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So a common method used to construct non-zero $\omega$-REA arithmetic degrees with various properties is to build an $\omega$-REA operator $J$ satisfying the constraints that (for all $X$)
$$\tag{1} J(X') \equiv_T J(X) \oplus X'$$
$$\tag{2} J(X) >_T X$$
Inductively, 1 implies that $J(X^n) \equiv_...