A less mathy idea for some extension world:
In a new chapter, you find that star keys are "exact single keys of the star effect", and unstar keys are exact-zero of them. Every star effect you have gives you the amount of keys that you just used to open the door, and you can now have multiple, negative, and complex star effects! Example: having 2 star effects gives you the costs of doors when you open them; having -1 star effect makes you lose them twice as fast.
Together with this is introduced the Star Door/Lock, which requires the amount of star effects to open. You can still shred the door with a master key, and combo locks spend keys with precedence over star effects.