Hello! welcome to my iOS 5.1.1/6.1.3 Game Center fix! if this works then please consider subscribing to my youtube channel at the very bottom!
1: you need to get the Certificates: "DigiCert Global Root CA G2" and "DigiCert Global Root CA G3" from https://tlsroot.litten.ca
2: Next you need to install the app called iFile, navigate to the directory: "/var/mobile/Library/Preferences"
3: look for anything folder that has anything EXTERMELY similar to "purplebuddy", "accountsettings", "itunesstored", "appstore", "gamecenter", "iTunesStore" and "gamed".
4: delete the files or move them after, next go to the directory: "/var/mobile/Library/Caches" and repeat the same progress, except instead of folders, its .plist files.
5: after you deleted/moved those files, you need to either reboot or respring, go through the Setup screen like normal.
6: once you get to the Apple ID screen, make sure you sign in correctly since you have the Certificates.
7: for the final result, open game center and see if it loaded, if it did not then please attempt again.
How to fix Game Center on iOS 7.1.2
credit to plackowy for supportz!!!! This tutorial is by @fifiIsAwesum. @2024
If you are having trouble following my guide, i made a video on how to do it! here!