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AxiomProver is credited for its formal certificate, not independently discovering that bound.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For every real bound X there are consecutive primes p < q with p ≥ X and q − p = 2; equivalently H₁ = 2.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every real bound X there are consecutive primes p < q with p ≥ X and q − p = 2; equivalently "},{"kind":"math","text":"H₁ = 2","latex":"H_1=2"},{"kind":"text","text":"."}]},"id":"claim-prime-gaps-twin-primes","slug":"prime-gaps-twin-primes","version":2,"kind":"conjecture","title":"Gaps of two infinitely often","assumptions":["Consecutive positive primes in increasing order."],"scope":"The twin-prime target. Finite upper bounds larger than two leave it unresolved.","source_entry_ids":["prime-gaps:entry:polymath:v1","prime-gaps:entry:212:v2"],"formal_declarations":[]},{"statement":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 70000000; in other words H₁ ≤ 70000000.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 70000000; in other words "},{"kind":"math","text":"H₁ ≤ 70000000","latex":"H_1\\le70{,}000{,}000"},{"kind":"text","text":"."}]},"id":"claim-prime-gaps-bound-70000000","slug":"prime-gaps-bound-70000000","version":3,"kind":"bound","title":"Gaps at most 70,000,000, infinitely often","assumptions":["Consecutive positive primes.","No additional conjectural hypothesis in the cited mathematical result."],"scope":"An unconditional upper bound on the recurring gap, attributed to Zhang; it does not imply every gap is this small.","source_entry_ids":["prime-gaps:entry:zhang:v1","prime-gaps:entry:212:v2"],"formal_declarations":[]},{"statement":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 600; in other words H₁ ≤ 600.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 600; in other words "},{"kind":"math","text":"H₁ ≤ 600","latex":"H_1\\le600"},{"kind":"text","text":"."}]},"id":"claim-prime-gaps-bound-600","slug":"prime-gaps-bound-600","version":3,"kind":"bound","title":"Gaps at most 600, infinitely often","assumptions":["Consecutive positive primes.","No additional conjectural hypothesis in the cited mathematical result."],"scope":"An unconditional upper bound on the recurring gap, attributed to Maynard; it does not imply every gap is this small.","source_entry_ids":["prime-gaps:entry:maynard:v1","prime-gaps:entry:212:v2"],"formal_declarations":[]},{"statement":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 246; in other words H₁ ≤ 246.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 246; 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it does not imply every gap is this small.","source_entry_ids":["prime-gaps:entry:212:v2"],"formal_declarations":[]},{"statement":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 212; in other words H₁ ≤ 212.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 212; in other words "},{"kind":"math","text":"H₁ ≤ 212","latex":"H_1\\le212"},{"kind":"text","text":"."}]},"id":"claim-prime-gaps-bound-212","slug":"prime-gaps-bound-212","version":2,"kind":"bound","title":"Gaps at most 212, infinitely often","assumptions":["Consecutive positive primes.","No additional conjectural hypothesis in the cited mathematical result."],"scope":"An unconditional upper bound on the recurring gap, attributed to the 2026 preliminary draft; it does not imply every gap is this small.","source_entry_ids":["prime-gaps:entry:212:v2"],"formal_declarations":[]},{"statement":"Assuming the rank-three Kloosterman estimate, the rank-two correlation estimate, and all integral and cap bounds in the precise forms specified by the cited proof, H₁ ≤ 186.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Assuming the rank-three Kloosterman estimate, the rank-two correlation estimate, and all integral and cap bounds in the precise forms specified by the cited proof, "},{"kind":"math","text":"H₁ ≤ 186","latex":"H_1\\le186"},{"kind":"text","text":"."}]},"id":"claim-prime-gaps-186-conditional","slug":"prime-gaps-186-conditional","version":3,"kind":"theorem","title":"A gap bound of 186 from explicit input assumptions","assumptions":["The rank-three Kloosterman bound in the source.","The rank-two correlation bound in the source.","All source physical-integral and cap bounds."],"scope":"A proved conditional deduction. The two literature estimates and numerical input remain project axioms; OpenMath has not verified or discharged them.","source_entry_ids":["prime-gaps:entry:186-lean:v1"],"formal_declarations":["PrimeGap186.primeGapLiminf_le_186"]},{"statement":"Assuming the precise BombieriVinogradov statement in PrimeGapsLib, infinitely many consecutive-prime gaps are at most 246.","id":"claim-prime-gaps-bv-implies-246","slug":"prime-gaps-bv-implies-246","version":1,"kind":"theorem","title":"Bombieri–Vinogradov implies recurring gaps at most 246","assumptions":["The source’s formal BombieriVinogradov premise."],"scope":"The certificate and downstream sieve deduction are formalized; the analytic distribution theorem is a premise in this endpoint.","source_entry_ids":["prime-gaps:entry:246-lean:v1"],"formal_declarations":["bombieriVinogradov_implies_frequently_prime_gap_le_246"]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions.","report_ids":["prime-gaps:report:target-open:v1"],"progress_claim_ids":["claim-prime-gaps-bound-70000000","claim-prime-gaps-bound-600","claim-prime-gaps-bound-246","claim-prime-gaps-bound-240","claim-prime-gaps-186-conditional","claim-prime-gaps-bv-implies-246"],"remaining":"Establish recurring gaps of two, or independently validate and strengthen the selected bounded-gap results and their formal evidence."},"statement_kind":"conjecture","problem_id":"problem-prime-gaps","slug":"prime-gaps","title":"How small can prime gaps be?","question":"How small can gaps between consecutive primes be, infinitely often? In particular, is liminfₙ(pₙ₊₁ − pₙ) = 2?","question_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"How small can gaps between consecutive primes be, infinitely often? In particular, is "},{"kind":"math","text":"liminfₙ(pₙ₊₁ − pₙ) = 2","latex":"\\liminf_{n\\to\\infty}(p_{n+1}-p_n)=2","display":true},{"kind":"text","text":"?"}]},"scope":"The primes pₙ are listed in increasing order. H₁ = liminfₙ(pₙ₊₁ − pₙ) measures the smallest gap recurring arbitrarily far out. A bound H₁ ≤ B does not bound every gap or prove twin primes unless B = 2.","scope_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"The primes pₙ are listed in increasing order. "},{"kind":"math","text":"H₁ = liminfₙ(pₙ₊₁ − pₙ)","latex":"H_1=\\liminf_{n\\to\\infty}(p_{n+1}-p_n)","display":true},{"kind":"text","text":" measures the smallest gap recurring arbitrarily far out. A bound H₁ ≤ B does not bound every gap or prove twin primes unless B = 2."}]},"aliases":["Bounded prime gaps","Twin prime conjecture","Bounded gaps between primes"],"fields":["Number theory"],"msc":["11N05","11N36"],"topics":[],"background":"graduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["prime-gaps:entry:polymath:v1","prime-gaps:entry:212:v2","prime-gaps:entry:246-lean:v1","prime-gaps:entry:186-lean:v1","prime-gaps:entry:zhang:v1","prime-gaps:entry:maynard:v1"],"status_report_ids":["prime-gaps:report:target-open:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"partly_resolved","scope":"Source comparison and page curation only; independent mathematical review is pending.","rationale":"Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["prime-gaps:report:target-open:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"Variants of the Selberg sieve, and bounded intervals containing many primes","url":"https://arxiv.org/abs/1407.4897v2","note":"Main bounded-gap results and introduction"},{"title":"A new bound for small gaps between primes","url":"https://primegaps.axiommath.ai/bgp212.pdf","note":"Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver"},{"title":"PrimeGapsLib: a formal deduction of the 246 bound","url":"https://github.com/AxiomMath/PrimeGapsLib/tree/1faa7b14e82ddebc2772dfb9153922f01b106477","note":"README main results; PrimeGaps/Bounded246.lean"},{"title":"Prime gaps at most 186: a conditional Lean development","url":"https://github.com/openai/PrimeGaps186/tree/61340d0b74163003b32756bb16e91d9209a5e330","note":"README input axioms and verification; PrimeGaps186.lean"}],"formalizations":[{"title":"Lean: 246 from Bombieri–Vinogradov","url":"https://github.com/AxiomMath/PrimeGapsLib/blob/1faa7b14e82ddebc2772dfb9153922f01b106477/PrimeGaps/Bounded246.lean","note":"Complete as an implication with the stated premise. This is the public 246 artifact, not the later reported 212 certificate.","target":{"id":"claim-prime-gaps-bv-implies-246","version":1},"proof_status":"reported_complete","source_entry_ids":["prime-gaps:entry:246-lean:v1"],"declaration":"bombieriVinogradov_implies_frequently_prime_gap_le_246","source_commit":"1faa7b14e82ddebc2772dfb9153922f01b106477","alignment":"staff_linked","checked_on":"2026-10-13"},{"title":"Lean: 186 with three input axioms","url":"https://github.com/openai/PrimeGaps186/blob/61340d0b74163003b32756bb16e91d9209a5e330/PrimeGaps186.lean","note":"Complete as a conditional deduction. The formal input names are PrimeGap186.kloosterman3_bound, PrimeGap186.kloosterman2_correlation_bound and PrimeGap186.physical_integral_bounds; they remain assumptions, not discharged proof obligations.","target":{"id":"claim-prime-gaps-186-conditional","version":3},"proof_status":"reported_complete","source_entry_ids":["prime-gaps:entry:186-lean:v1"],"declaration":"PrimeGap186.primeGapLiminf_le_186","source_commit":"61340d0b74163003b32756bb16e91d9209a5e330","alignment":"staff_linked","checked_on":"2026-10-13"}],"related":[{"slug":"prime-infinitude","title":"Infinitely many primes","relation":"prerequisite","note":"An infinite supply of primes does not establish infinitely many bounded gaps."}],"tasks":[],"insights":[{"id":"prime-gaps-approach","title":"Sieve weights, distribution estimates and exact certificates","summary":"The 212 argument improves a sieve construction descended from Stadlmann’s work and verifies rational inequalities. Different formal artifacts certify different deductions and inputs.","review":"pending","kind":"approach","source_entry_ids":["prime-gaps:entry:polymath:v1","prime-gaps:entry:212:v2","prime-gaps:entry:246-lean:v1","prime-gaps:entry:186-lean:v1"],"claims":[{"id":"claim-prime-gaps-bound-212","version":2},{"id":"claim-prime-gaps-bv-implies-246","version":1},{"id":"claim-prime-gaps-186-conditional","version":3}],"limitation":"The public 246 repository cannot be substituted for the 212 paper’s reported certificate."},{"id":"prime-gaps-impact","title":"Smaller recurring gaps, with the twin-prime target still open","summary":"If the 212 draft’s assertion holds, it improves the best proved unconditional bound, 240, in the selected literature. 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It does not contain the 212 formalization reported in the later paper.","revision":"1faa7b14e82ddebc2772dfb9153922f01b106477","content_sha256":"3c8edc561d416edcef84fe56f53fb2570abd9a837bb2de67766a3b2b279675ab","update_method":"Source comparison followed by a new immutable curation revision.","identifier_policy":"Stable source and entry identities; changed wording receives a new version.","observed_on":"2026-10-13","source_updated_on":"2026-08-18","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://github.com/AxiomMath/PrimeGapsLib/tree/1faa7b14e82ddebc2772dfb9153922f01b106477","attribution":"Axiom Math, PrimeGapsLib contributors","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90},{"work":{"authors":["OpenAI","PrimeGaps186 contributors"],"kind":"research_project","publication":"project","publication_note":"Public formal source; build and independent mathematical review are separate checks.","ai_role":"self_reported","ai_note":"The project is an OpenAI formal development. 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Its appendix reports a formal deduction with analytic and variational hypotheses.","wording_kind":"paraphrase","locator":"Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver","supersedes":"prime-gaps:entry:212:v1"},{"id":"prime-gaps:entry:246-lean:v1","source_version_id":"prime-gaps:source:246-lean:v1","item_id":"246-lean","url":"https://github.com/AxiomMath/PrimeGapsLib/tree/1faa7b14e82ddebc2772dfb9153922f01b106477","wording":"The public revision proves prime-gap bounds of 246 assuming Bombieri–Vinogradov and includes numerical certificates. It does not contain the 212 formalization reported in the later paper.","wording_kind":"paraphrase","locator":"README main results; PrimeGaps/Bounded246.lean","supersedes":null},{"id":"prime-gaps:entry:186-lean:v1","source_version_id":"prime-gaps:source:186-lean:v1","item_id":"186-lean","url":"https://github.com/openai/PrimeGaps186/tree/61340d0b74163003b32756bb16e91d9209a5e330","wording":"The endpoint derives a gap bound of 186 from two unproved-in-Lean exponential-sum estimates and an unproved-in-Lean physical integral input. A separate numerical certificate does not discharge those axioms.","wording_kind":"paraphrase","locator":"README input axioms and verification; PrimeGaps186.lean","supersedes":null},{"id":"prime-gaps:entry:zhang:v1","source_version_id":"prime-gaps:source:zhang:v1","item_id":"Abstract","url":"https://annals.math.princeton.edu/2014/179-3/p07","wording":"Zhang proves that lim inf (pₙ₊₁ − pₙ) < 7 × 10⁷, refining Goldston–Pintz–Yıldırım with a stronger Bombieri–Vinogradov estimate for moduli without large prime factors.","wording_kind":"paraphrase","locator":"Abstract","supersedes":null},{"id":"prime-gaps:entry:maynard:v1","source_version_id":"prime-gaps:source:maynard:v1","item_id":"Abstract","url":"https://arxiv.org/abs/1311.4600v3","wording":"Maynard shows that lim inf (pₙ₊₁ − pₙ) ≤ 600, and ≤ 12 assuming the Elliott–Halberstam conjecture.","wording_kind":"paraphrase","locator":"Abstract","supersedes":null}],"reports":[{"id":"prime-gaps:report:target-open:v1","entry_id":"prime-gaps:entry:polymath:v1","state":"partly_resolved","source_label":"Bounded gaps established; twin primes remain unresolved","scope":"H₁ = 2 remains unresolved despite finite upper bounds.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1407.4897v2","supersedes":null},{"id":"prime-gaps:report:246:v1","entry_id":"prime-gaps:entry:polymath:v1","state":"resolved","source_label":"Unconditional H₁ ≤ 246","scope":"Published Polymath8b bound.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1407.4897v2","supersedes":null},{"id":"prime-gaps:report:212:v2","entry_id":"prime-gaps:entry:212:v2","state":"resolved","source_label":"The preliminary draft proves H₁ ≤ 212","scope":"The paper’s unconditional assertion; formalization assumes analytic and variational inputs.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://primegaps.axiommath.ai/bgp212.pdf","supersedes":"prime-gaps:report:212:v1"},{"id":"prime-gaps:report:246-conditional:v1","entry_id":"prime-gaps:entry:246-lean:v1","state":"resolved","source_label":"Formal deduction from Bombieri–Vinogradov","scope":"The implication to the 246 bound, not a Lean proof of its analytic premise.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://github.com/AxiomMath/PrimeGapsLib/tree/1faa7b14e82ddebc2772dfb9153922f01b106477","supersedes":null},{"id":"prime-gaps:report:186-conditional:v1","entry_id":"prime-gaps:entry:186-lean:v1","state":"resolved","source_label":"Conditional formal endpoint for 186","scope":"The implication from the three stated input axioms.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://github.com/openai/PrimeGaps186/tree/61340d0b74163003b32756bb16e91d9209a5e330","supersedes":null},{"id":"prime-gaps:report:earlier:v2","entry_id":"prime-gaps:entry:212:v2","state":"resolved","source_label":"Historical bounds recorded in the paper","scope":"Zhang, Maynard and Stadlmann bounded-gap results.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://primegaps.axiommath.ai/bgp212.pdf","supersedes":"prime-gaps:report:earlier:v1"},{"id":"prime-gaps:report:twin-target-open:v1","entry_id":"prime-gaps:entry:polymath:v1","state":"reported_open","source_label":"Twin primes remain unresolved","scope":"H₁ = 2 remains unresolved despite finite upper bounds.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1407.4897v2","supersedes":null},{"id":"prime-gaps:report:zhang:v1","entry_id":"prime-gaps:entry:zhang:v1","state":"resolved","source_label":"Zhang: H₁ < 7 × 10⁷","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://annals.math.princeton.edu/2014/179-3/p07","supersedes":null},{"id":"prime-gaps:report:maynard:v1","entry_id":"prime-gaps:entry:maynard:v1","state":"resolved","source_label":"Maynard: H₁ ≤ 600","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/1311.4600v3","supersedes":null}]},"status":{"state":"partial","label":"Partly resolved","extent":"general_case","summary":"Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-13","progress_label":"Conditional result proved","progress_claim_ids":["claim-prime-gaps-bound-70000000","claim-prime-gaps-bound-600","claim-prime-gaps-bound-246","claim-prime-gaps-bound-240","claim-prime-gaps-186-conditional","claim-prime-gaps-bv-implies-246"],"remaining":"Establish recurring gaps of two, or independently validate and strengthen the selected bounded-gap results and their formal evidence.","views":["open","partial"],"knowledge_categories":["Bounds established","Conditional results"]}},"selected_context":null}}