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Math-2241-A1 · revision r1 · on Math-2241

OpenAI claim summary for Math-2241

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Automorphism Problem for the Turing Degrees
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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