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Math-2304 · mathematics · Logic

Finite-fiber normalization for elliptic Diophantine encodings

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Can elliptic curves replace Pell equations in the proof of the undecidability of Hilbert's tenth problem over $\mathbb Z$, allowing fewer variables or a smaller degree? Can this approach represent every recursively enumerable set by a Diophantine equation having only finitely many witnesses for each parameter?

Formulation references: [S1], [S2].

Short English statement

Can elliptic curves replace Pell equations in the proof of the undecidability of Hilbert's tenth problem over $\mathbb Z$, allowing fewer variables or a smaller degree? Can this approach represent every recursively enumerable set by a Diophantine equation having only finitely many witnesses for each parameter?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2304-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2304

The single-fold theorem also establishes finite-fold representability. Its cutoff still uses Pell equations, however, and it does not provide the requested full elliptic replacement or sharper variable and degree bounds.

Finite-fiber normalization for elliptic Diophantine encodings

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 selected statement includes the clause used for this problem.

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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