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Unrestricted pro-modularity at the prime two

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

Every continuous odd absolutely irreducible two-dimensional 2-adic representation of \(G_{\mathbb Q}\) unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.

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

Three current papers; two replaced by later editions.

The paper states

Theorem 1.1

In Fontaine–Mazur modularity at the prime 2, 6 October 2026

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

The other possible matches are Theorem 1.1 in Unrestricted pro-modularity at the prime two () and Theorem 1.1 in The Dimension of the Two-Adic Hecke Algebra at Odd Level ().

Let

r:G_{\mathbb Q}\longrightarrow\GL_2(\overline{\mathbb Q}_2)

be continuous and irreducible. Suppose that r is odd, is unramified outside finitely many primes, and is de Rham at 2 with distinct Hodge–Tate weights. Then r is isomorphic to a Tate twist of the 2-adic Galois representation associated with a classical cuspidal eigenform.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Three current papers, each readable here as a page beside its PDF.

Papers

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Unrestricted pro-modularity at the prime two

6 October 2026, current edition.

Read it herePDF

This version updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 4 October 2026

Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of \(G_{\mathbb Q}\) that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.

Fontaine–Mazur modularity at the prime 2

6 October 2026, current edition.

Read it herePDF

This version updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 23 September 2026

We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of \(G_{\mathbb Q}\), unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.

Previous editions

Lean statements

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Independent comparisons and proof-verification reports

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What people have said

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

    Author not stated. Source

  • Kevin Buzzard, Algebraic number theory results without Lean, 7 October 2026. Source

    An expert flags that the release gives no Lean for Fontaine-Mazur modularity at the prime two and suggests it may be the last mile of a programme already close to done rather than a new idea.

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Identify the theorem the catalogue describes

  • Family 010 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 010 The release has no Lean statement for this family. Open the form