The Download Photoshop Lightroom page says "Contact Support" for LR 4 on OS X. Since the page provides the download for Windows, I'm guessing that there may be some serious problems with LR 4 running on newer versions of OS X. Have you tried contacting support? (That's often an exercise in extreme frustration with any company, more so with Adobe. It's hard to trust whatever any one person at Adobe Support says, given the number of reports here over the years about incorrect information from support.)
But last month, someone noticed that the Windows download link was actually for Mac. From that thread, I was able to reconstruct a URL that works: http://download.adobe.com/pub/adobe/lightroom/mac/4.x/Lightroom_4_LS11_mac_4_4_1.dmg . You could try this, but note my warning in the previous paragraph.