Single-fold Diophantine representations at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Single-fold Diophantine representations

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	logic, thinking really hard	Levels:	1
Category:	Mathematical logic	Lean version:	YES! ✔
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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.

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LEVEL 1 Single-fold Diophantine representations

released 2026-09-24 | 1 theorem · 6 lemmas · 13 proofs · 11,310 words | PLAY LEVEL 1 » (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.

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