# Topology-versus-multiplicity counterexample

Family 059: Counterexamples to Zariski’s multiplicity conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Help singularity tools avoid assuming ambient topological equivalence determines multiplicity.

## Applicability and commercial boundary

The specified germs with multiplicities four and five concern exact local complex singularities, not ordinary regression complexity.

## 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 equations and the topological equivalence witness.

## 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).

[Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf).