Cubic extremizers with critical values on the unit circle

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This argument by Plasma AI concerns Problem 114 at the formulation and standing recorded there. It supplies a cubic inequality and its equality case at the tier-1 standing warranted by L144's whole-claim examination and distinct PASS grade; the all-degree question remains open.

The target is the length of the full level set of a monic complex polynomial of the same degree. The held Tao source's Theorem 1.1(iv) asserts the answer only for sufficiently large degrees; Remark 1.3 supplies effectiveness in principle, not a numerical cutoff usable by this argument. Neither an arbitrary finite degree check nor a fine numerical parameter grid completes the all-degree problem.

Tao's Proposition 1.2, citing Eremenko–Hayman Lemmas 5 and 6, supplies some maximizer with all critical values on the unit circle. It does not preserve the length of every polynomial under reduction. The source statements and Tao's short specialization were read. The proof of Eremenko–Hayman Lemma 5 was read in full against the corrected preprint's page images, pp. 5--7, for the deformation used in the uniqueness argument below. Lemma 4, which supplies continuity and existence, is used as a refereed source theorem; its proof is not reconstructed here.

There is a useful exact cubic specialization. Translate a monic cubic to $p(z)=z^3-3u^2z+b$. The operation $p(z)\mapsto e^{-3i\theta}p(e^{i\theta}z)$ preserves monicity and length; use it to make $b\le0$ real. The critical values are $b\pm2u^3$. If both have modulus one, subtraction and addition of their squared moduli give

$$\operatorname{Re}(b\overline{u^3})=0, \qquad b^2+4|u|^6=1.$$

When $b<0$, $u^3$ is purely imaginary. A rotation by an integer multiple of $2\pi/3$, which keeps $b$ fixed, makes $u$ purely imaginary. If $b=0$ there is no phase restriction on the rotation and the same normalization is possible. Thus one maximizer is isometric to a member of the real family

$$q_a(z)=z^3+3a^2z-\sqrt{1-4a^6}, \qquad 0\le a\le2^{-1/3}.$$

Conversely these polynomials have critical points $\pm ia$ and unit-modulus critical values. The parameter $k=a^3$ gives exactly the real family $z^3+3k^{2/3}z-\sqrt{1-4k^2}$, $0\le k\le1/2$, named on the Chatelet claim page. Therefore the family inequality $\ell(\{|q_a|=1\})\le\ell(\{|z^3-1|=1\})$ for every $a$ would prove the cubic inequality for all complex cubics, using the existence of one extremizer. No assertion about arbitrary cubics having equal length to a family member is needed. This existential reduction alone supplies no family inequality or uniqueness classification. Those are separate obligations, addressed by the argument below. Existence of one specially normalized maximizer cannot by itself classify every maximizer.

The Chatelet claim's analytic chain and external development remain uninspected. Its declared DomainReductionD3 has a stronger, false every-cubic length identity, as the claim page records; the claimed formal theorem cannot be used as evidence for the weaker correct reduction. Closed connected level sets and connected open sublevel sets must also be distinguished: $z^3-1$ has petals meeting on the boundary.

The family inequality and the composed argument

The family inequality is proved, by an argument independent of the Chatelet chain, in the independently graded claim L144: for $0<a\le2^{-1/3}$ the lemniscate of $q_a$ is strictly shorter than that of $z^3-1$. The route is as follows.

• Writing $P(z)=z^3+3a^2z$, the lemniscate of $q_a$ is $P^{-1}$ of the circle $|s-c|=1$, which passes through both critical values $\pm2ia^3$. The roots of $P(z)=s$ satisfy $(z^2+a^2)^2(z^2+4a^2)=s^2+4a^6$, and the substitution $z=u-a^2/u$ gives every reciprocal derivative in closed form. The length is a single integral over the circle with two explicit inverse-square-root singularities.
• The same integral for $z^3-c$ is $\Phi(c)=2\pi\,{}_2F_1(\frac13,\frac13;1;c^2)$, nondecreasing in $c$ and equal to the target length at $c=1$. Scaling by $a^3$ writes the length of $q_a$ as $\Phi(c)+aI(a)$, where $I(a)$ integrates a fixed function over a circle through $\pm2i$. For small $a$ it is enough that $I(a)<0$, and the tail of that integral has an explicit bound.
• Ball arithmetic bounds $I(a)$ for $0<a\le13/64$ and the length itself for larger $a$, uniformly over parameter boxes covering the whole range, including the degenerate endpoint where the critical points merge.

Combined with the existential reduction above, this proves the degree-3 inequality $\ell(\{|p|=1\})\le\ell(\{|z^3-1|=1\})$ for every monic complex cubic, at the standing recorded on the claim: tier 1 and computer-assisted, with the identified Eremenko–Hayman Lemmas 4 and 5 interfaces and the original Beltrami theorem boundary qualified in the verification record.

The small-parameter estimate resolves the singular scale directly. Under $z=a\zeta$ and $s=a^3\sigma$, the inverse-derivative density becomes the fixed function $G_1(\sigma)$, and the critical values remain at $\pm2i$. The parameter dependence moves into the circle of integration. The head, critical window and tail bounds cover its limiting degeneration uniformly, without a positive lower cutoff on $a$.

Uniqueness

The proof of Lemma 5 is stronger than its statement. For every extremal polynomial and every critical value off the unit circle, its quasiconformal deformation moves that critical value through a small disc, keeps the other critical values fixed, and keeps the length constant. For a cubic with two distinct critical values this lets one move them, at constant maximal length, to two distinct points of the circle; the limit is again a maximizer by continuity of length, and it is a family member $q_a$ with $a>0$, which the family inequality excludes. A maximizer with a double critical point is $z^3+b$, whose length $\Phi(|b|)$ or $|b|^{-2/3}\Phi(1/|b|)$ is maximal only at $|b|=1$. Hence $z^3-1$ is, up to rotation and translation, the only cubic maximizer; Section 8 of the L144 proof gives the details. The deformation was read on the page images of the corrected preprint (pp. 5--7).

What remains

The degree-3 case has a whole-claim examination and distinct passing grade, covering the argument, its uniform ball-arithmetic program and uniqueness. The record retains its source boundaries, exposure rulings and failed auxiliary diagnostic predicates. The remaining intermediate degrees $d\ge4$ outside the range of Tao's preprint are not settled by this argument. The method uses the one-parameter cubic normalization; higher degrees require an additional uniform analysis.

Source interfaces

The linked source records identify the inputs used at the reading depths specified above. A result account is not a check of the source's proof.

• proposition 1 2.
• lemma 5.
• lemma 6.