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Claim. For a monic polynomial $p\in\mathbb C[z]$ let $\ell(p)$ be the length of the curve $\{z\in\mathbb C:\lvert p(z)\rvert=1\}$. Every monic cubic $p\in\mathbb C[z]$ satisfies

$$\ell(p)\le\ell(z^3-1)=\frac{2\pi\,\Gamma(1/3)}{\Gamma(2/3)^2},$$

with equality exactly when $p(z)=(z-z_0)^3-w$ for some $z_0\in\mathbb C$ and some $w$ with $\lvert w\rvert=1$. The core is the strict family inequality $\ell(z^3+3a^2z-c)<\ell(z^3-1)$, with $c=\sqrt{1-4a^6}$, for every real $a$ with $0<a\le2^{-1/3}$.

Covers. The degree-3 case of the question of Problem 114, answered yes: $z^3-1$ maximizes the length among monic cubics and, up to rotation and translation, is the only maximizer. No other degree is settled, and the question in general degree remains open.

Credit. The passage from the family inequality to every cubic uses two results of Eremenko and Hayman's refereed 1999 paper, Lemma 4, the existence of a maximizer in each degree, and Lemma 5, that some maximizer has all its critical points on its lemniscate; the equality case also uses the deformation constructed in their proof of Lemma 5. Tao's Proposition 1.2 restates these results. The family inequality, its written reduction to explicit integrals, the ball-arithmetic bounds over the whole parameter range and the composition are Plasma AI's.

Standing. L144 is tier 1. Its whole-claim examination has verdict refutation-failed, and its distinct grade, dated 2026-10-15 UTC, passes the complete statement, including the equality case. The qualifications filed with the card remain in force: the result is computer-assisted, Lemma 4 is an external theorem whose supplied proof was only partly verified, and a cited book theorem on the Beltrami equation was not read in its original source. No Lean proof is claimed. Catalog acceptance is pending.

Depends on. L144 (tier 1), which consumes no other native claim; its external premises are the Eremenko–Hayman results credited above.