{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "hurwitz-sharp-universal-rational-approximation-1891",
    "title": "Hurwitz proves the sharp universal square-root-five rational-approximation bound",
    "summary": "Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant.",
    "boundary": "The universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.",
    "area": "Diophantine approximation and continued fractions",
    "topic": "Mathematics",
    "year": 1891,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "hurwitz-sharp-universal-rational-approximation-1891",
    "target": {
      "record_id": "urn:uuid:201d8341-ed47-4512-a433-c797fad41ff3",
      "version": 1,
      "content_hash": "sha256:9349dfca73f26ba46df1b63e47da03c6b676f81f686ff831b30ac35dc3a619aa"
    },
    "humans": [
      "A. Hurwitz"
    ],
    "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-hurwitz-rational-approximation-1891",
        "title": "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (Mathematische Annalen39,279–284)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0039/LOG_0023.pdf",
        "date": "1891"
      }
    ],
    "points": "220000",
    "published_grade": {
      "id": "hurwitz-sharp-universal-rational-approximation-1891",
      "entry_id": "hurwitz-sharp-universal-rational-approximation-1891",
      "band": "A",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:201d8341-ed47-4512-a433-c797fad41ff3",
        "version": 1,
        "content_hash": "sha256:9349dfca73f26ba46df1b63e47da03c6b676f81f686ff831b30ac35dc3a619aa"
      },
      "what_changed": "Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant.",
      "scope": "Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant. The universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.",
      "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:201d8341-ed47-4512-a433-c797fad41ff3",
        "version": 1,
        "content_hash": "sha256:9349dfca73f26ba46df1b63e47da03c6b676f81f686ff831b30ac35dc3a619aa"
      },
      "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": "A. Hurwitz",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (Mathematische Annalen39,279–284)",
            "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0039/LOG_0023.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (Mathematische Annalen39,279–284)",
            "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0039/LOG_0023.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.",
        "domain": "Diophantine approximation and continued fractions"
      },
      "sources": [
        {
          "citation": "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (Mathematische Annalen39,279–284)",
          "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0039/LOG_0023.pdf"
        }
      ],
      "statement": "# Hurwitz proves the sharp universal square-root-five rational-approximation bound\n\nHurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant.\n\n## Scientific scope\n\nThe universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.\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\nA. Hurwitz\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- [Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (Mathematische Annalen39,279–284)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0039/LOG_0023.pdf)\n",
      "summary": "Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant.",
      "title": "Hurwitz proves the sharp universal square-root-five rational-approximation bound"
    },
    "record_id": "urn:uuid:201d8341-ed47-4512-a433-c797fad41ff3",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:201d8341-ed47-4512-a433-c797fad41ff3",
      "version": 1,
      "content_hash": "sha256:9349dfca73f26ba46df1b63e47da03c6b676f81f686ff831b30ac35dc3a619aa"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:7d95e9ce0357060c1698af0dbddf630855ba51b3b9afa30cb8e6afdb08690b2d"
  },
  "target": {
    "record_id": "urn:uuid:201d8341-ed47-4512-a433-c797fad41ff3",
    "version": 1,
    "content_hash": "sha256:9349dfca73f26ba46df1b63e47da03c6b676f81f686ff831b30ac35dc3a619aa"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "hurwitz-sharp-universal-rational-approximation-1891",
    "entry_id": "hurwitz-sharp-universal-rational-approximation-1891",
    "revision": 1,
    "title": "Hurwitz proves the sharp universal square-root-five rational-approximation bound",
    "context": "Diophantine approximation and continued fractions",
    "track": "Exact theorem and reusable mathematical method",
    "band": "A",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant.",
    "scope": "Hurwitz proves that every real irrational alpha has infinitely many rational approximations p/q with |alpha−p/q|<1/(sqrt(5)q²). The constant sqrt(5) in the denominator cannot be uniformly increased: irrationals whose continued fractions are eventually all 1, including the golden ratio and its integral fractional-linear equivalents, obstruct every larger constant. The universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.",
    "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 universal quantifier is every real irrational, not every real including rationals and not a merely almost-everywhere statement. Infinitely many rational approximants have unbounded positive denominators; no finite precision search certifies the theorem. Sharpness is uniform over alpha, not a claim that every individual irrational has the same best approximation constant. Later refinements/Markoff classifications mentioned at the end are credited to Markoff rather than exported as wholly new Hurwitz theorems.",
    "references": [
      {
        "title": "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (Mathematische Annalen39,279–284)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0039/LOG_0023.pdf",
        "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:201d8341-ed47-4512-a433-c797fad41ff3",
      "version": 1,
      "content_hash": "sha256:9349dfca73f26ba46df1b63e47da03c6b676f81f686ff831b30ac35dc3a619aa"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "220000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}