---
title: "Scalar curvature and finite-time Ricci-flow singularities"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 351; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "differential-geometry"
  - "unformalized"
seed_rank: 2206
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup"
    url: "https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Bounded scalar curvature and smooth extension of four-dimensional Ricci flow"
    url: "https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Path selection and an elliptic inequality on degenerating Ricci-flat trees"
    url: "https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Scalar curvature and finite-time Ricci-flow singularities

## One-sentence takeaway

OpenAI's result family 351 (Differential geometry) claims: Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded.

## Why it matters here

Differential geometry is the deep background for geometry processing, curvature flows and mesh work in graphics. 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): A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.
- *A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup*: We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.
- *Bounded scalar curvature and smooth extension of four-dimensional Ricci flow*: We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.
- *Path selection and an elliptic inequality on degenerating Ricci-flat trees*: We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies $E=o(M)$. The root end has decay exponent greater than one, and every joined quotient group is nontrivial.

## 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: [A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup](https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf)
- Manuscript: [Bounded scalar curvature and smooth extension of four-dimensional Ricci flow](https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf)
- Manuscript: [Path selection and an elliptic inequality on degenerating Ricci-flat trees](https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/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