{"schema_version":"source-reader-v1","record":{"source_id":"openai-bsd-low-selmer","versions":[{"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}],"entries":[{"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}],"problems":[{"slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","publication_version":17}],"selected_version_id":"openai-bsd-low-selmer@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","available_versions":[{"id":"openai-bsd-low-selmer@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","revision":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","observed_on":"2026-10-10","title":"OpenAI · proposed full BSD from low Selmer corank","publication":"preprint","content_sha256":"64b46603e174462e5bb32d242852eb7f0afe6e519f26534e42fd3f178a16b885","used_in_current_publications":true}],"usages":[{"id":"source-use-29b064a38d7b255590f3","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","problem":{"slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","publication_version":17},"kind":"activity","title":"Company AI proposals: Selmer converses, leading terms and twist densities","claim":null,"fragment":"activity","assessment":null,"superseded":false,"formalization":null,"update":{"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"]}},{"id":"source-use-0ce9566fb8c08127e1d9","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","problem":{"slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","publication_version":17},"kind":"assessment","title":"Proposed source assertion awaiting assessment","claim":{"id":"claim-bsd-openai-low-selmer-proposal","version":2,"slug":"bsd-openai-low-selmer-proposal","title":"Proposed full BSD at Selmer corank zero or one"},"fragment":"claim-bsd-openai-low-selmer-proposal","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-630154c0840be6b43f81","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","problem":{"slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","publication_version":17},"kind":"milestone","title":"OpenAI’s proposed full low-Selmer-corank formula","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-05a8240d19d9c2579b71","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","problem":{"slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","publication_version":17},"kind":"question","title":"Birch and Swinnerton-Dyer conjecture","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-d0348c7770454c8daee4","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1","problem":{"slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","publication_version":17},"kind":"statement","title":"Proposed full BSD at Selmer corank zero or one","claim":{"id":"claim-bsd-openai-low-selmer-proposal","version":2,"slug":"bsd-openai-low-selmer-proposal","title":"Proposed full BSD at Selmer corank zero or one"},"fragment":"claim-bsd-openai-low-selmer-proposal","assessment":null,"superseded":false,"formalization":null,"update":null}],"linked_entries":[],"linked_sources":[]}}