bibkey: gallardo2025collatzlength
authors: Luis H. Gallardo and Olivier Rahavandrainy
year: 2025
title: A variant of Collatz's Conjecture over Binary Polynomials
doi: 10.48550/arXiv.2510.07530
url: https://arxiv.org/abs/2510.07530v1
claim: "However, several computations suggest that here also r_A is bounded by a polynomial in deg(A) (see Section 3)."
strata_touched:
  - D5/S1/Recurrence/GallardoRahavandrainyCollatzLengthBound
license: citation-only
triage: anchor

Polynomial bounds for binary-polynomial Collatz length

Verified locator

DOI: 10.48550/arXiv.2510.07530

Source: https://arxiv.org/abs/2510.07530v1

The journal version is Revista de la Unión Matemática Argentina 69(1), 295–302, DOI 10.33044/revuma.5555.

Source statements

The transformations in §1 (p. 4) include “A₀ = A” and “A₂ₖ = 1 + M₁A₂ₖ₋₁”, where M₁ = x² + x + 1. The odd polynomial A₁ satisfies “A₀ = x^{a₀}(x + 1)^{b₀}A₁”. Subsequent odd terms satisfy “A₂ₖ₊₁ = A₂ₖ/(x^{a₂ₖ}(x + 1)^{b₂ₖ})”.

Theorem 1.2 (p. 4): “Let A be a nonzero binary polynomial. Then the sequences of polynomials obtained from Collatz transformations are of finite length r_A.” It gives “where m ∈ ℕ* and r_A = m + 1 ≤ 2^{deg(A)−1}.” The terminal pair is x(x+1), 1 from index m onward.

The prediction in §1 (p. 5) reads: “However, several computations suggest that here also r_A is bounded by a polynomial in deg(A) (see Section 3).”

Encoding and result

The coefficient ring is ZMod 2. The existing declarations M, oddPart and oddSeq give the multiplier, removal of linear factors and odd-indexed sequence, with oddSeq A k = A₂ₖ₊₁. The even terms retain A at k=0 and 1+M*oddSeq A (k-1) at positive k. The length is one plus the least positive index from which the terminal pair persists.

For nonzero A the repository theorem proves r_A ≤ 2 + ⌊natDegree(A)²/4⌋. On each constant-degree plateau, the smaller root multiplicity at 0 and 1 of F+1 decreases by one; the sum of the degree budgets gives the quadratic bound. Exact plateau duration, sharpness examples and a formal equality for the source’s g(n) are not asserted by this module.

Prior-art completeness remains ASSUMED-UNVERIFIED; the literature screen is in https://github.com/the-omega-institute/trureturing/issues/20129.

The corroborating experiment is in https://github.com/the-omega-institute/trureturing-experiments at commit 8db3cff2b9136b3086be66583c454f3ca93fb369, path docs/reports/gallardo-rahavandrainy-2025-binary-collatz-length-bound/check.py, SHA-256 4f0db8a8d3cf2081c030bc3b8df0bb194c991b58c8831c2ef48f015531422238. The preregistration records exit 0. This finite computation corroborates the definitions and bound; the universal estimate is proved in Lean.