A Fibonacci theorem for Collatz trajectories via modular graph structure
Abstract
Let T(n)=n/2 if n is even and T(n)=(3n+1)/2 if n is odd. We prove that for each m≥1, exactly F(m+1) odd integers n in {1,…,2m} have the property that none of the iterates T(n),T2(n),…,Tm−1(n) lies in the residue class 4pmod6, where F(m+1) is the (m+1)-th Fibonacci number; the proportion decays at rate (φ/2)m, φ=(1+√5)/2. Equivalently, these are the odd n≤2m for which no two consecutive terms of n,T(n),…,Tm−1(n) are even. The proof uses the directed graph G of Collatz transitions modulo $6$ and its unique absorbing strongly connected component G'=G[{1,2,4,5}]. Removing vertex $4$ from G' yields a subgraph of spectral radius φ, against ρ(G')=2; the Fibonacci count follows from this spectral gap. We construct an explicit bijection Ψm:{1,…,6·2m}→𝒫m(G) onto the directed paths of length m in G. We further show that no vertex of G' is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy 1<√2<φ<2. In particular, every positive cycle of T must visit residue class 2pmod6, and a flow conservation identity forces this class to account for more than 18% of the steps in any such cycle.
Related
- The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits
- Quantitative synchronization of Collatz trajectories
- Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem
- Hecke structure of quaternionic modular forms mod p
- The parameterised complexity of generalised temporal domination on temporal graphs with modular structure
BibTeX
@misc{jimenez2026fibonacci,
title = {{A Fibonacci theorem for Collatz trajectories via modular graph structure}},
author = {Manuel-Alejandro Reyes Jiménez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.02621},
eprint = {2606.02621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02621},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Fibonacci theorem for Collatz trajectories via modular graph structure
AU - Manuel-Alejandro Reyes Jiménez
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2606.02621
UR - https://arxiv.org/abs/2606.02621
AB - Let T(n)=n/2 if n is even and T(n)=(3n+1)/2 if n is odd. We prove that for each m≥1, exactly F(m+1) odd integers n in {1,…,2^m} have the property that none of the iterates T(n),T^2(n),…,T^m−1(n) lies in the residue class 4pmod6, where F(m+1) is the (m+1)-th Fibonacci number; the proportion decays at rate (φ/2)^m, φ=(1+√5)/2. Equivalently, these are the odd n≤2^m for which no two consecutive terms of n,T(n),…,T^m−1(n) are even. The proof uses the directed graph G of Collatz transitions modulo $6$ and its unique absorbing strongly connected component G'=G[{1,2,4,5}]. Removing vertex $4$ from G' yields a subgraph of spectral radius φ, against ρ(G')=2; the Fibonacci count follows from this spectral gap. We construct an explicit bijection Ψ_m:{1,…,6·2^m}→𝒫_m(G) onto the directed paths of length m in G. We further show that no vertex of G' is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy 1<√2<φ<2. In particular, every positive cycle of T must visit residue class 2pmod6, and a flow conservation identity forces this class to account for more than 18% of the steps in any such cycle.
ER -