---
title: "The Solomon–Yau least-volume conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 349; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "differential-geometry"
  - "unformalized"
seed_rank: 2204
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The Solomon–Yau least-volume theorem"
    url: "https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The Solomon–Yau least-volume conjecture

## One-sentence takeaway

OpenAI's result family 349 (Differential geometry) claims: Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres.

## 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): For every m ≥ 2, a closed connected minimal immersion into the unit sphere $S^{m+1}$ with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.
- *The Solomon–Yau least-volume theorem*: We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m-dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.

## 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 Solomon–Yau least-volume theorem](https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-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