# Group-algebra zero-divisor counterexample fixture

Family 196: A counterexample to Kaplansky’s zero-divisor conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Symbolic algebra and formalization teams can use explicit nonzero products equaling zero to test group-ring representations.

## Applicability and commercial boundary

The characteristic-two torsion-free-group example does not refute results over every field or supply an efficient generic zero-divisor decision algorithm.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No practical implementation, recurring buyer need or profitability has been demonstrated; a weak business bridge is explicitly deferred.

## Next verification action

Read the finite presentation and exact zero-divisor elements before implementing a verifier.

## Evidence scope

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

[A Torsion-Free Group Algebra with Zero Divisors](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf).