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

Theorem 1.1: every r.e. $S\subseteq\mathbb N^n$ , $n\ge1$ , has a polynomial $P\in\mathbb Z[A_1..A_n,W_1..W_m]$ with $\#\{w\in\mathbb N^m:P(a,w)=0\}$ equal to 1 for $a\in S$ and 0 otherwise. Hence the finite-fold conjecture holds, and Diophantine solvability over $\mathbb N$ 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 $\mathbb N$ ; the paper does not claim the analogue over $\mathbb Z$ beyond what follows from it, nor any bound on degree or number of variables.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

Confidence: Not scored

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

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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 $\mathbb N$ . lean/formalization.yaml lists a main result (comparator SingleFold, declaration OAI.SingleFold.main). The comparator statement was read here: for every $n\ge1$ and every r.e. $S\subseteq\mathbb N^n$ (via Mathlib's REPred) there are $m\ge1$ and an integer polynomial in $n+m$ variables with $a\in S$ iff some $w\in\mathbb N^m$ is a zero, and the zero $w$ unique for each $a$ . This states the headline. The undecidability-under-promise corollary is not separately formalized (lean/docs/242.md). Not rebuilt here.