A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erdős's \$1000 conjecture -- that every finite Sidon set extends to a finite PDS -- was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025), via the size-5 counterexamples $\{1,2,4,8,13\}$ and Hall's earlier $\{1,3,9,10,13\}$; they then asked: what is the smallest size $s$ of a non-extending Sidon set? The trivial bounds give $3 \le s \le 5$. Our evidence points to $s = 4$. We exhibit two integer Sidon sets, \[ A = \{0, 1, 3, 11\}, \qquad B = \{0, 1, 4, 11\}, \] together with the apparent infinite family of dilations $kA$, $kB$ and their reflections, all of which fail to extend for every prime power $q \le 317$ via the Singer affine-orbit check (rigorous under Hall's 1947 uniqueness for Desarguesian cyclic planes through $q \le 40$ and under the prime-power conjecture beyond that), and unconditionally for every modulus $v \le 133$ via brute-force depth-first search. We also report the exact density $N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$ of non-extending size-4 Sidon sets in $[0, N]$ for $N \le 50$ -- the match is exact, which suggests the $kA, kB$ family is complete in this range. A complete proof, perhaps in the spirit of Alexeev--Mixon's polarity argument or via a multiplier descent, remains open.

翻译：有限集合 $S \subset \mathbb{Z}$ 称为西顿集，若其两两差互不相同。回顾 $n$ 阶完美差集 (PDS) 是指子集 $B \subset \mathbb{Z}_v$（其中 $v=n^2-n+1$），满足 $|B|=n$ 且每个非零剩余恰好出现一次为 $B$ 中两元素之差。Erdős 的 $1000$ 美元猜想——每个有限西顿集均可扩展为有限完美差集——已被 Alexeev 与 Mixon 通过大小5的反例 $\{1,2,4,8,13\}$ 及 Hall 先前提出的 $\{1,3,9,10,13\}$ 所否定（arXiv:2510.19804, 2025年10月）；随后他们提出问题：不可扩展西顿集的最小规模 $s$ 是多少？平凡界给出 $3\le s\le 5$。我们的证据指向 $s=4$。我们给出两个整数西顿集： \[ A = \{0,1,3,11\},\quad B = \{0,1,4,11\}, \] 连同明显的无限族 $kA$, $kB$ 及其反射变换。通过 Singer 仿射轨道检验（在 $q\le 40$ 时由 Hall 1947年关于 Desarguesian 循环平面的唯一性严格成立，在此之上则依赖于素数幂猜想），对于所有素数幂 $q\le 317$ 以及通过暴力深度优先搜索对所有模数 $v\le 133$，这些集合均无法扩展。我们还报告了 $[0,N]$ 中不可扩展大小4西顿集的精确密度 $N_{\text{ne}}(N)=4\lfloor N/11\rfloor$（当 $N\le 50$ 时）——该匹配完全精确，表明在此范围内 $kA,kB$ 族是完备的。完整证明（或许沿袭 Alexeev-Mixon 的对偶论证思路或通过乘子下降法）仍有待解决。