{
  "abstract": "Let $α>0$ and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height $\\left(a, b^2\\right)$, delete classes whose coefficient sum is zero, and let $m$ be the number of remaining classes. We prove the following dichotomy. If $m=0$, exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If $m \\geq 1$, there are at most $2 m-1$ critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov-Novikov-Shapiro, valid throughout their range and in fact for every $α>0$. The structural input is a projective paired Haar theorem: for finite $β>1$, the full $2 m$-dimensional space $\\sum L_j / Q_j^β$, with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most $2 m-1$ projective zeros counted with multiplicity. For positive charges, the substitution $p=2 α$ proves Conjecture 3 of Edelsbrunner-Fillmore-Oliveira throughout its stated range $p \\geq 1$ and extends the same conclusion to every $p>0$. For every $n$, an explicit positive $n$-charge configuration attains $2 n-1$ simple critical points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28376",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiuqing Duan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e751z2jx5sy13.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28376",
  "fulltext_url": null,
  "id": "e751z2jx5sy13",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28376",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204852843",
  "pdf_url": "https://arxiv.org/pdf/2608.28376",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e751z2jx5sy13.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2608.28376",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28376"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Critical Points of Line Restrictions of Signed Point-Charge Potentials",
  "title_text": "Critical Points of Line Restrictions of Signed Point-Charge Potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/e751z2jx5sy13",
  "venue": "arXiv",
  "versions": []
}