# Algebraic inverse and cellular-automaton scope audit

Family 197: A torsion-free group algebra that is not directly finite. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Researchers can distinguish left/right inverses, surjectivity and determinant conjectures across the specified group and coefficient regimes.

## Applicability and commercial boundary

The infinite group/configuration counterexamples do not show ordinary finite matrices have one-sided inverses without two-sided inverses. Integral and rational group-ring determinant claims have different inputs.

## 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

Map each manuscript separately and extract finite descriptions with exact test semantics.

## 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 That Is Not Directly Finite](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf).