A finite set $S \subset \mathbb{Z}$ is a \emph{Sidon set} if its pairwise differences are distinct. A \emph{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 raised the question of the smallest size $s$ of a non-extending Sidon set, with $3 \le s \le 5$. We present strong empirical evidence that $s = 4$. Specifically 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 give the exact density formula $N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$ of non-extending size-4 Sidon sets in $[0, N]$ for $N \le 50$, an exact match suggesting the $kA, kB$ family is complete in this range. A complete proof, in the spirit of Alexeev--Mixon's polarity argument or via a multiplier descent, remains the central open problem.

翻译：一个有限集合 $S \subset \mathbb{Z}$ 若其任意两元素之差互不相同，则称为\emph{Sidon集}。阶数为 $n$ 的\emph{完美差集}（PDS）是指集合 $B \subset \mathbb{Z}_v$（其中 $v = n^2 - n + 1$）满足 $|B| = n$，且每个非零剩余恰好作为 $B$ 中两个元素的差出现一次。Erdős 的 \$1000 猜想——每个有限 Sidon 集都能扩展为有限 PDS——已被 Alexeev 与 Mixon （arXiv:2510.19804，2025年10月）通过大小5的反例 $\{1,2,4,8,13\}$ 以及 Hall 早先的 $\{1,3,9,10,13\}$ 所否定；他们提出了不可扩展 Sidon 集的最小大小 $s$ 问题，其中 $3 \le s \le 5$。本文给出强有力的数值证据表明 $s = 4$。具体而言，我们构造了两个整数 Sidon 集 \[ A = \{0, 1, 3, 11\}, \qquad B = \{0, 1, 4, 11\}, \] 以及显然的无限仿射族 $kA$、$kB$ 及其反射集，这些集合在通过 Singer 仿射轨道检验（在 $q \le 40$ 时基于 Hall 1947年关于德萨格循环平面的唯一性严格成立，更大范围则在素幂猜想下成立）时，对所有 $q \le 317$ 的素幂均不可扩展；此外通过暴力深度优先搜索，对所有 $v \le 133$ 的模数无条件成立。我们给出 $N \le 50$ 时 $[0, N]$ 中不可扩展的大小4 Sidon 集的精确密度公式 $N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$，该精确匹配表明在此范围内 $kA, kB$ 族是完备的。遵循 Alexeev–Mixon 的极性论证思路或通过乘子降阶方法给出完整证明，仍是核心悬而未决的问题。