The partitions are strictly independent. Remember that.
Okay monsterer really. I must end this now. I am 100% certain my point is unarguable.
They are not independent when you combine them like that with the LCR rule. You must see that? Every block that partition A builds on top of partition B's block is essentially a merger of partitions A and B.
Make sure you are agreeing on the exact definition of a partition...