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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.