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A Fibonacci theorem for Collatz trajectories via modular graph structure

This paper establishes a connection between Fibonacci numbers and Collatz trajectories by demonstrating that the count of odd integers whose orbits avoid the residue class $4 \pmod 6$ for $m$ steps equals $F(m+1)$ , a result derived from the spectral properties of the Collatz transition graph modulo 6 and implying that any positive cycle must visit the residue class $2 \pmod 6$ .

Original authors: Manuel-Alejandro Reyes Jiménez

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Manuel-Alejandro Reyes Jiménez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Collatz Conjecture as a giant, chaotic maze where every number is a traveler. The rules are simple: if you're even, you divide by two; if you're odd, you triple yourself and add one. The big mystery is whether every traveler eventually gets trapped in a tiny loop (1 → 2 → 1) or if some get lost forever.

This paper doesn't solve the whole maze, but it builds a very specific, clever map to understand how these travelers move. The author, Manuel-Alejandro Reyes Jiménez, uses two different "languages" to describe the journey: Binary (like a light switch: on/off) and Modular (like a clock with only 6 hours: 0, 1, 2, 3, 4, 5).

Here is the breakdown of the paper's discoveries using simple analogies:

1. Two Ways to Track the Traveler

The author shows that you can track a number's journey in two parallel ways that mirror each other perfectly:

The Binary Code: You write down a string of 0s and 1s representing whether each step was even or odd.
The Modular Clock: You write down a string of numbers (0–5) representing the remainder when you divide by 6 at each step.

The paper proves that these two codes are locked together. If you know the "clock" path, you know the "switch" path, and vice versa. This allows the author to study the problem using the simpler "clock" (modular) system.

2. The "Forbidden Zone" and the Fibonacci Connection

The main discovery is about a specific "forbidden zone" on the clock: the number 4.

The author asks: How many odd numbers, when they start their journey, manage to avoid landing on the number 4 for a certain number of steps?

The answer is surprisingly beautiful: It's a Fibonacci number.

The Analogy: Imagine a tree growing. At every step, the number of paths that avoid the forbidden zone splits in a specific way, just like the famous Fibonacci sequence (1, 1, 2, 3, 5, 8...).
The Result: If you look at all odd numbers up to a certain size, the count of those that successfully dodge the number 4 for $m$ steps is exactly the $(m+1)$ -th Fibonacci number.
The Decay: However, as the journey gets longer, the proportion of numbers that can keep dodging this zone shrinks rapidly. It's like trying to walk through a forest without stepping on a specific type of leaf; the longer you walk, the harder it gets, and eventually, almost everyone steps on it.

3. The "Absorbing" Neighborhood

The paper maps out the "neighborhood" of the clock (the numbers 0–5).

Transient Zones: Numbers 0 and 3 are like dead ends or temporary stops. If a traveler starts there, they quickly leave and never come back.
The Safe House: Once a traveler is an odd number, they immediately enter a "safe house" neighborhood consisting of the numbers 1, 2, 4, and 5. From that point on, they can never leave this group.
The Spectral Gap: The author uses a concept called "spectral radius" (think of it as the "speed limit" or "growth rate" of the paths).
- The whole safe house has a growth rate of 2.
- If you remove the forbidden number 4 from the safe house, the growth rate drops to 1.618 (the Golden Ratio, $\phi$ ).
- This difference (the "spectral gap") is exactly why the Fibonacci numbers appear. The math of the "dodging" paths is governed by the Golden Ratio, while the total paths are governed by 2.

4. The Indispensable "Hub" (Number 2)

The paper investigates what happens if you try to remove any number from the safe house (1, 2, 4, or 5).

The Finding: You cannot remove any of them without slowing down the system. Every single number in this group is essential.
The Critical Hub: The number 2 is the most important.
- If you remove 2, the growth rate drops to 1 (the system stops growing).
- The paper proves that every possible loop (cycle) in the Collatz system must visit the number 2.
- The Flow: It's like a river system where the number 2 is the main dam. The math proves that in any loop, the traveler must spend more than 18% of their time at this "dam" (residue class 2). You can't have a cycle that avoids it.

Summary of the "Big Picture"

The paper doesn't prove the Collatz Conjecture (that everyone eventually reaches 1). Instead, it builds a rigorous framework that shows:

Structure: The chaotic movement of numbers follows a hidden, rigid structure when viewed through a 6-hour clock.
Counting: The number of "lucky" paths that avoid a specific trap (4) follows the Fibonacci sequence exactly.
Necessity: The number 2 is the heart of the system. No loop can exist without it, and it must be visited frequently.

The author concludes by asking three open questions, essentially asking: "Can we use this map to predict exactly when a number will hit the trap?" and "Does every loop have to visit all the numbers in the safe house?" These remain mysteries for future explorers.

Technical Summary: A Fibonacci Theorem for Collatz Trajectories via Modular Graph Structure

Problem Statement
The paper addresses the dynamics of the accelerated Collatz map $T(n)$ , defined as $n/2$ for even $n$ and $(3n+1)/2$ for odd $n$ . While the Collatz conjecture posits that all positive integers eventually reach the cycle $1 \leftrightarrow 2$ , no proof exists. The paper focuses on a specific structural property of trajectories: the frequency with which odd integers avoid the residue class $4 \pmod 6$ during their initial iterations. Specifically, it investigates the count of odd integers $n \in \{1, \dots, 2^m\}$ whose orbits under $T$ do not visit the vertex $4$ in the modular graph during steps $2, \dots, m$ .

Methodology
The author employs a dual-encoding framework, analyzing the Collatz dynamics simultaneously through binary and modular lenses, linked by specific bijections and structural symmetries.

Binary Encoding and Corrective Residue:
The paper utilizes the parity encoding $C_m(n) = (c_1, \dots, c_m)$ , where $c_i = T^{i-1}(n) \pmod 2$ . A key algebraic tool is the "corrective residue" $R(C_m(n))$ , defined such that $T^m(n) = (3^s n + R)/2^m$ , where $s$ is the Hamming weight of the encoding. The paper establishes that $R$ depends solely on the binary encoding, allowing the analysis of $T^m(n)$ to be conducted entirely at the level of binary strings without reference to the specific integer value $n$ .
Modular Encoding and Graph Construction:
The dynamics are also encoded modulo 6 via a directed graph $G$ with vertices $\{0, 1, 2, 3, 4, 5\}$ . The transitions are determined by the parity of the current iterate. The author constructs a bijection $\Psi_m: \{1, \dots, 6 \cdot 2^m\} \to \mathcal{P}_m(G)$ , mapping integers to directed paths of length $m$ in $G$ . This bijection shares structural properties (bijectivity, periodicity, and antisymmetry) with the binary encoding map $\Phi_m$ .
Spectral Graph Theory:
The analysis focuses on the subgraph $G' = G[\{1, 2, 4, 5\}]$ , which is the unique absorbing strongly connected component for odd integer orbits (after step 2). The paper investigates the spectral properties of $G'$ and its subgraphs obtained by removing specific vertices. The core mechanism relies on the spectral gap between the full effective subgraph $G'$ and the subgraph $H_4$ (obtained by removing vertex 4).

Key Contributions and Results

Exact Fibonacci Count (Theorem 4.7):
The primary result is an exact count of odd integers avoiding a specific residue. For each $m \ge 1$ , exactly $F(m+1)$ odd integers in $\{1, \dots, 2^m\}$ have orbits that avoid the residue class $4 \pmod 6$ during steps $2, \dots, m$ . Here, $F(m+1)$ is the $(m+1)$ -th Fibonacci number. This count is derived from the spectral radius of the avoiding subgraph $H_4$ , which is the golden ratio $\phi = (1+\sqrt{5})/2$ .
Exponential Decay Rate (Corollary 4.9):
The proportion of such integers decays exponentially at the rate $(\phi/2)^m$ . This rate is precisely the ratio of the spectral radius of the avoiding subgraph $\rho(H_4) = \phi$ to the spectral radius of the full effective subgraph $\rho(G') = 2$ .
Modular Path Theorem (Theorem 4.1):
The paper characterizes the action of the displacement $+2^m$ on modular encodings. It proves that for an integer $n$ , the modular path levels $i$ are preserved under this displacement if and only if $2^{i-1} \nmid n$ . If $2^{i-1} \mid n$ , the path shifts by $+2$ or $+4 \pmod 6$ depending on the parity of $m+i$ . This rigidity implies that cycles of length $m$ in arithmetic progressions with difference $2^m$ are separated by at least $6 \cdot 2^m$ .
Vertex Indispensability and Spectral Hierarchy (Theorem 4.15):
The paper demonstrates that no vertex in the effective subgraph $G'$ is dispensable. Removing any single vertex $v \in \{1, 2, 4, 5\}$ strictly reduces the spectral radius below 2. The resulting spectral radii form a strict hierarchy:
$\rho(H_2) = 1 < \rho(H_5) = \sqrt{2} < \rho(H_4) = \rho(H_1) = \phi < \rho(G') = 2$
This hierarchy governs the exponential decay rates of trajectories avoiding each respective vertex.
Constraints on Positive Cycles (Corollaries 4.17 & 4.19):
Based on the spectral hierarchy, the paper proves that every positive cycle of $T$ must visit the residue class $2 \pmod 6$ . Furthermore, using flow conservation identities within $G'$ , it is shown that vertex 2 must account for more than $18\%$ ( $> m(1-\log_3 2)/2$ ) of the steps in any positive cycle of length $m$ .

Significance and Claims
The paper claims to provide a structural explanation for the appearance of Fibonacci numbers in Collatz trajectories, distinct from previous combinatorial enumerations of permutations. The connection arises from a spectral argument on the modular graph, where the golden ratio appears as the Perron–Frobenius eigenvalue of the avoiding subgraph.

The author emphasizes that the Fibonacci count is exact and unconditional, not an approximation, driven by the spectral gap $\phi/2 < 1$ . The work introduces a new modular bijection $\Psi_m$ and its structural properties, which are not present in prior literature. While the paper does not resolve the Collatz conjecture, it establishes rigorous constraints on the distribution of residues in hypothetical cycles, specifically proving that residue 2 is unavoidable and must constitute a significant portion of any cycle's length. The paper concludes by identifying open questions regarding the effectivity of these bounds (bounding the first occurrence of residue 4) and whether all vertices of $G'$ are visited by non-trivial cycles.