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Math-2303-A1 · revision r1 · on Math-2303

OpenAI claim summary for Math-2303

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Rational undecidability can be interpreted in central quaternion or matrix variables. The bundle also includes integral-order equations, and those are not resolved merely by the rational-polynomial reduction.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. One-equation reductions for the rational Hamilton quaternions and Lipschitz order
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

One-equation reductions for the rational Hamilton quaternions and Lipschitz order

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 004; 2026-09-24), the asserted result is that rational solvability of integer-coefficient polynomials with an unrestricted number of variables is undecidable.

Mathematical statement of the claimed result

Manuscript 1.
With $n$ variable and integral polynomial coefficients explicitly encoded, the rational Diophantine decision claim is \[ \{f\in\mathbb Z[X_1,\ldots,X_n]:\exists x\in\mathbb Q^n,\ f(x)=0\} \quad\text{is undecidable}. \] There is no algorithm giving the correct yes/no answer for every such input; the coefficient ring is $\mathbb Z$, but sought solutions lie in $\mathbb Q$.

Outline of the argument

For manuscript 1 (family 004; 2026-09-24), integer solvability is represented by an effective sequence of finite rational tests. Elliptic indices, odd valuation contacts, and a height argument force an ordinary integer witness when every test succeeds; a hypothetical rational decision procedure would then decide the integer problem by two concurrent searches.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 004; 2026-09-24).

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