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

What OpenAI claims

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)

Paper

1. Single-fold Diophantine representations

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.