# Scalar curvature and topology assumption ledger

Family 335: Gromov’s integral scalar-curvature bound for simplicial volume. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometric researchers can relate negative scalar-curvature integrals, simplicial volume and positive-curvature topological restrictions.

## Applicability and commercial boundary

The dimension constant and exact global smooth/topological hypotheses are not finite mesh curvature certificates or shape-design algorithms.

## 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

Identify an exact manifold with computable simplicial volume and curvature integral.

## 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).

[An integral scalar curvature bound for real simplicial volume](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf).