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Statement

Setting (pp. 6--8). For a block $S$ of the minimal partition $p_*$, with $T_S$ and $\rho_S$ as on Proposition 6.1, the note puts $a_S=T_S^{1/n_S}$ $(=\lambda_S^{-1/n_S}$ by Lemma 7.1$)$, so $r(S)=2\rho_Sa_S$ and $\sum_{S\in p_*}r(S)\le2$ is equivalent to $\sum_{S\in p_*}\rho_Sa_S\le1$. For $u>0$ and $t>0$,

$$N(u,t)=\#\{S\in p_*:\rho_S\ge u,\ a_S\ge t\},$$

and the layer-cake formula gives $\sum_{S\in p_*}\rho_Sa_S=\int_0^\infty\int_0^\infty N(u,t)\,dt\,du$ (Section 7.3, pp. 7--8).

Reduction (p. 8). If $p_*$ has at least two blocks, every block is proper and Proposition 6.1 gives $\rho_S\le1$, so in this multi-block case $\sum_{S\in p_*}r(S)>2$ is equivalent to $\int_0^1\int_0^\infty N(u,t)\,dt\,du>1$. The note says the truncation to $0<u<1$ is not justified when $p_*$ is a single block, where Proposition 6.1 does not apply.

The remaining target (Section 8.4, p. 10). In the multi-block case the note lists what it has for $(u,t)$-dangerous blocks (blocks with $\rho_S\ge u$ and $a_S\ge t$): $r(S)\ge2ut$; distinct such centers are more than $2ut$ apart (Section 8.1); each has a root outside it within distance $t^{-n_S/(N-n_S)}$ of $K_S$ (Section 8.2); and, as expectations rather than results, that $\rho_S$ near $1$ should force near-segmental shape through the Barnard--Pearce--Solynin refinement of Faber's inequality unless the mean center is highly eccentric (Section 8.3, for connected $L_S$), and that split irreducibility should rule out highly eccentric mean centers. It states the remaining task as proving

$$\int_0^1\int_0^\infty N(u,t)\,dt\,du\le1$$

from this rigidity together with Pólya--Faber projection control. Section 9 (p. 10) recalls Pólya's projection theorem and says the projected optimal centers need not equal the projected root means, which is where it stops.

The note proves none of the target. Being conditional on Proposition 6.1, the reduction itself assumes Claim 3.1 in lower degrees.

Read depth

Claims checked: Sections 7.3 and 8.1 to 8.4 and Section 9 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Source. Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509, unpublished note (2026), 11 pp.; the edition read is named on the source card.

Bears on

• Problem 509: the note's own statement of the step its route to Claim 3.1 lacks; it is an open target, not a result, and settles no case of the problem.