{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "marques-neves-willmore-conjecture-2012",
    "title": "Marques and Neves: min–max proof of the Willmore conjecture",
    "summary": "A canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds.",
    "boundary": "The geometric setting, mean-curvature normalization, genus and embedding assumptions belong to the stated theorem. The immersed extension uses inherited Li–Yau results. This entry records the geometric theorem and its stated inherited implications.",
    "area": "Geometric analysis and minimal surfaces",
    "topic": "Mathematics",
    "year": 2012,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "marques-neves-willmore-conjecture-2012",
    "target": {
      "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
      "version": 1,
      "content_hash": "sha256:4c1fe998ee08ed131f8bd81f66d10aec049c86bb91a3ccb2f061779da48e3388"
    },
    "humans": [
      "Fernando C. Marques",
      "André Neves"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-arxiv-1202-6036v1",
        "title": "Min-Max theory and the Willmore conjecture",
        "url": "https://arxiv.org/abs/1202.6036v1",
        "date": "2012-02-27"
      }
    ],
    "points": "330000",
    "published_grade": {
      "id": "marques-neves-willmore-conjecture-2012",
      "entry_id": "marques-neves-willmore-conjecture-2012",
      "band": "S",
      "modifier": "-",
      "target": {
        "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
        "version": 1,
        "content_hash": "sha256:4c1fe998ee08ed131f8bd81f66d10aec049c86bb91a3ccb2f061779da48e3388"
      },
      "what_changed": "A canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds.",
      "scope": "A canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds. The geometric setting, mean-curvature normalization, genus and embedding assumptions belong to the stated theorem. The immersed extension uses inherited Li–Yau results. This entry records the geometric theorem and its stated inherited implications.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
        "version": 1,
        "content_hash": "sha256:4c1fe998ee08ed131f8bd81f66d10aec049c86bb91a3ccb2f061779da48e3388"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Fernando C. Marques",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Min-Max theory and the Willmore conjecture",
            "url": "https://arxiv.org/abs/1202.6036v1"
          }
        },
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "André Neves",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Min-Max theory and the Willmore conjecture",
            "url": "https://arxiv.org/abs/1202.6036v1"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Min-Max theory and the Willmore conjecture",
            "url": "https://arxiv.org/abs/1202.6036v1"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The geometric setting, mean-curvature normalization, genus and embedding assumptions belong to the stated theorem. The immersed extension uses inherited Li–Yau results. This entry records the geometric theorem and its stated inherited implications.",
        "domain": "Geometric analysis and minimal surfaces"
      },
      "sources": [
        {
          "citation": "Min-Max theory and the Willmore conjecture",
          "url": "https://arxiv.org/abs/1202.6036v1"
        }
      ],
      "statement": "# Marques and Neves: min–max proof of the Willmore conjecture\n\nA canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds.\n\n## Scientific scope\n\nThe geometric setting, mean-curvature normalization, genus and embedding assumptions belong to the stated theorem. The immersed extension uses inherited Li–Yau results. This entry records the geometric theorem and its stated inherited implications.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nFernando C. Marques; André Neves\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Min-Max theory and the Willmore conjecture](https://arxiv.org/abs/1202.6036v1)\n",
      "summary": "A canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds.",
      "title": "Marques and Neves: min–max proof of the Willmore conjecture"
    },
    "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
      "version": 1,
      "content_hash": "sha256:4c1fe998ee08ed131f8bd81f66d10aec049c86bb91a3ccb2f061779da48e3388"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:15a287e289a94523fcee021fc184d4620357d0ba46a6aaa8af83a08e73c752d9"
  },
  "target": {
    "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
    "version": 1,
    "content_hash": "sha256:4c1fe998ee08ed131f8bd81f66d10aec049c86bb91a3ccb2f061779da48e3388"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "marques-neves-willmore-conjecture-2012",
    "entry_id": "marques-neves-willmore-conjecture-2012",
    "revision": 1,
    "title": "Marques and Neves: min–max proof of the Willmore conjecture",
    "context": "Geometric analysis and minimal surfaces",
    "track": "Exact theorem and reusable mathematical method",
    "band": "S",
    "modifier": "-",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "A canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds.",
    "scope": "A canonical five-parameter family and Almgren–Pitts min–max methods prove the 2π² Willmore-energy lower bound for embedded closed positive-genus surfaces in the round three-sphere, with conformal Clifford-torus equality. The source explains how the Euclidean immersed-torus conjecture follows using existing nonembedded-surface bounds. The geometric setting, mean-curvature normalization, genus and embedding assumptions belong to the stated theorem. The immersed extension uses inherited Li–Yau results. This entry records the geometric theorem and its stated inherited implications.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The geometric setting, mean-curvature normalization, genus and embedding assumptions belong to the stated theorem. The immersed extension uses inherited Li–Yau results. This entry records the geometric theorem and its stated inherited implications.",
    "references": [
      {
        "title": "Min-Max theory and the Willmore conjecture",
        "url": "https://arxiv.org/abs/1202.6036v1",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:aab96ae3-138f-49f4-af83-34a87434eafc",
      "version": 1,
      "content_hash": "sha256:4c1fe998ee08ed131f8bd81f66d10aec049c86bb91a3ccb2f061779da48e3388"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "330000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}