---
title: "Unrestricted pro-modularity at the prime two"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 010; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1870
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Unrestricted pro-modularity at the prime two"
    url: "https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "The Dimension of the Two-Adic Hecke Algebra at Odd Level"
    url: "https://github.com/openai/math/blob/main/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Fontaine–Mazur modularity at the prime 2"
    url: "https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Unrestricted pro-modularity at the prime two

## One-sentence takeaway

OpenAI's result family 010 (Number theory) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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.
- *Unrestricted pro-modularity at the prime two*: 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.
- *The Dimension of the Two-Adic Hecke Algebra at Odd Level*: For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level $\Gamma_1(N)$ has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.
- *Fontaine–Mazur modularity at the prime 2*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Unrestricted pro-modularity at the prime two](https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf)
- Manuscript: [The Dimension of the Two-Adic Hecke Algebra at Odd Level](https://github.com/openai/math/blob/main/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf)
- Manuscript: [Fontaine–Mazur modularity at the prime 2](https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math