The site's label is OPEN; the derived standing is claimed, through a pending full claim on the site's proof-claims tab that would settle the gap questions if accepted. What the refereed sources give: a sum-free set has density zero, $\sum1/a_i<103$ and $\liminf A(x)x^{-(\sqrt5-1)/2}<\infty$, and some sum-free set has $A(x)>cx^{2/7}$ (Erdős 1962, Theorems I--III and the construction); a sum-free set has $A(n)\le403\sqrt{n\log n}$ for infinitely many $n$, and for every $\varepsilon>0$ some sum-free set has $A(n)\ge n^{1/2}(\log n)^{-1/2-\varepsilon}$ for all large $n$ (Łuczak and Schoen 2000); the reciprocal sum is below an absolute constant that the authors say "seems certain" to be below $10$ (Benkoski and Erdős 1974), below $100$ by Erdős's 1977 restatement, and below $4$ by Sullivan, who conjectured a maximum "only a little greater than $2$" (Erdős 1977, the site's figures; Sullivan's work is not held). None of these decides the gap questions: the site's near-linear gap statement $a_{n+1}-a_n<n^{1+o(1)}$ (Graham, as reported by Erdős 1998) and the $a_n\sim n^{3+o(1)}$ construction (Deshouillers, Erdős and Melfi 1999) rest on papers not held and are reported from the site. Two claims on the site's proof-claims tab, neither adopted by the site, bear on the gap questions and have claim pages: Price's partial claim of 18 July 2026, credited to GPT 5.6 Sol Pro, that every sum-free sequence has $\limsup(a_{n+1}-a_n)/n=\infty$, a negative answer to the second question; and Korsky's full claim of 22 September 2026, credited to GPT Astra, with two theorems: for a nondecreasing slowly varying $F\ge1$, a sum-free sequence with gaps of the order $nF(n)$ exists if and only if the integral $\int^\infty dx/(xF(x))$ is finite (its claim page states how that reading of the summary's $\Omega(nF(n))$ is reached); and the smallest $c$ for which a sum-free sequence can have gaps $O(n^c)$ and infinitely many bounded gaps is $(1+\sqrt{13})/2$. Both are unreviewed; the pending full claim makes the derived standing claimed, while the site's label stays OPEN. Neither touches the reciprocal-sum question. The refereed sources alone give a bounded negative finding, not a certificate of openness.