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Math-2573 · mathematics · Algebraic geometry

Completeness of original Pixton relations on closed moduli

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every stable (g,n), do the original Pixton dual-graph relations, together with their pullbacks and pushforwards, generate the entire kernel of the strata-algebra map to $R*(Mbar_{g,n})$? Only closed moduli are asserted; compact-type and rational-tail completeness are separate.

Short English statement

For every stable (g,n), do the original Pixton dual-graph relations, together with their pullbacks and pushforwards, generate the entire kernel of the strata-algebra map to $R*(Mbar_{g,n})$? Only closed moduli are asserted; compact-type and rational-tail completeness are separate.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2573-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2573

Manuscript 1 (family 053, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Completeness of original Pixton relations on closed moduli

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.

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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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