A short answer is "no": You can not restrict users to not move or not delete any part of your project (except copyright notices - the integrity of copyright notices are required by law, and also by most free software licenses).
The GPLv3 (sec. 7: Additional Terms) say that may add some additional terms to your license, and one of these may be:
Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it;
But as this is limited to "legal notices or author attributions" it doesn't let you require preservation of arbitrary materials such as "project stats".
And to make this absolutely clear, the license goes on to say:
If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term.
That is, given that you want to use a recognized (by the FSF or OSI) free software / open source license. Provided you use a license where you grant users the four freedoms that is inherent in free software - you cannot impose any additional restrictions on what the user can do. (You can of course concoct a your own non-free custom license and put whatever you like, including restrictions on what the user may alter or delete, in it.)
For the record, I know of no license that qualify as "free software" under the FSF definition, or "open source" under the OSI defintion, that will allow you to do this.
As pointed out by user490, you can insist on having unchangeable sections in a document that is part of your project if the document is licensed under the GFDL (instead of the GPL). However, this is a documentation license, not a software license, and besides: There is nothing that will stop someone downstream from you removing that particular document - including the "invariant section" - if they for some reason really wanted to remove this text from their fork of your project.