# Four-dimensional L2 homology contract

Family 315: The four-dimensional Singer conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Track middle-degree concentration for the specified closed aspherical manifold or finite integral Poincaré-duality complex.

## Applicability and commercial boundary

Asphericity and duality are global assumptions; L2 Betti numbers differ from finite ordinary homology.

## 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 explicit group and complex examples with independently known invariants.

## 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).

[The Singer conjecture in dimension four](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf).