{
  "abstract": "We construct a smooth, strictly positive, Gaussian-decaying density on $\\mathbb{R}^2$ for which Fisher information along the heat flow is not log-convex. This disproves the Cheng--Geng log-convexity conjecture in dimension two and, by tensorization, in every dimension $d\\ge2$. Consequently, the multidimensional forms of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture also fail, complementing the one-dimensional counterexample of Gu and Sellke. Our construction is a small hexagonal perturbation on the triangular torus, transferred to $\\mathbb{R}^2$ by a Gaussian envelope and supported by explicit two-dimensional numerics. We also initiate the study of the sharp constants $θ_d^*$ by proving $θ_1^*=1$, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant $θ_\\infty^*$ governed by the sign of $\\mathcal{D}$. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.18081",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiayang Zou"
    },
    {
      "name": "Luyao Fan"
    },
    {
      "name": "Jiayang Gao"
    },
    {
      "name": "Jia Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/764sf6nx6x4n2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.18081",
  "fulltext_url": null,
  "id": "764sf6nx6x4n2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.18081",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7161690483",
  "pdf_url": "https://arxiv.org/pdf/2605.18081",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/764sf6nx6x4n2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2605.18081",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.18081"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow",
  "title_text": "A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/764sf6nx6x4n2",
  "venue": "arXiv",
  "versions": []
}