{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "zhang-bounded-prime-gaps-2014",
    "title": "Zhang: bounded gaps between primes via smooth-modulus distribution estimates",
    "summary": "The paper proves unconditionally that the liminf of consecutive-prime gaps is less than seventy million. A refinement of the GPY sieve uses stronger averaged distribution estimates for primes in arithmetic progressions with moduli having only small prime factors.",
    "boundary": "This is an infinitely-often bound, not a bound on every prime gap and not the twin-prime conjecture. The restricted-modulus distribution result is not the unrestricted Elliott-Halberstam conjecture. The explicit bound is this paper's bound, not a claim about the current best constant.",
    "area": "Analytic number theory",
    "topic": "Mathematics",
    "year": 2014,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "zhang-bounded-prime-gaps-2014",
    "target": {
      "record_id": "urn:uuid:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
      "version": 1,
      "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
    },
    "humans": [
      "Yitang Zhang"
    ],
    "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-zhang-bounded-prime-gaps-annals2014",
        "title": "Bounded gaps between primes",
        "url": "https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf",
        "date": "2014-05-01"
      },
      {
        "id": "cur20261020-zhang-annals-publication-milestones",
        "title": "Bounded gaps between primes: publisher publication milestones",
        "url": "https://annals.math.princeton.edu/2014/179-3/p07",
        "date": "2014-05-01"
      }
    ],
    "points": "400000",
    "published_grade": {
      "id": "zhang-bounded-prime-gaps-2014",
      "entry_id": "zhang-bounded-prime-gaps-2014",
      "band": "S",
      "modifier": "-",
      "target": {
        "record_id": "urn:uuid:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
        "version": 1,
        "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
      },
      "what_changed": "The paper proves unconditionally that the liminf of consecutive-prime gaps is less than seventy million. A refinement of the GPY sieve uses stronger averaged distribution estimates for primes in arithmetic progressions with moduli having only small prime factors.",
      "scope": "The paper proves unconditionally that the liminf of consecutive-prime gaps is less than seventy million. A refinement of the GPY sieve uses stronger averaged distribution estimates for primes in arithmetic progressions with moduli having only small prime factors. This is an infinitely-often bound, not a bound on every prime gap and not the twin-prime conjecture. The restricted-modulus distribution result is not the unrestricted Elliott-Halberstam conjecture. The explicit bound is this paper's bound, not a claim about the current best constant.",
      "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:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
        "version": 1,
        "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
      },
      "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": "Yitang Zhang",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Bounded gaps between primes",
            "url": "https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Bounded gaps between primes",
            "url": "https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "Bounded gaps between primes: publisher publication milestones",
            "url": "https://annals.math.princeton.edu/2014/179-3/p07"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "This is an infinitely-often bound, not a bound on every prime gap and not the twin-prime conjecture. The restricted-modulus distribution result is not the unrestricted Elliott-Halberstam conjecture. The explicit bound is this paper's bound, not a claim about the current best constant.",
        "domain": "Analytic number theory"
      },
      "sources": [
        {
          "citation": "Bounded gaps between primes",
          "url": "https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf"
        },
        {
          "citation": "Bounded gaps between primes: publisher publication milestones",
          "url": "https://annals.math.princeton.edu/2014/179-3/p07"
        }
      ],
      "statement": "# Zhang: bounded gaps between primes via smooth-modulus distribution estimates\n\n## A finite gap, infinitely often\n\nPrime numbers become less common on average as numbers grow, yet small gaps keep appearing. The twin-prime question asks whether consecutive primes differ by two infinitely often. Zhang proved a different decisive statement: some fixed finite bound works infinitely often. His paper's bound is below seventy million. The qualitative change is the existence of an unconditional finite bound, rather than the numerical size of that first constant.\n\n## Finite examples are not the theorem\n\nThe consecutive primes 7, 11 and 13 have gaps four and two. These ordinary classroom calculations show what a gap means. Even a very long list of small gaps would remain a finite observation. Zhang's statement controls infinitely many positions in the prime sequence; it does not say every gap is below the bound.\n\n## Why distribution in progressions matters\n\nAn arithmetic progression has the form a, a + q, a + 2q and so on. Researchers study how primes are distributed among these sequences. Zhang obtains stronger averaged information for a restricted collection of q values whose prime factors are small. Combined with an inherited sieve framework, this information is enough for the bounded-gap conclusion. A sieve is a way of organizing divisibility restrictions to locate or count number patterns.\n\n## Separate the advance from later improvements\n\nThe paper builds on the work of Goldston, Pintz and Yildirim. Its restricted distribution theorem is not the unrestricted Elliott-Halberstam conjecture, and the bounded-gap result is not a proof of twin primes. Later smaller numerical bounds are separate developments. The publisher records receipt and acceptance in 2013 and online publication in May 2014; these dates describe different milestones.\n\n## Make the quantifiers visible\n\nWrite two sentences: infinitely many gaps are small, and all sufficiently late gaps are small. Explain why the second is a much stronger statement. Then trace the paper's actual formula and original bound to the publisher page. This small reading task can prevent a misleading headline without pretending to reproduce the sieve proof.\n\n## Scientific scope\n\nThis is an infinitely-often bound, not a bound on every prime gap and not the twin-prime conjecture. The restricted-modulus distribution result is not the unrestricted Elliott-Halberstam conjecture. The explicit bound is this paper's bound, not a claim about the current best constant.\n\n## Sources for this explanation\n\n- [Bounded gaps between primes](https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n- [Bounded gaps between primes: publisher publication milestones](https://annals.math.princeton.edu/2014/179-3/p07): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "The paper proves unconditionally that the liminf of consecutive-prime gaps is less than seventy million. A refinement of the GPY sieve uses stronger averaged distribution estimates for primes in arithmetic progressions with moduli having only small prime factors.",
      "title": "Zhang: bounded gaps between primes via smooth-modulus distribution estimates"
    },
    "record_id": "urn:uuid:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
      "version": 1,
      "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:dcc624160faadd41650bf80e849aefbd8a8f7884c1842e5d474bd02d68953bc6"
  },
  "target": {
    "record_id": "urn:uuid:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
    "version": 1,
    "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "zhang-bounded-prime-gaps-2014",
    "entry_id": "zhang-bounded-prime-gaps-2014",
    "revision": 1,
    "title": "Zhang: bounded gaps between primes via smooth-modulus distribution estimates",
    "context": "Analytic number theory",
    "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": "The paper proves unconditionally that the liminf of consecutive-prime gaps is less than seventy million. A refinement of the GPY sieve uses stronger averaged distribution estimates for primes in arithmetic progressions with moduli having only small prime factors.",
    "scope": "The paper proves unconditionally that the liminf of consecutive-prime gaps is less than seventy million. A refinement of the GPY sieve uses stronger averaged distribution estimates for primes in arithmetic progressions with moduli having only small prime factors. This is an infinitely-often bound, not a bound on every prime gap and not the twin-prime conjecture. The restricted-modulus distribution result is not the unrestricted Elliott-Halberstam conjecture. The explicit bound is this paper's bound, not a claim about the current best constant.",
    "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. This is an infinitely-often bound, not a bound on every prime gap and not the twin-prime conjecture. The restricted-modulus distribution result is not the unrestricted Elliott-Halberstam conjecture. The explicit bound is this paper's bound, not a claim about the current best constant.",
    "references": [
      {
        "title": "Bounded gaps between primes",
        "url": "https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Bounded gaps between primes: publisher publication milestones",
        "url": "https://annals.math.princeton.edu/2014/179-3/p07",
        "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:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
      "version": 1,
      "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "400000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "zhang-bounded-prime-gaps-2014",
    "target": {
      "record_id": "urn:uuid:44e4e657-7f28-4e8a-a3a4-603f8a8e03e3",
      "version": 1,
      "content_hash": "sha256:0572e77c14f1738477f5b4394375a9d901aaa4a87ac87aff606c607a94ffbcc5"
    },
    "sections": [
      {
        "title": "A finite gap, infinitely often",
        "text": "Prime numbers become less common on average as numbers grow, yet small gaps keep appearing. The twin-prime question asks whether consecutive primes differ by two infinitely often. Zhang proved a different decisive statement: some fixed finite bound works infinitely often. His paper's bound is below seventy million. The qualitative change is the existence of an unconditional finite bound, rather than the numerical size of that first constant."
      },
      {
        "title": "Finite examples are not the theorem",
        "text": "The consecutive primes 7, 11 and 13 have gaps four and two. These ordinary classroom calculations show what a gap means. Even a very long list of small gaps would remain a finite observation. Zhang's statement controls infinitely many positions in the prime sequence; it does not say every gap is below the bound."
      },
      {
        "title": "Why distribution in progressions matters",
        "text": "An arithmetic progression has the form a, a + q, a + 2q and so on. Researchers study how primes are distributed among these sequences. Zhang obtains stronger averaged information for a restricted collection of q values whose prime factors are small. Combined with an inherited sieve framework, this information is enough for the bounded-gap conclusion. A sieve is a way of organizing divisibility restrictions to locate or count number patterns."
      },
      {
        "title": "Separate the advance from later improvements",
        "text": "The paper builds on the work of Goldston, Pintz and Yildirim. Its restricted distribution theorem is not the unrestricted Elliott-Halberstam conjecture, and the bounded-gap result is not a proof of twin primes. Later smaller numerical bounds are separate developments. The publisher records receipt and acceptance in 2013 and online publication in May 2014; these dates describe different milestones."
      },
      {
        "title": "Make the quantifiers visible",
        "text": "Write two sentences: infinitely many gaps are small, and all sufficiently late gaps are small. Explain why the second is a much stronger statement. Then trace the paper's actual formula and original bound to the publisher page. This small reading task can prevent a misleading headline without pretending to reproduce the sieve proof."
      }
    ],
    "references": [
      {
        "title": "Bounded gaps between primes",
        "url": "https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      },
      {
        "title": "Bounded gaps between primes: publisher publication milestones",
        "url": "https://annals.math.princeton.edu/2014/179-3/p07",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}