{"payload":{"schema_version":"research-record-v0","publication":{"version":11,"sha256":"84ac216b24d2c9a8decbf0078757c76e9e828749a3900de652f34866a6d93c7e","published_at":"2026-10-11T06:54:13.222136+00:00","reason":"Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.","withdrawn":false,"draft":{"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"visual-birch-and-swinnerton-dyer-conjecture-overview","version":5,"subject":{"kind":"problem","id":"problem-birch-and-swinnerton-dyer-conjecture","sha256":"e49e18e788b4c838cc31dd2558d9c5398c2a7aac1a81af83fd5a195882933c3d"},"purpose":"coverage","title":"Which parts of BSD are covered","caption":"Rank equality with finite Sha, selected prime parts and the full formula have different coverage. Select a result to see the exact hypotheses.","description":"For elliptic curves over Q, analytic rank 0 or 1 implies rank equality and finite Sha, but does not alone prove the full leading-term formula. The converse assumes algebraic rank 1, finite Sha and stated reduction hypotheses. Two other results cover selected p-parts under analytic-rank and prime hypotheses. The full formula is covered for specified quadratic twists and a historical certified finite corpus. These scopes overlap and do not partition all curves.","assumptions":["Elliptic curves over Q.","Analytic rank, algebraic rank, prime conditions and curve-family restrictions are distinguished in each result."],"limitation":"Rank equality and finite Sha at analytic rank 0 or 1 do not alone prove the full leading-term formula. General analytic rank ≥ 2 remains open. Finite Sha in the converse is assumed. Selected p-parts are not the full formula; scopes overlap and the historical finite enumeration excludes its unresolved curve/prime pairs.","author":"OpenMath curation","authoring_method":"ai_assisted_curation","authored_on":"2026-10-11","independent_review":"pending","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2","birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1","birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2","birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1"],"payload":{"kind":"cases_map","rows":[{"id":"low","label":"Analytic rank 0 or 1"},{"id":"converse","label":"Conditional rank-one converse"},{"id":"semistable","label":"Semistable curves · specified primes"},{"id":"families","label":"Specified families or finite sets"}],"columns":[{"id":"rank","label":"Rank equality / finite Sha"},{"id":"parts","label":"Selected p-parts"},{"id":"full","label":"Full leading-term formula"}],"cells":[{"id":"low-rank","row":"low","column":"rank","label":"Rank equality + finite Sha · analytic rank 0 or 1","claim":{"id":"claim-bsd-analytic-rank-at-most-one","version":4}},{"id":"converse","row":"converse","column":"rank","label":"Algebraic rank 1 + finite Sha assumed","claim":{"id":"claim-bsd-rank-one-converse-finite-sha","version":3}},{"id":"rank-one-primes","row":"semistable","column":"parts","label":"Analytic rank 1 · good p ≥ 5 · irreducible E[p]","claim":{"id":"claim-bsd-semistable-rank-one-p-part","version":2}},{"id":"supersingular","row":"semistable","column":"parts","label":"Analytic rank 0 or 1 · supersingular p ≥ 5","claim":{"id":"claim-bsd-supersingular-p-part","version":2}},{"id":"twist-family","row":"families","column":"full","label":"Selected twists of 46a1","claim":{"id":"claim-bsd-46a1-quadratic-twists","version":2}},{"id":"finite-enumeration","row":"families","column":"full","label":"16,714 certified curves · exceptions excluded","claim":{"id":"claim-bsd-miller-finite-computations","version":2}}],"empty_label":"No result indexed in this selection"}}],"visuals":[{"representation":{"id":"visual-birch-and-swinnerton-dyer-conjecture-overview","version":5},"section":"research-state"}],"claim_relations":[{"claim":{"id":"claim-bsd-analytic-rank-at-most-one","version":4},"relation":"special_case","relationship_note":"The hypothesis is analytic rank ≤ 1, not merely algebraic rank ≤ 1. This does not alone prove the full leading-coefficient formula.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2:report"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-rank-one-converse-finite-sha","version":3},"relation":"conditional_result","relationship_note":"Runs in the converse direction to the low-analytic-rank theorem. General Sha finiteness is conjectural and is not supplied by assuming algebraic rank 1.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2:report"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-semistable-rank-one-p-part","version":2},"relation":"special_case","relationship_note":"Adds prime-specific leading-coefficient information beyond low-rank equality. Both good ordinary and supersingular reduction are permitted here; the p=3 extension is not included in this selected Claim.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1:report"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-supersingular-p-part","version":2},"relation":"special_case","relationship_note":"Overlaps the supersingular rank-one portion of the Jetchev–Skinner–Wan Claim and additionally includes analytic rank zero. Prime parts are not independent percentages of BSD solved.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1:report"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-46a1-quadratic-twists","version":2},"relation":"special_case","relationship_note":"A full-formula family within the low-analytic-rank case. The 46a1 family is retained after the 2026 Banwait–Huang check; four other base curves in the earlier list have an infinitude caveat.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1:report"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-miller-finite-computations","version":2},"relation":"special_case","relationship_note":"Strengthens the formula for a finite subset of the classical low-rank regime. This is computer-assisted proof in the source, not AI theorem discovery or a Lean kernel check.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1:report"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-rank-target","version":3},"relation":"primary_formulation","relationship_note":"The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.","result":"unproved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-target:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-full-target","version":3},"relation":"related_variant","relationship_note":"A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.","result":"unproved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-sha-finiteness-target","version":2},"relation":"related_variant","relationship_note":"A separately open finiteness assertion, proved in selected low-analytic-rank cases. It is included in the strong BSD scenario.","result":"unproved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-sha-finiteness-target:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-mordell-weil","version":2},"relation":"supporting_result","relationship_note":"An unconditional foundation. Knowing finite generation does not compute the rank of an arbitrary curve.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-strong-implies-rank","version":2},"relation":"conditional_implication","relationship_note":"A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-strong-implies-rank:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-finite-points-consequence","version":2},"relation":"conditional_implication","relationship_note":"A proved implication under the rank-equality premise, not an unconditional arbitrary-curve decision algorithm.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-congruent-number-consequence","version":2},"relation":"conditional_implication","relationship_note":"A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-sha-order-consequence","version":2},"relation":"conditional_implication","relationship_note":"Source-grounded deduction from the normalized strong formula, not an independently proved general formula for Sha.","result":"proved","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-sha-order-consequence:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-openai-low-selmer-proposal","version":2},"relation":"special_case","relationship_note":"A provider proposal depending on companion low-corank converse and two-primary inputs. Algebraic rank ≤1 alone does not establish this Selmer-corank hypothesis.","result":"proposed","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-low-selmer-proposal:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-bsd-openai-goldfeld-proposal","version":2},"relation":"related_variant","relationship_note":"A proposed family-frequency theorem, not a theorem for every individual twist. The mean-rank assertion needs separate tail control and is not included in this Claim.","result":"proposed","report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld-proposal:report:v1"],"independent_review":"pending"}],"connections":[{"id":"bsd-connection-strong","source":{"id":"claim-bsd-full-target","version":3},"target":{"id":"claim-bsd-rank-target","version":3},"relation":"implies","assertion":{"id":"claim-bsd-strong-implies-rank","version":2},"explanation":"The strong assertion includes the rank statement and additional arithmetic information.","limitation":"Proving rank equality alone does not prove finite Sha or the leading coefficient.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"independent_review":"pending"},{"id":"bsd-connection-sha","source":{"id":"claim-bsd-full-target","version":3},"target":{"id":"claim-bsd-sha-finiteness-target","version":2},"relation":"implies","assertion":{"id":"claim-bsd-strong-implies-rank","version":2},"explanation":"The full BSD scenario explicitly includes finiteness of the whole Tate–Shafarevich group.","limitation":"Finiteness is stronger than a finite p-primary check for one prime and does not follow from rank equality alone.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"independent_review":"pending"},{"id":"bsd-connection-converse","source":{"id":"claim-bsd-analytic-rank-at-most-one","version":4},"target":{"id":"claim-bsd-rank-one-converse-finite-sha","version":3},"relation":"related_to","assertion":null,"explanation":"The first starts from analytic rank ≤1; Skinner’s converse starts from algebraic rank one, finite Sha and specified semistable reduction.","limitation":"Do not reverse the first implication without the additional converse hypotheses.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2","birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2"],"independent_review":"pending"},{"id":"bsd-connection-p-parts","source":{"id":"claim-bsd-semistable-rank-one-p-part","version":2},"target":{"id":"claim-bsd-supersingular-p-part","version":2},"relation":"related_to","assertion":null,"explanation":"These formulas overlap at selected supersingular rank-one curves; the supersingular theorem also covers analytic rank zero in its scope.","limitation":"An equality of p-valuations leaves other primes and hypotheses to be checked; neither statement is the general full formula.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1"],"independent_review":"pending"},{"id":"bsd-connection-congruent","source":{"id":"claim-bsd-rank-target","version":3},"target":{"id":"claim-bsd-congruent-number-consequence","version":2},"relation":"related_to","assertion":null,"explanation":"Rank BSD specialized to E_n would supply the missing sufficient direction of Tunnell’s exact counting criterion.","limitation":"This implication is restricted to the stated odd squarefree congruent-number curves.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1","birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1"],"independent_review":"pending"}],"milestones":[{"id":"bsd-milestone-mordell","date":{"value":"1922"},"date_basis":"publication","category":"foundation","title":"Finite generation makes rank a precise invariant","summary":"Mordell’s theorem describes rational points as a finite-rank free group plus finite torsion.","limitation":"Finite generation does not determine the rank of each elliptic curve.","claims":[{"id":"claim-bsd-mordell-weil","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1"],"independent_review":"pending"},{"id":"bsd-milestone-origin","date":{"value":"1965"},"date_basis":"publication","category":"foundation","title":"The published conjecture grows from computer experiments","summary":"Birch–Swinnerton-Dyer’s Notes II records the rank and L-function prediction motivated by early computations.","limitation":"This is a cited publication year. Agreement in finite data did not establish the general formula.","claims":[{"id":"claim-bsd-rank-target","version":3}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-origin-history:v1"],"independent_review":"pending"},{"id":"bsd-milestone-gross-zagier","date":{"value":"1986"},"date_basis":"publication","category":"foundation","title":"Heegner points and central derivatives","summary":"The published Gross–Zagier formula provides the height/L-derivative connection later combined with Euler systems.","limitation":"The current analytic-rank-≤1 synthesis also uses Kolyvagin and modularity; the 1986 paper alone is not that full synthesis.","claims":[],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-gross-zagier-history:v1"],"independent_review":"pending"},{"id":"bsd-milestone-modularity","date":{"value":"2001"},"date_basis":"publication","category":"special_case","title":"The low-analytic-rank theorem applies to all E/Q","summary":"General modularity, published in 2001, completes the classical scope: analytic rank 0 or 1 gives rank equality and finite Sha for every E/Q.","limitation":"This is a synthesis of earlier Gross–Zagier/Kolyvagin inputs with modularity, not a proof of every leading coefficient or higher-rank case.","claims":[{"id":"claim-bsd-analytic-rank-at-most-one","version":4}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-modularity-history:v1","birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2"],"independent_review":"pending"},{"id":"bsd-milestone-miller","date":{"value":"2011"},"date_basis":"publication","category":"special_case","title":"Computer-assisted full formulas in a finite corpus","summary":"Miller’s published proof covers 16,714 of 16,725 enumerated analytic-rank-≤1 curves of conductor <5000.","limitation":"The 11 excluded curve/prime pairs describe that paper’s boundary, not today’s list of unsolved cases.","claims":[{"id":"claim-bsd-miller-finite-computations","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1"],"independent_review":"pending"},{"id":"bsd-milestone-jsw","date":{"value":"2015-12-21"},"date_basis":"preprint","category":"special_case","title":"A rank-one p-part formula under precise hypotheses","summary":"Jetchev–Skinner–Wan’s author preprint gives the p-part for semistable analytic-rank-one curves at good p ≥5 with irreducible E[p].","limitation":"The theorem is prime-specific and hypothesis-specific; its later journal publication is a separate bibliographic event.","claims":[{"id":"claim-bsd-semistable-rank-one-p-part","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1"],"independent_review":"pending"},{"id":"bsd-milestone-skinner","date":{"value":"2020-02-13"},"date_basis":"publication","category":"conditional_result","title":"A conditional algebraic-rank-one converse","summary":"Skinner’s published theorem obtains analytic rank one from algebraic rank one, finite Sha and the stated semistable reduction pattern.","limitation":"Finite Sha is assumed here; this converse cannot be used to prove its own premise.","claims":[{"id":"claim-bsd-rank-one-converse-finite-sha","version":3}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2"],"independent_review":"pending"},{"id":"bsd-milestone-zeta-elements","date":{"value":"2024-09-11"},"date_basis":"preprint","category":"special_case","title":"Supersingular p-parts and a conditioned full-formula family","summary":"Burungale–Skinner–Tian–Wan v2 gives selected supersingular p-parts and full BSD for the explicitly constrained 46a1 twist family.","limitation":"These are two different scopes. The selection omits p=3’s extra conditions and does not cover every quadratic twist.","claims":[{"id":"claim-bsd-supersingular-p-part","version":2},{"id":"claim-bsd-46a1-quadratic-twists","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1"],"independent_review":"pending"},{"id":"bsd-milestone-effective","date":{"value":"2026-06-04"},"date_basis":"preprint","category":"algorithm","title":"Effective enumeration of sufficient twist subfamilies","summary":"Banwait–Huang v3 supplies finite algorithms for subfamilies satisfying cited BSD criteria and clarifies the base-curve restrictions.","limitation":"The four-base list in the family result is not an unconditional full-BSD assertion for every listed curve; there is no enumeration of all BSD cases.","claims":[{"id":"claim-bsd-46a1-quadratic-twists","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1"],"independent_review":"pending"},{"id":"bsd-milestone-ai-selmer","date":{"value":"2026-10-07"},"date_basis":"preprint","category":"proposal","title":"OpenAI’s proposed full low-Selmer-corank formula","summary":"The provider proposes the complete leading-term formula and finite Sha when a full q-power Selmer corank is 0 or 1.","limitation":"The manuscript and its companion dependencies remain unassessed here. Mordell–Weil rank ≤1 is not the stated hypothesis.","claims":[{"id":"claim-bsd-openai-low-selmer-proposal","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1"],"independent_review":"pending"},{"id":"bsd-milestone-ai-density","date":{"value":"2026-10-07"},"date_basis":"preprint","category":"proposal","title":"Proposed half-densities for analytic ranks in twist families","summary":"A separate provider manuscript proposes Goldfeld’s 0/1 half-densities for every E/Q in the signed squarefree ordering.","limitation":"An asymptotic family statement allows exceptional twists; mean rank and full leading terms require additional results.","claims":[{"id":"claim-bsd-openai-goldfeld-proposal","version":2}],"source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1"],"independent_review":"pending"}],"progress":[{"claim_id":"claim-bsd-analytic-rank-at-most-one","aspect":"Rank equality & finite Sha","summary":"Every E/Q of analytic rank 0 or 1.","remaining":"No unrestricted analytic-rank ≥ 2 theorem follows.","overlaps_with":[],"overlap_note":"The hypothesis is analytic rank ≤ 1, not merely algebraic rank ≤ 1. This does not alone prove the full leading-coefficient formula.","category":"special_case"},{"claim_id":"claim-bsd-rank-one-converse-finite-sha","aspect":"Conditional rank converse","summary":"Algebraic rank 1 ⇒ analytic rank 1 in this family if Sha is finite.","remaining":"Finite Sha and the reduction hypotheses must be established for an application.","overlaps_with":["claim-bsd-analytic-rank-at-most-one"],"overlap_note":"Runs in the converse direction to the low-analytic-rank theorem. General Sha finiteness is conjectural and is not supplied by assuming algebraic rank 1.","category":"conditional_result"},{"claim_id":"claim-bsd-semistable-rank-one-p-part","aspect":"Leading coefficient · p-part","summary":"Good-reduction p ≥ 5 with irreducible E[p], for semistable analytic-rank-one curves.","remaining":"The missing primes and unrestricted curves require further results.","overlaps_with":["claim-bsd-analytic-rank-at-most-one"],"overlap_note":"Adds prime-specific leading-coefficient information beyond low-rank equality. Both good ordinary and supersingular reduction are permitted here; the p=3 extension is not included in this selected Claim.","category":"special_case"},{"claim_id":"claim-bsd-supersingular-p-part","aspect":"Leading coefficient · p-part","summary":"Semistable analytic-rank 0 or 1 at supersingular p ≥ 5.","remaining":"Other primes and higher analytic ranks are not covered by this selected theorem.","overlaps_with":["claim-bsd-analytic-rank-at-most-one","claim-bsd-semistable-rank-one-p-part"],"overlap_note":"Overlaps the supersingular rank-one portion of the Jetchev–Skinner–Wan Claim and additionally includes analytic rank zero. Prime parts are not independent percentages of BSD solved.","category":"special_case"},{"claim_id":"claim-bsd-46a1-quadratic-twists","aspect":"Leading coefficient · full formula","summary":"An infinite, explicitly conditioned family of twists of 46a1.","remaining":"Does not give full BSD for every twist, every rank-zero curve, or analytic rank ≥ 2.","overlaps_with":["claim-bsd-analytic-rank-at-most-one","claim-bsd-supersingular-p-part"],"overlap_note":"A full-formula family within the low-analytic-rank case. The 46a1 family is retained after the 2026 Banwait–Huang check; four other base curves in the earlier list have an infinitude caveat.","category":"special_case"},{"claim_id":"claim-bsd-miller-finite-computations","aspect":"Leading coefficient · finite corpus","summary":"Full formula certified in the paper for 16,714 enumerated curves.","remaining":"The proof does not cover all curves, all conductor ranges or higher analytic ranks.","overlaps_with":["claim-bsd-analytic-rank-at-most-one","claim-bsd-semistable-rank-one-p-part"],"overlap_note":"Strengthens the formula for a finite subset of the classical low-rank regime. This is computer-assisted proof in the source, not AI theorem discovery or a Lean kernel check.","category":"special_case"},{"claim_id":"claim-bsd-finite-points-consequence","aspect":"Rational-point consequence","summary":"Under rank equality, the central L-value detects whether E(Q) is finite or infinite.","remaining":"Rank equality for arbitrary E/Q remains open; numerical near-zero values are not exact certificates.","overlaps_with":[],"overlap_note":"A consequence under an open premise, not an additional resolved part of that premise.","category":"conditional_implication"},{"claim_id":"claim-bsd-congruent-number-consequence","aspect":"Diophantine consequence","summary":"Two finite counts would decide the odd squarefree congruent-number problem, assuming rank BSD for E_n.","remaining":"The sufficient direction uses the conjecture; the odd-squarefree restrictions and exact counts matter.","overlaps_with":[],"overlap_note":"A consequence under an open premise, not an additional resolved part of that premise.","category":"conditional_implication"},{"claim_id":"claim-bsd-sha-order-consequence","aspect":"Leading-term consequence","summary":"The leading-term formula would recover Sha’s order from compatible certified invariants.","remaining":"The full formula and finiteness remain unproved in general. A decimal comparison alone is insufficient.","overlaps_with":[],"overlap_note":"A consequence under an open premise, not an additional resolved part of that premise.","category":"conditional_implication"}],"activities":[{"id":"bsd-effective-families-2026","title":"2026 · effective families and an earlier-list caveat","kind":"source_correction","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1"],"summary":"Banwait–Huang give sufficient-condition algorithms for BSD twist subfamilies and list eligible curves of conductor <500,000. Section 2.7 retains 46a1 but identifies a nonempty-prime-support gap for 62a1, 66b1, 105a1 and 141c1 in an earlier infinitude list.","limitation":"The authors do not characterize every BSD-satisfying twist. The flagged four curves are not disproved BSD cases. The code and dataset have not been executed or reproduced by OpenMath.","related_claim_ids":["claim-bsd-46a1-quadratic-twists"]},{"id":"bsd-ai-project-kent","title":"AI-assisted exploration · code and Lean targets","kind":"formalization","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v2"],"summary":"This individual AI-augmented research project reports numerical experiments and proof-search scaffolding. Its phase record states that Lean verification has not started; RankEquality.lean contains sorry targets.","limitation":"No new BSD theorem, complete formal proof or OpenMath-verified execution is established by this project citation. Its rank-two regulator identity remains conjectural in the project’s own notes.","related_claim_ids":["claim-bsd-analytic-rank-at-most-one"]},{"id":"bsd-high-rank-witness-2026","title":"2026 · reported rank lower bound 30","kind":"computation","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1"],"summary":"The primary record supplies a specific elliptic curve over Q and 30 rational-point witnesses, with a reported rank lower bound of 30.","limitation":"A lower bound on algebraic rank does not establish equality with analytic rank or BSD. OpenMath has not reproduced the independence certificate. Secondary claims of exact rank under BSD+GRH cannot be used as evidence proving BSD.","related_claim_ids":[]},{"id":"bsd-openai-proposals","title":"Company AI proposals: Selmer converses, leading terms and twist densities","kind":"research_direction","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1"],"summary":"The October provider release proposes low-corank converses and exact leading terms, alongside a distinct analytic twist-density theorem. The pinned manuscripts state companion dependencies.","limitation":"These source-reported proposals are not established progress here. No corresponding external formal-code execution or proof-dependency audit is asserted; asymptotic twist densities do not resolve every curve.","related_claim_ids":["claim-bsd-openai-low-selmer-proposal","claim-bsd-openai-goldfeld-proposal"]}],"curation":{"checked_on":"2026-10-10","selection":"Rank equality and finite Sha at analytic rank ≤ 1; one conditional converse; selected p-parts; a corroborated infinite twist family; a historical finite-corpus proof. Recent algorithms and AI/Lean exploratory artifacts are separate context. Includes selected source-dated history, exact Claim connections, interpretations, approaches and grounded conditional impacts.","search_scope":"Clay/Wiles; published Skinner and Jetchev–Skinner–Wan papers; author arXiv revisions through June 2026; public code and datasets; Google DeepMind, Axiom, Harmonic, Anthropic-related and individual AI/formalization searches. Cutoff: 10 October 2026. Not exhaustive. Rechecked source revision dates and the October OpenAI low-Selmer-corank and Goldfeld manuscripts, including their distinct hypotheses and companion dependencies.","formal_status":"No complete Lean proof of the selected BSD Claims has been established here. The inspected individual Lean targets contain sorry; they are explicitly classified as placeholders. No external code has an OpenMath Run or reproduction attached.","ai_status":"Company AI manuscripts now indexed as proposals: full BSD from low full-Selmer corank and analytic half-densities in twist families. Their independent assessment and companion proof chains are unassessed. The existing individual AI program and high-rank witness stay separate activities; neither proves general BSD.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/Q) is finite.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For "},{"kind":"math","text":"E/Q","latex":"E/\\mathbb{Q}"},{"kind":"text","text":" with "},{"kind":"math","text":"ord_(s=1)L(E,s)","latex":"\\operatorname{ord}_{s=1}L(E,s)"},{"kind":"text","text":" equal to 0 or 1, "},{"kind":"math","text":"rank E(Q)","latex":"\\operatorname{rank}_{\\mathbb{Z}}E(\\mathbb{Q})"},{"kind":"text","text":" equals that analytic rank and "},{"kind":"math","text":"Sha(E/Q)","latex":"\\operatorname{Sha}(E/\\mathbb{Q})"},{"kind":"text","text":" is finite."}]},"id":"claim-bsd-analytic-rank-at-most-one","slug":"bsd-analytic-rank-at-most-one","version":4,"kind":"theorem","title":"Rank equality at analytic rank 0 or 1","assumptions":["E is an elliptic curve over Q.","The full Hasse–Weil L-function has order of vanishing 0 or 1 at s=1."],"scope":"Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2","birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1"],"formal_declarations":[]},{"statement":"For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.","id":"claim-bsd-rank-one-converse-finite-sha","slug":"bsd-rank-one-converse-finite-sha","version":3,"kind":"theorem","title":"Rank-one converse assuming finite Sha","assumptions":["E/Q is semistable.","E has nonsplit multiplicative reduction at at least one odd prime, or split multiplicative reduction at at least two odd primes.","Algebraic rank E(Q) = 1.","Assume Sha(E/Q) is finite."],"scope":"Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2"],"formal_declarations":[]},{"statement":"For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.","id":"claim-bsd-semistable-rank-one-p-part","slug":"bsd-semistable-rank-one-p-part","version":2,"kind":"theorem","title":"Rank-one formula · selected p-parts","assumptions":["E/Q is semistable (squarefree conductor N).","ord_(s=1)L(E,s) = 1.","p ≥ 5 is prime and p does not divide N.","The mod-p Galois representation on E[p] is irreducible."],"scope":"The p-part of the leading-coefficient formula, not an assertion at every prime. Irreducibility removes p-primary rational torsion.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1"],"formal_declarations":[]},{"statement":"For semistable E/Q of analytic rank 0 or 1 and a good supersingular prime p ≥ 5, the p-part of the BSD leading-coefficient formula holds.","id":"claim-bsd-supersingular-p-part","slug":"bsd-supersingular-p-part","version":2,"kind":"theorem","title":"Low-rank formula at supersingular p ≥ 5","assumptions":["E/Q is semistable.","ord_(s=1)L(E,s) is 0 or 1.","p ≥ 5 is a prime of good supersingular reduction."],"scope":"Theorem 1.5 restricted to p ≥ 5; its additional p=3 conditions and quadratic-twist extension are not asserted here.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1"],"formal_declarations":[]},{"statement":"Let E be Cremona curve 46a1. For squarefree M>1 coprime to 46, if L(E^M,1)≠0 and E has ordinary reduction at every prime dividing M, E^M satisfies full BSD. Infinitely many M meet these conditions.","id":"claim-bsd-46a1-quadratic-twists","slug":"bsd-46a1-quadratic-twists","version":2,"kind":"theorem","title":"Full BSD in a specified infinite twist family","assumptions":["E/Q is the specific Cremona curve 46a1.","M>1 is squarefree and gcd(M,46)=1; E^M is the twist by Q(√M).","L(E^M,1) is nonzero.","E has ordinary reduction at every prime dividing M."],"scope":"Analytic-rank-zero quadratic twists of one named non-CM base curve; rank, finite Sha and the entire leading-coefficient identity.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1"],"formal_declarations":[]},{"statement":"Miller’s computation-assisted proof establishes full BSD for 16,714 of the 16,725 elliptic curves of analytic rank at most one and conductor <5000 in the paper’s enumeration.","id":"claim-bsd-miller-finite-computations","slug":"bsd-miller-finite-computations","version":2,"kind":"theorem","title":"Computer-assisted full formula · 16,714 curves","assumptions":["E/Q has conductor <5000 and analytic rank 0 or 1.","E belongs to the successfully certified subset in the cited enumeration, excluding the 11 unresolved curve/prime pairs in Table 9.","Computer-assisted calculations are part of the cited proof; OpenMath has not repeated them."],"scope":"Historical finite-corpus proof. Table 9 exceptions are limitations of this paper, not a statement of current unresolved status.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1"],"formal_declarations":[]},{"statement":"For every elliptic curve E over Q, rank_Z E(Q) = ord_{s=1} L(E,s), where L is the full Hasse–Weil L-function with all finite Euler factors.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every elliptic curve "},{"kind":"math","text":"E","latex":"E"},{"kind":"text","text":" over "},{"kind":"math","text":"Q","latex":"\\mathbb{Q}"},{"kind":"text","text":", "},{"kind":"math","text":"rank_Z E(Q) = ord_{s=1} L(E,s)","latex":"\\operatorname{rank}_{\\mathbb{Z}}E(\\mathbb{Q})=\\operatorname{ord}_{s=1}L(E,s)","display":true},{"kind":"text","text":", where L is the full Hasse–Weil L-function with all finite Euler factors."}]},"id":"claim-bsd-rank-target","slug":"bsd-rank-target","version":3,"kind":"conjecture","title":"BSD rank equality holds for every E/Q","assumptions":["Elliptic curve over Q, with its rational identity point.","The full Hasse–Weil L-function, whose analytic continuation is known by modularity."],"scope":"The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1"],"formal_declarations":[]},{"statement":"For every E/Q, let r = rank_Z E(Q). Then ord_{s=1}L(E,s) = r, Sha(E/Q) is finite, and L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors².","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every "},{"kind":"math","text":"E/Q","latex":"E/\\mathbb{Q}"},{"kind":"text","text":", let "},{"kind":"math","text":"r = rank_Z E(Q)","latex":"r=\\operatorname{rank}_{\\mathbb{Z}}E(\\mathbb{Q})"},{"kind":"text","text":". Then "},{"kind":"math","text":"ord_{s=1}L(E,s) = r","latex":"\\operatorname{ord}_{s=1}L(E,s)=r"},{"kind":"text","text":", "},{"kind":"math","text":"Sha(E/Q)","latex":"\\operatorname{Sha}(E/\\mathbb{Q})"},{"kind":"text","text":" is finite, and "},{"kind":"math","text":"L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors²","latex":"\\begin{aligned}\\frac{L^{(r)}(E,1)}{r!}&=\\Omega_E\\operatorname{Reg}_E\\\\&\\quad\\cdot\\frac{\\#\\operatorname{Sha}(E/\\mathbb{Q})\\prod_{\\ell\\ \\mathrm{finite}}c_\\ell(E)}{\\bigl(\\#E(\\mathbb{Q})_{\\mathrm{tors}}\\bigr)^2}\\end{aligned}","display":true},{"kind":"text","text":"."}]},"id":"claim-bsd-full-target","slug":"bsd-full-target","version":3,"kind":"conjecture","title":"the full BSD formula holds for every E/Q","assumptions":["Full Hasse–Weil L-function; Ω_E = ∫_{E(R)} |ω_E| for a global minimal Néron differential, including both real components when present.","Reg_E is the Néron–Tate regulator of a basis of E(Q)/torsion, with Reg_E = 1 at rank zero.","c_ℓ are finite-prime Tamagawa numbers; E(Q)_tors is rational torsion."],"scope":"A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"formal_declarations":[]},{"statement":"The Tate–Shafarevich group Sha(E/Q) is finite for every elliptic curve E over Q.","id":"claim-bsd-sha-finiteness-target","slug":"bsd-sha-finiteness-target","version":2,"kind":"conjecture","title":"Sha is finite for every E/Q","assumptions":["Sha parametrizes everywhere locally soluble principal homogeneous spaces up to equivalence."],"scope":"A separately open finiteness assertion, proved in selected low-analytic-rank cases. It is included in the strong BSD scenario.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"formal_declarations":[]},{"statement":"For every elliptic curve E/Q, E(Q) ≅ Z^r ⊕ E(Q)_tors with r ≥ 0 and finite torsion. Consequently E(Q) is finite if and only if r = 0.","id":"claim-bsd-mordell-weil","slug":"bsd-mordell-weil","version":2,"kind":"theorem","title":"Rational points form a finitely generated group","assumptions":["An elliptic curve over Q with the rational identity defining its group law."],"scope":"An unconditional foundation. Knowing finite generation does not compute the rank of an arbitrary curve.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1"],"formal_declarations":[]},{"statement":"If the full BSD assertion holds for every E/Q, then BSD rank equality holds for every E/Q and Sha(E/Q) is finite for every E/Q.","id":"claim-bsd-strong-implies-rank","slug":"bsd-strong-implies-rank","version":2,"kind":"theorem","title":"Strong BSD implies its rank and finiteness assertions","assumptions":["Strong BSD includes rank equality, Sha finiteness and the normalized leading-term formula."],"scope":"A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"formal_declarations":[]},{"statement":"For an elliptic curve E/Q satisfying BSD rank equality, E(Q) is infinite if and only if L(E,1) = 0; if L(E,1) ≠ 0, E(Q) is finite.","id":"claim-bsd-finite-points-consequence","slug":"bsd-finite-points-consequence","version":2,"kind":"theorem","title":"Rank BSD would decide finite versus infinite rational points analytically","assumptions":["BSD rank equality for the particular E/Q.","Known analytic continuation and Mordell–Weil finite generation."],"scope":"A proved implication under the rank-equality premise, not an unconditional arbitrary-curve decision algorithm.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1"],"formal_declarations":[]},{"statement":"For every odd positive squarefree integer n, assuming BSD rank equality for E_n: y² = x³−n²x, n is the area of a right triangle with positive rational side lengths if and only if #{(x,y,z)∈Z³: 2x²+y²+8z²=n} = 2·#{(x,y,z)∈Z³: 2x²+y²+32z²=n}.","id":"claim-bsd-congruent-number-consequence","slug":"bsd-congruent-number-consequence","version":2,"kind":"theorem","title":"BSD would complete the odd squarefree congruent-number criterion","assumptions":["n is positive, odd and squarefree.","BSD rank equality for the congruent-number elliptic curve E_n."],"scope":"A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1"],"formal_declarations":[]},{"statement":"For an E/Q satisfying strong BSD, #Sha(E/Q) = [L^{(r)}(E,1)/r!] · #E(Q)_tors² / [Ω_E Reg_E ∏_{ℓ finite} c_ℓ(E)], with r = rank E(Q) and the stated full-L and Néron conventions.","id":"claim-bsd-sha-order-consequence","slug":"bsd-sha-order-consequence","version":2,"kind":"theorem","title":"Strong BSD would determine Sha’s order from the other invariants","assumptions":["Strong BSD for the particular E/Q, including Sha finiteness.","Compatible normalizations and certified values of the other inputs are required to recover an exact order."],"scope":"An algebraic rearrangement of the conjectural leading-term formula under its premise; approximating an L-value does not certify the integer.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"formal_declarations":[]},{"statement":"For E/Q and some prime q, if s_q(E) = corank_{Z_q} Sel_{q∞}(E/Q) belongs to {0,1}, then rank E(Q) = ord_{s=1}L(E,s) = s_q(E), Sha(E/Q) is finite, and the full normalized BSD leading-term formula holds.","id":"claim-bsd-openai-low-selmer-proposal","slug":"bsd-openai-low-selmer-proposal","version":2,"kind":"theorem","title":"Proposed full BSD at Selmer corank zero or one","assumptions":["Full q-power Selmer group with local Kummer conditions at every place, not only the finite q-Selmer group.","Some prime q; no extra reduction, CM, rational torsion, isogeny or residual-representation restriction is claimed.","Full L-function, total Néron real period and full point-lattice regulator of Theorem 1.1."],"scope":"A provider proposal depending on companion low-corank converse and two-primary inputs. Algebraic rank ≤1 alone does not establish this Selmer-corank hypothesis.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1"],"formal_declarations":[]},{"statement":"For every E/Q, let D(X)={d∈Z:0<|d|≤X, d squarefree}. For j=0,1 the provider claims lim_{X→∞} #{d∈D(X): ord_{s=1}L(E^(d),s)=j}/#D(X) = 1/2.","id":"claim-bsd-openai-goldfeld-proposal","slug":"bsd-openai-goldfeld-proposal","version":2,"kind":"theorem","title":"Proposed half-densities in every quadratic-twist family","assumptions":["A fixed E/Q; both positive and negative squarefree twist parameters are counted.","Order by |d| and take the asymptotic limit; no fixed residue-class or finite-X half split is asserted."],"scope":"A proposed family-frequency theorem, not a theorem for every individual twist. The mean-rank assertion needs separate tail control and is not included in this Claim.","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"Rank equality is proved at analytic rank 0 or 1. Further cited results give a conditional converse, selected p-parts and full BSD in specified twist families and finite computations.","report_ids":["birch-and-swinnerton-dyer-conjecture:report:0:v2"],"progress_claim_ids":["claim-bsd-analytic-rank-at-most-one","claim-bsd-rank-one-converse-finite-sha","claim-bsd-semistable-rank-one-p-part","claim-bsd-supersingular-p-part","claim-bsd-46a1-quadratic-twists","claim-bsd-miller-finite-computations"],"remaining":"Rank equality in general, especially analytic rank ≥ 2, remains open. General finiteness of Sha and the complete leading-coefficient formula are stronger unresolved assertions."},"statement_kind":"conjecture","problem_id":"problem-birch-and-swinnerton-dyer-conjecture","slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","question":"For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at s = 1?","question_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at "},{"kind":"math","text":"s = 1","latex":"s=1"},{"kind":"text","text":"?"}]},"scope":"Elliptic curves over Q. The canonical question is rank equality. The stronger BSD formula additionally predicts finite Sha and L^(r)(E,1)/(r! Ω_E Reg_E) = #Sha(E/Q) · ∏ c_l / #E(Q)_tors², with the standard full Hasse–Weil L-function, Néron real period, regulator and finite-prime Tamagawa factors. These aspects are tracked separately.","scope_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Elliptic curves over "},{"kind":"math","text":"Q","latex":"\\mathbb{Q}"},{"kind":"text","text":". The canonical question is rank equality. The stronger BSD formula additionally predicts finite Sha and "},{"kind":"math","text":"L^(r)(E,1)/(r! Ω_E Reg_E) = #Sha(E/Q) · ∏ c_l / #E(Q)_tors²","latex":"\\begin{aligned}\\frac{L^{(r)}(E,1)}{r!\\,\\Omega_E\\operatorname{Reg}_E}&=\\frac{\\#\\operatorname{Sha}(E/\\mathbb{Q})}{\\bigl(\\#E(\\mathbb{Q})_{\\mathrm{tors}}\\bigr)^2}\\\\&\\quad\\cdot\\prod_{\\ell\\ \\mathrm{finite}}c_\\ell(E)\\end{aligned}","display":true},{"kind":"text","text":", with the standard full Hasse–Weil L-function, Néron real period, regulator and finite-prime Tamagawa factors. These aspects are tracked separately."}]},"aliases":["Birch and Swinnerton-Dyer conjecture"],"fields":["Number theory","Algebraic geometry"],"msc":["11G05","11G40"],"topics":["Scoped progress","Conditional results"],"background":"specialist","scale":"major conjecture","depth":"source-linked","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:entry:0:v1","birch-and-swinnerton-dyer-conjecture:entry:0:v2","birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2","birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1","birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2","birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1","birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v2","birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1","birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1","birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1","birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1","birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1","birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1","birch-and-swinnerton-dyer-conjecture:bsd-origin-history:v1","birch-and-swinnerton-dyer-conjecture:bsd-gross-zagier-history:v1","birch-and-swinnerton-dyer-conjecture:bsd-modularity-history:v1"],"status_report_ids":["birch-and-swinnerton-dyer-conjecture:report:0:v1","birch-and-swinnerton-dyer-conjecture:report:0:v2"],"editorial":null,"literature":[{"title":"Clay problem page and official description","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","note":"Clay reports the general question unsolved, observed 10 October 2026. This current-state observation is separate from the dated theorem and artifact citations below."}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"bsd-interpretation-ranks","title":"Two ranks count different things","summary":"Mordell–Weil rank counts independent rational generators. Analytic rank counts the vanishing order of L(E,s) at 1. BSD predicts these integers agree for every E/Q.","review":"pending","kind":"interpretation","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1","birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1"],"claims":[{"id":"claim-bsd-mordell-weil","version":2},{"id":"claim-bsd-rank-target","version":3}],"limitation":"A set of independent points gives a rank lower bound; a rigorous equality also needs an upper bound. Approximate L-values do not certify an exact vanishing order."},{"id":"bsd-interpretation-formula","title":"Rank equality, p-parts and full BSD have separate scopes","summary":"The full formula additionally involves finite Sha, periods, regulator, Tamagawa factors and torsion. A p-part identifies one valuation under specific hypotheses; a family theorem fixes a class of curves.","review":"pending","kind":"interpretation","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1"],"claims":[{"id":"claim-bsd-full-target","version":3},{"id":"claim-bsd-semistable-rank-one-p-part","version":2}],"limitation":"These results overlap. Matching decimals, one valuation or rank equality alone does not establish the entire normalized formula."},{"id":"bsd-interpretation-selmer","title":"Selmer corank is not just algebraic rank","summary":"The new provider proposal assumes the full q-power Selmer corank is 0 or 1. Its exact Kummer sequence also contains the divisible q-primary contribution of Sha, so a Mordell–Weil rank bound alone is insufficient.","review":"pending","kind":"interpretation","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1"],"claims":[{"id":"claim-bsd-openai-low-selmer-proposal","version":2}],"limitation":"The proposal is not independently accepted here. Do not read its low-corank hypothesis as every curve of algebraic rank zero or one."},{"id":"bsd-approach-euler","title":"Heegner points and Euler systems at low analytic rank","summary":"Gross–Zagier relates L-derivatives to heights; Kolyvagin’s derived classes constrain ranks and Sha. Together with modularity they establish the selected analytic-rank-0/1 theorem.","review":"pending","kind":"approach","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-gross-zagier-history:v1","birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2"],"claims":[{"id":"claim-bsd-analytic-rank-at-most-one","version":4}],"limitation":"Higher analytic rank needs further constructions; the low-rank theorem does not identify every leading coefficient."},{"id":"bsd-approach-iwasawa","title":"Iwasawa and p-adic comparisons isolate local arithmetic","summary":"The selected rank-one and supersingular formulas compare analytic data with Selmer information and arithmetic factors prime by prime.","review":"pending","kind":"approach","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1"],"claims":[{"id":"claim-bsd-semistable-rank-one-p-part","version":2},{"id":"claim-bsd-supersingular-p-part","version":2}],"limitation":"Reduction type, residual irreducibility, rank and prime bounds are theorem hypotheses. Dropping them requires an additional argument."},{"id":"bsd-approach-computation","title":"Descent and certified finite computations","summary":"Descent, rigorous point lattices and theorem-backed computations can establish full BSD for particular curves. Effective family algorithms produce further explicitly sufficient cases.","review":"pending","kind":"approach","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1"],"claims":[{"id":"claim-bsd-miller-finite-computations","version":2},{"id":"claim-bsd-46a1-quadratic-twists","version":2}],"limitation":"The finite-corpus counts and sufficient-family tests are bounded guarantees. Exploratory numerics and Lean targets containing sorry are separate activities."},{"id":"bsd-impact-points","title":"Connect analytic vanishing to rational solutions","summary":"Rank BSD would make the central L-value an exact criterion for finite versus infinite rational points on every elliptic curve over Q.","review":"pending","kind":"impact","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1"],"claims":[{"id":"claim-bsd-finite-points-consequence","version":2}],"scenario":{"id":"claim-bsd-rank-target","version":3},"limitation":"This is a mathematical criterion; it does not make floating-point zero testing rigorous or imply a fast implementation."},{"id":"bsd-impact-congruent","title":"Complete a concrete rational-triangle criterion","summary":"For odd positive squarefree n, rank BSD for the congruent-number curve would make Tunnell’s two finite representation counts sufficient as well as necessary for a rational right triangle of area n.","review":"pending","kind":"impact","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1"],"claims":[{"id":"claim-bsd-congruent-number-consequence","version":2}],"scenario":{"id":"claim-bsd-rank-target","version":3},"limitation":"This selected formula is restricted to odd squarefree n; the even case has a different formula. It does not decide arbitrary Diophantine equations."},{"id":"bsd-impact-sha","title":"Determine the missing arithmetic factor","summary":"The full formula would tie Sha’s exact order to the central leading term, point-lattice regulator, periods, torsion and local factors.","review":"pending","kind":"impact","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1"],"claims":[{"id":"claim-bsd-sha-order-consequence","version":2}],"scenario":{"id":"claim-bsd-full-target","version":3},"limitation":"Exact compatible inputs are required. This concerns rational elliptic curves and gives no claimed algorithm for finite-field elliptic-curve discrete logarithms."}],"coverage_gaps":["Selected results, not an exhaustive literature survey or a percentage of the conjecture solved.","Source-attributed theorem statements; independent mathematical and relationship review is pending.","External papers, programs and Lean placeholders have no OpenMath execution or independent reproduction attached.","AI involvement is recorded only where the cited source reports it; absence from this selection does not establish absence of work.","Timeline dates refer to cited research events at their supported precision. Overlapping results and conditional consequences do not define a percentage solved."],"checked_on":"2026-10-10"},"sources":[{"id":"clay-millennium@observed-2026-10-01","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-01","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge","observed_on":"2026-10-01","source_updated_on":null,"rights":"metadata-link","license":"Not established; links and OpenMath paraphrases only","terms_url":"https://www.claymath.org/millennium-problems/","attribution":"Clay Mathematics Institute","rights_note":"Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.","check_interval_days":30},{"id":"clay-millennium@observed-2026-10-10","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-10","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"Not established; links and OpenMath paraphrases only","terms_url":"https://www.claymath.org/millennium-problems/","attribution":"Clay Mathematics Institute","rights_note":"Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.","check_interval_days":30},{"work":{"authors":["Andrew Wiles"],"kind":"official_description","publication":"published","publication_note":"Official Clay problem description; separates rank equality from its refined formula.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Official description PDF","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","version":"official-description","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"clay-wiles-bsd@official-description","source_id":"clay-wiles-bsd","title":"Wiles · official BSD description","owner":"Andrew Wiles","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","subject_scope":"Wiles · official BSD description","revision":"official-description","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","attribution":"Andrew Wiles","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Christopher Skinner"],"kind":"research_paper","publication":"published","publication_note":"Annals of Mathematics 191 (2020), 329–354; published online 13 February 2020. Introduction also states the classical low-analytic-rank theorem.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Journal proof PDF","url":"https://annals.math.princeton.edu/wp-content/uploads/annals-v191-n2-p01-s.pdf","version":"Annals 191 (2020), 329–354","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"skinner-annals-2020@published-online-2020-02-13","source_id":"skinner-annals-2020","title":"Skinner · converse to Gross–Zagier–Kolyvagin","owner":"Christopher Skinner","url":"https://doi.org/10.4007/annals.2020.191.2.1","subject_scope":"Skinner · converse to Gross–Zagier–Kolyvagin","revision":"published-online-2020-02-13","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2020-02-13","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://doi.org/10.4007/annals.2020.191.2.1","attribution":"Christopher Skinner","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Dimitar Jetchev","Christopher Skinner","Xin Wan"],"kind":"research_paper","publication":"published","publication_note":"Published in Cambridge Journal of Mathematics 5 (2017), 369–434; the cited author preprint supplies Theorem 1.2.1.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Author proof PDF · v1","url":"https://arxiv.org/pdf/1512.06894v1","version":"arXiv:1512.06894v1","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-1512.06894@arxiv-v1","source_id":"arxiv-1512.06894","title":"Jetchev–Skinner–Wan · rank-one BSD p-part","owner":"Dimitar Jetchev; Christopher Skinner; Xin Wan","url":"https://arxiv.org/abs/1512.06894v1","subject_scope":"Jetchev–Skinner–Wan · rank-one BSD p-part","revision":"arxiv-v1","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2015-12-21","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/1512.06894v1","attribution":"Dimitar Jetchev; Christopher Skinner; Xin Wan","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Ashay Burungale","Christopher Skinner","Ye Tian","Xin Wan"],"kind":"research_paper","publication":"preprint","publication_note":"Author preprint at the cited revision; journal acceptance has not been established by this curation.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Proof PDF · v2","url":"https://arxiv.org/pdf/2409.01350v2","version":"arXiv:2409.01350v2","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-2409.01350@arxiv-v2","source_id":"arxiv-2409.01350","title":"Burungale–Skinner–Tian–Wan · zeta elements","owner":"Ashay Burungale; Christopher Skinner; Ye Tian; Xin Wan","url":"https://arxiv.org/abs/2409.01350v2","subject_scope":"Burungale–Skinner–Tian–Wan · zeta elements","revision":"arxiv-v2","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2024-09-11","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2409.01350v2","attribution":"Ashay Burungale; Christopher Skinner; Ye Tian; Xin Wan","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Barinder S. Banwait","Xiaoyu Huang"],"kind":"research_paper","publication":"accepted","publication_note":"arXiv metadata reports acceptance for ANTS XVII. Algorithms enumerate sufficient subfamilies, not all curves or twists satisfying BSD.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Paper PDF · v3","url":"https://arxiv.org/pdf/2601.16044v3","version":"arXiv:2601.16044v3","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"code","title":"Sage algorithms · pinned commit","url":"https://github.com/cocoxhuang/ants_xvii/tree/31fae20c8df3f1f0383f41112b914d4995d5809d/ants_xvii/infinite_bsd","version":"31fae20c8df3f1f0383f41112b914d4995d5809d","assessment":"not_executed","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"dataset","title":"Eligible curve labels · pinned file","url":"https://github.com/cocoxhuang/ants_xvii/blob/31fae20c8df3f1f0383f41112b914d4995d5809d/ants_xvii/infinite_bsd/output/ec_labels_500k.txt","version":"31fae20c8df3f1f0383f41112b914d4995d5809d","assessment":"not_executed","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-2601.16044@arxiv-v3","source_id":"arxiv-2601.16044","title":"Banwait–Huang · effective BSD twist families","owner":"Barinder S. Banwait; Xiaoyu Huang","url":"https://arxiv.org/abs/2601.16044v3","subject_scope":"Banwait–Huang · effective BSD twist families","revision":"arxiv-v3","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-06-04","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2601.16044v3","attribution":"Barinder S. Banwait; Xiaoyu Huang","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Robert L. Miller"],"kind":"research_paper","publication":"published","publication_note":"LMS Journal of Computation and Mathematics 14 (2011), 327–350. This is the scope of that historical paper, not a current unsolved-case list.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Computer-assisted proof PDF · v3","url":"https://arxiv.org/pdf/1010.2431v3","version":"arXiv:1010.2431v3","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-1010.2431@arxiv-v3","source_id":"arxiv-1010.2431","title":"Miller · computer-assisted finite BSD cases","owner":"Robert L. Miller","url":"https://arxiv.org/abs/1010.2431v3","subject_scope":"Miller · computer-assisted finite BSD cases","revision":"arxiv-v3","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2011-12-21","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/1010.2431v3","attribution":"Robert L. Miller","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Owen Kent (repository owner: owenpkent)"],"kind":"research_project","publication":"project","publication_note":"Individual project documentation and code; no accepted proof of BSD is asserted by this curation. Repository-owner name follows the public GitHub profile; no institutional authorship is inferred.","ai_role":"self_reported","ai_note":"README calls the program AI-augmented and contains Claude agent specifications. A precise model version and independently assessed division of labor are not supplied.","artifacts":[{"kind":"project_notes","title":"Project phase and limitations","url":"https://github.com/owenpkent/birch-swinnerton-dyer/blob/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1/PHASE_STATE.md","version":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","assessment":"external_report","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"code","title":"Computational experiments","url":"https://github.com/owenpkent/birch-swinnerton-dyer/tree/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1/experiments","version":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","assessment":"not_executed","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"formal_code","title":"Lean rank-equality targets","url":"https://github.com/owenpkent/birch-swinnerton-dyer/blob/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1/lean/BSD/RankEquality.lean","version":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","assessment":"placeholder","note":"The inspected weakBSD and low-rank theorem bodies contain sorry. This is a target skeleton, not a completed formal proof."}]},"id":"kent-bsd-project@0e9519aaeb5c8b8c524f95925eecb970d86bc5b1-metadata-2026-10-10","source_id":"kent-bsd-project","title":"Kent · AI-augmented BSD research substrate","owner":"Owen P. Kent (repository owner)","url":"https://github.com/owenpkent/birch-swinnerton-dyer/tree/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","subject_scope":"Kent · AI-augmented BSD research substrate","revision":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-06-14","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://github.com/owenpkent/birch-swinnerton-dyer/tree/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","attribution":"Owen P. Kent (repository owner)","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["ICARM leaderboard; submitter ranksunbounded"],"kind":"research_project","publication":"project","publication_note":"Public individual curve and rational-point witness record. Verification and AI attribution are not independently established by OpenMath.","ai_role":"unassessed","ai_note":"Secondary reports associate this search with Claude and Anthropic researchers; the cited primary curve record itself does not establish that attribution, so it is not treated as verified company authorship.","artifacts":[{"kind":"dataset","title":"Curve and thirty-point witness","url":"https://elliptic-rank.icarm.cloud/curve/273","version":"record updated 2026-08-20; observed 2026-10-10","assessment":"external_report","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"icarm-curve-273@observed-2026-10-10","source_id":"icarm-curve-273","title":"ICARM · individual high-rank curve record","owner":"ICARM leaderboard; submitter ranksunbounded","url":"https://elliptic-rank.icarm.cloud/curve/273","subject_scope":"ICARM · individual high-rank curve record","revision":"observed-2026-10-10","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-08-20","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://elliptic-rank.icarm.cloud/curve/273","attribution":"ICARM leaderboard; submitter ranksunbounded","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["OpenAI"],"kind":"research_paper","publication":"preprint","publication_note":"Provider preprint in the October 2026 public collection. The selected manuscript is a proposal here; a release announcement is not independent expert acceptance.","ai_role":"self_reported","ai_note":"The provider attributes this manuscript to its internal AI model. Mathematical acceptance, paper-to-formal correspondence and independent reproduction are not established by OpenMath.","artifacts":[{"kind":"pdf","title":"OpenAI · proposed full BSD from low Selmer corank — manuscript","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","version":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","assessment":"not_executed","note":"External manuscript. No OpenMath proof execution, formalization or independent mathematical acceptance is inferred.","sha256":"64b46603e174462e5bb32d242852eb7f0afe6e519f26534e42fd3f178a16b885"}],"published":{"value":"2026-10-07"}},"id":"openai-bsd-low-selmer@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","source_id":"openai-bsd-low-selmer","title":"OpenAI · proposed full BSD from low Selmer corank","owner":"OpenAI","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","subject_scope":"OpenAI · proposed full BSD from low Selmer corank","revision":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","content_sha256":"64b46603e174462e5bb32d242852eb7f0afe6e519f26534e42fd3f178a16b885","update_method":"Manual primary-source revision check; changes require a new immutable source version.","identifier_policy":"Stable bibliographic identity, explicit revision and exact source-entry locators.","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"External full-text redistribution rights not asserted","terms_url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","attribution":"OpenAI","rights_note":"Only metadata and original OpenMath paraphrases are published; full text stays upstream.","check_interval_days":30},{"work":{"authors":["OpenAI"],"kind":"research_paper","publication":"preprint","publication_note":"Provider preprint in the October 2026 public collection. The selected manuscript is a proposal here; a release announcement is not independent expert acceptance.","ai_role":"self_reported","ai_note":"The provider attributes this manuscript to its internal AI model. Mathematical acceptance, paper-to-formal correspondence and independent reproduction are not established by OpenMath.","artifacts":[{"kind":"pdf","title":"OpenAI · proposed Goldfeld analytic densities and 2-converse — manuscript","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","version":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","assessment":"not_executed","note":"External manuscript. No OpenMath proof execution, formalization or independent mathematical acceptance is inferred.","sha256":"c417858c9fe566daacb7738741dc66e211eea148b3da587eb5ee68eaad208d64"}],"published":{"value":"2026-10-07"}},"id":"openai-goldfeld@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","source_id":"openai-goldfeld","title":"OpenAI · proposed Goldfeld analytic densities and 2-converse","owner":"OpenAI","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","subject_scope":"OpenAI · proposed Goldfeld analytic densities and 2-converse","revision":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","content_sha256":"c417858c9fe566daacb7738741dc66e211eea148b3da587eb5ee68eaad208d64","update_method":"Manual primary-source revision check; changes require a new immutable source version.","identifier_policy":"Stable bibliographic identity, explicit revision and exact source-entry locators.","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"External full-text redistribution rights not asserted","terms_url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","attribution":"OpenAI","rights_note":"Only metadata and original OpenMath paraphrases are published; full text stays upstream.","check_interval_days":30}],"entries":[{"id":"birch-and-swinnerton-dyer-conjecture:entry:0:v1","source_version_id":"clay-millennium@observed-2026-10-01","item_id":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","wording":"For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at s = 1?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:entry:0:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","wording":"For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at s = 1?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":"birch-and-swinnerton-dyer-conjecture:entry:0:v1"},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2","source_version_id":"skinner-annals-2020@published-online-2020-02-13","item_id":"Introduction, journal page 330 (PDF page 2), classical theorem","url":"https://doi.org/10.4007/annals.2020.191.2.1","wording":"For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/Q) is finite.","wording_kind":"paraphrase","locator":"Introduction, journal page 330 (PDF page 2), classical theorem","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF pages 2 and 4, conjecture/refinement and low-rank theorem","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"Wiles distinguishes the rank statement from the refined leading-coefficient formula and records the analytic-rank-zero-or-one theorem.","wording_kind":"paraphrase","locator":"PDF pages 2 and 4, conjecture/refinement and low-rank theorem","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2","source_version_id":"skinner-annals-2020@published-online-2020-02-13","item_id":"Abstract and Theorem A, journal pages 329–331","url":"https://doi.org/10.4007/annals.2020.191.2.1","wording":"For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.","wording_kind":"paraphrase","locator":"Abstract and Theorem A, journal pages 329–331","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","source_version_id":"arxiv-1512.06894@arxiv-v1","item_id":"Theorem 1.2.1, PDF pages 1–2","url":"https://arxiv.org/abs/1512.06894v1","wording":"For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.","wording_kind":"paraphrase","locator":"Theorem 1.2.1, PDF pages 1–2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","source_version_id":"arxiv-2409.01350@arxiv-v2","item_id":"Theorem 1.5, PDF page 4","url":"https://arxiv.org/abs/2409.01350v2","wording":"For semistable E/Q of analytic rank 0 or 1 and a good supersingular prime p ≥ 5, the p-part of the BSD leading-coefficient formula holds.","wording_kind":"paraphrase","locator":"Theorem 1.5, PDF page 4","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","source_version_id":"arxiv-2409.01350@arxiv-v2","item_id":"Theorem 1.7, PDF page 4, restricted to base curve 46a1","url":"https://arxiv.org/abs/2409.01350v2","wording":"Let E be Cremona curve 46a1. For squarefree M>1 coprime to 46, if L(E^M,1)≠0 and E has ordinary reduction at every prime dividing M, E^M satisfies full BSD. Infinitely many M meet these conditions.","wording_kind":"paraphrase","locator":"Theorem 1.7, PDF page 4, restricted to base curve 46a1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1","source_version_id":"arxiv-2601.16044@arxiv-v3","item_id":"Lemma 2.18, Algorithms 1–2, Remark 2.20 and Section 2.7, v3 PDF pages 10–13","url":"https://arxiv.org/abs/2601.16044v3","wording":"Banwait–Huang give sufficient-condition algorithms for BSD twist subfamilies and list eligible curves of conductor <500,000. Section 2.7 retains 46a1 but identifies a nonempty-prime-support gap for 62a1, 66b1, 105a1 and 141c1 in an earlier infinitude list.","wording_kind":"paraphrase","locator":"Lemma 2.18, Algorithms 1–2, Remark 2.20 and Section 2.7, v3 PDF pages 10–13","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1","source_version_id":"arxiv-2601.16044@arxiv-v3","item_id":"Section 2.7, v3 PDF page 13","url":"https://arxiv.org/abs/2601.16044v3","wording":"The 2026 eligible-family list retains Cremona curve 46a1; the discrepancy concerns four other base curves.","wording_kind":"paraphrase","locator":"Section 2.7, v3 PDF page 13","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","source_version_id":"arxiv-1010.2431@arxiv-v3","item_id":"Theorem 1.2 and Table 9, v3 PDF pages 2 and 23","url":"https://arxiv.org/abs/1010.2431v3","wording":"Miller’s computation-assisted proof establishes full BSD for 16,714 of the 16,725 elliptic curves of analytic rank at most one and conductor <5000 in the paper’s enumeration.","wording_kind":"paraphrase","locator":"Theorem 1.2 and Table 9, v3 PDF pages 2 and 23","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v2","source_version_id":"kent-bsd-project@0e9519aaeb5c8b8c524f95925eecb970d86bc5b1-metadata-2026-10-10","item_id":"Pinned README, PHASE_STATE.md and lean/BSD/RankEquality.lean","url":"https://github.com/owenpkent/birch-swinnerton-dyer/tree/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","wording":"This individual AI-augmented research project reports numerical experiments and proof-search scaffolding. Its phase record states that Lean verification has not started; RankEquality.lean contains sorry targets.","wording_kind":"paraphrase","locator":"Pinned README, PHASE_STATE.md and lean/BSD/RankEquality.lean","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1","source_version_id":"icarm-curve-273@observed-2026-10-10","item_id":"Curve #273: a-invariants, rank (lower bound), witness and update date","url":"https://elliptic-rank.icarm.cloud/curve/273","wording":"The primary record supplies a specific elliptic curve over Q and 30 rational-point witnesses, with a reported rank lower bound of 30.","wording_kind":"paraphrase","locator":"Curve #273: a-invariants, rank (lower bound), witness and update date","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF page 2: rank conjecture; all finite Euler factors restored for the standard L-function","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"For every elliptic curve E/Q, rank E(Q) equals ord_{s=1} L(E,s).","wording_kind":"paraphrase","locator":"PDF page 2: rank conjecture; all finite Euler factors restored for the standard L-function","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF page 2, refinement in Remark 1; standard full-L, total-real-period convention","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"Strong BSD predicts rank equality, finite Sha, and the leading coefficient formula with real period, regulator, Tamagawa factors and torsion.","wording_kind":"paraphrase","locator":"PDF page 2, refinement in Remark 1; standard full-L, total-real-period convention","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF page 1: Mordell theorem and rank definition","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"E(Q) is isomorphic to Z^r plus a finite torsion group; it is finite exactly when r = 0.","wording_kind":"paraphrase","locator":"PDF page 1: Mordell theorem and rank definition","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF pages 1–2: Mordell theorem and “In particular” consequence of rank BSD","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"Under BSD rank equality, L(E,1) = 0 exactly when E(Q) is infinite.","wording_kind":"paraphrase","locator":"PDF pages 1–2: Mordell theorem and “In particular” consequence of rank BSD","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"§1 Early History, PDF page 3: odd squarefree Tunnell criterion","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"For odd positive squarefree n, BSD for y²=x³−n²x would make a finite ternary-quadratic count criterion equivalent to being a congruent number.","wording_kind":"paraphrase","locator":"§1 Early History, PDF page 3: odd squarefree Tunnell criterion","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","source_version_id":"openai-bsd-low-selmer@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","item_id":"Theorem 1.1 and normalization, PDF page 3; companion dependencies §2","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","wording":"The provider claims full BSD for E/Q whose full q-power Selmer group has corank 0 or 1 for some prime q, without additional reduction or residual hypotheses.","wording_kind":"paraphrase","locator":"Theorem 1.1 and normalization, PDF page 3; companion dependencies §2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1","source_version_id":"openai-goldfeld@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","item_id":"Theorems 1.1–1.2, PDF page 2; separation of mean-rank assertion, page 3","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","wording":"The provider claims analytic ranks 0 and 1 each have density 1/2 among signed squarefree quadratic twists of every E/Q ordered by absolute parameter.","wording_kind":"paraphrase","locator":"Theorems 1.1–1.2, PDF page 2; separation of mean-rank assertion, page 3","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-origin-history:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"§2 Recent History, PDF page 3; reference [4], Notes on elliptic curves II (1965)","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"Computer experiments in the early 1960s motivated the conjecture; Notes on elliptic curves II was published in 1965.","wording_kind":"paraphrase","locator":"§2 Recent History, PDF page 3; reference [4], Notes on elliptic curves II (1965)","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-gross-zagier-history:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"§2, PDF pages 3–4; reference [11], Gross–Zagier (1986)","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"The Gross–Zagier height formula links Heegner points to central L-derivatives and supplies a key low-rank input.","wording_kind":"paraphrase","locator":"§2, PDF pages 3–4; reference [11], Gross–Zagier (1986)","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-modularity-history:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF page 4, low-rank synthesis; references [1], [24], [25]","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"General modularity of elliptic curves over Q removes the modularity restriction from the low-analytic-rank rank theorem.","wording_kind":"paraphrase","locator":"PDF page 4, low-rank synthesis; references [1], [24], [25]","supersedes":null}],"reports":[{"id":"birch-and-swinnerton-dyer-conjecture:report:0:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:entry:0:v1","state":"reported_open","source_label":"Unsolved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-01","evidence_url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:report:0:v2","entry_id":"birch-and-swinnerton-dyer-conjecture:entry:0:v2","state":"reported_open","source_label":"Unsolved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":"birch-and-swinnerton-dyer-conjecture:report:0:v1"},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2","state":"resolved","source_label":"Rank equality at analytic rank 0 or 1","scope":"Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.","source_date":"2020-02-13","observed_on":"2026-10-10","evidence_url":"https://doi.org/10.4007/annals.2020.191.2.1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2","state":"resolved","source_label":"Rank-one converse assuming finite Sha","scope":"Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.","source_date":"2020-02-13","observed_on":"2026-10-10","evidence_url":"https://doi.org/10.4007/annals.2020.191.2.1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","state":"resolved","source_label":"Rank-one formula · selected p-parts","scope":"The p-part of the leading-coefficient formula, not an assertion at every prime. Irreducibility removes p-primary rational torsion.","source_date":"2015-12-21","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/1512.06894v1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","state":"resolved","source_label":"Low-rank formula at supersingular p ≥ 5","scope":"Theorem 1.5 restricted to p ≥ 5; its additional p=3 conditions and quadratic-twist extension are not asserted here.","source_date":"2024-09-11","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2409.01350v2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","state":"resolved","source_label":"Full BSD in a specified infinite twist family","scope":"Analytic-rank-zero quadratic twists of one named non-CM base curve; rank, finite Sha and the entire leading-coefficient identity.","source_date":"2024-09-11","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2409.01350v2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","state":"resolved","source_label":"Computer-assisted full formula · 16,714 curves","scope":"Historical finite-corpus proof. Table 9 exceptions are limitations of this paper, not a statement of current unresolved status.","source_date":"2011-12-21","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/1010.2431v3","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-target:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1","state":"reported_open","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-full-target:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","state":"reported_open","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-sha-finiteness-target:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","state":"reported_open","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1","state":"resolved","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-strong-implies-rank:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","state":"resolved","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1","state":"resolved","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1","state":"resolved","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-sha-order-consequence:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1","state":"resolved","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-low-selmer-proposal:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","state":"proposed_solution","source_label":"Proposed source assertion awaiting assessment","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld-proposal:report:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1","state":"proposed_solution","source_label":"Proposed source assertion awaiting assessment","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","supersedes":null}]},"status":{"state":"open","label":"Open","extent":"general_case","summary":"Rank equality is proved at analytic rank 0 or 1. Further cited results give a conditional converse, selected p-parts and full BSD in specified twist families and finite computations.","basis":"source","attributed_to":"Clay · Millennium Problems","as_of":"2026-10-10","progress_label":"Special cases & conditional results","progress_claim_ids":["claim-bsd-analytic-rank-at-most-one","claim-bsd-rank-one-converse-finite-sha","claim-bsd-semistable-rank-one-p-part","claim-bsd-supersingular-p-part","claim-bsd-46a1-quadratic-twists","claim-bsd-miller-finite-computations"],"remaining":"Rank equality in general, especially analytic rank ≥ 2, remains open. General finiteness of Sha and the complete leading-coefficient formula are stronger unresolved assertions.","views":["open","partial"],"knowledge_categories":["Special cases proved","Conditional results","Conditional implications"]}},"data_mode":"published catalog snapshot; historical Phase 0 checks only","context_packs":[],"reference_fixture_checks":[],"repeat_instructions":"No verification task is attached to this discovery record."},"payload_sha256":"82202957a8c90d8af34aab9e6d8d90f850f634b0bed752c79a772614c19adcaa","canonicalization":"RFC 8785 JSON Canonicalization Scheme. Hash the payload, excluding this envelope. Immutable embedded domain hashes retain their original profile."}