# Conformal domain representation research toolkit

Family 071: Koebe’s circle-domain conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry researchers can organize circle-domain representations and their rigidity under removable-boundary assumptions.

## Applicability and commercial boundary

Existence of a conformal equivalence gives neither a cheap numerical map nor certified approximation for arbitrary digital domains. Rigidity has additional boundary assumptions.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Inspect a constructive quantitative restricted case and compare existing conformal mapping methods.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Removable Boundaries and Rigidity of Circle Domains](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf).