{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "friedmann-positive-curvature-dynamics-1922",
    "title": "Friedmann: a changing radius in positively curved cosmology",
    "summary": "Friedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants.\n\nA universe governed by relativity need not remain static. Friedmann lets the common spatial curvature radius change with time and derives an equation for its evolution under a simplified matter model. Depending on the cosmological constant and initial conditions, the solutions can expand monotonically or expand and contract. This supplies a theoretical possibility to be compared with observation, while leaving the actual universe and several later geometric developments unresolved.",
    "boundary": "The target is the 1922 positively curved, pressureless matter construction with Einstein equations and a cosmological constant, plus explicit coordinate and metric assumptions. It is not the complete modern FLRW geometry, arbitrary-pressure cosmology, spatially flat models, or the 1924 negative-curvature extension. A later scholarly note flags incomplete geometric precision; the full tensor reduction was not independently certified. Positive spatial curvature does not alone force recollapse: sufficiently large positive cosmological constant permits a monotone branch. Periodic-radius descriptions do not independently prove regular spacetime continuation through zero-radius singularities. The paper does not derive cosmic redshift, fit galaxy recession data, prove our actual universe expands, or determine its age; its numerical period is explicitly illustrative. Lemaître’s later observation connection and Robertson/Walker developments remain separate. Extracted formula glyphs were not visually certified; the normalized first-integral check confirms a narrow root interpretation, not the full proof or singular boundary behavior.",
    "area": "Time-dependent cosmological solutions",
    "topic": "Physics",
    "year": 1922,
    "lane": "HUMAN",
    "result": "THEORY",
    "package_id": "friedmann-positive-curvature-dynamics-1922",
    "target": {
      "record_id": "urn:uuid:b1148720-bb15-4799-a211-6d59d9c82e4e",
      "version": 1,
      "content_hash": "sha256:16b4458e02a4a5783ac94f004ee13a93a9a4088a3597ae6714e2fa8fe32a0534"
    },
    "humans": [
      "Alexander Friedmann"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.",
    "agents": [],
    "related": [
      "lemaitre-expanding-universe",
      "hubble-distance-velocity"
    ],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "wrs24-friedmann-curvature-1922-original",
        "title": "A. Friedman, Über die Krümmung des Raumes (Zeitschrift für Physik10,377-386,1922), original German scan",
        "url": "https://web.phys.ntnu.no/~mika/friedmann1.pdf"
      },
      {
        "id": "wrs24-friedmann-editor-scope-1999",
        "title": "Ellis and Krasiński, Editor’s Note on Friedmann 1922/1924 (General Relativity and Gravitation31,1985-1990,1999), selected scholarly scope commentary",
        "url": "https://users.camk.edu.pl/akr/friededibio.grg.pdf"
      },
      {
        "id": "wrs24-friedmann-leiden-archive-identification",
        "title": "Instituut-Lorentz, Friedmann papers: institutional manuscript catalogue and work identification",
        "url": "https://lorentz.leidenuniv.nl/history/Friedmann_archive/"
      }
    ],
    "points": "330000",
    "published_grade": {
      "id": "friedmann-positive-curvature-dynamics-1922",
      "entry_id": "friedmann-positive-curvature-dynamics-1922",
      "band": "S",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:b1148720-bb15-4799-a211-6d59d9c82e4e",
        "version": 1,
        "content_hash": "sha256:16b4458e02a4a5783ac94f004ee13a93a9a4088a3597ae6714e2fa8fe32a0534"
      },
      "what_changed": "Friedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants.",
      "scope": "Friedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants. The target is the 1922 positively curved, pressureless matter construction with Einstein equations and a cosmological constant, plus explicit coordinate and metric assumptions. It is not the complete modern FLRW geometry, arbitrary-pressure cosmology, spatially flat models, or the 1924 negative-curvature extension. A later scholarly note flags incomplete geometric precision; the full tensor reduction was not independently certified. Positive spatial curvature does not alone force recollapse: sufficiently large positive cosmological constant permits a monotone branch. Periodic-radius descriptions do not independently prove regular spacetime continuation through zero-radius singularities. The paper does not derive cosmic redshift, fit galaxy recession data, prove our actual universe expands, or determine its age; its numerical period is explicitly illustrative. Lemaître’s later observation connection and Robertson/Walker developments remain separate. Extracted formula glyphs were not visually certified; the normalized first-integral check confirms a narrow root interpretation, not the full proof or singular boundary behavior.",
      "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:b1148720-bb15-4799-a211-6d59d9c82e4e",
        "version": 1,
        "content_hash": "sha256:16b4458e02a4a5783ac94f004ee13a93a9a4088a3597ae6714e2fa8fe32a0534"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-03T00:00:00Z",
    "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": "Alexander Friedmann",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "formulator"
          ],
          "source": {
            "citation": "A. Friedman, Über die Krümmung des Raumes (Zeitschrift für Physik10,377-386,1922), original German scan",
            "url": "https://web.phys.ntnu.no/~mika/friedmann1.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "A. Friedman, Über die Krümmung des Raumes (Zeitschrift für Physik10,377-386,1922), original German scan",
            "url": "https://web.phys.ntnu.no/~mika/friedmann1.pdf"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "Ellis and Krasiński, Editor’s Note on Friedmann 1922/1924 (General Relativity and Gravitation31,1985-1990,1999), selected scholarly scope commentary",
            "url": "https://users.camk.edu.pl/akr/friededibio.grg.pdf"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-2",
          "kind": "source_reference",
          "source": {
            "citation": "Instituut-Lorentz, Friedmann papers: institutional manuscript catalogue and work identification",
            "url": "https://lorentz.leidenuniv.nl/history/Friedmann_archive/"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The target is the 1922 positively curved, pressureless matter construction with Einstein equations and a cosmological constant, plus explicit coordinate and metric assumptions. It is not the complete modern FLRW geometry, arbitrary-pressure cosmology, spatially flat models, or the 1924 negative-curvature extension. A later scholarly note flags incomplete geometric precision; the full tensor reduction was not independently certified. Positive spatial curvature does not alone force recollapse: sufficiently large positive cosmological constant permits a monotone branch. Periodic-radius descriptions do not independently prove regular spacetime continuation through zero-radius singularities. The paper does not derive cosmic redshift, fit galaxy recession data, prove our actual universe expands, or determine its age; its numerical period is explicitly illustrative. Lemaître’s later observation connection and Robertson/Walker developments remain separate. Extracted formula glyphs were not visually certified; the normalized first-integral check confirms a narrow root interpretation, not the full proof or singular boundary behavior.",
        "domain": "Time-dependent cosmological solutions"
      },
      "sources": [
        {
          "citation": "A. Friedman, Über die Krümmung des Raumes (Zeitschrift für Physik10,377-386,1922), original German scan",
          "url": "https://web.phys.ntnu.no/~mika/friedmann1.pdf"
        },
        {
          "citation": "Ellis and Krasiński, Editor’s Note on Friedmann 1922/1924 (General Relativity and Gravitation31,1985-1990,1999), selected scholarly scope commentary",
          "url": "https://users.camk.edu.pl/akr/friededibio.grg.pdf"
        },
        {
          "citation": "Instituut-Lorentz, Friedmann papers: institutional manuscript catalogue and work identification",
          "url": "https://lorentz.leidenuniv.nl/history/Friedmann_archive/"
        }
      ],
      "statement": "# Friedmann: a changing radius in positively curved cosmology\n\nFriedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants.\n\nA universe governed by relativity need not remain static. Friedmann lets the common spatial curvature radius change with time and derives an equation for its evolution under a simplified matter model. Depending on the cosmological constant and initial conditions, the solutions can expand monotonically or expand and contract. This supplies a theoretical possibility to be compared with observation, while leaving the actual universe and several later geometric developments unresolved.\n\n## Scientific scope\n\nThe target is the 1922 positively curved, pressureless matter construction with Einstein equations and a cosmological constant, plus explicit coordinate and metric assumptions. It is not the complete modern FLRW geometry, arbitrary-pressure cosmology, spatially flat models, or the 1924 negative-curvature extension. A later scholarly note flags incomplete geometric precision; the full tensor reduction was not independently certified. Positive spatial curvature does not alone force recollapse: sufficiently large positive cosmological constant permits a monotone branch. Periodic-radius descriptions do not independently prove regular spacetime continuation through zero-radius singularities. The paper does not derive cosmic redshift, fit galaxy recession data, prove our actual universe expands, or determine its age; its numerical period is explicitly illustrative. Lemaître’s later observation connection and Robertson/Walker developments remain separate. Extracted formula glyphs were not visually certified; the normalized first-integral check confirms a narrow root interpretation, not the full proof or singular boundary behavior.\n\n## People and systems\n\nAlexander Friedmann\n\n\n\n## Original sources\n\n- [A. Friedman, Über die Krümmung des Raumes (Zeitschrift für Physik10,377-386,1922), original German scan](https://web.phys.ntnu.no/~mika/friedmann1.pdf)\n- [Ellis and Krasiński, Editor’s Note on Friedmann 1922/1924 (General Relativity and Gravitation31,1985-1990,1999), selected scholarly scope commentary](https://users.camk.edu.pl/akr/friededibio.grg.pdf)\n- [Instituut-Lorentz, Friedmann papers: institutional manuscript catalogue and work identification](https://lorentz.leidenuniv.nl/history/Friedmann_archive/)\n",
      "summary": "Friedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants.\n\nA universe governed by relativity need not remain static. Friedmann lets the common spatial curvature radius change with time and derives an equation for its evolution under a simplified matter model. Depending on the cosmological constant and initial conditions, the solutions can expand monotonically or expand and contract. This supplies a theoretical possibility to be compared with observation, while leaving the actual universe and several later geometric developments unresolved.",
      "title": "Friedmann: a changing radius in positively curved cosmology"
    },
    "record_id": "urn:uuid:b1148720-bb15-4799-a211-6d59d9c82e4e",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:b1148720-bb15-4799-a211-6d59d9c82e4e",
      "version": 1,
      "content_hash": "sha256:16b4458e02a4a5783ac94f004ee13a93a9a4088a3597ae6714e2fa8fe32a0534"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:5cd4f8c8e0a904260bc91470f92f363660eafc82dc05a7ba643fe18e030deb0a"
  },
  "target": {
    "record_id": "urn:uuid:b1148720-bb15-4799-a211-6d59d9c82e4e",
    "version": 1,
    "content_hash": "sha256:16b4458e02a4a5783ac94f004ee13a93a9a4088a3597ae6714e2fa8fe32a0534"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "friedmann-positive-curvature-dynamics-1922",
    "entry_id": "friedmann-positive-curvature-dynamics-1922",
    "revision": 1,
    "title": "Friedmann: a changing radius in positively curved cosmology",
    "context": "Time-dependent cosmological solutions",
    "track": "Time-dependent cosmological solutions",
    "band": "S",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-04",
    "evidence_cutoff": "2026-10-04",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Friedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants.",
    "scope": "Friedmann shows that the Einstein equations under a positively curved pressureless cosmological ansatz allow a time-dependent curvature radius, and classifies monotone and expanding-contracting possibilities depending on constants. The target is the 1922 positively curved, pressureless matter construction with Einstein equations and a cosmological constant, plus explicit coordinate and metric assumptions. It is not the complete modern FLRW geometry, arbitrary-pressure cosmology, spatially flat models, or the 1924 negative-curvature extension. A later scholarly note flags incomplete geometric precision; the full tensor reduction was not independently certified. Positive spatial curvature does not alone force recollapse: sufficiently large positive cosmological constant permits a monotone branch. Periodic-radius descriptions do not independently prove regular spacetime continuation through zero-radius singularities. The paper does not derive cosmic redshift, fit galaxy recession data, prove our actual universe expands, or determine its age; its numerical period is explicitly illustrative. Lemaître’s later observation connection and Robertson/Walker developments remain separate. Extracted formula glyphs were not visually certified; the normalized first-integral check confirms a narrow root interpretation, not the full proof or singular boundary behavior.",
    "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.",
    "references": [
      {
        "title": "A. Friedman, Über die Krümmung des Raumes (Zeitschrift für Physik 10,377-386,1922), original German scan",
        "url": "https://web.phys.ntnu.no/~mika/friedmann1.pdf",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      },
      {
        "title": "Ellis and Krasiński, Editor’s Note on Friedmann 1922/1924 (General Relativity and Gravitation 31,1985-1990,1999), selected scholarly scope commentary",
        "url": "https://users.camk.edu.pl/akr/friededibio.grg.pdf",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      },
      {
        "title": "Instituut-Lorentz, Friedmann papers: institutional manuscript catalogue and work identification",
        "url": "https://lorentz.leidenuniv.nl/history/Friedmann_archive/",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      }
    ],
    "roles": [
      {
        "who": "Alexander Friedmann",
        "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.",
        "credit": "SCIENCE"
      }
    ],
    "target": {
      "record_id": "urn:uuid:b1148720-bb15-4799-a211-6d59d9c82e4e",
      "version": 1,
      "content_hash": "sha256:16b4458e02a4a5783ac94f004ee13a93a9a4088a3597ae6714e2fa8fe32a0534"
    },
    "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"
}