1 · The essentials
Some collections of lists of whole numbers can be listed by a computer procedure, even when deciding whether a particular list belongs is difficult. The main manuscript states that every such collection can be described by a polynomial equation using additional unknown whole numbers. For each list in the collection, there is exactly one complete choice of those additional numbers that makes the equation hold. For each list outside the collection, there is no such choice. Here, whole numbers include zero. This result has a Lean formalization and settles what the source calls the single-fold conjecture, also giving the weaker conclusion that the number of additional solutions is always finite. The repository summary further states that no algorithm can always decide whether a polynomial equation has a solution, even when it is promised to have at most one.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a collection of entries that a computer can gradually print, and a lock assigned to each possible entry. Every printed entry has exactly one working combination, while every other entry has none. Having only one possible working combination does not, by itself, supply a general way to decide whether one exists.
Why it matters
The result connects collections that can be enumerated by computation with polynomial equations whose additional solutions are uniquely determined whenever they exist. The repository's undecidability claim highlights a separate distinction: uniqueness of a possible solution does not guarantee an algorithm for deciding existence. Thus, restricting the number of solutions need not remove the difficulty of the decision problem.
2 · Going further
The precise representation problem starts with a recursively enumerable set of tuples of natural numbers. A single-fold Diophantine representation uses a polynomial equation , where is the input tuple and is a complete auxiliary tuple, all over the nonnegative integers. For each , exactly one tuple satisfies the equation; for each , none does. Uniqueness concerns the entire auxiliary tuple, not merely one selected coordinate.
The main manuscript states that every recursively enumerable set of natural-number tuples admits such a representation. The source identifies this as settling the single-fold conjecture and, consequently, the finite-fold conjecture: exactly one solution for members and none for nonmembers is in particular a finite number of solutions for every input. The main representation result has a Lean formalization.
The repository summary also states that Diophantine solvability over the nonnegative integers remains undecidable under an at-most-one-solution promise. Here the question is whether a solution exists, even though multiple solutions are ruled out by the promise. The supplied abstract states the representation theorem but gives no construction or argument outline, so the method cannot be described from this source. Only the main manuscript is supplied; no companion-manuscript abstracts are available to explain further relationships.
Useful notions
- Natural numbers, including zero
- Polynomials and equations
- Tuples of numbers
- Enumeration by a computer procedure
- Existence versus uniqueness of a solution
What was known before / what is new
Before
The source names the single-fold and finite-fold conjectures as the questions being settled. It provides no account of earlier partial results, their history, or previously known representation methods.
New
The main manuscript states a universal single-fold representation result: every recursively enumerable set of natural-number tuples has exactly one complete auxiliary solution for each member and none for nonmembers. This main result has a Lean formalization, and the abstract states the finite-fold conclusion as a consequence. The repository summary additionally claims undecidability of solvability under an at-most-one-solution promise.
Limits and assumptions
The representation theorem applies to recursively enumerable sets of natural-number tuples, with solutions taken over the nonnegative integers. Its uniqueness assertion is for the complete auxiliary tuple after the input tuple is fixed; it does not assert that the polynomial has only one solution across all inputs. The main result has a Lean formalization, but the supplied material does not specify its exact coverage of the finite-fold consequence or the separate undecidability claim. The abstract supplies no proof method, and the source identifies no remaining open questions.
Mini glossary
- Recursively enumerable set —
- A collection whose members can be listed by a computer procedure.
- Diophantine representation —
- A description of membership in a set by the existence of solutions to a polynomial equation in additional unknowns, here over the nonnegative integers.
- Auxiliary tuple —
- The complete list of additional unknown numbers used alongside a fixed input tuple in the representing equation.
- Single-fold representation —
- A Diophantine representation with exactly one auxiliary solution for each member and none for each nonmember.
- Finite-fold representation —
- A Diophantine representation with finitely many auxiliary solutions for each input.
- Undecidability under an at-most-one-solution promise —
- The absence of an algorithm that always decides whether a solution exists for equations guaranteed to have at most one solution.
Keywords : Single-fold representations · Diophantine equations · Undecidability
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Single-fold Diophantine representations
24/09/2026Every 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.