Dataset for "Necessary and sufficient conditions for correctness of complex Langevin"

We derive a family of correctness conditions for complex Langevin simulations. In particular, we show that if in a given theory the expectation values of all observables within a particular space satisfy the theory's Schwinger--Dyson equations as well as certain bounds, then these expectation values are necessarily correct. In fact, these findings are not only valid in the context of complex Langevin simulations, but they also hold for general probability densities on complex manifolds, given an initial complex density on a real manifold. We test these criteria in a few simple one- and two-dimensional toy models and find that they are indeed capable of ruling out incorrect results without the need of exact solutions. If you use this data, please cite the corresponding paper:2508.14512 [hep-lat] or J. Phys. A 58 495202 (2025) This work was funded by the Austrian Science Fund (FWF) [10.55776/P36875].我们为复朗之万（complex Langevin）模拟推导了一类正确性条件。具体而言，我们证明：若某一理论中，特定空间内所有可观测量的期望值满足该理论的施温格-戴森（Schwinger-Dyson）方程以及特定约束条件，则这些期望值必然是正确的。事实上，这些结论不仅适用于复朗之万模拟场景，且在实流形上存在初始复密度分布的前提下，同样适用于复流形上的一般概率密度分布。我们通过数个简单的一维与二维玩具模型对上述判据进行了测试，结果表明，无需精确解即可有效排除错误结果。 若使用本数据集，请引用对应论文：2508.14512 [hep-lat] 或 J. Phys. A 58 495202 (2025)