View source for Pixton's conjecture on tautological relations

'''Pixton's conjecture on tautological relations''' asks whether the graph-formula relations that Pixton proposed in 2012 generate all relations in the tautological ring of the moduli space ''M̄''<sub>''g'',''n''</sub> of stable curves, in rational Chow and in rational cohomology. A manuscript in OpenAI's public mathematics collection, dated 24 September 2026, claims a counterexample: an explicit class in genus 10<sup>60</sup>, with three markings for each triple of colors, that vanishes in rational Chow but does not lie in Pixton's original relation span. A Lean file in the collection checks only numerical bookkeeping of the construction, not the main theorem.

== Background ==
* As recounted in the manuscript's introduction:
** ''Tautological relations.'' Mumford began the systematic study of natural classes in the Chow ring of the moduli space of curves. Faber proposed a description of the tautological ring of ''M''<sub>''g''</sub>, including its Gorenstein property, and Faber and Zagier gave explicit relations among kappa classes, which Pandharipande and Pixton proved in Chow, conjecturing their completeness on ''M''<sub>''g''</sub>. In genus zero Keel gave a presentation of the whole Chow ring by boundary divisors.
** ''Pixton's relations.'' Pixton's 2012 graph formula extends the Faber–Zagier system to ''M̄''<sub>''g'',''n''</sub> and proposes a complete set of relations. Pandharipande, Pixton and Zvonkine proved the relations in cohomology using the 3-spin cohomological field theory, and Janda proved them in Chow using equivariant Gromov–Witten theory of the projective line. Completeness in Chow and in cohomology is recorded as Pixton's conjecture by Canning.
** ''Completeness and the Gorenstein property.'' Pixton proved that the tautological ring of ''M''<sub>25</sub> is not Gorenstein while establishing Faber–Zagier completeness there, and Canning, Larson and Schmitt proved completeness together with non-Gorenstein behavior on several compact-type moduli spaces.

== The original result ==
* '''Claim, 24 September 2026.''' The main theorem of ''A high-arity counterexample to Pixton completeness in Chow'' states that for ''g'' = 10<sup>60</sup>, ''D'' the binomial coefficient of ''g'' over 3 and ''n'' = 3''D'', an explicitly defined formal class ''Y'' of degree 2''D'' in the strata algebra of decorated stable graphs maps to zero in the rational Chow ring of ''M̄''<sub>''g'',''n''</sub> but does not lie in Pixton's original relation span, so that span is strictly smaller than the kernel of the realization map. A corollary gives the same strict inclusion in rational cohomology. The class uses ''g'' colors and one factor for each increasing triple of colors, a product of two rational-tail divisor sums corrected by a class supported on two-node rational bridges; the factors are multiplied and alternated over relabelings that preserve the color of each slot.
* ''Scope, as stated:'' the comparison is with Pixton's original span as fixed in the paper, which by Pixton's Proposition 1 is already an ideal stable under gluing and forgetting. The large parameters are not proposed as minimal. Completeness is a different question from the Gorenstein property, and the relation lies on the full stable compactification.
* ''Method, as described:'' nonmembership is detected by a formal test that keeps only rational-tails graphs and evaluates rational branches in a super Frobenius algebra whose coefficients form an exterior algebra on one odd symbol for each triple of colors; every element of Pixton's span has zero test, while the test of ''Y'' has a nonzero coefficient proportional to the product of all the odd symbols. For the vanishing, the correspondences of a smooth curve split into two even summands and an odd symplectic summand, following O'Sullivan's tensor reconstruction, and the small diagonal projected to odd summands gives an alternating cubic related to the Gross–Schoen modified diagonal. A relative Quot scheme with fixed determinant on the square of the curve, with virtual localization and a nested-Hilbert recursion of Ellingsrud, Göttsche and Lehn, gives a nilpotence identity for long products of these cubics. Weighted stable curves and a rational-bridge correction with coefficient −1/2 carry the identity to the compactification, and an induction on the support removes the remaining errors.
* '''Sources.''' Commit <code>adc7f12</code> of the openai/math repository: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026 ''A high-arity counterexample to Pixton completeness in Chow''].
* '''Formalization.''' The collection lists no Lean scope note or Comparator challenge for this family. The Lean file [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/StableGraphs/ExteriorBounds.lean <code>ExteriorBounds.lean</code>] proves the numerical parameters, color counts, a count of color-preserving permutations, the nonvanishing of a factorial multiple of an exterior monomial and finite stable-graph bounds. Its header states that the class ''Y'', its realizations, the Pixton relation span, the detector pairing and the geometric vanishing are not defined there, so the file does not assert the main theorem or its cohomological corollary.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 31 October 2026)
|-
| 2026-10-12 || M. Bintz, [https://bsky.app/profile/spurious-bows.bsky.social/post/3mxpl2opuvs2i thread on recurring techniques in the release], [https://bsky.app/profile/spurious-bows.bsky.social/post/3mxpl2oqt6k2i first pair] || Reading, comparison of techniques || A thread on techniques that recur across manuscripts of the collection without cross-citation. Its first pair matches this manuscript with the collection's paper on [[Bloch's conjecture for surfaces with geometric genus zero]]: Lemmas 3.3 and 5.1 there with Lemmas 5.3 and 5.5 here, and broadly Sections 5.1 to 5.4 there with Sections 5.5 to 5.7 here, with the equality sign used loosely. It suggests that the shared ideas may come from the proof systems sharing a workspace. || Social media thread on Bluesky. Read on 31 October 2026.
|-
| 2026-10-12 || ''AI Proved What?'', episode 053 [https://aiprovedwhat.com/episodes/053-counterexample-pixton-completeness-chow/ A relation Pixton missed?], remade version of 26 October 2026 || Reading, exposition || Video explainer of the 61-page preprint. It recalls Pixton's 2012 relations, their proofs in cohomology by Pandharipande, Pixton and Zvonkine and in Chow by Janda, and the completeness results for compact-type spaces of Canning, Larson and Schmitt. It states that the comparison is with Pixton's original span as the paper defines it, that the example has genus 10<sup>60</sup> and about 5 × 10<sup>179</sup> markings, that the paper gives no counterexample in small genus, and that completeness differs from the Gorenstein property, citing Pixton's result that the tautological ring of ''M''<sub>25</sub> is not Gorenstein while the known relations are complete there. || Episode status, last checked 30 October 2026: posted as a preprint, released 6 October, not withdrawn or revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, main theorem no, one Lean file of 446 lines checks bookkeeping only; no public assessment by specialists found. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_390.json referee review rank_390] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] || Review, AI-assisted audit || Reports no gap. It states that the reviewer read Sections 1 to 4, 6 and 8 in full and Sections 5 and 7 in part, without redoing the Quot obstruction theory, the marked recursion or the residue computation, and checked by hand the detector counts, a relative Pfaffian identity and the numerical estimates of Section 8. Its three minor points: the controlled-lifts proposition of Section 7 is proved only at sketch level across all descendant charts; the lemma extending the relative recursion of Ellingsrud, Göttsche and Lehn to any finite set of marked outputs goes beyond the one- and two-output cases of Kong and has a compressed proof; and the support-advancement lemma should be applied to each irreducible component of the base image. || Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]