Deterministic Upper Bound on Expansive Phases in the Collatz Dynamics

Deterministic Upper Bound on Expansive Phases in the Collatz Dynamics Daniele Grosso Department of Physics, University of Genoa, Italy [email protected] December 2025 Abstract We analyze the duration of consecutive expansive steps in the Collatz (3x+1) trajectory. By examining the binary structure of integers, specifically the runs of trailing ones, we prove that the length of any maximal sequence of consecutive odd steps in the reduced map T(n) = (3n+ 1)/2is strictly bounded by the binary length L(n)of the starting integer. For a number with ktrailing ones (e.g., Mersenne numbers), the expansive phase lasts exactly k−1steps, reaching a maximum duration of k−1=L(n)−1in the boundary case n= 2k−1. This establishes a sharp, deterministic O(L(n)) limit on how long a trajectory can remain in a strictly growth-favourable regime before a mandatory parity switch leads to a division by 2. 1 Introduction The Collatz conjecture posits that the orbit of any n∈Nunder the map f(n)reaches 1. A key difficulty lies in bounding the potential divergence of trajectories. While probabilistic models suggest that expansive steps are balanced by divisions, deterministic bounds are rare. In this work, we prove that continuous ascent under the reduced Collatz map is structurally impossible to sustain beyond a fixed and predictable limit. We show that the duration of any continuous expansive phase is bounded by the information content (number of bits) of the starting value. 2 Preliminaries Let L(n) = ⌊log2n⌋+ 1 denote the binary length of n≥1. We consider the Reduced Collatz Map: T(n) = 3n+ 1 2, n odd. An “expansive phase” is a sequence of iterations n07→ n1=T(n0)7→ n2=T(n1)7→ · · · in which all the niremain odd. 1 Definition 2.1 (Trailing ones).For an odd integer n≥1, let τ(n)denote the length of the terminal block of ones in the binary expansion of n. That is, if n=m·2k+ (2k−1), with modd and k≥1, then τ(n)=k. Note that τ(n)≤L(n), with equality if and only if nis a Mersenne number 2L−1. 3 Main Structural Lemma Lemma 3.1 (Decay of trailing ones under T).Let nbe an odd integer with τ(n)=k≥1. Then: 1. T(n)is odd if and only if k≥2; 2. when k≥2, one has τ(T(n))=k−1. Consequently, starting from nwith τ(n) = k, exactly k−1consecutive odd values are produced by iterating Tbefore an even output necessarily appears. Proof. Write n=m·2k+ (2k−1) with modd and k=τ(n)≥1. Then: 3n+ 1 = 3m·2k+ 3(2k−1) + 1 = 3m·2k+(3·2k−2). The term (3 ·2k−2) equals (11 ...10)2(with ktrailing zeros). The binary addition of 3m·2kand (3 ·2k−2) induces a carry chain that propagates exactly through the ktrailing ones of n, producing: - divisibility by 2but not by 4when k≥2, and - divisibility by at least 4when k= 1. Thus: v2(3n+ 1) = (1, k ≥2, ≥2, k = 1. If k= 1, then T(n)is even. If k≥2, division by 2removes precisely one trailing one: τ(T(n))=k−1. This proves both claims. 4 Main Theorem Theorem 4.1 (Length of Expansive Phases).Let n0be an odd integer. Let Kexp(n0)denote the number of consecutive odd values generated by iteratively applying T. Then: Kexp(n0) = max{τ(n0)−1,0} ≤ L(n0)−1. Moreover, Kexp(n0)=L(n0)−1if and only if n0is a Mersenne number 2L(n0)−1. Proof. By Lemma 3.1, an odd value nwith τ(n)=kproduces exactly k−1more odd outputs before the first even value appears, and τ(T(n))=k−1for each of these steps. Thus: Kexp(n0)=τ(n0)−1. Since τ(n0)≤L(n0)with equality if and only if n0is Mersenne, the bound follows. Corollary 4.2 (No Infinite Uninterrupted Growth).No Collatz trajectory can remain in an uninterrupted growth regime indefinitely. Every expansive phase has finite duration, bounded by the current binary length of the integer. Any divergent trajectory would therefore require an infinite sequence of Mersenne-like reconstructions. 2 5 Discussion The main theorem shows that the “fuel” for continuous ascent—the terminal block of ones— decays deterministically and cannot exceed the binary length L(n). Remark (Dissipative residues). Once the expansive phase ends, the integer enters a regime of divisions by 2whose intensity depends on the residue class of nmod 8: -ifn≡1 (mod 8) then 3n+ 1 is divisible by 4, - if n≡5 (mod 8) then 3n+ 1 is divisible by 8, - if n≡3 (mod 8) then T(n)returns to ≡1or 5 (mod 8). These classes tend to accelerate dissipation after the expansive phase. This remark is not needed for the proof of Theorem 4.1 but helps clarify why long-term runaway behaviour would require highly structured oscillations. Acknowledgements To Sara, Victor and Fiamma — for being the constants in my chaotic variables. Luna, for making everything converge. References [1] J. C. Lagarias, “The 3x+ 1 problem and its generalizations,” Amer. Math. Monthly 92 (1985), 3–23. [2] T. Tao, “Almost all orbits of the Collatz map attain almost bounded values,” 2019, arXiv:1909.03562. 3