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arxiv: 2502.00948 · v5 · pith:XSV2EMXUnew · submitted 2025-02-02 · 🧮 math.GM

Paradoxical behavior in Collatz sequences

Olivier Rozier , Claude Terracol This is my paper

Pith reviewed 2026-05-23 04:07 UTC · model grok-4.3

classification 🧮 math.GM
keywords Collatz conjectureTerras conjectureparadoxical sequencesstopping timeodd terms proportioniterative process
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The pith

Paradoxical sequences in Collatz iterations are closely tied to the conjecture and likely finite in number.

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

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.

What would settle it

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

Figures

Figures reproduced from arXiv: 2502.00948 by Claude Terracol, Olivier Rozier.

Figure 1
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… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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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Forward citations

Cited by 2 Pith papers

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

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

    math.NT 2026-05 accept novelty 6.0

    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...

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

    math.NT 2026-05 unverdicted novelty 6.0

    Proves unconditional theorems on sharp finitary parity-vector density, closed-form counts of paradoxical sequences of fixed length k, and density zero for bounded-length paradoxical sequences in the accelerated Collat...

Reference graph

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