we-solve (beta)

CS-131-A1 · revision r1 · on CS-131

OpenAI claim summary for CS-131

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: open. 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.

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 141; 2026-10-04).

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

Manuscript

Contents
  1. Counting-hierarchy upper bound for existential-universal real sentences
  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

Counting-hierarchy upper bound for existential-universal real sentences

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 141; 2026-10-04), the asserted result is that existential real feasibility, and more generally existential--universal real sentences with circuit-given integer polynomials, lie in a fixed counting-hierarchy level.

Mathematical statement of the claimed result

Manuscript 1.
Let $\mathcal T_{\exists\forall}$ consist of true real sentences \[ \exists x\in\mathbb R^r\ \forall y\in\mathbb R^s:\Phi(x,y), \] where $r,s\ge0$, $\Phi$ is an explicitly encoded Boolean combination of $p=0$ and $p>0$, and each $p$ is an acyclic arithmetic circuit with binary integer constants and $+,-,\times$ gates, with no division or exponentiation gates. Malformed encodings are rejected. Define $\mathsf C_0=\mathrm P$, $\mathsf C_{j+1}=\bigcup_{A\in\mathsf C_j}\mathrm{PP}^A$. The claim is \[ \exists j\in\mathbb N:\quad\mathcal T_{\exists\forall}\in\mathsf C_j, \] with one absolute level independent of input size, variable counts, degrees and coefficient sizes. These are uniform bit-time complexity classes.

Outline of the argument

For manuscript 1 (family 141; 2026-10-04), critical-point deformation turns quantified fibers into short algebraic witnesses. Modular evaluation accesses huge controlled polynomials without expansion, while residue--trace determinant formulas and compressed derivative-sign labels recover the needed root signs inside the counting hierarchy.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 141; 2026-10-04).

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