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Paradoxical behavior in Collatz sequences

T0 review · 1 major / 0 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Paradoxical sequences in Collatz iterations are closely tied to the conjecture and likely finite in number.

desk verdict The paper defines paradoxical Collatz sequences and ties their finiteness to the main conjecture, but that makes the support for Terras conditional rather than independent. read the letter →

arxiv 2502.00948 v5 pith:XSV2EMXU submitted 2025-02-02 math.GM

Olivier Rozier , Claude Terracol This is my paper · ORCID
classification math.GM
keywords CollatzconjectureTerrasparadoxicalsequencesstoppingtimeoddtermsproportioniterativeprocess
topics The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a variant of the Collatz process where each step halves or applies (3n+1)/2 based on parity. It identifies paradoxical sequences that, after the stopping time, exceed their starting value in a way not predicted by the proportion of odd terms encountered. These sequences are shown to be intimately connected to the validity of the Collatz conjecture. The authors conclude that such paradoxical behavior most likely occurs only finitely many times. This lends support to Terras' conjecture that the proportion of odd terms determines the stopping time.

What carries the argument

Paradoxical sequences of finite length, where the first term is unexpectedly exceeded given the proportion of odd terms.

What would settle it

Discovery of infinitely many paradoxical sequences starting from arbitrarily large integers would falsify the finiteness claim.

Watch

Extended reading notes

Core claim

We identify paradoxical sequences of finite length in the Collatz iteration that exceed their initial term contrary to expectation from the odd-term proportion when iterating beyond the stopping time. This non-typical behavior is closely related to the Collatz conjecture. It most likely occurs finitely many times, thus lending support to Terras' conjecture.

Load-bearing premise

The finiteness argument for paradoxical sequences rests on the assumption that the Collatz conjecture holds or on density properties that would fail precisely when the conjecture fails.

Editorial extensions

If this is right

  • If the Collatz conjecture holds, paradoxical sequences occur only finitely often.
  • This provides support for Terras' conjecture that the proportion of odd terms determines stopping times.
  • Non-typical behaviors become exceptional rather than recurrent in the iteration process.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The finiteness result could be used to derive explicit upper bounds on the size of any paradoxical sequence.
  • Similar finite-anomaly arguments might apply to other parity-based iterative maps on the integers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript studies a variant of the Collatz iteration (n maps to (3n+1)/2 or n/2 by parity) and identifies finite-length 'paradoxical' sequences whose first term is exceeded after the stopping time, contrary to the proportion of odd terms. It claims to show that this non-typical behavior is closely related to the Collatz conjecture and that such sequences occur only finitely many times, thereby lending support to Terras' conjecture that the odd-term proportion determines stopping time.

Significance. An unconditional demonstration that paradoxical sequences are finite would supply evidence for Terras' conjecture by showing that atypical behavior is rare. The manuscript provides no machine-checked proofs, reproducible code, or parameter-free derivations; the reported finiteness result is explicitly conditioned on the Collatz conjecture, so any support for Terras remains conditional and does not resolve the open question independently.

major comments (1)
  1. Abstract: the assertion that paradoxical sequences 'most likely occur finitely many times, thus lending support to Terras' conjecture' is derived from density properties of the stopping-time map that hold only under the assumption that every trajectory reaches the (1,2) cycle. If a counterexample to Collatz exists, those densities fail precisely on divergent trajectories, rendering the finiteness argument inapplicable and the claimed support conditional rather than independent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript. Below we respond point-by-point to the major comment.

read point-by-point responses

  1. Referee: Abstract: the assertion that paradoxical sequences 'most likely occur finitely many times, thus lending support to Terras' conjecture' is derived from density properties of the stopping-time map that hold only under the assumption that every trajectory reaches the (1,2) cycle. If a counterexample to Collatz exists, those densities fail precisely on divergent trajectories, rendering the finiteness argument inapplicable and the claimed support conditional rather than independent.

    Authors: We agree that the density properties used to establish finiteness of paradoxical sequences, and hence the claimed support for Terras' conjecture, hold only under the assumption that the Collatz conjecture is true. The full manuscript text already states the results as conditional on every trajectory reaching the (1,2) cycle. The abstract's wording 'most likely' was chosen to signal the conjectural setting, but we accept that it does not sufficiently emphasize the dependency. We will revise the abstract to read that paradoxical sequences 'occur finitely many times under the Collatz conjecture, thereby lending conditional support to Terras' conjecture.' This change makes the conditional character explicit while preserving the logical relation shown in the paper. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: claims are conditional on external Collatz conjecture without self-referential reduction

full rationale

The abstract explicitly conditions the finiteness of paradoxical sequences on the Collatz conjecture and uses that to lend support to Terras' conjecture, but this is a conditional statement rather than a derivation that reduces to its own inputs by construction. No equations, fitted parameters, or self-citations are quoted that would make any prediction equivalent to the input data or prior results within the paper. The derivation chain remains self-contained against external benchmarks because the linkage is stated as an assumption rather than a closed loop, and no load-bearing step matches the enumerated circularity patterns.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on the standard definition of the Collatz map and on Terras' earlier conjecture; no new free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • domain assumption The iterative Collatz map is well-defined on positive integers and the stopping time is finite for every starting value under the conjecture.
    Invoked when relating paradoxical behavior to the conjecture.

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Cite this review

Pith. "Pith review of Paradoxical behavior in Collatz sequences." pith.science (2026). https://pith.science/paper/XSV2EMXU

@misc{pith2026250200948,
  author       = {Pith},
  title        = {Pith review of: Paradoxical behavior in Collatz sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSV2EMXU}},
  note         = {Machine review of arXiv:2502.00948}
}
abstract

On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial cycle $(1,2)$. In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting with an integer $n\geq2$ is sufficient to determine its stopping time, namely, the number of iterations needed to descend below $n$. However, when iterating beyond the stopping time, there exist "paradoxical" sequences of finite length whose first term is unexpectedly exceeded, given the proportion of odd terms. In the present study, we show that this non-typical behavior is closely related to the Collatz conjecture. Furthermore, we find that it most likely occurs finitely many times, thus lending support to Terras' conjecture.

Figures

Figures reproduced from arXiv: 2502.00948 by the authors.

Figure 1
Figure 1. Hasse diagrams of (a) the total order on parity vectors of length 3 and ones-ratio 1 3 , and of (b-d) the partial orders on parity vectors of length 4, 5, 6 and ones-ratios 1 2 , 3 5 , 1 2 , in that order. not ordered. Similarly, in Figure 1c, the vector ⟨01110⟩ cannot be compared with any of the vectors ⟨11001⟩, ⟨10101⟩ and ⟨10011⟩. More generally, when considering two parity vectors V = (v0, . . . , vj−1) and V ′ … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Christoffel words as extremal structures in Collatz dynamics

    math.DS 2026-07 conditional novelty 7.0 of 10

    Christoffel words are, up to rotation, the unique maximizers of the rotation-invariant Collatz functional C_min on binary words of fixed length and density.

  2. Parity vectors and paradoxical sequences in the accelerated Collatz map

    math.NT 2026-05 unverdicted novelty 6.0 of 10

    Three unconditional theorems give a sharp finitary parity-vector density, a closed-form count of paradoxical sequences of fixed length k, and a density-zero result with explicit constant for bounded-length paradoxical...

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 2 Pith papers

  1. [1]

    Albert, B

    M. Albert, B. Gudmundsson, B., H. Ulfarsson, Collatz meets Fibonacci, Math. Mag. 95 (2022), 130–136

    work page 2022

  2. [2]

    Baker, Transcendental Number Theory, Cambridge University Press, 1975

    A. Baker, Transcendental Number Theory, Cambridge University Press, 1975

    work page 1975

  3. [3]

    Barina, Convergence verification of the Collatz problem, J

    D. Barina, Convergence verification of the Collatz problem, J. Supercomput., 77 (2021), 2681–2688

    work page 2021

  4. [4]

    Eliahou, The 3 x + 1 problem: new lower bounds on nontrivial cycle lengths, Discrete Math

    S. Eliahou, The 3 x + 1 problem: new lower bounds on nontrivial cycle lengths, Discrete Math. 118 (1993), 45–56

    work page 1993

  5. [5]

    C. J. Everett, Iteration of the number theoretic function f(2n) = n, f(2n+ 1) = 3n + 2, Adv. Math. 25 (1977), 42–45

    work page 1977

  6. [6]

    L. E. Garner, On the Collatz 3 n + 1 algorithm, Proc. Amer. Math. Soc. 82 (1981), 19–22

    work page 1981

  7. [7]

    A. V. Kontorovich, J. C. Lagarias, Stochastic models for the 3 x + 1 and 5x + 1 problems and related problems, published in [9], 131–188

    work page

  8. [8]

    J. C. Lagarias, The 3 x+1 problem and its generalizations, Amer. Math. Monthly 92 (1985), 3–23. Reprinted in [9]

    work page 1985

Show all 20 references
  1. [9]

    J. C. Lagarias, editor, The Ultimate Challenge: The 3x+ 1 Problem , Amer. Math. Soc., 2010

    2010

  2. [10]

    A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of Majorization and Its Applications (2nd ed.), Springer, 2011

    2011

  3. [11]

    M. B. Nathanson, Permutation patterns of the iterated Syracuse function, IN- TEGERS 24A, paper A14 (2024), 1–24

    2024

  4. [12]

    OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, https://oeis.org/

  5. [13]

    Oliveira e Silva, Empirical verification of the 3 x + 1 and related conjectures, published in [9], 189–207

    T. Oliveira e Silva, Empirical verification of the 3 x + 1 and related conjectures, published in [9], 189–207

  6. [14]

    Rhin, Approximants de Pad´ e et mesures effectives d’irrationalit´ e, in S´ eminaire de Th´ eorie des Nombres, Paris 1985–86, Progr

    G. Rhin, Approximants de Pad´ e et mesures effectives d’irrationalit´ e, in S´ eminaire de Th´ eorie des Nombres, Paris 1985–86, Progr. Math. 71 Birkh¨ auser Verlag, 1987, 155–164. 20

    1985

  7. [15]

    Roosendaal, On the 3x+1 problem, web site available at http://www.ericr.nl/wondrous/

    E. Roosendaal, On the 3x+1 problem, web site available at http://www.ericr.nl/wondrous/

  8. [16]

    Rozier, The 3 x + 1 problem: a lower bound hypothesis, Funct

    O. Rozier, The 3 x + 1 problem: a lower bound hypothesis, Funct. Approx. Comment. Math. 56 (2017), 7–23

    2017

  9. [17]

    Rozier, D´ emonstration de l’absense de cycles d’une certain forme pour le probl` eme de Syracuse,Singularit´ e1 No

    O. Rozier, D´ emonstration de l’absense de cycles d’une certain forme pour le probl` eme de Syracuse,Singularit´ e1 No. 3 (1990), 9–10, available at https://www.researchgate.net/profile/Olivier-Rozier

    1990

  10. [18]

    Simons, B

    J. Simons, B. de Weger, Theoretical and computational bounds for m-cycles of the 3n + 1 problem, Acta Arith. 117 (2005), 51–70

    2005

  11. [19]

    Tao, Almost all orbits of the Collatz map attain almost bounded values, Forum Math

    T. Tao, Almost all orbits of the Collatz map attain almost bounded values, Forum Math. Pi 10 (2022), 1–56

    2022

  12. [20]

    Terras, A stopping time problem on the positive integers, Acta Arith

    R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976), 241–252. Institut de Physique du Globe de Paris, Universit´e Paris Cit´e, France E-mail: rozier@ipgp.fr Bri´e et Angonnes, France E-mail: claudeterracol34@gmail.com 21

    1976

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