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The rational Hodge conjecture for CM abelian varieties

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne’s theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

Five current papers; three withdrawn on 6 October 2026 and three replaced by later editions.

Nine problems have been reported by others; the atlas has not judged them.

The paper states

Theorem 1.1

In The rational Hodge conjecture for CM abelian varieties, 6 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Nine other possible matches are listed below the card, among them Theorem 1.2 (Quadratic actions in dimension at most five) in Weil classes and Hodge classes on abelian powers and Theorem 1.1 (Powers of abelian-cover Jacobians) in Abelian covers, Gale correspondences, and the Hodge conjecture for powers.

Let A be a complex CM abelian variety. For every integer p\geq0, the Betti cycle-class map

\operatorname{cl}_B:\CH^p(A)_{\Q} \longrightarrow H^{2p}(A,\Q)\cap H^{p,p}(A)

is surjective, where \CH^p(A)_{\Q}=\CH^p(A)\otimes_{\Z}\Q and the intersection is taken in H^{2p}(A,\C). In particular, the rational Hodge conjecture holds in every codimension on every finite product of complex CM abelian varieties and on every power of such a product.

Nine more possible matches

  • Theorem 1.2 (Quadratic actions in dimension at most five) in Weil classes and Hodge classes on abelian powers, 6 October 2026.
  • Theorem 1.1 (Powers of abelian-cover Jacobians) in Abelian covers, Gale correspondences, and the Hodge conjecture for powers, 6 October 2026.
  • Theorem 1.1 (Split Weil sixfolds) in Weil classes and Hodge classes on abelian powers, 6 October 2026.
  • Theorem 1.1 (Recovery at a point with full Hodge group) in A Conditional Reduction for Algebraic Kuga–Satake Correspondences, 10 September 2026.
  • Theorem 1.2 (Powers of diagonal complete intersections) in Abelian covers, Gale correspondences, and the Hodge conjecture for powers, 6 October 2026.
  • Theorem 1.2 in Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus, 30 September 2026.
  • Theorem 1.2 (Specialization in a polarized component) in A Conditional Reduction for Algebraic Kuga–Satake Correspondences, 10 September 2026.
  • Theorem 1.1 in Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus, 30 September 2026.
  • Corollary 1.3 in A Conditional Reduction for Algebraic Kuga–Satake Correspondences, 10 September 2026.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Five current papers, each readable here as a page beside its PDF. One explanation by others.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

The rational Hodge conjecture for CM abelian varieties

6 October 2026, current edition.

Read it herePDF

This version updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 30 September 2026

We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.

Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus

30 September 2026, current edition.

Read it herePDF

We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in \(\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4\). This includes every ample P-polarized K3 surface, including all Picard jumps, for \(P=\mathbb U\oplus D_8(-1)\oplus D_4(-1)\). On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.

Weil classes and Hodge classes on abelian powers

6 October 2026, current edition.

Read it herePDF

This version updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 30 September 2026

We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature \((3,3)\). We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms. The proof uses the companion theorem on the rational Hodge conjecture for CM abelian varieties.

Abelian covers, Gale correspondences, and the Hodge conjecture for powers

6 October 2026, current edition.

Read it herePDF

This version updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 30 September 2026

We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus. We also prove the conjecture on every self-power of a very general member of the full smooth labelled family of diagonal complete intersections cut out by at most two equations of a common degree, in every dimension and every degree at least two.

A Conditional Reduction for Algebraic Kuga–Satake Correspondences

10 September 2026, current edition.

Read it herePDF

For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover. If this input holds very generally in a polarized component, the prescribed correspondence is algebraic throughout that component, for every choice of standard data on the transcendental part. The existence of the initial correspondence remains a hypothesis.

Previous editions

Withdrawn from this family

These papers were listed under this family when the release was published (CONTENTS.md at its first commit), when it was titled “Hodge and Kuga–Satake results for all projective K3 surfaces”. OpenAI has since withdrawn them; each notice is quoted below.

Algebraicity of Weil classes on split abelian eightfolds

The proof contains a sign error in the stabilization-trace argument used to obtain negative double points. In the manuscript’s signed double-point convention, write $I(f_1)=-m$ with $m>0$. The proof assigns each reverse stabilization trace sign $+1$ and therefore claims that inserting $m$ such traces makes the signed count zero.

Accounting for the opposite source orientations of the two branches of the standard cusp gives sign $-1$ for each reverse trace in this convention. The resulting count is therefore

$$I_{\mathrm{new}}=I(f_1)-m=-2m\ne0.$$

The Eliashberg-Murphy cancellation theorem invoked at this step requires zero signed double-point count, so its hypothesis is not met. The subsequent oriented-surgery and embedded-brane construction is therefore unsupported, and the paper does not establish its claimed algebraicity theorem.

This withdrawal concerns the proof; it does not assert that the mathematical statement is false.

Withdrawn on 6 October 2026.

The rational Hodge conjecture for products of K3 surfaces

The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper’s claimed rational Hodge theorem for products of projective K3 surfaces unproved.

This withdrawal concerns the proof; it does not assert that the mathematical statement is false.

Withdrawn on 6 October 2026.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

Other reports of proof verification

Other people’s accounts of running proof-verification software, with links to their reports.

  • ityonemo, openai-wiles, 9 October 2026. Source

    In a non-Lean checker the withdrawn Weil-classes argument fails at the stabilisation-trace count when the sign given in the withdrawal notice is used and passes only with the manuscript’s sign, matching OpenAI’s account of the error.

Problems reported by others

Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.

  • VibeMathed, VibeMathed: math problems solved with AI, 11 October 2026. Source

    Two entries (Hodge conjecture for products of K3 surfaces; Kuga-Satake Hodge conjecture) are marked retracted following OpenAI’s withdrawal notices.

  • Elliot Glazer, Tweet reporting a flaw in the Weil classes manuscript, 8 October 2026. Source

    A tweet says an AI check of the Weil classes manuscript, which had no Lean, found it flawed, about two and a half hours before OpenAI’s withdrawal notice; whether OpenAI acted on it is not stated.

  • Hacker News readers, OpenAI withdraws three mathematical results, 8 October 2026. Source

    The withdrawn papers all trace to a sign error in the Weil classes on split abelian eightfolds manuscript.

  • From retractionwatch.com, 8 October 2026. Source

    OpenAI withdrew three manuscripts on 7 October (Weil classes on split abelian eightfolds, Kuga-Satake correspondences for K3 surfaces, and the rational Hodge conjecture for products of K3 surfaces) because a sign error invalidated a stabilisation-trace argument and the construction used by two dependent papers; 14 other manuscripts were revised.

  • From github.com/openai/math, 8 October 2026. Source

    Withdrawal notice: Algebraicity of Weil classes on split abelian eightfolds (sign error in a stabilisation-trace cancellation argument).

  • From github.com/openai/math, 8 October 2026. Source

    Withdrawal notice: Algebraicity of Kuga-Satake Correspondences for K3 Surfaces (depends on the flawed construction).

  • From github.com/openai/math, 8 October 2026. Source

    Withdrawal notice: The rational Hodge conjecture for products of K3 surfaces (depends on the flawed construction).

  • From scottaaronson.blog, 7 October 2026. Source

    Comment on 8 Oct notes OpenAI withdrew the rational Hodge conjecture for products of K3 surfaces paper after a sign error; Aaronson had not seen it and expects other bugs.

  • mrnathanhumphrey-droid, mrnathanhumphrey-droid/openai-split-eightfold-gap: sign-convention check of Proposition 3.1, 7 October 2026. Source

    Sign-convention mismatch in Proposition 3.1 identified by exact scripts on 7 Oct 2026.

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source

    Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

What these papers cite in the release, and what cites them

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Within this family

Papers elsewhere in the release that this family’s papers cite

Papers elsewhere in the release that cite this family’s papers

Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Landmark of the century”. The page says that nothing on it is verified.

    Author not stated. Source

  • From github.com/jkbennitt/math-release-atlas, 9 October 2026. Source

    The math-release-atlas shows a fixed caution on this family: it is the Hodge conjecture for CM abelian varieties, not the full Hodge conjecture.

  • Kevin Buzzard, Algebraic number theory results without Lean, 7 October 2026. Source

    An expert flags that the release gives no Lean for the Hodge conjecture for CM abelian varieties and suggests it may be the last mile of a programme already close to done rather than a new idea.

  • Raemon, Are OpenAI’s math results “creative” in an important way?, 7 October 2026. Source

    Pepin Lehalleur singled out the Kuga-Satake algebraicity paper as the most original they had skimmed; that paper was among the three withdrawn on 7 Oct.

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 032 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 032 The release has no Lean statement for this family. Open the form