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Eisenbud–Green–Harris and lex-plus-powers

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

What the papers state

Theorem 1.1

In The Artinian Lex-Plus-Powers Betti Theorem, 23 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

For every n\ge1, every ordered degree sequence 2\le a_1\le\cdots\le a_n, and every homogeneous ideal I\subseteq\C[x_1,\ldots,x_n] containing a regular sequence of these degrees, the spaces in (1.1) exist and J=\bigoplus_{d\ge0}J_d is a homogeneous ideal. For this single ideal,

\beta^S_{p,j}(S/I)\le\beta^S_{p,j}(S/J) \qquad\text{for every \(p\ge0\) and \(j\in\Z\)}.

There is no restriction on the degrees or number of the additional generators of I.

Corollary 1.3 (Eisenbud–Green–Harris in characteristic zero)

In Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture, 23 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let F be a field of characteristic zero, let S_F=F[x_1,\ldots,x_n] be standard graded, and let

1\le c\le n,\qquad 2\le a_1\le\cdots\le a_c.

If a homogeneous ideal I\subseteq S_F contains a homogeneous regular sequence f_1,\ldots,f_c with \deg f_i=a_i, there is a monomial ideal J\subseteq S_F containing P_F=(x_1^{a_1},\ldots,x_c^{a_c}) such that

\dim_F(S_F/J)_d=\dim_F(S_F/I)_d\qquad(d\ge0).

Moreover, J may be chosen lex-plus-powers: in each degree its monomials outside P_F are the lexicographically greatest \dim_F I_d-\dim_F(P_F)_d monomials outside P_F, for the order x_1>\cdots>x_n.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Three more possible matches

  • Corollary 1.2 (Eisenbud–Green–Harris and lex-plus-powers in characteristic zero) in The Artinian Lex-Plus-Powers Betti Theorem, 23 September 2026.
  • Theorem 1.1 (Commuting-form construction) in Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture, 23 September 2026.
  • Corollary 1.2 (Artinian Eisenbud–Green–Harris) in Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture, 23 September 2026.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF. One explanation by others.

Papers

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The Artinian Lex-Plus-Powers Betti Theorem

23 September 2026, current edition.

Read it herePDF

We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • mo271, Paper/EisenbudGreenHarris: the conjecture holds in characteristic zero for degrees at least two (issue 6913), 7 October 2026. Source

    The release proves a large special case (characteristic zero, all degrees at least two).

What these papers cite in the release, and what cites them

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Within this family

Papers elsewhere in the release that this family’s papers cite

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Papers elsewhere in the release that cite this family’s papers

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References outside the release

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 200 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 200 The release has no Lean statement for this family. Open the form