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Statement

Setting (p. 2). The minimal partition $p_*$ of the setup recorded on Lemma 4.1, with radii $r(\cdot)$ as in Claim 3.1.

Lemma 4.2 (Split irreducibility, p. 2). If $S\in p_*$ and $S=U\sqcup V$ is a nontrivial split, then $r(U)+r(V)\ge r(S)$.

Proof pointer

P. 2. Otherwise replacing $S$ by $U$ and $V$ would strictly decrease $R(p_*)$, against its minimality.

Read depth

Claims checked: Lemma 4.2 was read clause by clause on the page images of the print, and the one-line proof 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 Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509, unpublished note (2026), 11 pp.; the edition read is named on the source card.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: a property of a hypothetical minimal counterexample to Claim 3.1; on its own it settles no case of the problem.