How To Fix Mavericks Mail Unable To Connect Error

Mavericks Mail.app stopped being able to authenticate to any IMAP or POP mail servers -- my own OS X Server.app, iCloud, Gmail, etc. iOS clients could connect just fine, and I see that many, many others are having this problem as well. I solved this, but essentially by throwing in the towel and using these posts to bring Mavericks Mail and the login Keychain to their factory settings.

The following worked for me, and blowing away the Mail settings and Keychain were both necessary. If anyone has a less drastic solution, please post it. Make sure you backup any necessary certificates or messages before wiping this all out.

sudo rm -fr ~/Library/Caches/com.apple.mail ~/Library/Saved\ Application\ State/com.apple.mail.savedState* ~/Library/Application\ Support/AddressBook/MailRecents-v4.abcdmr* ~/Library/Containers/com.apple.mail ~/Library/Mail ~/Library/Preferences/com.apple.mail*

Keychain Access>Preferences>Reset My Default Keychain

I also used Disk Utility to fix permissions and did a safe boot to clear out a bunch of cache, though these steps didn't fix this specific problem.