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Birational geometry · cubic fourfolds · Hodge theory · surface monodromy

Very General Cubic Fourfold Irrationality — Retained Monodromy Models

Collaboration beta

The packet tries to rule out a rational parametrization by forcing a Hodge-theoretic surface carrier and eliminating possible carrier models. Although it postdates the cited 2025/2026 proof sources, its distinct route remains conditional and is not that external proof.

X⊂PC5very general smooth cubic⇒X≁birP4
Known results and sources
An abstract cubic fourfold beside a breaking rational parameter grid, labeled Very General Cubic Fourfolds and Nonrational 2025.
Current literature reports very general cubic fourfolds nonrational; the image is explanatory and not proof evidence.

Research problem

Exact mathematical statement

For a very general smooth cubic hypersurface X⊂PC5X\subset\mathbf P^5_{\mathbf C}, prove

Xis not rational;∄φ:P4⇢Xbirational.X\text{ is not rational};\qquad \nexists\,\phi: \mathbf P^4\dashrightarrow X\text{ birational}.

The retained v6 source explicitly says its full irrationality theorem is open and its carrier/monodromy argument is conditional. External literature published before the source reports the theorem solved; that earlier literature does not convert this separate packet into proof evidence.

Problem infographic

Problem at a glance

Explainer defining a smooth cubic fourfold, recording the 2025/2026 nonrationality sources, and separating them from the later conditional packet dated 2026-08-13.
The 2025 proof preprint and 2026 revision predate the retained 2026-08-13 conditional packet; the packet is not proof evidence for the external result.

Current mathematical picture

Where work on Very General Cubic Fourfold Irrationality — Retained Monodromy Models stands

Solved

Current primary literature treats the question as solved: Katzarkov, Kontsevich, Pantev, and Yu state and prove that a very general cubic fourfold is nonrational; the March 2026 revision retains that claim; Schreieder’s ICM 2026 survey presents it as a theorem; and Guéré’s March 2026 follow-up explicitly builds on their irrationality work. This is a literature-status record, not independent ProofAtlas verification or proof acceptance.

Reader boundaryHistorical research record

The retained argument stages below document an earlier investigation. They do not reopen this solved question or advertise new contribution tasks.

Archived quantitative snapshot

The retained investigation in numbers

This reconstructs the source's normalized mathematical content at the archived current snapshot. It describes historical material, not current work or progress toward an already solved question.

1,234current normalized lines of mathematical investigationArchived source snapshot
Argument development
1,035 · 84%
Explored or eliminated routes
31 · 3%
Computational analysis
46 · 4%
Open obligations
43 · 3%
Definitions and setup
79 · 6%
How to interpret the archived roles

The role composition is an exact reconstruction from the retained content map. Labels such as open obligations describe the historical source classification; they are not current ProofAtlas tasks or invitations to continue this solved question.

Archived route inventory

Historical routes retained from the investigation

These records preserve the approaches explored in the retained investigation. They are archival: none is a current ProofAtlas route or an invitation to continue the solved question.

2 retained records
Historical route recordSimple discriminant mismatch

The source gives an explicit index-three gluing and says the contradiction is closed as false. Use equisingular anti-invariant monodromy with a valid transfer theorem or a mixed second-order period obstruction.

Archived disposition · Narrowed route
Historical route recordAmbient monodromy without transfer

The v6 audit says the actual carrier-family image may lie in a monodromy-reducing locus. Prove dominance, generic finiteness, or sufficiently large monodromy for the carrier-family map.

Archived disposition · Narrowed route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusSolved

Current primary literature treats the question as solved: Katzarkov, Kontsevich, Pantev, and Yu state and prove that a very general cubic fourfold is nonrational; the March 2026 revision retains that claim; Schreieder’s ICM 2026 survey presents it as a theorem; and Guéré’s March 2026 follow-up explicitly builds on their irrationality work. This is a literature-status record, not independent ProofAtlas verification or proof acceptance.

[1][2][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintGuéré’s follow-up derives a Hodge-structure restriction for every rational smooth complex cubic fourfold, explicitly following the Katzarkov–Kontsevich–Pantev–Yu work.[3]
  2. Authoritative summarySchreieder’s ICM 2026 survey presents the result as Theorem 1.4 and identifies stable rationality of cubic fourfolds as a next open case.[2]
  3. PreprintKatzarkov, Kontsevich, Pantev, and Yu introduced Hodge atoms and stated a proof that a very general cubic fourfold is not rational.[1]
3 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusVery General Cubic Fourfold Irrationality
Related problemstable rationality of cubic fourfolds

The ICM survey identifies stable rationality, a weaker property than rationality, as a separate remaining open direction. Proving stable irrationality would be stronger than proving irrationality.

[2]
Solved special caserational special cubic fourfolds

The survey notes that many special smooth cubic fourfolds remain rational; the theorem concerns the very general member.

[2]
Related problemHodge structure of rational cubic fourfolds

Guéré proves that rational smooth complex cubic fourfolds have primitive cohomology matching the twisted middle cohomology of a projective K3 surface.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetFormalize smooth cubic hypersurfaces over the complex numbers, birationality, very-general parameter statements, and the Hodge-atom obstruction used by the proof.
  • Formalization targetFormalize the relevant quantum multiplication, F-bundle, Hodge-structure, and blowup-invariance machinery in a pinned proof-assistant ecosystem.
  • Formalization targetCompile an exact no-sorry theorem with full axiom and dependency-cone audits and independently review alignment to the very-general irrationality statement.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Birational Invariants from Hodge Structures and Quantum Multiplicationpreprint · Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev, Tony Yue Yu · arXiv · 2025; revised 2026-03-06 · ARXIV 2508.05105 · accessed Aug 14, 2026
  2. 2
    Rationality of hypersurfacessurvey or monograph · Stefan Schreieder · Proceedings of the International Congress of Mathematicians 2026 · 2025; ICM 2026 proceedings survey · ARXIV 2510.13679 · accessed Aug 14, 2026
  3. 3
    On the irrationality of cubic fourfoldspreprint · Jérémy Guéré · arXiv · 2026-03-04 · ARXIV 2603.04518 · accessed Aug 14, 2026

Important qualifications

  • The proof source is an arXiv preprint, revised in March 2026; no peer-reviewed publication was located in the bounded review.
  • The solved status reports current primary literature: the proof preprint states the theorem, Schreieder’s ICM 2026 survey presents it as Theorem 1.4, and Guéré’s 2026 follow-up builds on it. ProofAtlas has not independently checked the proof.
  • The retained private packet is dated 2026-08-13, after the cited 2025 proof preprint and March 2026 revision. It remains a distinct source-reported conditional route, not proof evidence for the external theorem.
  • Scoped official-repository searches did not locate a checked formalization of the result; this negative result does not establish nonexistence elsewhere.
  • External metadata grants no proof acceptance, review, credit, publication, or rights authority.

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