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 $\mathbf 0'$ .

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Rigidity of the Turing degrees: is every automorphism of the Turing degrees the identity? · Rigidity of the Turing degrees

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Registry verification: lean checked · announcement · candidate

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

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the rigidity problem; it states that every order automorphism of the full Turing degrees is the identity, in ZFC, with no definability hypothesis on the automorphism. Lean: lean/formalization.yaml lists comparator DegreeRigidity with declaration OAI.TuringRigidity.ManuscriptMain.rigidity. The statement ComparatorChallenges/DegreeRigidity.lean was read here: it defines oracles as $\mathbb N\to$ Bool, reduction by Mathlib's TuringReducible on their characteristic functions, Degree as the antisymmetrization, and asserts that every order isomorphism Degree to Degree fixes every degree. This is the headline claim. The Lean development (OAI/Computability/DegreeRigidity, with forcing, constructibility and set-model subdirectories) derives the Borel representation inside Lean rather than assuming the Slaman-Woodin input. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.