Rigidity of the Turing degrees: is every automorphism of the Turing degrees the identity?

Theorem 1.1: every order automorphism of $(\mathcal D_T,\le_T)$ is the identity. The proof represents an arbitrary automorphism by an arithmetic, hence Borel, map on reals (Slaman-Woodin), recovers any irrational from four values of such a map at rational affine combinations of it and two auxiliary reals, for a comeager set of auxiliary pairs, and removes the auxiliaries by Sacks-type category cone avoidance. As a corollary it settles the Slaman-Woodin biinterpretability conjecture positively. It does not address rigidity of other degree structures such as the computably enumerable degrees or the degrees below $\mathbf 0'$.

Theorem 1.1: every order automorphism of $(\mathcal D_T,\le_T)$ is the identity. The proof represents an arbitrary automorphism by an arithmetic, hence Borel, map on reals (Slaman-Woodin), recovers any irrational from four values of such a map at rational affine combinations of it and two auxiliary reals, for a comeager set of auxiliary pairs, and removes the auxiliaries by Sacks-type category cone avoidance. As a corollary it settles the Slaman-Woodin biinterpretability conjecture positively. It does not address rigidity of other degree structures such as the computably enumerable degrees or the degrees below $\mathbf 0'$.