Math-2241 · mathematics · Logic

Automorphism Problem for the Turing Degrees

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.

$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.

Statement recorded in the supplied source.

Determine the automorphism group of the partial order of Turing degrees.

Formulation references: [S1], [S2].

Short English statement

Determine the automorphism group of the partial order of Turing degrees.

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500003 (AMR-074-0003). Catalogue attribution under CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
S2 Original problem formulation and mathematical context. https://mathoverflow.net/questions/227083/what-are-some-important-but-still-unsolved-problems-in-mathematical-logic#227108.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2241-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2241

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 241; 2026-09-24).

Automorphism Problem for the Turing Degrees

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 241; 2026-09-24), the asserted result is that an order-preserving bijection of Turing degrees acts trivially on every degree.

Mathematical statement of the claimed result

Manuscript 1.

Let $\mathcal D_T$ be equivalence classes of subsets of $\mathbb N$ under mutual Turing reducibility, ordered by $\le_T$. The claimed rigidity statement is \[ \forall\pi\in\operatorname{Aut}(\mathcal D_T,\le_T)\ \forall\mathbf a\in\mathcal D_T:\quad\pi(\mathbf a)=\mathbf a. \] It concerns the full degree structure, rather than only degrees above a distinguished jump or on a cone.

Outline of the argument

For manuscript 1 (family 241; 2026-09-24), an arithmetic representation theorem turns an arbitrary degree automorphism into a Borel function on sets. Category cone avoidance removes auxiliary parameters from bounds on comeager pairs, and recovery from four values identifies the original degree.

Formal verification

For manuscript 1 (family 241; 2026-09-24), OpenAI lists Lean statements for this assertion: an order-preserving bijection of Turing degrees acts trivially on every degree.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are DegreeRigidity.lean.

Their named declarations include OAI.TuringRigidity.ManuscriptMain.rigidity.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

OpenAI, Rigidity of the Turing degrees, 2026-09-24; citation key OAI:Rigidity-of-the-Turing-degrees-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:rigidity.
Family 241: Lean scope.
Comparator DegreeRigidity.json.
Comparator DegreeRigidity.lean.
OpenAI catalogue entry.

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Record details

Subjects: Logic
Collection: Top Open Problems
Mathematics rank: 2241
UnsolvedMath: Catalogue entry