Result 053, Algebraic and complex geometry

A counterexample to Pixton completeness in Chow

Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.

Disproof or counterexample

The bigger picture

Why it matters

A proposed complete list of geometric identities appears to miss one, even when tested in two different mathematical frameworks. The counterexample challenges how researchers organize universal relationships among families of curves.

What changes?

The unreviewed manuscript reports a relation among tautological classes, standard geometric building blocks on spaces parameterizing stable curves with marked points. It is zero in rational Chow, where algebraic cycles are identified by rational equivalence, and therefore in rational cohomology, but is not generated by Pixton's original relations. The example has genus ten to the sixtieth power. Its marking count is given by an incomplete formula in the supplied abstract, so that count cannot be stated unambiguously.

What does that help mathematicians do?

If established, this example rules out treating Pixton's original relation system as a complete account of tautological identities in either rational Chow or rational cohomology. Passing from algebraic cycles to cohomology does not remove the gap: the constructed expression still vanishes without belonging to the proposed span. Researchers seeking a complete system must therefore accommodate additional relations. The example does not establish where incompleteness first occurs.

Are there practical applications?

The immediate value is foundational: the result identifies a limitation in a proposed framework for describing the geometry of families of curves. It would guide work on more complete relation systems and caution against assuming that Pixton's original relations capture every identity. The supplied material offers no practical algorithm or claim about small-genus cases.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A high-arity counterexample to Pixton completeness in Chow

September 24, 2026 61 pages

We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus 106010^{60} and 3(10603)3\binom{10^{60}}3 markings.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026,
  author = {{OpenAI}},
  title = {{A high-arity counterexample to Pixton completeness in Chow}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf}{OAI:A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.