{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0019",
  "title": "Uniform Weak Robin Comparisons on Triangles and a Cubic Moment Formula",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give a quantitative comparison for the first Robin eigenvalue among triangles of fixed area. There exists an absolute positive constant epsilon such that, whenever 0 < |alpha| sqrt(A) < epsilon, the eigenvalue difference from the equilateral triangle, divided by alpha, is at least one twenty-fourth of the perimeter deficit divided by A. The parameter interval is common to all triangle shapes, including degenerating families. The proof combines analytic perturbation near the equilateral triangle, compactness on bounded-perimeter families, and a one-dimensional estimate on thin triangles. Earlier attractive-Robin comparisons and width estimates are explicitly credited. Separately, for each fixed domain satisfying (x-I) dot nu = r almost everywhere on its boundary, the cubic Robin coefficient equals Var(|X-I|^2)/r^3 for X uniform in the domain. Neither result settles the triangle optimization problem at arbitrary nonzero Robin parameter; frozen AIM-ANALYSIS-0019 remains partial. This is an unrefereed preprint with no absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.SP",
    "math.AP"
  ],
  "keywords": [
    "AIM-ANALYSIS-0019",
    "Robin Laplacian",
    "triangle",
    "shape optimization",
    "weak coupling",
    "circumsolid",
    "eigenvalue perturbation"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0019/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0019/paper.pdf?v=7f5ba85a520d",
  "doi": "10.5281/zenodo.23404478",
  "zenodo_record_url": "https://zenodo.org/records/23404478",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proofs of a uniform two-sided weak-Robin comparison on all triangle shapes and a fixed-circumsolid cubic variance coefficient. The parameter threshold is existential, not numerical. The all-parameter frozen AIM-ANALYSIS-0019 conjecture remains partial. KLV's negative comparisons, Bucur-Fragala's slicing method and ACH's quadratic corrector are credited. Unrefereed and AI-assisted; no independent review, formal verification or absolute priority is certified.",
  "files": {
    "paper.pdf": {
      "sha256": "7f5ba85a520d86a0482c646373dba63b6380d562763aa608c84ec62895fe7801"
    },
    "source.zip": {
      "sha256": "6c6824c3bf526c82b4046518653be6eee60691ddee3027b4dc984fe9ed0742cf"
    },
    "verification_report.md": {
      "sha256": "336e8cefa046774e2d8df1434c93fcdfcd9d9fe00bc75df621894f1e04397d8e"
    }
  },
  "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."
}