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Let $p$ be a local maximizer as in the claim, and suppose that $a$ is a critical value with $|a|\ne1$. Choose a disc $U=D(a,r)$ whose closure misses the unit circle and all other distinct critical values.

The source premise

We use the construction in the proof of Eremenko and Hayman's Lemma 5, pp. 5--6 of the authors' corrected preprint identified on the source card. Choose a smooth function $\Phi$ supported in $U$ with $\Phi(a)=1$ and put $\phi_\lambda(w)=w+\lambda\Phi(w)$. For $\lambda$ in a sufficiently small complex disc about zero, the source constructs quasiconformal maps $\psi_\lambda$, with $\psi_0(z)=z$ and

$$\psi_\lambda(z)=z+o(1)\quad(z\to\infty),\qquad p_\lambda=\phi_\lambda\circ p\circ\psi_\lambda^{-1}.$$

The $p_\lambda$ are monic quartics, continuous in their coefficients, and their critical values are those of $p$ with $a$ replaced by $a+\lambda$. Multiplicities are preserved; all critical points with value $a$ move in this way. The maps $\psi_\lambda$ are conformal off $p^{-1}(\overline U)$, depend holomorphically on $\lambda$ at each such point, and are conformal on a common neighborhood of $E(p)$. The source gives

$$E(p_\lambda)=\psi_\lambda(E(p)),\qquad \ell(p_\lambda)=\int_{E(p)}|\psi_\lambda'(z)|\,|dz|. \tag{1}$$

The function $h_z(\lambda)=\psi_\lambda'(z)$ is holomorphic in $\lambda$ for each $z$ in that neighborhood, with $h_z(0)=1$. On compact subsets the needed parameter and spatial continuity follow from the source's analytic-dependence construction and the Cauchy formula in $z$.

This local construction is used as an external source premise. Its formulas, normalization, support, analytic dependence and length formula were read in proof on 2026-10-15 against pp. 5--7 of the corrected preprint identified on the source card, including the page images and transcription; the Beltrami existence and analytic-dependence theorems cited there are not re-proved here. The source's continuation along arcs is not used.

Constancy of length forces a trivial deformation

By (1), $\lambda\mapsto\ell(p_\lambda)$ is subharmonic. Local maximality of $p$, and continuity of the coefficients of $p_\lambda$, make it at most $\ell(p)$ in a small disc. The maximum principle makes it constant there.

Fix a circle $|\lambda|=\rho$ contained in that disc. For each $z\in E(p)$ the mean-value inequality gives

$$\frac1{2\pi}\int_0^{2\pi}|h_z(\rho e^{i\theta})|\,d\theta\ge1.$$

Integrating over $E(p)$ and using (1) and constancy shows equality for arclength-almost every $z\in E(p)$. Equality forces $h_z\equiv1$ in $|\lambda|<\rho$: the mean of $h_z$ is $h_z(0)=1$, and equality in the triangle inequality says that its boundary values lie on the nonnegative real axis almost everywhere. Continuity makes the boundary imaginary part zero everywhere, the harmonic maximum principle makes it zero in the disc, and the open mapping theorem makes $h_z$ constant.

We next check the spatial component on which this equality holds. The open set

$$\Omega=\mathbb C\setminus p^{-1}(\overline U) =\{z:|p(z)-a|>r\}$$

is connected. Indeed it contains a neighborhood of infinity, and a bounded component would have $|p-a|=r$ on its boundary and $|p-a|>r$ inside, contradicting the maximum modulus principle for the polynomial $p-a$. Since $\overline U$ misses the unit circle, $E(p)\subset\Omega$.

Fix $|\lambda|<\rho$. The holomorphic function $\psi_\lambda'-1$ on $\Omega$ vanishes at arclength-almost every point of $E(p)$. The latter contains a regular arc: take any unit value that is not one of the finitely many critical values and apply the inverse function theorem. Zeros on that arc have an accumulation point in $\Omega$. The identity theorem therefore gives $\psi_\lambda'\equiv1$ on $\Omega$. Connectedness gives $\psi_\lambda(z)=z+b_\lambda$ there, and the principal normalization at infinity forces $b_\lambda=0$.

For $z\in\Omega$ we have $\phi_\lambda(p(z))=p(z)$, so the defining identity for $p_\lambda$ now gives $p_\lambda(z)=p(z)$. Polynomial identity implies $p_\lambda=p$ everywhere. For a sufficiently small nonzero $\lambda$, however, $a+\lambda$ is a critical value of $p_\lambda$ and is not a critical value of $p$, since $U$ contains no other such value. This is a contradiction.

Thus a local maximizing quartic has no critical value off the unit circle, as claimed. The argument is author-recorded; it does not assert an independent audit of the source construction.