Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.
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In Existential–universal real sentences in the counting hierarchy, 4 October 2026
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There is an absolute integer j such that
The level j is independent of the input length, the numbers of variables, the degrees of the circuit polynomials, and their coefficient magnitudes.
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Alex Meiburg, $\exists \mathbb{R} \subseteq \textsf{CH}$, 7 October 2026. Source
Meiburg posted a 114-page ChatGPT-derived proof that the existential theory of the reals lies in the fourth level of the counting hierarchy, noting OpenAI’s very similar result has a slightly weaker bound.
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Theorem 1.1
In Existential–universal real sentences in the counting hierarchy, 4 October 2026