# Einstein four-manifold classification reference

Family 348: Nonnegative-curvature Einstein classification and an <i>L</i><sup>2</sup> topological gap. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry researchers can relate Einstein/nonnegative curvature models to exact universal-cover classifications and a small trace-free Ricci norm gap.

## Applicability and commercial boundary

The universal positive gap constant and exact curvature hypotheses do not supply a practical numerical classification tolerance.

## Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

## Next verification action

Inspect effective constants and a known exact model before software classification.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and formal-scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Zero-Plane Rigidity for Einstein Four-Manifolds](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf).