{
  "abstract": "Let $A$ be a finite abelian group. A nowhere-zero $A$-flow of a graph gives every edge a direction and a non-zero value in $A$, so that at every vertex the sum of the values on the entering edges equals the sum on the leaving edges. Such a flow can be changed along a cycle: going around the cycle, add a fixed non-zero element of $A$ to the value of every edge that points forward and subtract it from the value of every edge that points backward. The Cranston--Li--Su--Wang--Xu (2026) conjecture states that if $A$ has at least four elements, then in every graph in which all degrees are even, any two nowhere-zero $A$-flows are joined by a chain of such changes through nowhere-zero $A$-flows. It also states that, for every integer $k\\ge 4$, the same holds for integer flows whose values are between $1$ and $k-1$ in absolute value, where each change adds a non-zero integer. We prove this conjecture. The core of the proof is an induction on the number of edges: if a flow enters a vertex $v$ along an edge $uv$ and leaves $v$ along an edge $vw$ with the same value, the two edges can be replaced by one edge $uw$, and the induction hypothesis applies to the smaller graph. So the main task is to change every flow into a flow with such a pair of edges. For nowhere-zero $A$-flows with $|A|\\ge 4$, replacing vertices of degree more than $4$ by triangles reduces the problem to graphs with all degrees at most $4$. For integer flows, a counting argument joins any two nowhere-zero flows with values in the cyclic group $\\mathbb{Z}_k$, $k\\ge 4$, by changes that add $1$, $-1$, $2$ or $-2$; every such chain gives a chain of integer flows with values between $1$ and $k-1$ in absolute value, and every integer flow of this chain reduces modulo $k$ to a flow of the given chain. All results of this paper are also proved in the Lean 4 proof assistant, from Lean's standard axioms alone.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40863",
  "author_total": 1,
  "authors": [
    {
      "name": "Dave Goel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7rxeek2yx9nx2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40863",
  "fulltext_url": null,
  "id": "7rxeek2yx9nx2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40863",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40863",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7rxeek2yx9nx2.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40863",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40863"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Cranston--Li--Su--Wang--Xu (2026) conjecture is true",
  "title_text": "The Cranston--Li--Su--Wang--Xu (2026) conjecture is true",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/7rxeek2yx9nx2",
  "venue": "arXiv",
  "versions": []
}