# Universal-cover convexity counterexample library

Family 046: Shafarevich counterexamples in dimension two and with large fundamental group. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Prevent researchers from inferring Stein or holomorphic convexity solely from a large fundamental group and absence of compact positive-dimensional analytic subspaces.

## Applicability and commercial boundary

The specific projective fourfold and surface examples do not rule out convexity in all restricted geometries.

## 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 an explicit counterexample and each implication it refutes.

## 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).

[A projective fourfold with large fundamental group and non-Stein universal cover](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf).