{
  "schema_version": 1,
  "problem_number": "OWR-14297736-026",
  "title": "Sharp Growth Exponents for Positive Robin Eigenvalues at Fixed Volume",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Fix a positive Robin parameter alpha and a volume M. We prove that the kth Robin eigenvalue of every bounded convex domain in R^n with volume M is at least a k^(2/(2n-1)), where a>0 depends only on n,alpha,M. The proof combines a one-dimensional trace estimate, a John ellipsoid and convex-domain Poincare cube counting. For general bounded Lipschitz sets, the bound b k^(1/n) is recorded as a direct consequence of van den Berg and Bucur's established 2014 heat-trace estimate, with explicit constants and attribution. Thin boxes and unions of equal balls show both exponents are optimal over their respective classes; dimension one is also covered. This answers the two-part Euclidean Robin conjecture on p.2094 of Oberwolfach Report36/2023. The separately bundled manifold question in UnsolvedMath record OWR-14297736-026 remains unaddressed. This English preprint includes an AI-assisted analytic proof, source comparison, self-audit and supplementary exact/numerical checks. No optimal-constant, extremizer, historical-priority, independent-review or formal-verification claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.SP",
    "math.AP",
    "math.MG"
  ],
  "keywords": [
    "OWR-14297736-026",
    "Robin eigenvalue",
    "convex domain",
    "spectral geometry",
    "John ellipsoid",
    "heat trace",
    "sharp exponent"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14297736-026/",
  "pdf_url": "https://eulersolve.org/papers/owr-14297736-026/paper.pdf?v=6b60b7b00aa7",
  "doi": "10.5281/zenodo.23668477",
  "zenodo_record_url": "https://zenodo.org/records/23668477",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every fixed n, alpha>0 and volume M, positive Robin eigenvalues satisfy rho_k>=a*k^(2/(2n-1)) on every bounded open convex domain and rho_k>=b*k^(1/n) on every bounded Lipschitz open set, with explicit positive constants depending only on n,alpha,M. Both exponents are sharp over the respective classes. The one-dimensional case is included. The general-domain half is explicitly credited as a consequence of van den Berg--Bucur (2014). Complete proof of the two-part Euclidean Robin conjecture on OWR2023/36 p.2094, with sharp exponents. The general-domain bound is explicitly credited as a consequence of van den Berg and Bucur (2014). The separate manifold question bundled into the same UnsolvedMath record remains unaddressed. No optimal-constant, extremizer or historical-priority claim is made. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "6b60b7b00aa738acad0e1b02d4a67366910e8c4e11743326ae02eb1788964aa3"
    },
    "source.zip": {
      "sha256": "e3be7a56e394f7afb0a9379e87b18deb59db6e88e13cf5bf5ab540782f0fe24d"
    },
    "verification_report.md": {
      "sha256": "281ba1e2176a2d2a49ce8ad67d3ae0c93091815088edd9794ade94cb87c3285c"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}