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Claim. Theorem 1.1 of B. E. Dahlke, The cubic Erdős–Herzog–Piranian lemniscate inequality (Zenodo, 2026): for every monic cubic $p$ the length $L(p)$ of the lemniscate $\{z:\lvert p(z)\rvert=1\}$ satisfies $L(p)\le L(z^3-1)=2^{1/3}B(\tfrac16,\tfrac12)$. This is the degree-3 inequality of Problem 114; the manuscript proves the inequality only and has no uniqueness clause. The argument takes the Eremenko–Hayman reduction as an external theorem, in the form of Proposition 1.2 of Tao 2025: some monic cubic maximizing the length has, after the length-preserving normalizations, no $z^2$ term, a real non-positive constant term, a connected lemniscate and all its critical points on the lemniscate. Up to those symmetries such cubics form the one-parameter family $p_\phi(z)=z^3-3c_\phi^2z-\cos\phi$ with $2c_\phi^3=i\sin\phi$ and $0\le\phi\le\pi/2$, where $\phi=0$ gives $z^3-1$. The scaling $z=2c_\phi x$ turns $\lvert p_\phi(z)\rvert=1$ into a condition on the Chebyshev polynomial $T_3(x)$, the substitution $x=(u+u^{-1})/2$ (the Joukowski map) turns that condition into two problems on rays, and a three-branch estimate followed by a one-variable comparison shows that every $\phi>0$ gives a strictly shorter lemniscate than $\phi=0$. The author announced the manuscript on the site's discussion thread on 2026-05-21; the post drew no reply.

Covers. Degree $3$ only, the inequality $L(p)\le L(z^3-1)$ for monic cubics, resting on the Eremenko–Hayman reduction as Tao states it. Degree $2$ is settled by Wang 1998 and Eremenko–Hayman 1999, and all sufficiently large degrees are claimed, pending, by Tao 2025. Chatelet 2026 later claimed the same degree, with a uniqueness clause.

Depends on. No page of this wiki; the reduction is cited from Proposition 1.2 of Tao's preprint.

Acceptance. None recorded. The manuscript is a Zenodo deposit with no journal version, so no refereed evidence exists. The site's label is FALSIFIABLE, an open label, and its commentary does not mention the manuscript, so no reviewed evidence is listed. No formal proof is recorded. Nothing here is this project's own review.