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Statement

Lemma 4.1 (p. 5), titled "Projection shadow". Let $H$ be a Euclidean subspace and let points $x_c\in H$, $c\in C$, all have squared norm $R^2$. Let $x_v$ be a vector of a larger ambient space and $u=P_Hx_v$ its orthogonal projection onto $H$. Suppose that for constants $\alpha>\beta$ every $c\in C$ has $\langle u,x_c\rangle=\langle x_v,x_c\rangle\in\{\alpha,\beta\}$. Fix $D>0$ and choose $t>0$ with

$$t^2\|u\|^2+R^2-2t\beta=D^2. \tag{17}$$

Then $z=tu$ satisfies $\|z-x_c\|^2=D^2-2t(\alpha-\beta)<D^2$ when $\langle x_v,x_c\rangle=\alpha$, and $\|z-x_c\|^2=D^2$ when $\langle x_v,x_c\rangle=\beta$.

The lemma assumes a positive $t$ satisfying (17); it does not assert that one exists. Section 5 applies it on p. 6, citing it as "theorem 4.1".

Source. Yibo Ji, An AI Generated Counterexample to Borsuk Problem in Dimension 63, arXiv:2608.12561v1 (12 August 2026), withdrawn by version 2 of 14 August 2026; Lemma 4.1 on p. 5. The source card records the withdrawal and provenance.

Read depth. Claims checked: the statement was read clause by clause on the PDF.

Proof pointer

Since each $x_c$ lies in $H$, projecting does not change its inner product with $x_v$. Expanding $\|tu-x_c\|^2=t^2\|u\|^2+R^2-2t\langle u,x_c\rangle$ and substituting (17) gives both cases; the strict inequality uses $t>0$ and $\alpha>\beta$ (p. 5).

Dependencies

None.

Bears on

• E0505: only through Theorem 6.2, whose added point is this lemma's $z$ with $R^2=15/4$, $\alpha=3/4$, $\beta=-1/4$ and $D^2=8$ (p. 6). The lemma alone says nothing about the number of parts.