We report on discovering new record-breaking Condorcet domains on $n=10$ and n=11 alternatives, challenging long-standing voting theory results. Our work presents new records with sizes of 1082 (previous record 1069) for n=10 and 2349 (previous record 2324) for $n=11$, which appear sporadic and do not fit into the existing alternating schema discovered in 1996. While the method used to discover these domains was inspired by the application of value functions in reinforcement learning, a subcategory of artificial intelligence, the current version of the method is somewhat ad-hoc and unstable. Therefore, we will not expound on the search method in this paper. Instead, we outline the key components that contribute to the success of our approach. We will also discuss the theoretical implications of our findings and explore the structure of the new Condorcet domains, raising several open questions related to them. Our results contribute to the ongoing investigation of Condorcet domains and their mathematical properties, potentially demonstrating the power of artificial intelligence-inspired problem-solving methods in advancing mathematical research.

翻译：我们报告在$n=10$和$n=11$个备选方案上发现了新的打破纪录的Condorcet域，挑战了长期存在的投票理论结果。我们的工作呈现了新纪录：对于$n=10$，大小为1082（此前纪录为1069）；对于$n=11$，大小为2349（此前纪录为2324）。这些纪录似乎是零星的，并不符合1996年发现的现有交替模式。虽然用于发现这些域的方法受到强化学习中价值函数应用的启发——强化学习是人工智能的一个分支——但目前版本的方法略显特设且不稳定。因此，我们将不在本文中详细阐述搜索方法，而是概述促成我们方法成功的关键组成部分。我们还将讨论我们发现的理论意义，探索新Condorcet域的结构，并提出与之相关的若干未解决问题。我们的成果有助于对Condorcet域及其数学性质的持续研究，可能展示了受人工智能启发的问题解决方法在推动数学研究方面的潜力。