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Statement

Setting (pp. 3--5). For a block $S$ of the minimal partition $p_*$ (setup on Lemma 4.1), $f=P_SQ_S$ with $P_S$ carrying the roots of $f$ in $S$, as on Lemma 5.1; $T_S=\sup_{z\in K_S}\lvert P_S(z)\rvert$ and $\lambda_S=\min_{z\in K_S}\lvert Q_S(z)\rvert$, which is positive because $Q_S$ has no zeros on $K_S$. On $K_S$, $\lvert P_SQ_S\rvert=\lvert f\rvert\le1$, so $T_S\le\lambda_S^{-1}$.

Lemma 7.1 (p. 5). For every block $S\in p_*$, $T_S=\lambda_S^{-1}$.

Hence $T_S^{1/n_S}=\lambda_S^{-1/n_S}$ (p. 6), and the note rewrites the radius sum as $\sum_{S\in p_*}r(S)=2\sum_{S\in p_*}\rho_S\lambda_S^{-1/n_S}$ (Section 7.2, p. 7).

Proof pointer

P. 6. The minimum of $\lvert Q_S\rvert$ on $K_S$ is attained at a boundary point $z_0$, where $\lvert f(z_0)\rvert=1$, so $\lvert P_S(z_0)\rvert=\lambda_S^{-1}$ and $T_S\ge\lambda_S^{-1}$.

Read depth

Claims checked: the definitions and Lemma 7.1 were read clause by clause on the page images of the print, and the proof on p. 6 was followed. 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 Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509, unpublished note (2026), 11 pp.; the edition read is named on the source card.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: an identity used to restate the radius-sum bound of Claim 3.1 for a hypothetical minimal counterexample; it settles no case of the problem.