Unrestricted pro-modularity at the prime two

What OpenAI claims

Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G_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.

Papers (3)

1. Unrestricted pro-modularity at the prime two

   6 October 2026 · Preprint, not peer reviewed · PDF

   Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of G_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.

2. The Dimension of the Two-Adic Hecke Algebra at Odd Level

   5 October 2026 · Preprint, not peer reviewed · PDF

   For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level Γ₁(N) has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.

3. Fontaine–Mazur modularity at the prime 2

   6 October 2026 · Preprint, not peer reviewed · PDF

   We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of G_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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.