带回访的延拓实数线：在克隆茨层级中分离US与KC

我们构造带回访的延拓实数线(ERI)：将$\{-\infty, 0, +\infty\}$识别为单个点$\ast$于$\overline{\mathbb{R}}$中，要求每个关于$\ast$的邻域具有稠密原像。 resulting space(结果空间)为紧凑、路径连通且 sober；它为$\ T_1$和US(唯一序列紧致)，但不为弱Hausdorff、KC或Hausdorff。 In the refined hierarchy of Clontz, ERI sits at the $k_2$-Hausdorff level. A search of pi-Base for compact US-not-KC spaces returns three entries -- $\mathbb{Q}^{\ast} \times \mathbb{Q}^{\ast}$, $\omega_1+1$ with doubled endpoint (S37), and the one-point compactification of the Arens-Fort space (S165) -- all totally disconnected. ERI is the first compact path-connected example. The same density condition on a general compact Hausdorff base without isolated points defines a Filter-Modified Quotient (FMQ). We prove that the density modifier $\mathcal{D}_Y$ is the least restrictive admissible modifier preserving US, and that the hierarchy level $k_2\mathrm{H}$-not-$\mathrm{wH}$ is invariant under infinite closed nowhere-dense collapse sets, iteration of the construction, and arbitrary products. The only remaining direction toward a US-not-$k_2\mathrm{H}$ level runs through non-first-countable base spaces.