---
title: "Milne’s rationality conjecture and algebraic specialization"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 001; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1861
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Milne's rationality conjecture for abelian varieties"
    url: "https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
see:
  - "1892-hodge-and-kugasatake-results-for-all-projective-k3-surfaces"
---

# Milne’s rationality conjecture and algebraic specialization

## One-sentence takeaway

OpenAI's result family 001 (Number theory) claims: Proves Milne's rationality conjecture for abelian varieties over $\overline{\mathbb Q}$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory.

## 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): Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
- *Milne's rationality conjecture for abelian varieties*: We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.

## 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: [Milne's rationality conjecture for abelian varieties](https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-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