# Curved-space isoperimetric and filling reference

Family 337: Sharp Cartan–Hadamard isoperimetry and rigidity. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry and transport researchers can compare area/volume bounds in complete simply connected nonpositive-curvature or proper CAT0 settings.

## Applicability and commercial boundary

A sharp inequality does not provide a minimizer algorithm for an arbitrary discrete mesh; global curvature and finite-perimeter hypotheses matter.

## 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

Extract an exact model-ball reference and a separately justified mesh approximation.

## 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).

[Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf).