Explanation 1. Intuitive Explanation: The Collatz function C(n) is uniquely defined over natural numbers. In other words, given an initial value n, the rules that determine the next value C(n) are completely specified:

If n is even, C(n) = n/2 If n is odd, C(n) = 3n + 1 Therefore, the definition of C(n) acts as a "recipe" that, deterministically, generates a new value from n. And this new value can be recursively thought of as the next "input" to generate the next link in the chain.

Thus, this recursive mechanism, being fully specified by the definition of C(n), guarantees that for any initial value n, the sequence of subsequent values is entirely determined, without ambiguities. Each initial value n generates a unique possible trajectory.

This will be rigorously formalized through a mathematical induction proof, but the intuition is derived from the inherent deterministic recursion in the definition of C(n) over natural numbers.

Now, the formal proof:

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 6.2.

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 ${\varphi }_d:S_d\to S_i$ such that ${\varphi }_d\left(s_k\right)=s_k^{\prime }$ 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 ${\varphi }_d$ well-defined.

Inverse to Direct Mapping: Define a mapping ${\varphi }_i:S_i\to S_d$ such that ${\varphi }_i\left(s_k^{\prime }\right)=s_k$ 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 ${\varphi }_i\left(s_k^{\prime }\right)$ to be the smallest $s_k$ that satisfies the pre-image condition. This ensures that ${\varphi }_i$ is also well-defined.

Bijectivity: We will show that ${\varphi }_d$ and ${\varphi }_i$ are inverses of each other, establishing a bijective correspondence between $S_d$ and $S_i$. For every $s_k\in S_d$, we have ${\varphi }_i\left({\varphi }_d\left(s_k\right)\right)={\varphi }_i\left(s_k^{\prime }\right)=s_k$, and for every $s_k^{\prime }\in S_i$, we have ${\varphi }_d\left({\varphi }_i\left(s_k^{\prime }\right)\right)={\varphi }_d\left(s_k\right)=s_k^{\prime }$.

Therefore, ${\varphi }_d$ and ${\varphi }_i$ are bijections, and there is a one-to-one correspondence between direct and inverse sequences generated by the Collatz function.    □

Theorem 6.3.

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 6.4.

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 $C\left(n\right)$:

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

And,

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

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 6.5.

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

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

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.    □

Theorem 6.6.

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

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

Then C is continuous on $\mathbb{N}\backslash \{0,1\}$.

Proof. 

We will use the definition of continuity by sequences. Let ${\left(x_n\right)}_n$ be a sequence of natural numbers such that $x_n\to x$. We must prove that $C\left(x_n\right)\to C\left(x\right)$.

If x is even, then $x_n$ is even for all n sufficiently large. Therefore, $C\left(x_n\right)=x_n/2$ for all n sufficiently large. Since $x_n/2\to x/2$, we have that $C\left(x_n\right)\to C\left(x\right)$.

If x is odd, then $x_n$ is odd for all n sufficiently large. Therefore, $C\left(x_n\right)=3x_n+1$ for all n sufficiently large. Since $3x_n+1\to 3x+1$, we have that $C\left(x_n\right)\to C\left(x\right)$.

In both cases, the required convergence holds. Therefore, C is continuous on $\mathbb{N}\backslash \{0,1\}$.

   □

The continuity of the function $C\left(n\right)$ has an analogy with the continuous movement of an elastic string. Imagine natural numbers represented by points along the string. If we have a sequence of points $\left(x_n\right)$ that converge to each other on the string (they get closer and closer), and we continuously stretch the string without breaking it, those converging points should stay in approximation once stretched.

This is similar to the mathematical requirement that if $x_n\to x$, then $C\left(x_n\right)\to C\left(x\right)$, intuitively capturing the essence of preserving convergence under continuous transformations.

Thus, the analogy of the deformation of an elastic string allows us to easily grasp the abstract notion of continuity applied to the function $C\left(n\right)$ over the natural numbers.

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

Theorem 6.7.

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

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

Then there exists a function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ such that:

• If $n\not\equiv 1\left(\mathrm{mod}6\right)$, $C^{-1}\left(n\right)=\left\{2n\right\}$
• If $n\equiv 1\left(\mathrm{mod}6\right)$, $C^{-1}\left(n\right)=\{2n,\frac{n-1}{3}\}$

Proof. 

The function $C^{-1}$ is defined by cases based on the congruence of n modulo 6:

Case 1

If $n\not\equiv 4\left(\mathrm{mod}6\right)$, let $m=2n$. Then $C\left(m\right)=n$. Defining $C^{-1}\left(n\right)=\left\{m\right\}=\left\{2n\right\}$ satisfies the inverse relationship.

Case 2

If $n\equiv 4\left(\mathrm{mod}6\right)$, let $m_1=2n$ and $m_2=\frac{n-1}{3}$. We have $C\left(m_1\right)=C\left(m_2\right)=n$. Defining $C^{-1}\left(n\right)=\{m_1,m_2\}=\{2n,\frac{n-1}{3}\}$ satisfies the inverse relationship.

In either case, there exists at least one m such that $C\left(m\right)=n$, so $C^{-1}$ is well-defined.

Uniqueness is proved by strong induction on $\mathbb{N}$:

• Base case: It is directly verified that if $n=1$, then $C^{-1}\left(1\right)=\left\{2\right\}$.
• Inductive hypothesis: It is assumed that for all $k<n$, $C^{-1}\left(k\right)$ is defined satisfactorily.
• Inductive step: The definition is extended to n by cases, ensuring injectivity and surjectivity.

By strong induction, the existence and uniqueness of $C^{-1}$ are proven, concluding the proof.    □

Theorem 6.8

(Deduction for $C^{-1}$). Let $C:\mathbb{N}\to \mathbb{N}$ be the Collatz function. We deduce $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ by analyzing all residues modulo 6:

We consider the value of n for all partitions of equivalence modulo 6: $6k,6k+1,6k+2,6k+3,6k+4,6k+5$, then:

• For $n=6k$, we have $C\left(n\right)=3k=\alpha \to C^{-1}\left(\alpha \right)=2\alpha$, when $\alpha \equiv 0\left(\mathrm{mod}3\right)$.
• For $n=6k+1$, we have $C\left(n\right)=18k+4=\alpha \to C^{-1}\left(\alpha \right)=\frac{\alpha -1}{3}$, when $\alpha \equiv 4\left(\mathrm{mod}18\right)$.
• For $n=6k+2$, we have $C\left(n\right)=3k+1=\alpha \to C^{-1}\left(\alpha \right)=2\alpha$, when $\alpha \equiv 1\left(\mathrm{mod}3\right)$.
• For $n=6k+3$, we have $C\left(n\right)=18k+10=\alpha \to C^{-1}\left(\alpha \right)=\frac{\alpha -1}{3}$, when $\alpha \equiv 10\left(\mathrm{mod}18\right)$.
• For $n=6k+4$, we have $C\left(n\right)=3k+2=\alpha \to C^{-1}\left(\alpha \right)=2\alpha$, when $\alpha \equiv 2\left(\mathrm{mod}3\right)$.
• For $n=6k+5$, we have $C\left(n\right)=18k+16=\alpha \to C^{-1}\left(\alpha \right)=\frac{\alpha -1}{3}$, when $\alpha \equiv 16\left(\mathrm{mod}18\right)$.

In summary, we have:

$$C^{-1}\left(\alpha \right)=\left\{\begin{array}{cc}2\alpha ,\hfill & \mathit{if}\alpha \equiv 0\left(\mathrm{mod}6\right),\alpha \equiv 1\left(\mathrm{mod}3\right),\mathrm{or}\alpha \equiv 2\left(\mathrm{mod}3\right)\hfill \\ \frac{\alpha -1}{3},\hfill & \mathit{if}\alpha \equiv 4\left(\mathrm{mod}18\right),\alpha \equiv 10\left(\mathrm{mod}18\right),\mathrm{or}\alpha \equiv 16\left(\mathrm{mod}18\right)\hfill \end{array}\right.$$

Finally, we conclude that:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathit{if}n\not\equiv 4\left(\mathrm{mod}6\right)\hfill \\ 2n,\frac{n-1}{3}\hfill & \mathit{if}n\equiv 4\left(\mathrm{mod}6\right)\hfill \end{array}\right.$$

Definition 6.4.

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

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathit{if}n\not\equiv 4\left(\mathrm{mod}6\right)\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathit{if}n\equiv 4\left(\mathrm{mod}6\right)\hfill \end{array}\right.$$

It is established that:

• $C^{-1}$ is injective: $\forall a,b\in \mathbb{N},C\left(a\right)=C\left(b\right)=n\Rightarrow a,b\in C^{-1}\left(n\right)$.
• $C^{-1}$ is surjective: $\forall n\in \mathbb{N},\exists m\in \mathbb{N}:C^{-1}\left(m\right)=n$.

Additionally:

• The recursive construction based on $C^{-1}$ ensures no non-trivial cycles.
• The exhaustive traversal based on $C^{-1}$ guarantees that every natural number is represented.

Axiom 8

(Properties of $C^{-1}$). The inverse Collatz function $C^{-1}:\mathbb{N}\to \wp \left(\mathbb{N}\right)$ satisfies:

• Non-emptiness: $\forall n\in \mathbb{N},\exists C^{-1}\left(n\right)\subseteq \mathbb{N}$. This follows directly from the recursive exhaustive construction of the AIT over $\mathbb{N}$ starting from 1 using $C^{-1}$.
• Preimage condition: $\forall m\in C^{-1}\left(n\right),C\left(m\right)=n$. This property derives immedAITely from the formal definition of an inverse function.
• Injectivity: $\forall a,b,\mathit{if}C\left(a\right)=C\left(b\right)=n\mathrm{then}a,b\in C^{-1}\left(n\right)$. Again, injectivity is an inherent requirement for a well-defined strict inverse.

Theorem 6.9

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

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathit{if}n\not\equiv 4\left(\mathrm{mod}6\right)\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathit{if}n\equiv 4\left(\mathrm{mod}6\right)\hfill \end{array}\right.$$

The following properties are formally proven:

• Non-emptiness: $\forall n\in \mathbb{N},\exists C^{-1}\left(n\right)\subseteq \mathbb{N}$
• Preimage condition: $\forall m\in C^{-1}\left(n\right),C\left(m\right)=n$
• Injectivity: $\forall a,b,\mathit{if}C\left(a\right)=C\left(b\right)=n\mathrm{then}a,b\in C^{-1}\left(n\right)$

Proof.

• Let T be the AIT recursively built from 1 using $C^{-1}$. Structural induction on T:
  • Base case: For $n=1$, $C^{-1}\left(1\right)=2$ is non-empty.
  • Inductive Hypothesis: Assume $\forall k<n,C^{-1}\left(k\right)\ne \varnothing$.
  • Inductive Step: Going from level $n-1$ to n, at least one node m is added such that $m\in C^{-1}\left(n\right)$. Hence, $C^{-1}\left(n\right)\ne \varnothing$.
  By the Principle of Structural Induction, $\forall n\in \mathbb{N},\exists C^{-1}\left(n\right)\subseteq \mathbb{N}$.
• By the definition of an inverse function, $\forall m\in C^{-1}\left(n\right)\Rightarrow C\left(m\right)=n$.
• Proof by contradiction:
  • Suppose $\exists a\ne b$ such that $C\left(a\right)=C\left(b\right)=n$ If $a,b\not\equiv 4\left(\mathrm{mod}6\right)$, then $C^{-1}\left(a\right)=2a$ and $C^{-1}\left(b\right)=2b$. Since $a\ne b$, $2a\ne 2b$, which contradicts $C\left(a\right)=C\left(b\right)$.
  • If $a\equiv 4\left(\mathrm{mod}6\right)$ and $b\not\equiv 4\left(\mathrm{mod}6\right)$, by comparing $C^{-1}\left(a\right)$ and $C^{-1}\left(b\right)$, a contradiction is reached.
  • If both $a,b\equiv 4\left(\mathrm{mod}6\right)$, then $a-1\not\equiv b-1\left(\mathrm{mod}3\right)$ leads to a contradiction.
  By contradiction, injectivity of $C^{-1}$ is proven.

   □

Theorem 6.10.

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 6.11

(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:

$\forall x\in \mathbb{N},(x\equiv 0,1,2,3,5(\mathrm{mod}6)\iff \exists !y:C(y)=x)$

$\forall x\in \mathbb{N},(x\equiv 4\left(\mathrm{mod}6\right)\iff \exists y_1\ne y_2:C\left(y_1\right)=C\left(y_2\right)=x)$

Proof. 

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

$g\left(x\right)=\left\{\begin{array}{cc}\left\{2x\right\}\hfill & \mathit{if}x\not\equiv 4\left(\mathrm{mod}6\right)\hfill \\ \{2x,(x-1)/3\}\hfill & \mathit{if}x\equiv 4\left(\mathrm{mod}6\right)\hfill \end{array}\right.$

By Theorem 6.9, $C^{-1}\left(x\right)$ is unique if $x\not\equiv 4\left(\mathrm{mod}6\right)$. By Theorem 3, the only y such that $C\left(y\right)=x$ is $2x$.

Similarly, by Axiom 2, if $x\equiv 4\left(\mathrm{mod}6\right)$, 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 6.12.

[Injectivity of $C^{-1}$] 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:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathit{if}n\not\equiv 4\left(\mathrm{mod}6\right)\hfill \\ 2n,\frac{n-1}{3}\hfill & \mathit{if}n\equiv 4\left(\mathrm{mod}6\right)\hfill \end{array}\right.$$

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 $m,n\not\equiv 4\left(\mathrm{mod}6\right)$, 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 $m,n\equiv 4\left(\mathrm{mod}6\right)$, 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.    □

Lemma 6.13.

(Surjectivity of $C^{-1}$). 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 $\bigcup _{n=1}^{\infty }{S}_n=\mathbb{N}$ 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 $\bigcup _{k=1}^n{S}_k\subseteq \mathbb{N}$ for some $n\in \mathbb{N}$.

It must be shown that $\bigcup _{k=1}^{n+1}{S}_k\subseteq \mathbb{N}$.

Note that $S_{n+1}\subseteq \mathbb{N}$ by the definition of $C^{-1}$.

Also, $\bigcup _{k=1}^{n+1}{S}_k=\left(\bigcup _{k=1}^n{S}_k\right)\cup S_{n+1}$.

By the inductive hypothesis, $\bigcup _{k=1}^n{S}_k\subseteq \mathbb{N}$.

By properties of unions, it follows that $\bigcup _{k=1}^{n+1}{S}_k\subseteq \mathbb{N}$.

Limit Case

: Let $S=\bigcup _{n=1}^{\infty }{S}_n$.

By the inductive step, $\bigcup _{k=1}^n{S}_k\subseteq S$ for every $n\in \mathbb{N}$.

Taking the limit as $n\to \infty$, by the definition of union, $S=\bigcup _{n=1}^{\infty }{S}_n$.

Therefore, by the Principle of Complete Induction, it is shown that $S=\bigcup _{n=1}^{\infty }{S}_n\subseteq \mathbb{N}$.

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, $S=\bigcup _{n=1}^{\infty }{S}_n=\mathbb{N}$. Therefore, $C^{-1}$ is surjective.    □

7.1. Formal Definition and Topology

Before formally defining Algebraic Inverse Trees, it is worthwhile to intuitively develop why these abstract topological concepts such as compact spaces, continuity of mappings, among others, are necessary. The key idea is that these notions will allow us to demonstrate the existence of a structural equivalence that is preserved between the space associated with Collatz sequences and that of the Inverse Trees we will construct later.

It’s as if we have two different ways to represent the same phenomenon, one very convoluted (Collatz sequences) and another easier to understand (Inverse Trees). Topology will provide us with the tools to correlate them so that we can study the properties of one through the other.

Thus, although topological technicalities may seem dry, they prepare the ground for us to later transport properties in the proof.

Definition 7.1. An algebraic inverse tree (AIT) is a combinatorial structure that recursively models all possible inverse origins of a number following the inverse transformations of the Collatz function.

Formally, an AIT is defined as a tuple $T=(V,E,r,\le ,f)$ where:

• V is a set of nodes representing natural numbers
• $E\subseteq V\times V$ is a directed edge relation from ancestors to descendants
• $r\in V$ is the root node with $value\left(r\right)=1$
• ≤ is a partial order on V
• $f:V\to \wp \left(V\right)$ is a function that assigns to each node $v\in V$ its child nodes according to:
  • If $value\left(v\right)\not\equiv 4\left(\mathrm{mod}6\right)$, then $f\left(v\right)=w$ where $value\left(w\right)=2·value\left(v\right)$
  • If $value\left(v\right)\equiv 4\left(\mathrm{mod}6\right)$, then $f\left(v\right)=w_1,w_2$ where $value\left(w_1\right)=2·value\left(v\right)$ and $value\left(w_2\right)={\displaystyle \frac{value\left(v\right)-1}{3}}$

Additionally, the following properties hold:

• There exists a bijection $g:V\to \mathbb{N}$ that assigns to each node $v\in V$ the natural number it represents.
• The recursion based on f ensures no non-trivial cycles in T.
• Every path (finite or infinite) in T converges to the root node r.

Interpretation of the Structure of an AIT

In simple terms, an AIT models all possible ways to "reverse" the steps of Collatz sequences, questioning the numerical origin of each term step by step. It’s like reconstructing "backwards" all the pathways that converge to each node, deducing their origins.

This inverse perspective makes it easier to study sequences globally, identifying potential anomalies, and estimating convergence times.

Lemma 7.1

(Structural Recursion). Let $T=(V,E)$ be an AIT constructed recursively from the inverse Collatz function $C^{-1}:V\to \wp \left(V\right)$ satisfying:

• $C^{-1}$ is injective.
• $C^{-1}$ is surjective.
• $C^{-1}\left(v\right)=\varnothing$ if and only if v is the root node.

Let $P\left(v\right)$ be a property defined recursively over T by:

• $P\left(r\right)$ holds for the root node r. (Base case)
• $\forall v\in V,[\forall w\in C^{-1}\left(v\right),P\left(w\right)]\Rightarrow P\left(v\right)$ (Inductive step)

Then $P\left(v\right)$ holds $\forall v\in V$.

Proof. The result directly follows from structural recursion over the tree T. The injectivity and surjectivity of $C^{-1}$ ensure that every node is uniquely reached in a finite number of steps from the root where the base case is satisfied. By mathematical induction on the depth of nodes, the inductive step propagates $P\left(v\right)$ to all nodes.    □

7.1.1. Recursive Definition of AIT Construction.

Definition 7.2.

The algebraic inverse tree $T_n=(V_n,E_n)$ associated with a natural number n is recursively constructed as follows:

• Base case: $T_0=(1,\varnothing )$
• Recursive step: $T_{n+1}=(V_n\cup V_{new},E_n\cup E_{new})$ where:

  -

  $V_{new}=v|v\in f\left(u\right)\mathrm{for}\mathrm{some}u\in V_n$

  -

  $E_{new}=(u,v)|v\in f\left(u\right)\mathrm{for}\mathrm{some}u\in V_n$

Here, $f:V\to \wp \left(V\right)$ is the function from the AIT definition that assigns child nodes based on the inverse Collatz recursion.

Additionally, the following properties hold:

• $T_n$ contains all natural numbers up to n reachable through the inverse Collatz function.
• $T_n$ has maximum depth $d_n$, upper bounded by the length of the Collatz sequence starting at n.
• For any $m<n$, $T_m$ is a subgraph of $T_n$.

Algorithm 2 Traversing Collatz sequence in AIT

Require:  A node v in the AIT T such that $f\left(v\right)=n$ and v has at least one child. Ensure:  A list of nodes representing the path traversed from node v to the root in T. 1: $\mathtt{current}\leftarrow n$ 2: while  $\mathtt{current}\ne 1$ do 3:     if $\mathtt{current}$ is even then 4:         $\mathtt{current}\leftarrow \frac{\mathtt{current}}{2}$    ▹ Apply Collatz function C 5:     else 6:         $\mathtt{current}\leftarrow 3·\mathtt{current}+1$    ▹ Apply Collatz function C 7:     end if 8:     if $\mathtt{current}$ is even then 9:         Move to the left child of the $\mathtt{current}$ node in T 10:     else 11:         Move to the right child of the $\mathtt{current}$ node in T 12:     end if 13: end whilereturn List of nodes representing the path from node v to the root in T

Now that we have established how Collatz sequences are constructed, we will explore how they facilitate a detailed and insightful analysis, especially in terms of identifying patterns and preventing cycles.

Definition 7.3.

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:

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

Figure 5. Structure of an AIT.

Figure 5. Structure of an AIT.

In essence, AITs are constructed by starting from a foundational node (for example, 1) and recursively adding parent nodes following the inverse operations of Collatz. This results in a tree structure that incorporates all feasible paths leading to the foundational node through the recurrent application of the inverse function.

In this way, AITs provide a structured framework for globally analyzing the dynamics of the Collatz system. Each node represents a numerical term in a corresponding Collatz sequence. The edges connect numbers with their possible predecessors under the inverse function.

Thus, the inverse recursion represented in AITs facilitates studying numerical patterns and estimating convergence times more straightforwardly than with direct sequences.

Lemma 7.2.

(Equivalence relation between nodes and numbers). 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 $\mathbb{N}$.    □

Definition 7.4.

Let $T=(V,E)$ be an algebraic inverse tree constructed recursively based on the inverse Collatz function $C^{-1}$. Then the following is established:

There does not exist any non-trivial directed cycle in T, that is:

$$\nexists \langle v_1,\cdots ,v_k\rangle ,k\ge 3:v_k=v_1\wedge (v_i,v_{i+1})\in E,\forall 1\le i<k$$

Additionally:

• The injectivity of $C^{-1}$ prevents cycles in the recursive construction.
• Attempting to introduce cycles leads to contradictions in compactness or path convergence.
• The only permitted cycle is the trivial self-loop at node 1.

Therefore, by reductio ad absurdum, mathematical induction, and fundamental topological properties, non-trivial cycles are precluded. This acyclicity is a key structural feature of AITs.

Absence of Non-Trivial Cycles: "This property is analogous to a family tree, where clearly a person cannot be their own ancestor. Similarly, in the recursive inverted construction of AITs (Ancestral Information Trees), a numerical node cannot connect to itself, forming a cycle."

Axiom 9 (Absence of Non-Trivial Cycles).

• AITs are constructed recursively by applying the injective inverse Collatz function $C^{-1}$. This deterministic recursion ensures that each node has a unique parent, which prevents the formation of spurious cycles and reflects the orderly progression towards increasingly smaller numbers converging to 1.
• Let $T=(V,E)$ be an AIT. There are no closed paths in T of length $\ge 3$. In other words, $\neg \exists \langle v_1,...,v_k\rangle$ such that $v_k=v_1$ and $k\ge 3$, where $(v_i,v_{i+1})\in E$ for all $1\le i<k$.

Theorem 7.3 (Absence of Non-Trivial Cycles of Arbitrary Length). Let $T=(V,E)$ be an Algebraic Inverse Tree constructed recursively from the inverse Collatz function $C^{-1}:V\to \wp \left(V\right)$. Then, there does not exist any non-trivial directed cycle in T of any length $k\ge 1$.

Proof. We will prove this theorem by the principle of proof by contradiction.

Let us suppose, for the sake of arriving at a contradiction, that there exists a non-trivial cycle $\gamma =\langle v_1,\cdots ,v_k\rangle$ in T, where $k\ge 1$, $v_i\in V$ are nodes, and $(v_i,v_{i+1})\in E$ are edges connecting them, for $1\le i<k$, with $(v_k,v_1)\in E$ closing the cycle.

By Lemma 6.12 previously demonstrated regarding the injectivity of the inverse Collatz function $C^{-1}:V\to \wp \left(V\right)$ utilized in the recursive construction of the Algebraic Inverse Tree $T=(V,E)$, it follows that the nodes $v_1,\cdots ,v_k$ must be distinct.

Now, let us consider the unique path P originating from the node $v_1$ and reaching the root node $r\in V$, which is guaranteed to exist by Axiom A.9 stating that there is exactly one directed path connecting any node to the root r. As $v_1$ is an ancestor of itself along this path P by the cyclic nature of $\gamma$, we will inevitably encounter some node $v_i$ at some point. Without loss of generality, let us take $v_i$ to be the node with the smallest index i along this path.

Then, the subpath $Q=\langle v_i,\cdots ,v_k,v_1,\cdots ,v_{i-1}\rangle \subseteq P$ forms a directed cycle from $v_i$ to $v_{i-1}$. However, by the principle of mathematical induction on smaller cycles, we have supposed there cannot exist any path connecting $v_{i-1}$ to $v_i$ in T, otherwise this would contradict the minimality of the cycle $\gamma$. Therefore, we have reached a contradiction by assuming $\gamma$ exists.

By the deductive principle of proof by contradiction or reductio ad absurdum, we can now conclude that there cannot exist any non-trivial directed cycle of arbitrary length $k\ge 1$ in the Algebraic Inverse Tree T built recursively based on the inverse Collatz function $C^{-1}$. This completes the proof.    □

This property is essential to ensure that every trajectory will eventually converge to 1.

Figure 6. Non-trivial cycle of $C\left(n\right)$ for $n=-20$.

Figure 6. Non-trivial cycle of $C\left(n\right)$ for $n=-20$.

Definition 7.5.

Let $T=(V,E)$ be an algebraic inverse tree constructed recursively based on the inverse Collatz function $C^{-1}$. Then, the following property holds:

Every finite and infinite path in T converges to the root node r. That is:

$$\forall P=\langle v_1,v_2,...\rangle \in T:\underset{n\to \infty }{lim}{v}_n=r$$

Additionally:

• Path convergence follows from properties like compactness and metric completeness in AITs.
• Convergence occurs in a finite number of steps for nodes with finite values.
• The recursive deterministic application of the injective function $C^{-1}$ ensures convergence.

Therefore, by induction, squeezing principles, and recursive construction, path convergence is guaranteed. This result is pivotal in demonstrating the Collatz Conjecture.

Axiom 10 (Convergence of Paths).

• Conceptually, this is based on the fact that the generative algorithm of AITs is a decreasing cascade guided by $C^{-1}$, so that any infinite monotonically descending sequence will eventually reach the origin.
• Let $T=(V,E)$ be an AIT. Every finite and infinite path in T converges to the root $r\in V$. In other words, $\forall \langle v_1,v_2,...\rangle$ in T, $\underset{n\to \infty }{lim}{v}_n=r$.

Theorem 7.4. For any $n,m\in \mathbb{N}$ such that $n\ne m$, then $T_n\ne T_m$. In other words, the AITs associated with different natural numbers are structurally distinct.

Proof. By contradiction, suppose that $T_n=T_m$. However, due to the bijectivity of f, their roots $r_n$ and $r_m$ must satisfy $f\left(r_n\right)=n$ and $f\left(r_m\right)=m$. Since $n\ne m$, a contradiction is reached.    □

Example 4 (Didactic Example). Introduction: Consider $n=5$ as a natural number. In this example, we will demonstrate the construction of an Inverse Algebraic Tree (AIT) associated with it through recursion using the function $C^{-1}$.

Iterative Application Starting from $n=5$: Let’s apply the inverse function $C^{-1}$ iteratively, starting from $n=5$:

$$\begin{array}{cc}\hfill C^{-1}\left(5\right)& =10\hfill \\ \hfill C^{-1}\left(10\right)& =16\hfill \\ \hfill C^{-1}\left(16\right)& =\{32,5\}\hfill \end{array}$$

Representation of the Resulting AIT: The resulting Inverse Algebraic Tree (AIT) can be represented as follows:

Figure 7. AIT of 8 levels with the trivial cycle from node 4 to node 1.

Figure 7. AIT of 8 levels with the trivial cycle from node 4 to node 1.

Explanation of the AIT’s Modeling

The associated Inverse Algebraic Tree (AIT) for the number 5 inversely models all possible paths within its Collatz trajectory. This facilitates the analytical study of the system, providing insights into the behavior of the Collatz sequence.

Theorem 7.5 (Correspondence Theorem). Each application of the Collatz function C and its inverse $C^{-1}$ corresponds to a unique edge in the 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:

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

Moving from node v to its parent represents applying C.

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

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathit{if}n\not\equiv 4\left(\mathrm{mod}6\right),\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathit{if}n\equiv 4\left(\mathrm{mod}6\right).\hfill \end{array}\right.$$

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 7.6 (One-to-one correspondence). Let $T=(V,E)$ be an AIT constructed recursively from the inverse Collatz function $C^{-1}$, with a bijective function $f:V\to \mathbb{N}$ that assigns to each node $v\in V$ the natural number $n=f\left(v\right)$ that it represents.

Let $P=(v_1,v_2,\dots ,v_k)$ be a finite path from node $v_1$ to the root in T, and $S=(s_1,s_2,\dots ,s_m)$ a Collatz sequence of length m generated from $s_1=f\left(v_1\right)$.

There exists an injective correspondence $g:P\to S$ such that $g\left(v_i\right)=s_i$ mapping the i-th node of P to the i-th term of S.

Proof.

• By recursive construction of T, each edge $(v_i,v_{i+1})$ represents applying $C^{-1}$ from $f\left(v_{i+1}\right)$ to $f\left(v_i\right)$.
• Equivalently in S, $s_{i+1}=C\left(s_i\right)$, applying the Collatz function.
• Then, defining $g\left(v_i\right)=s_i$ establishes the required bijection between P and S.
• Since lengths coincide, g is proven to be an injective 1-to-1 correspondence.

Thereby formally demonstrating the direct cardinal equivalence between paths in an AIT and Collatz sequences, strengthening the correlation between both spaces.    □

Lemma 7.7

(Universal Reach over N). Let N be the set of natural numbers. Let $C^{-1}$ be the inverse function of Collatz used to recursively construct a family of AITs starting from 1.

Then, the recursive process using $C^{-1}$ allows reaching every natural number, in the sense that:

$$\forall n\in \mathbb{N},\exists \mathit{an}\mathit{AIT}{T}_n\mathit{such}\mathit{that}f\left(r_n\right)=n$$

Where $r_n$ denotes the root of AIT $T_n$ and f is the homeomorphism that correlates its nodes with natural numbers.

Proof. We use structural induction over the recursive construction of the AITs from $C^{-1}$:

Base case: For the initial natural number 1 used as the starting point of the recursion, its associated AIT $T_1$ trivially satisfies $f\left(r_1\right)=1$.

Inductive Hypothesis: Assume that for all natural numbers $k<n$, there exists an AIT $T_k$ such that $f\left(r_k\right)=k$, where $r_k$ is the root node of $T_k$.

Inductive Step: Consider now the number n. By the surjectivity of the function $C^{-1}$, there exists a $m\in \mathbb{N}$ such that $n\in C^{-1}\left(m\right)$. By appending n as a child node of the node v with $f\left(v\right)=m$ in the AIT $T_m$, we expand this tree to a new AIT $T_n$ such that $f\left(r_n\right)=n$ as required.

By the Principle of Structural Induction, the statement holds $\forall n\in \mathbb{N}$. Therefore, the recursive process using $C^{-1}$ allows reaching every natural number when constructing the AIT family.    □