You don't have to cut them right away.
In order to qualify for the comp equation, a player must have been a true Unrestricted Free Agent whose contract had expired or was voided after the previous season (i.e., he cannot have been released by his old team); he must sign during the UFA signing period (which ended July 27 last year); if he signs after June 1[*], he must have been tendered a June 1 qualifying offer by his old team; his compensatory value or contract value must be above a specific minimum amount; and he cannot have been permanently released by his new team before a certain point in the season (which seems to be after Week 10) or, possibly, before getting a certain amount of playing time, unless he was claimed off waivers by another team.