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Math-2387 · mathematics · Algebraic geometry

Dual-Graph Generation of Tautological Relations

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $\mathcal S_{g,n}$ be the strata algebra and let $\mathcal R\subset\mathcal S_{g,n}$ be the ideal generated by pullbacks and pushforwards of the dual-graph relations $R(g,n,r)$. Is $\mathcal R$ exactly the kernel of the natural surjection $\mathcal S_{g,n}\to R^*(\overline{\mathcal M}_{g,n})$? Do the analogous completeness statements hold after restriction to compact-type and rational-tail loci?

Formulation references: [S1], [S2].

Short English statement

Let $\mathcal S_{g,n}$ be the strata algebra and let $\mathcal R\subset\mathcal S_{g,n}$ be the ideal generated by pullbacks and pushforwards of the dual-graph relations $R(g,n,r)$. Is $\mathcal R$ exactly the kernel of the natural surjection $\mathcal S_{g,n}\to R^*(\overline{\mathcal M}_{g,n})$? Do the analogous completeness statements hold after restriction to compact-type and rational-tail loci?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2387-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2387

Manuscript 1 (family 053, 2026-09-24): The main relation lies outside the original Pixton closed-moduli span, refuting the closed-moduli completeness branch. The problem also asks separate compact-type and rational-tails tautological-relation questions, which this one counterexample does not settle.

Dual-Graph Generation of Tautological Relations

OpenAI's claim

A formal tautological class at genus $10^{60}$ with $3\binom{10^{60}}3$ markings is claimed to vanish in Chow and rational cohomology while lying outside the original Pixton relation span. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Put $g=10^{60}$, $D=\binom g3$ and $n=3D$. In the formal decorated stable-graph strata algebra $\mathcal S_{g,n}$, OpenAI constructs a degree-$2D$ class $Y$ as follows. Concatenate all increasing triples of colors $1,\ldots,g$ into a color word $\mathbf i$ of length $n$, with three-slot blocks $J=(a,b,c)$. Let $\overline D_{ab}$ sum all rational-tail divisors carrying $a,b$. Let $\beta_J$ sum one-edge unmarked genus-$g$ graphs with the edge subdivided by a genus-zero vertex carrying $J$, weighted by inverse automorphism orders and pulled back by forgetting other labels. Then \[ F_J=\overline D_{ab}\overline D_{ac}-\tfrac12\beta_J,\qquad Y=\sum_{\tau\mathbf i=\mathbf i}\operatorname{sgn}(\tau)\,\tau\left(\prod_JF_J\right). \] OpenAI claims $q_{g,n}(Y)=0$ but $Y\notin\mathcal P_{g,n}$, where $q$ realizes graph classes in rational Chow of $\overline{\mathcal M}_{g,n}$ and $\mathcal P_{g,n}$ is Pixton's original relation span. Thus that span is strictly smaller than the Chow kernel; the same class also lies in the rational-cohomological kernel outside the span.

Outline of the argument

Manuscript 1 (family 053, 2026-09-24): Color-indexed three-label factors combine tail-divisor products with bridge corrections, followed by alternating relabelings. A faithful coefficient-algebra treatment of relative curve correspondences yields the required odd cubic identity in Chow; the high-arity construction separates that identity from the original relation system.

Formal verification

Family 053 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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