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Finite upper bounds larger than two leave it unresolved.","source_entry_ids":["prime-gaps:entry:polymath:v1","prime-gaps:entry:212:v1"],"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.","id":"claim-prime-gaps-bound-70000000","slug":"prime-gaps-bound-70000000","version":1,"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:212:v1"],"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.","id":"claim-prime-gaps-bound-600","slug":"prime-gaps-bound-600","version":1,"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:212:v1"],"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.","id":"claim-prime-gaps-bound-246","slug":"prime-gaps-bound-246","version":1,"kind":"bound","title":"Gaps at most 246, 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 Polymath8b; it does not imply every gap is this small.","source_entry_ids":["prime-gaps:entry:polymath:v1"],"formal_declarations":[]},{"statement":"For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 240; in other words H₁ ≤ 240.","id":"claim-prime-gaps-bound-240","slug":"prime-gaps-bound-240","version":1,"kind":"bound","title":"Gaps at most 240, 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 Stadlmann; it does not imply every gap is this small.","source_entry_ids":["prime-gaps:entry:212:v1"],"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.","id":"claim-prime-gaps-bound-212","slug":"prime-gaps-bound-212","version":1,"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:v1"],"formal_declarations":[]},{"statement":"If PrimeGap186.kloosterman3_bound, PrimeGap186.kloosterman2_correlation_bound and PrimeGap186.physical_integral_bounds hold in the precise forms specified by the pinned development, then H₁ ≤ 186.","id":"claim-prime-gaps-186-conditional","slug":"prime-gaps-186-conditional","version":1,"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. 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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:v1","prime-gaps:entry:246-lean:v1","prime-gaps:entry:186-lean: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":"Bounded gaps between primes: the 212 bound","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":"Reported complete as a conditional theorem. Its numerical receipt does not turn the input axioms into Lean proofs.","target":{"id":"claim-prime-gaps-186-conditional","version":1},"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."},{"slug":"riemann-hypothesis","title":"Riemann hypothesis","relation":"related_result","note":"Global zero-distribution information and recurring small gaps are different questions; no implication to twin primes is asserted."}],"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:v1","prime-gaps:entry:246-lean:v1","prime-gaps:entry:186-lean:v1"],"claims":[{"id":"claim-prime-gaps-bound-212","version":1},{"id":"claim-prime-gaps-bv-implies-246","version":1},{"id":"claim-prime-gaps-186-conditional","version":1}],"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":"The 212 paper’s assertion would improve the unconditional recurring-gap bound in the selected literature. It gives arbitrarily large prime pairs at that distance, without controlling all successive gaps.","review":"pending","kind":"impact","source_entry_ids":["prime-gaps:entry:212:v1"],"claims":[{"id":"claim-prime-gaps-bound-212","version":1}],"scenario":{"id":"claim-prime-gaps-bound-212","version":1},"limitation":"The manuscript is a preliminary draft; independent OpenMath review is pending. A bound larger than two does not prove twin primes."}],"coverage_gaps":["Independent mathematical review and local execution of the cited formal proofs are pending."],"checked_on":"2026-10-13"},"sources":[{"work":{"authors":["D. H. J. 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Its appendix reports a formal deduction with analytic and variational hypotheses.","revision":"2026-09-03","content_sha256":"2b307ae26046dffd2153fe7466e5a9a302976dc3ba5142151a8a8104a37b3d80","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":"2026-09-03","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://primegaps.axiommath.ai/bgp212.pdf","attribution":"François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin A. 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A separate numerical certificate does not discharge those axioms.","wording_kind":"paraphrase","locator":"README input axioms and verification; PrimeGaps186.lean","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:v1","entry_id":"prime-gaps:entry:212:v1","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":null},{"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:v1","entry_id":"prime-gaps:entry:212:v1","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":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. 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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":null}],"sources":[{"work":{"authors":["D. H. J. Polymath"],"kind":"research_paper","publication":"published","publication_note":"Peer-reviewed historical result.","ai_role":"not_reported","ai_note":"No AI contribution is asserted by this source record.","artifacts":[],"published":{"value":"2014"}},"id":"prime-gaps:source:polymath:v1","source_id":"prime-gaps-polymath","title":"Variants of the Selberg sieve, and bounded intervals containing many primes","owner":"D. H. J. Polymath","url":"https://arxiv.org/abs/1407.4897v2","subject_scope":"The published Polymath8b argument establishes an unconditional recurring prime gap at most 246; the twin-prime target remains out of reach.","revision":"2014","content_sha256":null,"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":null,"rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://arxiv.org/abs/1407.4897v2","attribution":"D. H. J. Polymath","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90},{"work":{"authors":["François Charton","Letong Hong","Kenny Lau","Ken Ono","Guillaume Remy","Ho Chung Siu","Ashvin A. Swaminathan","Jesse Thorner","Yunzhou Xie"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"self_reported","ai_note":"The nine authors supply the mathematics, building on Julia Stadlmann and earlier sieve work. AxiomProver is credited with generating a formal deduction; this is not a claim that the model independently discovered the mathematical bound.","artifacts":[],"published":{"value":"2026-09-03"}},"id":"prime-gaps:source:212:v1","source_id":"prime-gaps-212","title":"Bounded gaps between primes: the 212 bound","owner":"François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin A. Swaminathan, Jesse Thorner, Yunzhou Xie","url":"https://primegaps.axiommath.ai/bgp212.pdf","subject_scope":"The preliminary draft states H₁ ≤ 212 and explains the analytic and exact rational calculations. Its appendix reports a formal deduction with analytic and variational hypotheses.","revision":"2026-09-03","content_sha256":"2b307ae26046dffd2153fe7466e5a9a302976dc3ba5142151a8a8104a37b3d80","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":"2026-09-03","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://primegaps.axiommath.ai/bgp212.pdf","attribution":"François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin A. Swaminathan, Jesse Thorner, Yunzhou Xie","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90}]}}}