ProbXiv
sign in
unchecked

A Counterexample to Nivat's Conjecture for a Non-Convex Window

Nothing has been published against this problem here, and nobody has checked anything. That is the ordinary condition of an open problem, not a defect in the record.

nivat-conjecture-non-convex-window-counterexampleCombinatoricsposed by Maurice Nivat; the specific question by Jarkko Kari and Etienne Moutot, 2023recorded: disproved

1 attempt · no person has looked

Statement

The paper constructs an exact cluster F⊆Z2F\subseteq\mathbb{Z}^2 of cardinality 8 with full affine span and an FF-tiling whose orbit closure contains no 1-periodic FF-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows. This answers negatively a question of Kari and Moutot from 2023.

Context

Nivat's conjecture is a well-known problem on low-complexity two-dimensional configurations; this closes the non-convex window case in the strong form Kari and Moutot asked about.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    constructionGPT-5 + Claude Opus 4 with Abhishek Khetan ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5 + Claude Opus 4
    people
    Abhishek Khetan

    The paper's note on the use of AI credits the construction of Section 2 - the cluster and tiling that constitute the counterexample - to GPT-5, alongside use of Claude Opus 4.

    Reviews

    0 human reviews · 0 machine checks

    No person has reviewed this attempt. It has not been checked at all.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.