亲吻数构造的大规模自主发现
Large-Scale Autonomous Discovery of Kissing Number Constructions
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中文总结 AI 辅助
本研究使用自主多智能体系统Qiushi Engine，在19个维度中发现亲吻数新下界，通过多种结构机制获得显著改进的构造，展示了自主研究系统在数学发现中的潜力。
中文摘要 AI 辅助
亲吻数问题是离散几何中的一个经典问题，其精确解仅在少数维度中已知。近期的人工智能方法已开始通过大规模数值与组合搜索发现改进的构型，但将此类搜索转化为一般数学构造和严格证明仍然具有挑战性。在此，我们使用Qiushi Engine（一个自主多智能体研究系统）来研究亲吻数，并在十九个维度中获得新的下界：$25$、$27$、$32$--$39$、$43$、$45$ 和 $49$--$55$。所得构造源于不同的结构机制，包括接触层的协调运动、标记方向重用、联合支撑交换、符号码替换、跨壳格构造、低重叠格等距以及球面设计矩证书。它们为参数化符号码模型提供了尖锐容量、确定性图像并集保证，以及由嵌入根系和锚图统计控制的精确截面与投影计数。这些方法尤其给出了 $K(25)\ge197580$、$K(27)\ge201567$、$K(38)\ge591900$、$K(43)\ge2553792$、$K(45)\ge7380090$ 和 $K(55)\ge53301140$。自主系统执行了构造搜索、数学分析和计算验证，而每个最终结果都被简化为明确的数学论证和可独立检查的有限证书。我们的结果展示了自主研究系统如何超越固定公式内的优化，以大规模发现新的数学表示和构造。
英文摘要
The kissing-number problem is a classical problem in discrete geometry whose exact solution is known in only a few dimensions. Recent artificial-intelligence approaches have begun to discover improved configurations through large-scale numerical and combinatorial search, but converting such searches into general mathematical constructions and rigorous proofs remains challenging. Here we use Qiushi Engine, an autonomous multi-agent research system, to investigate kissing numbers and obtain new lower bounds in nineteen dimensions: $25$, $27$, $32$--$39$, $43$, $45$, and $49$--$55$. The resulting constructions arise from distinct structural mechanisms, including coordinated motions of contact layers, labelled direction reuse, joint support exchanges, signed-code replacements, cross-shell lattice constructions, low-overlap lattice isometries, and spherical-design moment certificates. They yield sharp capacities for parameterized signed-code models, deterministic image-union guarantees, and exact section and projection counts controlled by embedded root systems and anchor-graph statistics. These methods yield, among others, $K(25)\ge197580$, $K(27)\ge201567$, $K(38)\ge591900$, $K(43)\ge2553792$, $K(45)\ge7380090$, and $K(55)\ge53301140$. The autonomous system carried out the construction searches, mathematical analysis and computational verification, while each final result was reduced to explicit mathematical arguments and independently checkable finite certificates. Our results illustrate how autonomous research systems can move beyond optimization within fixed formulations to discover new mathematical representations and constructions at scale.
发表机构
- College of Information Science and Electronic Engineering, Zhejiang University（浙江大学信息与电子工程学院）
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