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So my question is...

Are there any two irrational numbers that are entirely independent, that add together to make a rational? By "independent" I mean that their generation is not related.

This is a far deeper question as it is not always easy to determine if the irrational nature of one number is dependent upon the irrational nature of another. I suppose two such irrational numbers can only be said to be "independent" if their relationship is itself irrational, with all three irrationalities being distinct and unrelated.

For example, can the irrational number (4-pi) be generated in a way that is not related to pi?

If all mirrors to irrational numbers have no independent generators, then although in theory the addition of two irrational numbers can make a rational, in practice there are none where the irrationality of the second isn't spawned directly from the irrationality of the first.