Previously, Lemma 3.2 claimed that if $C$ and $C$ are two circles in $\mathbb{R}^3$ that intersect at two points $x$ and $y$, then under some mild assumptions, the projections of the shorter circular arcs between $x$ and $y$ to the $(x_1,x_2)$ plane will always intersect an even number of times. In particular, Lemma 3.2 claimed that the arcs must lift to a depth cycle in $\mathbb{R}^4$. While this claim was wrong as stated, it is true after applying a suitable orthogonal transformation, provided the arcs are sufficiently short. The statement and proof of Lemma 3.2, and the proof of Corollary 3.2 have been updated to fix this problem. The rest of the paper remains unchanged.

中文翻译：

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勘误：打破三维单位距离的 3/2 障碍

以前，引理 3.2 声称，如果 $C$ 和 $C$ 是 $\mathbb{R}^3$ 中的两个圆，它们在 $x$ 和 $y$ 两点相交，那么在一些温和的假设下， $x$ 和 $y$ 之间到 $(x_1,x_2)$ 平面的较短圆弧将始终相交偶数次。特别是，引理 3.2 声称弧必须提升到 $\mathbb{R}^4$ 中的深度循环。虽然这种说法是错误的，但在应用适当的正交变换后，只要弧足够短，它就是正确的。Lemma 3.2 的陈述和证明以及 Corollary 3.2 的证明已更新以解决此问题。论文的其余部分保持不变。