AI 中文总结
针对从原点出发需击中未知直线的确定性搜索路径，研究人员给出了其竞争比的新无条件下界，还通过相同方法得到一维最优牛路径常数9，并借助Arb球算术完成常数的严格数值包围。
AI 中文摘要
单位速度搜索者从欧氏平面原点出发，必须击中一条方向和距原点距离均未知的未知直线。我们证明，每条确定性搜索路径的竞争比至少为$C_{\text{log}} \approx 12.5937096701246675$。该下界是无条件的：路径无需是循环的、自相似的、螺旋状的或角度单调的。对于每个投影方向，我们将该路径与通过对其交替记录转弯排序得到的锯齿形路径进行比较。由此产生的完成约束被解释为具有尺度相关截止时间的作业，并通过有限窗口调度论证在对数时间内得到下界。对这些方向下界取平均时使用精确的欧氏速度预算。在一维情况下，相同方法可得到最优的牛路径常数9。最后，Arb 球算术为该常数提供了严格的数值包围。
英文摘要
A unit-speed searcher starts at the origin of the Euclidean plane and must hit an unknown straight line whose direction and distance from the origin are both unknown. We prove that every deterministic search path has competitive ratio at least $C_{\log}\approx 12.5937096701246675$. The bound is unconditional: the path need not be cyclic, self-similar, spiral-like, or monotone in angle. For each projection direction, we compare the path with a zigzag obtained by sorting its alternating record turns. The resulting completion constraints are interpreted as jobs with scale-dependent deadlines and lower-bounded through a finite-window scheduling argument in logarithmic time. Averaging these directional bounds then uses the exact Euclidean velocity budget. In one dimension, the same method recovers the optimal cow-path constant $9$. Finally, Arb ball arithmetic provides a rigorous numerical enclosure of the constant.
Comments13 pages, 3 figures. Ancillary files include the Arb numerical certificate and supporting Lean/mathlib checks
