We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by B\"{u}rgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.

翻译：我们证明了在实数理论更高层次上的完备性是一个稳健的概念（在改变符号和限制量词定义域的情况下）。这修补了层次结构中公认的缺口，并为各类计算问题带来了更强的完备性结果。我们展示了若干完备问题族，可用于未来实数层次结构中的完备性证明。作为应用，我们强化了Bürgisser和Cucker关于半代数集性质复杂性的若干结果，包括Jungeblut、Kleist和Miltzow亦研究过的豪斯多夫距离问题。