Can an AI Agent Rediscover a Blaschke-Curve Invariant?

AI 智能体能否重新发现布拉施克曲线（Blaschke-Curve）的不变量？

Abstract: We study generalized Blaschke curves as a controlled environment for AI-assisted mathematical rediscovery. For one fixed degree-four Blaschke product, an agent receives numerical coordinates of the six pair-lines determined by each of 80 boundary configurations. The target theorem is withheld from the task instructions.

摘要： 我们研究了广义布拉施克曲线，将其作为 AI 辅助数学重新发现的受控环境。针对一个固定的四次布拉施克乘积，智能体接收由 80 种边界配置中每一种所确定的六条配对线的数值坐标。任务指令中并未包含目标定理。

The saved research log reports rejected geometric hypotheses and a homogeneous cubic fitted to polygon sides. Its frozen coefficients predict 480 lines from 80 unseen parameter values, with a recorded RMS scale-free residual of $8.88\times10^{-17}$. Discovery-set diagonals provide an out-of-fit consistency check, not a fully held-out test.

保存的研究日志记录了被否决的几何假设，以及拟合多边形边长的齐次三次方程。其固定的系数根据 80 个未见过的参数值预测了 480 条线，记录的均方根（RMS）无标度残差为 $8.88\times10^{-17}$。发现集中的对角线提供了一种拟合外一致性检查，而非完全独立的测试。

A separate one-configuration run reports insufficient evidence for invariance. A post-review deterministic degree-search baseline also recovers the cubic, so the experiment does not establish an advantage over polynomial fitting. We present this single-instance case study as a protocol for separating conjecture, numerical validation, and proof, with explicit limitations concerning agent metadata, prior knowledge, and reproducibility.

另一项单配置运行报告显示，没有足够的证据证明其不变性。事后审查的确定性次数搜索基准也恢复了该三次方程，因此该实验并未证明其优于多项式拟合。我们提出这一单实例案例研究，作为区分猜想、数值验证和证明的协议，并明确了在智能体元数据、先验知识和可重复性方面的局限性。