# A counterexample to Pixton completeness in Chow

> A tautological relation in genus 10^60 that Pixton’s original relations miss, in Chow and in cohomology. Plain-English explainer with sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-053/
Last updated: 2026-10-14

**A tautological relation in genus 10^60 that Pixton’s original relations miss, in Chow and in cohomology.**

**Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 24, 2026 · **Field:** Algebraic and complex geometry · **Entry:** 053

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

The space of stable curves carries natural classes, and mathematicians want every relation among them. Pixton proposed a system of relations and conjectured that it gives all of them. This paper claims to build a class that is zero in the Chow ring, and so in rational cohomology. It also claims that this class does not come from Pixton’s original system. The example has genus 10^60, far from any small case.

## At a glance

- **First step:** 2012 (A. Pixton, who proposed the relation system and conjectured completeness)
- **Significance:** Major. If correct, it disproves the Chow and cohomological forms of a central conjecture about relations on moduli of curves.
- **Trust:** Claimed, not checked
- **Field:** Algebraic and complex geometry (1 paper)

## The explanation

Suppose you know three facts about some numbers a, b and c. You know a − b = 0 and b − c = 0. From those two rules you can derive a − c = 0. A list of rules is complete when every true relation can be derived this way. Proving rules is one problem. Knowing that the list is complete is a different and harder one.

Algebraic geometers face this problem for curves. A moduli space is a space whose points are themselves shapes. Here each point is a stable curve. That is a curve of some genus g, maybe with simple pinches called nodes, and n marked points. This space, written M̄_{g,n}, has a family of natural classes. These are building blocks for its Chow ring, a way of doing algebra with subspaces and how they intersect. The natural classes form the tautological ring. Describing every relation among them is a central problem in this area[1].

Pixton proposed a uniform system of relations in 2012 and conjectured that it is complete[2,3]. The relations themselves were later shown to hold. Pandharipande, Pixton and Zvonkine proved them in cohomology. Janda proved them in Chow[4]. Those results establish the relations. Completeness asks whether any further relations exist[5].

This paper claims a negative answer. It builds an explicit formal class Y outside Pixton’s original span[6]. Yet the image of Y in the rational Chow ring is zero[6]. So Y is a true relation that Pixton’s list does not produce. Applying the cycle-class map then gives the same failure in rational cohomology[7]. Cohomology is a coarser way of measuring a space, so a relation that holds in Chow also holds there.

The example is enormous. The genus is g = 10^60. The construction uses g colours and one factor for each set of three colours[8]. The number of such triples is D = g(g − 1)(g − 2)/6. That is about 1.67 × 10^179. The number of marked points is n = 3D, about 5.0 × 10^179[6]. To see the pattern at small size, take 5 colours. Then there would be 10 triples and 30 marked points. The paper does not claim the result for such small numbers. The large sizes leave room in the estimates and are not proposed as minimal[9].

The difficulty is that two separate claims are needed about the same class. One is that Y is not in Pixton’s span. The other is that Y is still zero in Chow. The paper claims to prove them with two different algebras[10]. A formal algebra can carry independent cubic coefficients and still kill everything in Pixton’s span. That shows Y lies outside the span[10]. The actual Chow correspondences of a curve satisfy an extra identity that forces Y to vanish[10]. That identity comes from a Quot scheme, a space that parametrizes quotients of a fixed sheaf on a surface.

Completeness is not the same as the Gorenstein property, which concerns pairings between classes of complementary degree[11]. Pixton showed that a related ring in genus 25 is not Gorenstein, while still proving completeness there[12]. Canning, Larson and Schmitt found a similar split on some compact-type spaces. This paper’s relation instead lives on the full space of stable curves[13]. It concerns Pixton’s original system, fixed precisely in the paper. The comparison throughout is with that explicit span[14].

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

**What it doesn't show.** This does not show that Pixton’s system fails in small genus, or give a corrected complete list.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major only if the proof holds. No Lean formalization is listed and no specialist review is published. The example is astronomically large and says nothing about small genus.

## Words used

- **Moduli space:** A space whose points are themselves shapes, here curves with marked points.
- **Stable curve:** A curve that may have simple pinches, called nodes, with finitely many symmetries once marked.
- **Chow ring:** An algebra built from subspaces of a space, where multiplication records how they intersect.
- **Tautological ring:** The part of the Chow ring generated by the natural classes of the moduli space of curves.
- **Complete:** A list of relations is complete when every true relation follows from it.
- **Cohomology:** A coarser algebraic measurement of a space's shape; relations true in Chow stay true there.
- **Quot scheme:** A space that parametrizes quotients of a fixed sheaf, used here to produce an identity.

## The original paper

- [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf) - September 24, 2026

## Quoted lines from the paper (14)

[1] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Describing all relations among these natural classes is a central problem in the intersection theory of moduli spaces.

[2] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Pixton proposed a uniform system of stable-graph relations and conjectured that it is complete.

[3] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Pixton's 2012 graph formula extends the Faber--Zagier system to \(\Mbar_{g,n}\) and proposes a complete set of relations.

[4] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Pandharipande--Pixton--Zvonkine proved the cohomological validity of this system using the \(3\)-spin CohFT; Janda proved its Chow validity using equivariant Gromov--Witten theory of \(\mathbb P^1\).

[5] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> These results establish relations, while completeness asks whether any further relations exist.

[6] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Consequently Pixton's original relation span does not exhaust the tautological Chow relations: the equality \(\ker q_{g,n}=\Pspan_{g,n}\) fails for this stable pair.

[7] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> The same class gives the rational-cohomological consequence.

[8] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> The construction uses \(g\) colors and one three-label factor for each increasing triple of colors. Each factor is a product of two rational-tail divisor sums, corrected by a class supported on two-node rational bridges.

[9] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> The large parameters leave ample room in the support and color estimates; they are not proposed as minimal parameters.

[10] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> The first algebra proves nonmembership; the second proves vanishing.

[11] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Completeness is distinct from the Gorenstein property.

[12] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Pixton proved that \(R^*(\mathcal M_{25})\) is not Gorenstein while establishing Faber--Zagier completeness there.

[13] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> Our relation lies on the full stable compactification.

[14] From [A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf):

> The comparison throughout this paper is with the explicitly fixed span \(\Pspan_{g,n}\).

## Other sources

- [openai/math release notes (2026): model provenance](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”
- [A. Pixton, “Conjectural relations in the tautological ring of M̄_{g,n}”, arXiv:1207.1918 (2012)](https://arxiv.org/abs/1207.1918v1)
- [R. Pandharipande, A. Pixton and D. Zvonkine, “Relations on M̄_{g,n} via 3-spin structures”, Journal of the AMS 28 (2015), 279–309](https://arxiv.org/abs/1303.1043v3)
- [F. Janda, “Relations on M̄_{g,n} via equivariant Gromov–Witten theory of P¹”, Algebraic Geometry 4 (2017), 311–336](https://doi.org/10.14231/AG-2017-018)
- [S. Canning, “The tautological ring of M̄_{g,n} is rarely Gorenstein”, Geometry & Topology 29 (2025), 3905–3919](https://doi.org/10.2140/gt.2025.29.3905)
- [S. Canning, H. Larson and J. Schmitt, “The Gorenstein property and Pixton's conjecture for compact type moduli”, arXiv:2607.02249 (2026)](https://arxiv.org/abs/2607.02249v2)

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