# Knot surgery distinctness research contract

Family 306: The purely cosmetic surgery conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Track orientation-preserving distinctness of Dehn surgeries along distinct slopes for a nontrivial knot in the three-sphere.

## Applicability and commercial boundary

Including the meridian and fixing orientation matter; the theorem is not a knot-recognition or homeomorphism algorithm.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No recurring buyer need, practical implementation or profitability has been demonstrated. Defer a standalone commercial product until a constructive example and a measurable workflow improvement exist. These records also identify unsupported inferences that an evidence platform could flag.

## Next verification action

Extract finite knot examples and check independently computed surgery invariants.

## Evidence scope

The repository catalog statement was individually read. Main-paper construction, effectiveness and selected formal-scope comparison remain queued for this record. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Purely cosmetic surgery on knots in the three-sphere](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf).