So mysteriously shut (4)
I believe the answer is:
thus
'so' is the definition.
(synonyms)
'mysteriously shut' is the wordplay.
'mysteriously' indicates an anagram.
'shut' with letters rearranged gives 'THUS'.
(Other definitions for thus that I've seen before include "Therefore, hence" , "As a result" , "as indicated" , "Consequently" , "an original Christmas gift" .)