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  "slug": "hartree-small-charge-rigidity",
  "title": "Small-charge rigidity for the quartic Hartree shape energy",
  "shortTitle": "Weak repulsion, exact roundness",
  "url": "https://evidencepress.org/releases/hartree-small-charge-rigidity/",
  "oneLine": "For a sufficiently weak repulsive interaction, the ball is the unique energy-minimising shape at fixed volume, up to translation and the stated negligible-set convention.",
  "abstract": "In three dimensions, a written proof candidate establishes one uniform small positive charge interval on which the volume-constrained quartic Hartree shape energy is minimised uniquely by a ball. The proof combines the global all-minimiser nearsphericity theorem of Mazzoleni–Muratov–Ruffini with local state-branch selection, an invertible projected boundary-flux map, and a fixed-Hölder-topology entrance argument. Quasi-open equality is up to zero H1 capacity; the measurable support relaxation uses equality almost everywhere. No numerical charge threshold or quantitative deficit is established.",
  "datePublished": "2026-10-09",
  "dateModified": "2026-10-09",
  "version": "1.0.0-candidate",
  "doi": "10.5281/zenodo.23269957",
  "doiUrl": "https://doi.org/10.5281/zenodo.23269957",
  "conceptDoi": "10.5281/zenodo.23269956",
  "pdfUrl": "https://github.com/ipitchford/hartree-small-charge-rigidity/releases/download/v1.0.0-candidate/hartree-small-charge-rigidity-v1.0.0-candidate.pdf",
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  "authors": [
    "Anonymous"
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    "detail": "Unrefereed analytic proof candidate. Internal proof inspection and finite sanity checks are not external peer review, independent reproduction or formal verification. Priority remains uncertain. The supplied review recommends minor revisions; its identity and independence are not established."
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      "evidenceUrl": "https://zenodo.org/records/23269957",
      "note": "Public GitHub and Zenodo assets verified by full downloads against the frozen archive; availability is not proof or peer review."
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      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "partial",
      "note": "Clean-extraction integrity and elementary checks pass normally and under optimized Python; these do not verify the analytic theorem.",
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      "question": "Has someone else reached the result from an independent implementation?",
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      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
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      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
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      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
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      "note": "No qualifying evidence establishing this dimension."
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      "dimension": "dataEnvironmentReproducibility",
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      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/hartree-small-charge-rigidity/blob/v1.0.0-candidate/ENVIRONMENT.txt",
      "note": "Standalone LaTeX and standard-library finite checks supplied; no independent rerun established."
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  "provenance": {
    "aiGenerated": true,
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    "generatedBy": [
      "AI-assisted supplied research; exact originating model not established",
      "OpenAI Codex assistant: review actions and publication"
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    "humanRole": "Publication authority only; no human research contribution asserted.",
    "reviewStatus": "Unrefereed analytic proof candidate. Internal proof inspection and finite sanity checks are not external peer review, independent reproduction or formal verification. Priority remains uncertain."
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  "attributionBoundary": null,
  "problem": {
    "name": "Small-charge rigidity for the repulsive quartic Hartree shape energy",
    "url": "https://pagine.dm.unipi.it/muratov/hartree.pdf"
  },
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  "metadataProvenance": "Supplied v2 manuscript and review; publication revisions and source checks on 9 October 2026. Discovery time and reviewer independence are not reconstructed.",
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  "keywords": [
    "Hartree energy",
    "shape optimization",
    "small charge",
    "ball rigidity",
    "quasi-open domains"
  ],
  "keyResults": [
    "One uniform sufficiently small positive charge interval gives ball minimality among all prescribed-volume competitors.",
    "Equality is a translated ball modulo zero H1 capacity for quasi-open domains, or null measure for the support relaxation.",
    "The local-to-global argument explicitly selects the minimising state branch and proves the full Hölder-space inverse."
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  "relatedWorks": [
    {
      "citation": "Mazzoleni, Muratov and Ruffini (2025), An optimal design problem for a charge qubit. Theorem 1.2 supplies global existence and all-minimiser nearsphericity; Remark 1.3 motivates exact rigidity.",
      "url": "https://pagine.dm.unipi.it/muratov/hartree.pdf",
      "doi": null
    },
    {
      "citation": "Cavallina (2024), Families of nondegenerate critical domains in shape optimization. Established implicit-function machinery, with precise Banach-space hypotheses.",
      "url": "https://arxiv.org/abs/2109.12559",
      "doi": "10.48550/arXiv.2109.12559"
    },
    {
      "citation": "Mazzoleni, Moraschi and Ruffini (2026), large-charge Hartree nonexistence result. A distinct charge regime, not the present small-charge classification.",
      "url": "https://arxiv.org/abs/2606.16503",
      "doi": "10.48550/arXiv.2606.16503"
    },
    {
      "citation": "Ruffini, conference abstract, 8 September 2026. Mentions rigidity but supplies no exact theorem or proof for comparison; priority overlap remains unresolved.",
      "url": "https://events.wias-berlin.de/event/1/contributions/82/",
      "doi": null
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