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Statement

Setting (p. 3). $D$ is the set of all nontrivial divisors of $720=2^4\cdot3^2\cdot5$ (the paper's equation (3); the divisor $1$ is left out):

$$D=\{2,3,4,5,6,8,9,10,12,15,16,18,20,24,30,36,40,45,48,60,72,80,90,120,144,180,240,360,720\},$$

listed as $d_1<d_2<\cdots<d_{29}$, with prefixes $D_j=\{d_1,\ldots,d_j\}$. $\alpha$ is the independence number of the unit-fraction hypergraph of Lemma 1.

Proposition 1 (Finite certificate, p. 3). For the prefixes $D_j$ of (3), the paper's table gives $\alpha(D_j)$. For $j=1,\ldots,29$ its values are

$$1,2,3,4,4,5,6,7,7,7,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,$$

so the deficiencies $j-\alpha(D_j)$ run

$$0,0,0,0,1,1,1,1,2,3,3,3,4,5,6,6,7,8,9,10,10,11,12,13,14,15,16,17,18,$$

and in particular $\alpha(D)=11$.

Source. Xinjun Wang, A 667/806 Upper Bound for Erdős Problem #301 on Unit-Fraction-Free Sets, unpublished manuscript dated May 27, 2026 on its title page, posted on ResearchGate (2026), identified on the source card: Proposition 1 and its table on p. 3, the set (3) on p. 3, and the verification script in Appendix A (pp. 6--9). An unrefereed manuscript, which the site's Problem 301 discussion thread describes as AI-generated.

Read depth. Claims checked: the statement and every row of the table were read on the page image, and the values above agree with the script's expected-value list on p. 6. The script was read as text and not rerun, so the values themselves are the manuscript's claim. Nothing here is independently reviewed.

Proof pointer

P. 3 and Appendix A (pp. 6--9). Each lower bound $\alpha(D_j)\ge$ the tabulated value is witnessed by an explicit independent subset of $D_j$ listed in the script. For the upper bounds the script enumerates the inclusion-minimal hyperedges of the full $D$ (each identity $1/d=\sum_{e\in E}1/e$ checked as the integer identity $720/d=\sum_{e\in E}720/e$), restricts them to prefixes, and runs an exact branch-and-bound search only at the last prefix of each run of equal values; since $\alpha$ cannot grow on passing to a subset, that settles the earlier prefixes of the run. Remark 1 (p. 3) explains why discarding non-minimal hyperedges leaves the independent sets unchanged.

Dependencies

• Lemma 1 for the hypergraph and $\alpha$; otherwise a finite computation.

Bears on

• Problem 301: the table is the finite input to the manuscript's Theorem 1; weighted over disjoint dilates it gives the manuscript's missing density $139/806$. The proposition is a finite statement about $D$ and by itself gives no bound for $f(N)$.