{
  "abstract": "We give a general criterion for transferring first-order responses on an underlying probability space to a closed differential calculus on the law of an observable. An integration-by-parts identity removes presentation ambiguity and yields a closable gradient, its divergence, and a closed Markov form. We apply this scheme to the subcritical Liouville quantum gravity metric throughout the range $0<γ<2$. Sequential compactness of Weyl-perturbed geodesics identifies the derivative of a logarithmic distance ratio with the Sobolev Riesz representative of the difference of two normalized geodesic occupation measures. The induced form on the projective metric law is the image of a Gaussian directional form and has the energy-image-density property. Finally, fixed-target confluence and a leaf-elimination argument make the response Gram matrix positive definite for every finite forest of marked pairs. The corresponding vector of logarithmic distance ratios therefore has a Lebesgue density.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27861",
  "author_total": 1,
  "authors": [
    {
      "name": "Chunhao Cai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jvh79j04jdrk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27861",
  "fulltext_url": null,
  "id": "7jvh79j04jdrk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27861",
  "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.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7214190330",
  "pdf_url": "https://arxiv.org/pdf/2609.27861",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jvh79j04jdrk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27861",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27861"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric",
  "title_text": "Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/7jvh79j04jdrk",
  "venue": "arXiv",
  "versions": []
}