# [Taming implies compatibility on four-manifolds](paper.pdf)

**Author:** OpenAI

**Date:** October 6, 2026

## Citation

```bibtex
@misc{OAI:Taming-implies-compatibility-on-four-manifolds-October-6-2026,
  author = {{OpenAI}},
  title = {{Taming implies compatibility on four-manifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf}{OAI:Taming-implies-compatibility-on-four-manifolds-October-6-2026}},
  year = {2026}
}
```

## Version note

This version characterizes the compatible cone by intersection positivity, restricts the cone-sum equality to $h_J^-=b_2^+-1$, and adds a four-torus example of strict inclusion; see the [previous version](../Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf).