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Single-fold Diophantine representations
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Abstract
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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Single-fold Diophantine representations and undecidability under an at-most-one-solution promise: Lean scope note (lean/docs/242.md, 2026-10-06). The scope note states that the Lean development covers the single-fold representation theorem for r.e. subsets of $\mathbb N^n$, $n\ge1$ (comparator statement
SingleFold.lean), and that the at-most-one-solution undecidability corollary is not separately formalized. [family-level]
Possibly relevant literature (found by the agent, not checked)
- A. Tyszka, Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers? (arXiv:1404.5975; Open Computer Science 7, 17–23, 2017). Per its abstract, its conjecture on a bound $\xi(n)$ would imply that sets with single-fold Diophantine representations are computable, which bears directly on Theorem 1.1 and the at-most-one-solution undecidability corollary.
- A. Tyszka, Small systems of Diophantine equations with a prescribed number of solutions in non-negative integers (arXiv:1110.3549, 2011; revised 2014-10-20). Assuming the single-fold conjecture that the paper's Theorem 1.1 asserts, it obtains for every computable $f$ and sufficiently large $n$ an $n$-variable system of elementary arithmetic equations with exactly $f(n)$ nonnegative integer solutions.
- D. Cantone, L. Cuzziol, E. G. Omodeo, Six equations in search of a finite-fold-ness proof (arXiv:2303.02208, v3 30 Sep 2024). Gives six quaternary quartic equations and shows that finite-foldness of any one of them would give finite-fold representations of all r.e. sets, an alternative conditional route to the finite-fold conjecture that the paper derives from Theorem 1.1.
- D. Cantone, L. Cuzziol, E. G. Omodeo, A Brief History of Singlefold Diophantine Definitions (CILC 2023, CEUR Workshop Proceedings Vol-3428, paper 5). Historical survey of the single-fold problem from Matiyasevich's 1974 work onward, the history the paper's introduction summarizes.
- D. Cantone, A. Casagrande, F. Fabris, E. Omodeo, The quest for Diophantine finite-fold-ness (Le Matematiche 76(1), 2021). Surveys the single-fold and finite-fold questions as open problems and outlines a hoped-for route to a positive answer, the conjectures that Theorem 1.1 claims to settle.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 242
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/242.md. The Lean statements have not been checked by people against the paper's claims.
Family 242: 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.
This is the only paper in this family.