Problem 114: Low-degree lemniscate extremizers

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cubic_extremizer_reduction: An existential cubic reduction, a uniform ball-arithmetic family inequality, and a deformation argument give the cubic bound and uniqueness at tier 1; the all-degree problem remains open.

quartic_critical_values: A compact reduction for monic quartics with unit critical values, an exact obstruction to comparison at the same constant term, a rotated comparison and uniform estimates through collisions and flattening. Bounds on both collision strata and a variance criterion isolate the remaining quartic inequality, which is unproved on the whole family.

quartic_evidence/: Ball-arithmetic proof that the rotated deficit integral is below -1 on every circle and line of both quartic collision pencils, so every polynomial of the two collision strata is shorter than z^4-1 by at least its scale.

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The [[research/erdos_114/cubic_extremizer_reduction|argument on cubic extremizers with critical values on the unit circle]] records its exact scope, source premises, and remaining gap toward Problem 114. With the family inequality of L144 and the deformation in Eremenko and Hayman's proof of Lemma 5, it gives a tier-1, computer-assisted proof that $z^3-1$ is the unique cubic maximizer up to rotation and translation; the other open degrees remain.

The quartic research sketch combines the compact coordinates of L186 with the explicit harmonic-line deficit of L179, its uniform small-curvature extension L180, and the full collision-pencil bounds of L181. These are author-recorded results at tier 0; the collision bounds use ball-arithmetic certificates. They cover the two collision strata and an explicit, very small curvature range near the unicritical limit. Practical two-dimensional neighborhoods of the collision pencils and the remaining regular compact family still need a uniform estimate. No full quartic inequality or quartic uniqueness theorem follows yet.