Family 053 — A relation beyond Pixton’s original system

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Family 053 · Algebraic and complex geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

A ring can be described by generators and relations. Giving many true relations is useful; proving that they are all the relations is a much stronger achievement. The distinction is central to the intersection theory of moduli spaces of curves.

These moduli spaces carry natural geometric classes associated with marked points, differential forms, and degenerations into nodal curves. Pixton proposed a uniform graph-based system of relations among them. The relations are known to be valid. The completeness conjecture says that every relation among these natural classes follows from that system.

Family 053 contains one manuscript claiming a counterexample to completeness. It constructs a formal expression that becomes zero as an algebraic cycle but is not generated by the original Pixton relations, even after their standard gluing, forgetting, and multiplication consequences are included.

The example has extraordinarily large parameters. Its role is to disprove a universal assertion, not to provide a small relation ready for everyday computation.

2. Background and essential definitions

A space whose points are curves

The moduli space $\mathcal M_{g,n}$ parametrizes smooth complex curves of genus $g$ with $n$ distinct labelled marked points. Its stable compactification $\overline{\mathcal M}_{g,n}$ also includes connected curves with ordinary nodes, where two branches cross.

Stability requires only finitely many automorphisms preserving the markings. For example, each rational component needs at least three special points, counting both markings and node branches. The condition $2g-2+n>0$ gives the stable range.

The natural object is a moduli stack, which remembers finite automorphism groups. For this digest, one can think of an orbifold-like parameter space with rational coefficients accounting for those symmetries. Its complex dimension is

$$3g-3+n.$$

Boundary strata describe the different ways a curve can break into components. A stable graph records them: vertices represent components and their genera, edges represent nodes, and labelled legs represent markings.

Natural classes and their relations

At the $i$-th marked point, the cotangent line has a first Chern class $\psi_i$. Other classes, called $\kappa$-classes, arise by forgetting a marking and pushing forward powers of a cotangent class. Together with boundary gluing operations, they generate the tautological ring.

The ambient Chow ring consists of algebraic cycles modulo rational equivalence, with multiplication given by intersection. We use rational coefficients. A codimension-$d$ cycle also has a cohomology class in degree $2d$, but different Chow classes can have the same cohomology class.

To keep formulas separate from their geometric values, introduce a strata algebra $\mathcal S_{g,n}$. Its elements are formal linear combinations of decorated stable graphs, with the standard dimension truncations. Realizing a graph as its geometric boundary class gives a map

$$q_{g,n}:\mathcal S_{g,n}\longrightarrow A^*(\overline{\mathcal M}_{g,n};\mathbb Q).$$

The image is the tautological ring. The kernel consists of formal expressions that vanish geometrically: exactly the relations one wants to understand.

Validity and completeness

Let $\mathcal P_{g,n}$ be Pixton’s original relation span, including its stated formal consequences. Validity means

$$\mathcal P_{g,n}\subseteq\ker q_{g,n}.$$

Completeness means equality. A counterexample must do two things for the same formal class $Y$: show $q_{g,n}(Y)=0$, and show $Y\notin\mathcal P_{g,n}$. Showing only that a new-looking formula vanishes is insufficient; it might already be a complicated combination of known relations.

3. Development history and previous partial results

Mumford initiated the systematic intersection-theoretic study of natural classes on moduli spaces of curves. Faber’s calculations and geometric constructions led to conjectural descriptions of the tautological ring on the smooth unpointed space $\mathcal M_g$.

Faber–Zagier supplied explicit hypergeometric relations among $\kappa$-classes. Pandharipande–Pixton proved those relations in Chow using stable quotients. The compactification requires additional boundary information, developed through the gluing and forgetting calculus of Arbarello–Cornalba and Graber–Pandharipande. In genus zero, Keel gave a presentation of the whole Chow ring by boundary divisors.

Pixton’s 2012 formula extended the relation system to stable pointed curves through decorated graphs and proposed completeness. Pandharipande–Pixton–Zvonkine proved cohomological validity using three-spin cohomological field theory. Janda proved Chow validity through equivariant Gromov–Witten theory of the projective line.

These are validity theorems: they establish that the proposed expressions vanish. They do not automatically identify the entire kernel of the realization map. Likewise, Janda’s comparison of relations produced by certain semisimple field-theory reconstructions compares those methods with Pixton’s system, not every possible source of Chow relations.

A nearby question asks whether the tautological ring is Gorenstein, meaning its multiplication pairings in complementary degrees are nondegenerate in the proposed manner. That is logically different from completeness. Pixton’s September 2026 paper proves that the tautological ring of $\mathcal M_{25}$ is not Gorenstein while also establishing Faber–Zagier completeness there. Other compact-type examples similarly separate the two properties.

M1 therefore targets the original full relation span directly on the stable compactification. It does not infer incompleteness merely from a failure of duality in the ring.

4. What the new manuscripts claim

The parameters and the two conclusions

Set

$$g=10^{60},\qquad D_*={g\choose3},\qquad n=3D_*.$$

Theorem 1.1 of M1 claims an explicitly defined class

$$Y\in\mathcal S_{g,n}^{2D_*}$$

for which

$$q_{g,n}(Y)=0, \qquad Y\notin\mathcal P_{g,n}.$$

The superscript is codimension. It lies below the dimension $3g-3+n$, so the assertion is not simply vanishing because the requested degree exceeds the dimension of the moduli space.

The same class yields a cohomological counterexample in degree $4D_*$. The cycle-class map sends its Chow realization to zero, while formal nonmembership in $\mathcal P_{g,n}$ remains unchanged. This implication needs no injectivity of the cycle-class map: Chow vanishing already implies cohomological vanishing.

What “explicit” means at this scale

The construction uses $g$ colors. For each increasing triple of colors, it allocates three labelled slots, producing $n=3{g\choose3}$ slots in total.

For a three-slot block $J=(a,b,c)$, the manuscript combines two rational-tail divisor sums and a correction supported on rational bridges:

$$F_J^{\mathrm{st}} =\overline D_{ab}\,\overline D_{ac} -\frac12\beta_J^{\mathrm{st}}.$$

Here $\overline D_{ab}$ sums boundary divisors cutting off a genus-zero tail containing $a,b$. The bridge class $\beta_J^{\mathrm{st}}$ inserts a genus-zero component carrying the three labels into an edge, giving two nodes, with specified graph-automorphism coefficients and forgetting pullbacks.

M1 multiplies one factor for every color triple and alternates over all relabelings preserving each slot’s color:

$$Y=\sum_{\tau\text{ preserves colors}} \operatorname{sgn}(\tau)\, \tau\left(\prod_JF_J^{\mathrm{st}}\right).$$

Every operation is finite, though the expression is immense. The formula specifies a class without listing all of its terms. “Cubic” in the manuscript refers to its three marked inputs; each factor has Chow codimension two.

5. Why the advance matters

Importance: 4/5. If correct, this would disprove a central completeness proposal in the intersection theory of moduli spaces, in both rational Chow and rational cohomology.

The existing Pixton relations would remain true and useful. What changes is the belief that their formal closure accounts for every tautological relation. A missing relation would require a broader presentation of the ring or additional geometric principles generating relations.

The scale of the example affects practical usefulness but not the logic. A universal statement covering all genera and all marking counts fails if even one finite pair fails. The manuscript makes no claim that $10^{60}$ is close to the smallest possible genus.

The proposed mechanism is also informative: a formal algebra can satisfy all the tested graph relations while lacking a further identity forced by actual Chow correspondences. This would distinguish formal consistency with a relation system from geometric realizability.

6. The main idea

Use different tools for nonmembership and vanishing

M1 deliberately separates the two required conclusions. To show nonmembership, it constructs a linear detector on formal decorated graphs. To show vanishing, it uses actual algebraic cycles in families of curves and surfaces.

A detector is analogous to a linear functional that vanishes on a subspace but not on a candidate vector. If $\ell$ vanishes on every element of $\mathcal P_{g,n}$ and $\ell(Y)\ne0$, then $Y$ cannot belong to that span. This is an algebraic obstruction to derivability, rather than an attempt to enumerate all possible combinations of relations.

A formal algebra preserves an alternating product

The detector uses a super Frobenius algebra, a multiplication-and-pairing structure with even and odd parts. Odd coefficients anticommute, as in an exterior algebra. There is an independent odd generator $\theta_{ijk}$ for each color triple.

The manuscript checks the graph identities needed to annihilate Pixton’s full original span, including the relevant boundary and forgetting operations. The test of $Y$ retains a nonzero coefficient proportional to the product of the distinct $\theta_{ijk}$. This supplies the claimed formal nonmembership.

The detector is not asserted to be the actual Chow theory of a family of curves. That distinction is exactly what allows it to separate the original relation system from the additional geometric identity.

Actual curve correspondences obey a further vanishing law

On the geometric side, the small diagonal of a curve links three point positions. Projecting to the appropriate odd part of the curve’s correspondence structure produces a modified diagonal of the Gross–Schoen type, with corrections needed in families.

M1 studies quotient-sheaf moduli spaces on the square of the curve. Virtual localization and a Hilbert-scheme recursion retain marked outputs rather than integrating everything to a scalar. The resulting relation is intended to force sufficiently long products of the cubic coefficients to vanish in relative Chow theory.

Passing to the stable compactification introduces boundary errors. The rational-bridge correction with coefficient $-1/2$ removes a designated local error. The remaining terms either involve simpler operations or are supported over subsets of smaller dimension in the base.

An induction repeatedly pushes that support into higher codimension. The enormous color budget leaves enough marked inputs for each step until no support remains. This is the claimed Chow vanishing of $Y$. The argument retains arbitrary Chow correspondences; replacing them with their cohomology classes would only establish a weaker result.

7. Scope, limitations, and open questions

The target is Pixton’s original relation system with its specified full formal closure, not a deliberately weakened selection of seed relations. The manuscript does not claim that any enlarged relation system is incomplete.

The example lives on $\overline{\mathcal M}_{g,n}$ at the displayed enormous parameters. It does not supply a small-genus counterexample, determine the first failure, or automatically disprove every completeness statement on the smooth unpointed or compact-type loci.

It also does not invalidate the known Pixton relations or establish that every non-Gorenstein phenomenon arises from a missing relation. Completeness, validity, and nondegeneracy of multiplication pairings are separate questions.

The remaining program would be to identify smaller examples, understand which additional relations are required, and formulate a replacement completeness conjecture with a geometric explanation for the new identities.

8. Sources and further reading

M1’s Definition 2.3 specifies the alternating class; Theorem 1.1 gives Chow vanishing and formal nonmembership; Corollary 1.2 gives the rational-cohomological consequence. Its introduction distinguishes the two proof mechanisms and the nearby Gorenstein questions.

The primary sources linked in Section 3 explain the original graph relations and their established validity. Optional background on tautological rings and moduli of algebraic curves can help with the vocabulary. The essential algebraic question is whether a known relation subspace equals the full kernel of geometric realization.

Source manuscripts

• M1. OpenAI, A high-arity counterexample to Pixton completeness in Chow (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.