1 · The essentials
The Turing degrees organize information according to relative computability: what can be computed with access to what. Their ordering records these relationships. The rigidity question asks whether the degrees can be relabeled without changing any of those relationships. The main manuscript states that the only such relabeling leaves every degree exactly where it was. This concerns the full ordering of Turing degrees, not just a selected part. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a chart of labeled boxes with arrows recording their relationships. You try to exchange the labels while keeping every recorded relationship unchanged; rigidity means that the only successful exchange moves no label at all. This illustrates the relabeling question, not how computability itself works.
Why it matters
The result answers the rigidity question positively for the full ordering of Turing degrees. It says that preserving relative-computability relationships everywhere leaves no freedom to rearrange the degrees.
2 · Going further
The precise statement concerns the full partial order of Turing degrees, ordered by relative computability. An order automorphism is a bijective relabeling that preserves the ordering in both directions. The main result says that every such automorphism is the identity: for every Turing degree .
The supplied abstract describes this as a positive resolution of the rigidity conjecture for the Turing degrees. It states the conclusion but gives no argument outline, so the main idea of the argument cannot be explained from this source alone. Only one manuscript is supplied, labeled Lean-formalized; no companion manuscripts or relationships among them are described.
Useful notions
- Relative computability
- Turing degrees
- Partial orders
- Bijective relabelings
- Order automorphisms
What was known before / what is new
Before
The source identifies the question as the rigidity conjecture for the Turing degrees. It provides no account of earlier partial results or approaches.
New
The main manuscript states that every order automorphism of the full partial order of Turing degrees is the identity. This rules out any nontrivial relabeling that preserves the relative-computability ordering, and the main result has a Lean formalization.
Limits and assumptions
The statement applies to order automorphisms of the full partial order of Turing degrees: the relabeling must be bijective and preserve the ordering in both directions. The supplied statement does not address arbitrary maps or automorphisms of selected parts of that order. The main result is identified as Lean-formalized, but no details of the formalization or argument are supplied. The source identifies no remaining open questions.
Mini glossary
- Relative computability —
- The relationship describing what can be computed with access to other information.
- Turing degrees —
- The objects in the ordering by relative computability studied in the manuscript.
- Order automorphism —
- A bijective relabeling of an ordered structure that preserves its ordering in both directions.
- Identity —
- The map that leaves every object unchanged.
- Rigidity —
- The property that the only order automorphism is the identity.
Keywords : Turing degrees · Order automorphisms · Rigidity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Rigidity of the Turing degrees
24/09/2026We 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.