Result 241, Mathematical logic

Rigidity of the Turing degrees

Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.

Lean formalization Proof

The bigger picture

Why it matters

A manuscript claims that the hierarchy of computational unsolvability has no hidden symmetries: its levels cannot be rearranged while preserving their relationships. This would show that those relationships uniquely fix every level's position.

What changes?

Turing degrees group decision problems that can each be solved given access to answers for the other. Their ordering records relative computability: one degree lies below another if the latter supplies enough information to solve the former's problems. The manuscript reports that every order automorphism of this full partial order is the identity. In plain language, any bijective relabeling that preserves these comparisons in both directions must leave every degree unchanged, not merely some special collection.

What does that help mathematicians do?

The claimed result rules out any global symmetry exchanging distinct degrees, even degrees that are incomparable because neither can compute the other. A researcher proposing such a symmetry would therefore know that it must fail somewhere in the ordering. This distinguishes genuine structural differences from arbitrary naming, without asserting the stronger conclusion that each degree has an individual description in a chosen logical language.

Are there practical applications?

The immediate value is foundational: Turing degrees organize the information needed to solve problems, including problems beyond ordinary algorithmic computation. Rigidity would constrain how researchers describe and compare that entire hierarchy. The reported conclusion is about its mathematical structure, not a procedure for solving undecidable problems or a demonstrated improvement to practical computation.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Rigidity of the Turing degrees

September 24, 2026 10 pages Main result formalized in Lean

We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Rigidity-of-the-Turing-degrees-September-24-2026,
  author = {{OpenAI}},
  title = {{Rigidity of the Turing degrees}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf}{OAI:Rigidity-of-the-Turing-degrees-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/241.md.

Rigidity of the Turing degrees

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The rigidity problem asks whether the ordering of Turing degrees by relative computability has any nontrivial automorphism. The formalized result gives a negative answer: every order automorphism of the full Turing degrees of subsets of N\mathbb N fixes every degree, with no definability or genericity assumption. The additional representation corollaries are not included.

Comparator links

Result Comparator statement
Rigidity of the Turing degrees DegreeRigidity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.