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  "slug": "regular-heptagon-octagon-local-minimality",
  "title": "Regular seven- and eight-sided drums resist small shape changes",
  "shortTitle": "Certified local minima for two regular polygons",
  "url": "https://evidencepress.org/releases/regular-heptagon-octagon-local-minimality/",
  "oneLine": "A computer-assisted proof candidate certifies that small genuine shape changes increase the first vibration frequency at fixed area.",
  "abstract": "For n=7 and n=8, the regular polygon strictly locally minimises area times its first Dirichlet Laplacian eigenvalue, modulo similarities. A written transfer theorem and outward-rounded residual certificates give reduced Hessian bounds 3/5 and 7/20 at unit circumradius in all 10 and 12 nonsimilarity directions. This is local, not global optimality; no neighbourhood radius is supplied.",
  "datePublished": "2026-09-30",
  "dateModified": "2026-09-30",
  "version": "1.0.1-candidate",
  "doi": "10.5281/zenodo.23066520",
  "doiUrl": "https://doi.org/10.5281/zenodo.23066520",
  "conceptDoi": "10.5281/zenodo.23066519",
  "pdfUrl": "https://github.com/ipitchford/regular-heptagon-octagon-local-minimality/releases/download/v1.0.1-candidate/paper.pdf",
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  "zenodoUrl": "https://zenodo.org/records/23066520",
  "repoUrl": "https://github.com/ipitchford/regular-heptagon-octagon-local-minimality",
  "releaseUrl": "https://github.com/ipitchford/regular-heptagon-octagon-local-minimality/releases/tag/v1.0.1-candidate",
  "markdownUrl": "https://evidencepress.org/releases/regular-heptagon-octagon-local-minimality/index.md",
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      "name": "Regular Seven- and Eight-Sided Drums Resist Small Shape Changes",
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
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      "note": "Complete frozen-package macOS replay and Linux CI passed: canonical certificates, exact matrix check, 13,763 interval cases, seven mutations; optimized Python rejected.",
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      "state": "partial",
      "note": "Pinned dependencies and producer-coordinated macOS/Linux replay; no unaffiliated reproduction or formal verification.",
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    "disclosure": "Scholarly creator Anonymous; producer-coordinated checking is not unaffiliated reproduction."
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  "attributionBoundary": null,
  "problem": {
    "name": "Polygonal Faber–Krahn local-minimality problem",
    "url": "https://arxiv.org/abs/2406.11575v1"
  },
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  "corrections": [],
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  "metadataProvenance": "Supplied candidate and model-assisted review, revised 30 September 2026; reviewer identity and external independence are not authenticated.",
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  "keywords": [
    "polygonal Faber-Krahn problem",
    "heptagon",
    "octagon",
    "Dirichlet Laplacian",
    "shape optimisation",
    "verified numerics"
  ],
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    "Strict local minimality modulo similarities for n=7,8.",
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  "evidencePackage": "Written continuous-to-finite transfer theorem, canonical trial arrays, outward-rounded residual checks, exact final matrix check and deliberate corruption tests.",
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    "Obtain an explicit neighbourhood radius without confusing Hessian positivity at the regular polygon with a quantitative radius.",
    "Reduce the dominant ground-state error and examine additional side counts.",
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  ],
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  "relatedWorks": [
    {
      "citation": "Bogosel and Bucur (2024), On the polygonal Faber–Krahn inequality: volume Hessians and the finite-computation route.",
      "url": "https://doi.org/10.5802/jep.250",
      "doi": "10.5802/jep.250"
    },
    {
      "citation": "Bogosel and Bucur, Polygonal Faber–Krahn inequality: Local minimality via validated computing: validated n=5,6 results and remaining error obstacles.",
      "url": "https://arxiv.org/abs/2406.11575v1",
      "doi": "10.48550/arXiv.2406.11575"
    },
    {
      "citation": "Bogosel (2026), Computer-assisted local maximality of regular polygons for torsional rigidity: closely related error-product methods for a different functional.",
      "url": "https://arxiv.org/abs/2609.14785v1",
      "doi": "10.48550/arXiv.2609.14785"
    },
    {
      "citation": "Bogosel, Bucur and Fragalà (2026), Proof of the local version of the Pólya–Szegő conjecture for the torsional rigidity of polygons: all-n torsion result, not this eigenvalue theorem.",
      "url": "https://arxiv.org/abs/2609.23051v1",
      "doi": "10.48550/arXiv.2609.23051"
    },
    {
      "citation": "Cancès et al. (2017), Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: residual-estimation background.",
      "url": "https://doi.org/10.1137/15M1038633",
      "doi": "10.1137/15M1038633"
    }
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