Yeah, while analytics would be great, I think that competing with your friends would work pretty well ("Wow, I am behind with 30 productivity points today - I need to focus!").
Analytics would be a differentiator from other such services, being able to give actual advice, would be a great premium service.
A competition among friends is a good idea, but I think without a driver at the end - say a virtual brag award, it doesn't drive the competition. Look at foursquare and their badges - that in itself is a driver for a lot of folks - it would also give the service, in an initial phase where friends might not be present on the service, a better driver (indirectly the badges aim at bettering your productivity).
Hmmm, I think FB apps are a hard added value sell with the current placement of apps on the FB website, perhaps an option to simply post from the your cherry app (i.e. a FB button) to showcase your productivity - when you want.
My sense is auto posting through an FB app, might be counter productive - of course it is easier, but doesn't a post demand that you were somehow productive? And how do you keep the frequency up, in order for FB to show that you are "a productive person" - to far between posts, and you are automatically not productive - or are you?.
If you work on the badges concept, that might be a better approach to FB app, simply having the badges post vs. a index number or other "simple" indicators.
A competition among friends is a good idea, but I think without a driver at the end - say a virtual brag award, it doesn't drive the competition. Look at foursquare and their badges - that in itself is a driver for a lot of folks - it would also give the service, in an initial phase where friends might not be present on the service, a better driver (indirectly the badges aim at bettering your productivity).