Chen 2025 bounds two distance sets euclidean space unit sphere

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theorem_3_1: Records the source's valid undivided inequality and the conditional bound obtained when its denominator is positive.

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Wei-Chun Chen and Wei-Hsuan Yu, Bounds on two-distance sets in Euclidean space and Unit Sphere, arXiv:2509.00858v1 (31 August 2025), 24 pages. The title page is dated 3 September 2025; the source record is https://arxiv.org/abs/2509.00858. The arXiv record (https://arxiv.org/abs/2509.00858, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

This folder records only the Euclidean ratio-dependent source lead. For an $N$-point two-distance set in $\mathbb{R}^{d}$ with distances $1$ and $\delta>1$, put

$$\gamma=\frac{1+\delta^2}{\delta^2-1}>0.$$

The proof on printed p. 9 displays the undivided inequality

$$(\gamma^2-(d+1))(N-1)\leq(d+1)(\gamma^2-1). \tag{3.6}$$

When the additional condition $\gamma^2>d+1$ holds, division gives the conditional bound

$$N\leq\frac{(d+1)(\gamma^2-1)}{\gamma^2-(d+1)}+1.$$

The printed Theorem 3.1 on p. 8 omits this sign condition and presents the rational expression as an unrestricted bound. That statement is false as printed: with $d=10$ and $\delta=\sqrt2$, one has $\gamma=3$ and the printed right-hand side is $-43$, while the regular-simplex midpoint construction gives 55 points in $\mathbb{R}^{10}$. Theorem 3.1 is therefore retained as a defect-bearing lead; no unrestricted Chen--Yu bound is asserted here.

The source's preceding spectral argument also contains a rank-index typo and states a strict smallest-eigenvalue multiplicity that its displayed argument does not establish. Those issues, together with the missing denominator hypothesis, are why the result page records the source inequality and its qualified algebraic consequence without claiming a complete corrected proof. The paper's spherical bounds and its inconsistent numerical table are outside this compilation.

Source verification. The retained v1 PDF is the canonical 209,140-byte, 24-page artifact. The abstract (p. 1), Theorem 3.1 (p. 8), and equation (3.6) in its proof (p. 9) were rendered and visually inspected; every formula and qualification recorded here was checked against those images.

Bears on. #502.

Results.

• [[distance_problems/chen_2025_bounds_two_distance_sets_euclidean_space_unit_sphere/theorem_3_1|Theorem 3.1, qualified source lead]]: the valid undivided inequality, its denominator-sign condition, and the source defect.