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Math-2723 · mathematics · Logic

Unique natural-number witnesses for every enumerable set

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $n\geq 1$ and recursively enumerable $S\subset N^{n}$, do there exist $m\geq 1$ and $P\in Z[A_{1},\dots ,A_{n},W_{1},\dots ,W_m]$ such that each $a\in S$ has exactly one tuple $w\in N^m$ with $P(a,w)=0$ and each $a\notin S$ has none?

Short English statement

For every $n\geq 1$ and recursively enumerable $S\subset N^{n}$, do there exist $m\geq 1$ and $P\in Z[A_{1},\dots ,A_{n},W_{1},\dots ,W_m]$ such that each $a\in S$ has exactly one tuple $w\in N^m$ with $P(a,w)=0$ and each $a\notin S$ has none?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2723-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2723

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 242; 2026-09-24).

Unique natural-number witnesses for every enumerable set

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 242; 2026-09-24), the asserted result is that every recursively enumerable relation on natural-number tuples has an integer-polynomial representation with exactly one complete auxiliary witness for each member.

Mathematical statement of the claimed result

Manuscript 1.
Use $\mathbb N$ including zero. For each $n\ge1$ and recursively enumerable $S\subseteq\mathbb N^n$, the claim gives an integer $m\ge1$ and an integral polynomial $P(a,w)$ with \[ \forall a\in\mathbb N^n:\quad |\{w\in\mathbb N^m:P(a,w)=0\}|= \begin{cases}1,&a\in S,\\0,&a\notin S.\end{cases} \] The number of witness variables is fixed for the set $S$. This asserts a unique Diophantine witness for every member, rather than merely a finite number of witnesses.

Outline of the argument

For manuscript 1 (family 242; 2026-09-24), pell equations recover exponentiation, but their indices are selected using a uniquely coded rank-one elliptic curve. Bounded height error gives an exact growth cutoff, and canonical arithmetic codes then encode each halting computation with a unique full tuple.

Formal verification

For manuscript 1 (family 242; 2026-09-24), OpenAI lists Lean statements for this assertion: every recursively enumerable relation on natural-number tuples has an integer-polynomial representation with exactly one complete auxiliary witness for each member.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are SingleFold.lean.

Their named declarations include OAI.SingleFold.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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