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Matiyasevich's single-fold conjecture: every r.e. set has a single-fold Diophantine representation

By the Davis-Putnam-Robinson-Matiyasevich theorem every recursively enumerable set S⊆NnS\subseteq\mathbb N^n has a Diophantine representation: a polynomial PP with integer coefficients such that a∈Sa\in S iff P(a,w)=0P(a,w)=0 for some w∈Nmw\in\mathbb N^m. The representation is single-fold if such ww is unique when it exists, finite-fold if there are finitely many. Matiyasevich (1974) proved single-fold exponential Diophantine representations and posed the polynomial case, reducing it to a single-fold representation of exponentiation; the finite-fold conjecture is the weaker form. Does every recursively enumerable set have a single-fold Diophantine representation?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Computability; Hilbert's tenth problem
Posed by
Yuri Matiyasevich
Year posed
1974
Years open
52y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every r.e. S⊆NnS\subseteq\mathbb N^n, n≥1n\ge1, has a polynomial P∈Z[A1..An,W1..Wm]P\in\mathbb Z[A_1..A_n,W_1..W_m] with #{w∈Nm:P(a,w)=0}\#\{w\in\mathbb N^m:P(a,w)=0\} equal to 1 for a∈Sa\in S and 0 otherwise. Hence the finite-fold conjecture holds, and Diophantine solvability over N\mathbb N stays undecidable under an at-most-one-solution promise (Section 8). The construction bounds a Pell-type exponential with a cutoff from rational points on a rank-one elliptic curve. Witnesses range over N\mathbb N; the paper does not claim the analogue over Z\mathbb Z beyond what follows from it, nor any bound on degree or number of variables.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 24, 2026.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Matiyasevich's conjecture; it counts all auxiliary variables and gives exactly one complete witness tuple for members and none for nonmembers, over N\mathbb N. lean/formalization.yaml lists a main result (comparator SingleFold, declaration OAI.SingleFold.main). The comparator statement was read here: for every n≥1n\ge1 and every r.e. S⊆NnS\subseteq\mathbb N^n (via Mathlib's REPred) there are m≥1m\ge1 and an integer polynomial in n+mn+m variables with a∈Sa\in S iff some w∈Nmw\in\mathbb N^m is a zero, and the zero ww unique for each aa. This states the headline. The undecidability-under-promise corollary is not separately formalized (lean/docs/242.md). Not rebuilt here.

Sources

Changelog1 change

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