I've searched but can't find this exact problem. I have OS X Mail (Lion) for a gmail account and an iCloud account. The latter works perfectly. For the gmail, I am using all of google's recommended settings, and it works great, except that if I archive or delete items out of the Mail inbox, those changes are not reflected in the gmail web interface or iOS mail app. Any change to any other folder updates properly, and if I make a change to the inbox on the web or on my iPhone, that change does show up in the Mail inbox. It's just changes to the Mail inbox not showing up elsewhere. Any help much appreciated!