Counting-hierarchy upper bound for existential-universal real sentences
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Does one fixed finite level of the counting hierarchy decide every existential-universal sentence over the reals encoded by Boolean combinations of sign tests on integer arithmetic-circuit polynomials, with signed binary constants and only addition, subtraction and multiplication gates, independent of variable counts, degrees and coefficient magnitudes?Short English statement
Does one fixed finite level of the counting hierarchy decide every existential-universal sentence over the reals encoded by Boolean combinations of sign tests on integer arithmetic-circuit polynomials, with signed binary constants and only addition, subtraction and multiplication gates, independent of variable counts, degrees and coefficient magnitudes?Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/2407.18006.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for CS-131
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).
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
- OpenAI, Existential–universal real sentences in the counting hierarchy, 2026-10-04; citation key
OAI:Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.