---
title: "The four-dimensional Singer conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 315; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "topology"
  - "unformalized"
seed_rank: 2170
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The Singer conjecture in dimension four"
    url: "https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The four-dimensional Singer conjecture

## One-sentence takeaway

OpenAI's result family 315 (Topology) claims: Proves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two.

## Why it matters here

Topology results are background for knot, surface and configuration-space reasoning in geometry code. 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): More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.
- *The Singer conjecture in dimension four*: We prove the four-dimensional Singer conjecture: the L2-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.

## 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: [The Singer conjecture in dimension four](https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/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