The Application Bundle for 5.1.0 for Mac OS X has the wrong Executable inside under jEdit.app/Contents/MacOS. The file should be named "jEdit" instead of "jedit" because if you run your OS on a Case-Sensitive Filesystem. OS X looks for a file named "jEdit" when launching the application and on a case sensitive filesystem it's not found and the launcher throws an exception ("Damaged Application").
This is not true. The name of the executable is defined by the Info.plist of the Application bundle. The problem is Gatekeeper blocking downloaded Application Bundles here. With 5.2pre1 the DMG will give clear advice of how to come over this false error message.