= Solution
There is no contradiction. The domain on which the field is smooth is the <punctured plane>, which is not simply connected. A loop enclosing the origin cannot be spanned by a planar region entirely inside that domain, so the smoothness hypothesis needed to apply <Green's theorem> to its full interior fails. A loop that bounds a region avoiding the origin does have zero circulation.
Locally the field is the gradient of the <polar angle>, but that angle cannot be made a single-valued continuous potential on the whole <punctured plane>. The nonzero period $2\pi$ around the hole measures this global obstruction, despite the field having zero local curl.
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