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A lot of books mention that Scale Invariance is a property of Self-Organized critical processes, but fail to mention why.

Why is Scale Invariance a property of Self-Organized processes?

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You got the wrong keyword: it's critical systems that are scale invariant, while the self organized part means their own dynamics can take them to a critical state.

As Wolphram pointed out, you can find more details in the answer to the question Scale invariance and self organized criticality.

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