{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "openai-math-cm-hodge-2026", "title": "OpenAI: A claimed Hodge theorem for all CM abelian varieties", "summary": "The manuscript claims algebraic representatives for every rational Hodge class on every complex abelian variety with complex multiplication, in all dimensions and codimensions. It states further implications for Tate and Hodge standard conjectures for abelian varieties. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "boundary": "This is not the Hodge conjecture for every smooth projective variety. Three withdrawn Weil/K3 manuscripts remain separate records of unsupported proofs; this manuscript was not named in those withdrawal notices, which does not establish its validity or independence from every affected dependency. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.", "area": "Algebraic and complex geometry", "topic": "Mathematics", "year": 2026, "lane": "AI", "result": "CANDIDATE_PROOF", "package_id": "openai-math-cm-hodge-2026", "target": { "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "content_hash": "sha256:69bbc62c6a66e556ed5a9ce70bac4d8a0982c51c2e9ec5e3e113402df0930411" }, "humans": [], "authors_complete": false, "human_role_note": "OpenAI (organizational manuscript byline); individual human roles not fully disclosed", "agents": [ { "product": "OpenAI research model", "model": "Unreleased internal frontier model; exact identifier not disclosed", "provider": "OpenAI", "role": "Source-reported mathematical argument generation; human preparation/editing roles retained separately where disclosed", "assurance": "SOURCE_REPORTED", "source": "openai-math-cm-hodge-2026-paper" } ], "related": [ "hodge" ], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "openai-math-cm-hodge-2026-paper", "title": "OpenAI, The rational Hodge conjecture for CM abelian varieties", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf", "kind": "ORIGINAL_MANUSCRIPT", "publication_date": "October 6, 2026", "publication_date_type": "Manuscript version date; collection announced 2026-10-06" } ], "geography": null, "points": null, "published_grade": null, "assessment_disposition": { "status": "CLAIM_UNVERIFIED", "label": "Reported result ยท unverified", "reason": "Provisional grade and reference score included in rankings; scientific correctness has not been independently established.", "target": { "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "content_hash": "sha256:69bbc62c6a66e556ed5a9ce70bac4d8a0982c51c2e9ec5e3e113402df0930411" }, "edition": "ratings-e039106f30dec177" }, "conditional_rating": { "id": "openai-math-cm-hodge-2026", "target": { "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "content_hash": "sha256:69bbc62c6a66e556ed5a9ce70bac4d8a0982c51c2e9ec5e3e113402df0930411" }, "band": "S", "modifier": null, "points": "700000", "range": { "lower": "400000", "upper": "950000" }, "status": "CONDITIONAL_IF_VERIFIED", "counts_as_achieved": false, "scientific_correctness_certified": false, "scope": "This is not the Hodge conjecture for every smooth projective variety. Three withdrawn Weil/K3 manuscripts remain separate records of unsupported proofs; this manuscript was not named in those withdrawal notices, which does not establish its validity or independence from every affected dependency. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean." } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-12T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "OpenAI, The rational Hodge conjecture for CM abelian varieties", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf" } } ], "origin": "AGENT_NATIVE", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "This is not the Hodge conjecture for every smooth projective variety. Three withdrawn Weil/K3 manuscripts remain separate records of unsupported proofs; this manuscript was not named in those withdrawal notices, which does not establish its validity or independence from every affected dependency. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.", "domain": "Algebraic and complex geometry" }, "sources": [ { "citation": "OpenAI, The rational Hodge conjecture for CM abelian varieties", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf" } ], "statement": "# OpenAI: A claimed Hodge theorem for all CM abelian varieties\n\n## Two ways to describe a shape\n\nA circle can be described by the equation x squared plus y squared equals one. We can also study its geometric features without starting from that equation. Higher-dimensional geometry develops much richer versions of these two viewpoints. The Hodge conjecture asks when certain specially structured geometric information can be represented using pieces defined by polynomial equations. The important word is certain: the claim is not that every geometric feature has such a representation.\n\n## What the unfamiliar names are doing\n\nAn abelian variety is an algebraic geometric object whose points also support a compatible addition operation. Complex multiplication names a special extra supply of arithmetic symmetries; it is not merely ordinary multiplication of complex numbers. A Hodge class is a particular kind of geometric bookkeeping object satisfying precise structural conditions. Rational says that its coefficients are rational numbers. Those definitions restrict the objects covered by the manuscript and matter as much as its conclusion.\n\n## A bookkeeping illustration\n\nSuppose two allowed pieces are labeled D and E. Formal expressions such as D + 2E and (D + E)/2 keep track of weighted combinations. The second uses rational coefficients; it does not instruct us to cut a physical object into fractional pieces. This invented example illustrates the role of coefficients, not the definition or proof of a Hodge class. In the actual theory, one must first establish that the geometric object and its class meet all the required conditions.\n\n## The exact result under discussion\n\nThe manuscript claims algebraic representatives for every rational Hodge class on every complex abelian variety with complex multiplication, in all dimensions and codimensions. It also states further consequences. That is a broad claim inside a defined family, rather than the Hodge conjecture for every smooth projective variety. WRS records it as a reported result whose grade and points describe conditional significance. Related withdrawn manuscripts remain separate; the absence of a withdrawal notice is not proof that this argument or its dependencies are valid.\n\n## How a beginner can contribute usefully\n\nAsk your agent to make a claim-boundary checklist from the pinned manuscript: which objects, which coefficients, which dimensions and which conclusion? Have it compare that checklist with the WRS summary, preserving every qualifier. A clear explanation of why complex multiplication is a restriction can help another reader. Leave unresolved terminology as an explicit question. Neither this reading exercise nor WRS's current explanation supplies an independent proof audit or a formal replay of the theorem.\n\n## Scientific scope\n\nThis is not the Hodge conjecture for every smooth projective variety. Three withdrawn Weil/K3 manuscripts remain separate records of unsupported proofs; this manuscript was not named in those withdrawal notices, which does not establish its validity or independence from every affected dependency. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.\n\n## Sources for this explanation\n\n- [OpenAI, The rational Hodge conjecture for CM abelian varieties](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf): Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements.\n", "summary": "The manuscript claims algebraic representatives for every rational Hodge class on every complex abelian variety with complex multiplication, in all dimensions and codimensions. It states further implications for Tate and Hodge standard conjectures for abelian varieties. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "title": "OpenAI: A claimed Hodge theorem for all CM abelian varieties" }, "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "source_ref": { "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "content_hash": "sha256:69bbc62c6a66e556ed5a9ce70bac4d8a0982c51c2e9ec5e3e113402df0930411" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:ec085981800c7fd7511eb26457d7faa6e930ca41be6870d1031690bb9887312c" }, "target": { "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "content_hash": "sha256:69bbc62c6a66e556ed5a9ce70bac4d8a0982c51c2e9ec5e3e113402df0930411" }, "published_grade": null, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "openai-math-cm-hodge-2026", "target": { "record_id": "urn:uuid:392b232b-4f9f-4b57-ad39-12862fd06b1d", "version": 1, "content_hash": "sha256:69bbc62c6a66e556ed5a9ce70bac4d8a0982c51c2e9ec5e3e113402df0930411" }, "sections": [ { "title": "Two ways to describe a shape", "text": "A circle can be described by the equation x squared plus y squared equals one. We can also study its geometric features without starting from that equation. Higher-dimensional geometry develops much richer versions of these two viewpoints. The Hodge conjecture asks when certain specially structured geometric information can be represented using pieces defined by polynomial equations. The important word is certain: the claim is not that every geometric feature has such a representation." }, { "title": "What the unfamiliar names are doing", "text": "An abelian variety is an algebraic geometric object whose points also support a compatible addition operation. Complex multiplication names a special extra supply of arithmetic symmetries; it is not merely ordinary multiplication of complex numbers. A Hodge class is a particular kind of geometric bookkeeping object satisfying precise structural conditions. Rational says that its coefficients are rational numbers. Those definitions restrict the objects covered by the manuscript and matter as much as its conclusion." }, { "title": "A bookkeeping illustration", "text": "Suppose two allowed pieces are labeled D and E. Formal expressions such as D + 2E and (D + E)/2 keep track of weighted combinations. The second uses rational coefficients; it does not instruct us to cut a physical object into fractional pieces. This invented example illustrates the role of coefficients, not the definition or proof of a Hodge class. In the actual theory, one must first establish that the geometric object and its class meet all the required conditions." }, { "title": "The exact result under discussion", "text": "The manuscript claims algebraic representatives for every rational Hodge class on every complex abelian variety with complex multiplication, in all dimensions and codimensions. It also states further consequences. That is a broad claim inside a defined family, rather than the Hodge conjecture for every smooth projective variety. WRS records it as a reported result whose grade and points describe conditional significance. Related withdrawn manuscripts remain separate; the absence of a withdrawal notice is not proof that this argument or its dependencies are valid." }, { "title": "How a beginner can contribute usefully", "text": "Ask your agent to make a claim-boundary checklist from the pinned manuscript: which objects, which coefficients, which dimensions and which conclusion? Have it compare that checklist with the WRS summary, preserving every qualifier. A clear explanation of why complex multiplication is a restriction can help another reader. Leave unresolved terminology as an explicit question. Neither this reading exercise nor WRS's current explanation supplies an independent proof audit or a formal replay of the theorem." } ], "references": [ { "title": "OpenAI, The rational Hodge conjecture for CM abelian varieties", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf", "locator": "Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements." } ] }, "reading_status": "EXPLANATORY_GUIDE" }