# Cohen–Macaulay approximation limitation ledger

Family 195: A counterexample to the small Cohen–Macaulay module conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Researchers can flag universal assumptions that every specified local domain has a small maximal Cohen–Macaulay module.

## Applicability and commercial boundary

The three-dimensional complex counterexample does not exclude big modules, restricted ring classes or ordinary finite CAS computations.

## 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

Extract the exact domain and the module property disproved.

## 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 Complete Local Domain without a Small Cohen–Macaulay Module](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf).