x C y, and y P z. Therefore, x C z.
No, you need x C y and z P y to get x C z (be careful about which way the P relation is going).
The intuition is "Anything which is a cause of the whole is a cause of the part", not "Anything which is a cause of the part is a cause of the whole". Again, there are intuitive examples here. (Compare me baking a cake for a child's birthday party vs just buying the cake from a shop, and putting a few sprinkles and candles on the top. In the second case, I am a cause of some part of the cake as presented to the child, but not the whole cake, and if someone says "Wow that cake tasted delicious!" I'd have to admit I didn't make it, only decorated it).
I agree that "x C y, and y P z. Therefore, x C z" is wildly unintuitive, causes problems, and is just plainly wrong. But...
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...actually, looking back, you're right. I apologise; I misread the definition of C* (I read w P y instead of y P w).
I'm going to have to look through it again before I can comment further.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.