AI Proved What?

Episode 065, October 2026

Virasoro everywhere?

Virasoro constraints for complete intersections and projective-bundle towers

Covers result 065 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.

Play Episode 065: Virasoro everywhere? Blue outlines of a genus-two surface, a sphere and a torus, above the equation L k of Z X equals zero.

5:47. English captions available. The video loads from YouTube when you press play.

Chapters

  1. 0:00Equations for curve counts
  2. 1:28The claims
  3. 2:49Degenerations and positivity
  4. 4:04Projective bundles
  5. 5:03Where it stands

The claims, in plain words

Gromov–Witten invariants count curves of a given genus and degree in a variety X, passing through chosen cycles, with extra ψ classes at the marked points. These descendant invariants fit into one generating function ZX. For a single point as the target, Witten conjectured and Kontsevich proved that Z satisfies the KdV hierarchy, which Dijkgraaf, Verlinde and Verlinde rewrote, together with the string equation, as Virasoro constraints LkZ = 0 for k ≥ −1. They pin down every number, like 1/24 in genus one or 1/1152 in genus two.

In 1997 Eguchi, Hori and Xiong conjectured such constraints for Fano varieties, with operators built from the Poincaré pairing, a Hodge grading and the first Chern class; a grading proposed by Katz, in Getzler’s formulation, extends the conjecture to every smooth projective variety. It was known in genus zero (Liu and Tian), for curves (Okounkov and Pandharipande), for Calabi–Yau targets of dimension at least three, and for semisimple quantum cohomology (Givental, Teleman). But a variety with off-diagonal Hodge classes, like a cubic threefold, never has semisimple quantum cohomology, so for many complete intersections the higher-genus constraints stayed open.

Result 065 is two OpenAI preprints. The first (September 24, 2026; 48 pages) claims the Virasoro conjecture for every smooth complete intersection in projective space: every genus, every curve class, and arbitrary insertions, including primitive and odd cohomology classes, with no semisimplicity assumption. The second (October 5, 2026; 41 pages) claims that the constraints pass from any smooth projective base to the projectivization of any vector bundle of rank at least two, with no splitting or positivity assumed, and so to towers of projective bundles.

What they don’t claim

The results concern the ordinary (unreduced) descendant theory of complete intersections in projective space and of projective bundles over bases where the constraints are already known; they do not prove the conjecture for every smooth projective variety.

Where it stands

Posted as preprints
OpenAI, dated September 24 and October 5, 2026, released October 6
Peer reviewed
Not yet
Checked in Lean
No: OpenAI published no Lean proof for this result

Last checked October 19, 2026. Neither preprint has been revised, and neither cites another paper from the release. Their inputs include degeneration formulas and the reconstruction of Argüz, Bousseau, Pandharipande and Zvonkine, Steenbrink’s limiting Hodge structures, Coates, Givental and Tseng’s toric-bundle transfer, Iritani’s shift operators, and quantum Riemann–Roch. We found no public assessment by specialists.

How the proofs go

  1. The Hodge grading. The operators weight each class in Hp,q by p − dim X/2, and in both papers’ normalization the zero-mode operator carries the constant χ/16 − str(μ2)/4: 1/16 for a cubic surface, −1 for the cubic threefold (with five classes of type (2,1)). Primitive classes like these need not extend one by one through a degeneration.
  2. Degeneration. Write one defining degree as a + b and use the family gagb = t fr: after resolving, X degenerates into U ∪D BlSM, with U and M of smaller degree and D, S of smaller dimension. This sets up an induction on dimension and degrees, with projective-bundle towers from the toric-bundle transfer.
  3. Testing the error. The failure of each equation is treated as an error tensor and tested by contractions on the first page of the weight spectral sequence; relative caps built from Hu–Li–Ruan fiber invariants reproduce the contact pairing at each seam, and an alternating sum over switches along a chain of ruled pieces cancels. Pairing the error with its own conjugate is itself a passed test, and Hodge–Riemann positivity makes it a squared norm, so the error vanishes.
  4. Projective bundles. Put B and X = ℙB(E) inside W = ℙB(E ⊕ 𝒪), with a ℂ* action whose fixed parts are B and X. Localization relates the theories; Iritani’s shift operator has spectral values λ + h with hr(h + λ) = y (r the rank): near y = 0 one branch belongs to B and r to X. The covering is connected away from its branch points, so monodromy carries the equations from the base branch to the bundle branches (for r = 2, λ = 1, a loop around y = 4/27 swaps them). Quantum Riemann–Roch and a scalar normalization finish.

The numbers in the video

  • A short Python script computes Witten–Kontsevich numbers from the string equation and the Dijkgraaf–Verlinde–Verlinde recursion (1/24, 1/1152, 1/82944, and ⟨τ3g−2⟩g = 1/(24g g!) up to genus 5), Hodge numbers of hypersurfaces from Griffiths’ residues, the zero-mode constants, and the root paths of h2(h + 1) = y around y = 4/27.

Read the source

The 2 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

Transcript

Read the full script

0:00Equations for curve counts

Gromov–Witten invariants count curves of a given genus and degree in a variety X, passing through chosen cycles, with extra psi classes at the marked points. These are the descendant invariants, and all of them fit into one generating function, Z. For a single point as the target, Witten conjectured and Kontsevich proved that Z satisfies the KdV hierarchy, which Dijkgraaf, Verlinde and Verlinde rewrote, together with the string equation, as Virasoro constraints: L_k Z = 0, for k from minus one up. They pin down every number, like 1/24 in genus one, or 1/1152 in genus two. In 1997, Eguchi, Hori and Xiong conjectured such constraints for Fano varieties, with operators built from the Poincaré pairing, a Hodge grading and the first Chern class; a grading proposed by Katz, in Getzler’s formulation, extends the conjecture to every smooth projective variety. It is known in genus zero by Liu and Tian, for curves by Okounkov and Pandharipande, for Calabi–Yau targets of dimension three or more, where counting dimensions suffices, and for semisimple quantum cohomology by Givental and Teleman. But a variety with off-diagonal Hodge classes, like a cubic threefold, never has semisimple quantum cohomology, so for many complete intersections the higher-genus constraints stayed open.

1:28The claims

Result 065 is two OpenAI preprints. The first, 48 pages, dated September 24, claims the Virasoro conjecture for every smooth complete intersection in projective space: every genus, every curve class, and arbitrary insertions, including primitive and odd cohomology classes, with no semisimplicity assumption. The second, 41 pages, dated October 5, claims that the constraints pass from any smooth projective base to the projectivization of any vector bundle of rank at least two, with no splitting or positivity assumed, and so to towers of projective bundles. Here the Hodge grading matters. The operators weight each cohomology class by its first Hodge index minus half the dimension, and in both papers’ normalization the zero-mode operator carries the constant: the Euler characteristic over sixteen, minus a quarter of the supertrace of that grading squared. For a cubic surface, with seven classes in the middle, this constant is 1/16; for the cubic threefold, with five classes of type (2,1), it is minus one, and no earlier theorem covered it in higher genus. Primitive classes like these need not extend one by one through a degeneration, and that is the first obstacle.

2:49Degenerations and positivity

The first paper argues by degeneration. Factor one defining equation and resolve the family: the complete intersection degenerates into two pieces, U and a blowup of M, glued along D, where U and M have smaller degree and D has smaller dimension. As in work of Argüz, Bousseau, Pandharipande and Zvonkine, this sets up an induction on dimension and degrees, with projective-bundle towers handled by the toric-bundle transfer of Coates, Givental and Tseng. Two things block a direct transfer: primitive classes need not extend through the degeneration, and the Virasoro contractions weight indices by Hodge degree. So the paper treats the failure of each equation as an error tensor, and tests it. Contractions happen on the first page of the weight spectral sequence; relative caps, built from fiber invariants of Hu, Li and Ruan, reproduce the contact pairing at each seam; and an alternating sum over switches along a long chain of ruled pieces cancels. Finally, Hodge–Riemann positivity: pairing the error with its own conjugate gives a squared norm; since that is itself a passed test, it vanishes, so the error is zero.

4:04Projective bundles

The second paper puts the base B and the bundle’s projectivization X inside one space, the projectivization of E plus a trivial line, with a circle action whose fixed parts are B and X. Localization relates the three theories. Iritani’s shift operator then has spectral values λ + h, where hʳ(h + λ) = y, and r is the rank: r + 1 branches. Near y = 0, one branch belongs to the base, and the other r belong to the bundle. The equations hold on the base branch. The covering is connected away from its branch points, so moving y around a loop can carry the base branch onto a bundle branch: with rank two and λ = 1, a loop around y = 4/27 swaps them. Monodromy transports the equations to every branch. Quantum Riemann–Roch identifies them with the ordinary constraints on X, and a final step fixes the scalar normalization.

5:03Where it stands

Neither preprint is peer reviewed yet, and neither has been revised. There is no Lean formalization, and neither cites another paper from the release. Their inputs include degeneration formulas and the reconstruction of Argüz, Bousseau, Pandharipande and Zvonkine, Steenbrink’s limiting Hodge structures, Coates, Givental and Tseng, Iritani’s shift operators, and quantum Riemann–Roch. We found no public assessment. If they hold, every smooth complete intersection in projective space, and every tower of projective bundles over a base where the constraints are known, has its whole descendant theory organized by the Virasoro equations, in every genus.

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.