# Tangent-sheaf smoothness contract counterexample

Family 048: A characteristic-zero counterexample to Lipman–Zariski. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Flag a false universal smoothness inference from a free rank-two tangent sheaf on a normal affine complex surface.

## Applicability and commercial boundary

The result refutes a specified singularity implication; it does not make differential geometric regularity tests universally unreliable.

## 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 the coordinate ring, freeness witness and singularity certificate.

## 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 singular normal affine surface with free tangent sheaf](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf).