# Homological conjecture dependency and counterexample graph

Family 199: Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Formalization teams can record which characteristic-two examples refute each named module/algebra implication.

## Applicability and commercial boundary

Several distinct conjectures and associated algebras share a family; a single blanket counterexample or formal badge is inaccurate. Field-extension persistence is a specific theorem.

## 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 assumptions, algebras and implications paper by paper.

## 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).

[An explicit counterexample to the Auslander-Reiten conjecture](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf).