Result 242, Mathematical logic

Single-fold Diophantine representations and undecidability under an at-most-one-solution promise

Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise.

Lean formalization Proof

The bigger picture

Why it matters

An equation can have at most one solution and still resist every general algorithm for deciding whether that solution exists. The manuscript claims that uniqueness does not remove this fundamental obstacle to solving polynomial equations.

What changes?

The manuscript reports that every recursively enumerable set of natural-number tuples, meaning a set whose members an algorithm can list, has a single-fold Diophantine representation. This is a polynomial equation with integer coefficients whose unknowns include the input tuple and auxiliary nonnegative integers. For each input belonging to the set, exactly one complete auxiliary tuple satisfies the equation; for every other input, none does. The claim covers all such enumerable sets, not just a special family.

What does that help mathematicians do?

If established, the result would show that multiple arithmetic witnesses are unnecessary for representing any recursively enumerable set. It would also imply the finite-fold conjecture, which asks for finitely many auxiliary solutions per member. The reported undecidability consequence rules out a general solvability algorithm even when promised at most one nonnegative-integer solution. Researchers could therefore separate two issues sharply: how many solutions exist and whether existence can be decided.

Are there practical applications?

The immediate value is foundational: the claim would sharpen the boundary of what polynomial equations can express and what algorithms can decide. For automated equation solving, the reported consequence is a limitation, not a faster method. A uniqueness guarantee alone cannot make every promised instance decidable; useful procedures would still need further restrictions.

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

Single-fold Diophantine representations

September 24, 2026 27 pages Main result formalized in Lean

Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.

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

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/242.md.

Single-fold Diophantine representations and undecidability under an at-most-one-solution promise

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

Scope

The single-fold Diophantine conjecture asks whether every recursively enumerable set has a polynomial representation with a unique auxiliary witness for each member. The formalization establishes this for every recursively enumerable subset of Nn\mathbb N^n, n≥1n\ge1: an integer polynomial has exactly one complete natural-number witness tuple for members and none for nonmembers.

The consequence that solvability remains undecidable under an at-most-one-solution promise is not separately formalized.

Comparator links

Result Comparator statement
Single-fold Diophantine representations SingleFold.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 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.