what is the bare minimum that needs to be on a paper to count as a "character sheet"?
I would say anything that you have to make a choice for should be on there. Nothing that can be calculated based on those choices would be required to be written down.
So let's see...race, class(s) and levels in each, deity, gender, alignment, faction, any alternate racial traits, ability scores, class option choices, skill ranks, feats, traits, all equipment, hit points (since that depends on your FCB choices).
I'm assuming you're asking because you want to fulfill the PFS requirement of a physical character sheet but you use an electronic tool of some kind to play from (Herolab, spreadsheet, etc).
As long as you can justify the things that Herolab (or whatever you're using) says using what's on the physical sheet, then you should be set.