It looks like the permissions for the President's Office Project are set to 'Managed By Owner'
This means that the permissions for workbooks in the project can be different to the Project that they reside in.
Please take a look at the workbook permissions to see if they are set differently and whether your user (Eric) has been explicitly added.
Hope this helps
Also it looks like the 'All user' group has permissions to that Project too.
Eric is a member of All Users (as is everyone else on your server) so will have permissions that way
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If you want to guarantee that only the people in the groups specified have access to workbooks in those projects, this is what I would do.
1. Go to the permissions of the Default Project, and remove all access to the All Users Group. This means that any new projects you create will by default not be permissioned to anyone. Also, ensure that there are no other users / groups permissioned against the default project.
2. On each of the projects, remove all permissions from the All Users groups, and set the permissions to be 'locked to project'. This will remove access from All Users, and ensure all content in the project are set at the project level. Also, when workbooks are published to that project they will inherit the permissions of that project and these cant be changed by the publisher.
The only warning I would give is that the current permissions may give be giving access which when removed may stop the users from seeing what they are expecting to see. So test and ensure users can see what they expecting to have access to.