The audit trail will show the date a period was closed. Search using OrganizationOptions in the field under Area.
Click the Expand button to see the details or select the option, Preview Report, to view several expanded items at once.
This example shows that on 08/13/2019 the month-end was run for the 6th fiscal period to close out December 2018. Normally it would have been run within one or two months of the actual month-end.
If the period has ever been closed and then re-opened, the audit will show this too. Here is an example of the Audit Report showing period 7 being re-opened and then closed again, all on 08/15/2019.
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