# Diffeomorphic deformation feasibility research toolkit

Family 368: The three-dimensional Ball–Evans approximation problem. First source-informed review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Registration and mesh researchers can study how Sobolev homeomorphisms in three dimensions can be approximated by smooth diffeomorphisms while retaining the same target domain.

## Applicability and commercial boundary

Strong W1p existence of an approximating sequence does not supply a practical algorithm, finite-stage approximation budget or an automatic fold-free guarantee for a computed mesh. The initial map must already be a homeomorphism.

## Initial business decision

Conditional research. Commercial demand and profitability remain hypotheses. Research usefulness is not evidence of a buyer budget.

## Next verification action

Read the construction quantitatively and test an explicit map with certified injectivity before considering a registration backend.

## Evidence scope

The catalog statement and main-paper abstract or introductory theorem passages were individually reviewed. Available family-level scope documents were inspected, with attention to version differences and exclusions. This is a first application triage, not a full proof audit or independent Lean verification. Family 376 has nine paper abstracts inspected; selected scopes are not assumed to cover all nine papers in full. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Pinned sources

- [Strong diffeomorphic approximation in three dimensions for 1≤p≤2](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf)
- [Strong diffeomorphic approximation in three dimensions for p>2](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf)