From Fallen London Wiki

Hi, I think there is a minimum of +1 CP gain on progress on some/all actions. I had page of lures, page of papers, priest of magpies (the group did not work out well) and tolerance >= 5 near the end. By the formulas given, basic actions should not had made progress but they did, 1 CP each. I can't say if this applies to all or just those what I had.