{
  "abstract": "We study the one-parameter family of Hessian metrics $g_α=αh_++(1-α)h_-$, $α\\in[0,1]$, formed by convexly combining the log-partition functions (not the densities) of the multinomial and negative multinomial Fisher information Hessians. Four results are proved in full. (1) $g_α\\succ0$ for all $α\\in[0,1]$; its Gaussian curvature ($n=2$) is $-\\tfrac14$ at $α=0$ and $+\\tfrac14$ at $α=1$, but non-constant for intermediate $α$, changing sign at a closed-form critical value $α^\\ast(w)$, with an analytic uniqueness proof. (2) Viewing $g_α$ as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity $β\\le0$, while a second, independent quantity $C=4\\det K$ governs the equation's branch (bounded oscillatory, parabolic, or unbounded growth) and is not fixed by $β$ or positive-definiteness alone; we show this discriminant dictionary is a theorem about the abstract matrix equation, to which $g_α$ supplies initial data of either sign of $C$ without the trajectories remaining on $\\{g_α(θ)\\}$ itself. (3) $Ψ_α$ is identified as the log-partition function of a compound distribution: a negative-binomial latent count $M$ (parameter $r=1-α$), augmented by a Bernoulli parity bit and split multinomially, with non-negative base measure for every $α$. (4) The associated Bregman divergence obeys a generalised Pythagorean theorem whose projection onto the symmetric locus is the arithmetic mean, independent of $α$; the endpoints' curvature sign matches the Gauss--Bonnet excess of genuine geodesic triangles. Numerical checks are kept separate from analytic proofs throughout.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26819",
  "author_total": 1,
  "authors": [
    {
      "name": "Shintaro Yoshizawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mg3h17aapks2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26819",
  "fulltext_url": null,
  "id": "9mg3h17aapks2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26819",
  "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.DG"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.26819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mg3h17aapks2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials",
  "title_text": "A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/9mg3h17aapks2",
  "venue": "arXiv",
  "versions": []
}