Polynomials

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L144_cubic_lemniscate_length/: A uniform ball-arithmetic family inequality and Eremenko-Hayman's deformation prove the cubic lemniscate bound and its equality case at tier 1 on an independently graded whole-claim examination.

L176_quartic_constant_term_comparison/: An explicit family of centered monic quartics with three distinct unit critical values tends to z^4+1 while its lemniscate is longer than that of the unicritical quartic with the same constant term. A positive scaled limiting difference is proved by a geometric estimate on a harmonic curve.

L177_quartic_maximizer_critical_values/: Every monic quartic that locally maximizes lemniscate length in coefficient space has all its critical values on the unit circle. The equality case in the holomorphic mean-value inequality strengthens the local deformation used in Eremenko and Hayman's Lemma 5.

L178_quartic_length_variation/: A local motion of three distinct unit critical values gives exact first and second variations of quartic lemniscate length in terms of arclength moments of two explicit polynomials. A further formula expresses convexity in the angular gaps as a pointwise moment inequality.

L179_quartic_harmonic_deficit/: A centered monic quartic with a nonzero quadratic or linear coefficient has renormalized harmonic-level length at most minus its coefficient scale divided by 2500. Two elementary real-root tests give the explicit uniform constant in a projection argument.

L180_quartic_small_curvature_deficit/: Explicit root-window and far-field estimates transfer a quantitative harmonic-line deficit to every sufficiently flat circle, uniformly over centered quartic shapes whose largest critical-point modulus is one. The curvature cutoff is explicit and very small.

L181_quartic_collision_pencil_deficits/: Certified ball arithmetic bounds the rotated length-deficit integral strictly below -1 on every circle and line through the critical values of either normalized quartic collision stratum. The strict bound is uniform up to the flattening limit.

L182_quartic_critical_value_mixtures/: Arbitrarily near a unicritical quartic, a quartic with three distinct unit critical values admits no pointwise upper bound by a convex combination of unicritical inverse densities centered at those values. Two opposite tails of an explicit collision limit give the obstruction.

L183_quartic_variance_coefficient_bound/: Five Laurent modes and an orthogonal projection give an explicit lower bound for the arclength variance of the quartic length-variation polynomial. The bound is strictly below the needed value at the unicritical extremizer, so it does not settle the full sign condition.

L184_quartic_petal_length_obstruction/: The component containing zero for z^4+4*3^(-3/4)z has all three critical points on its boundary. Their equilateral triangle gives a perimeter bound greater than one quarter of the length of |z^4-1|=1.

L186_quartic_critical_value_parameters/: Explicit compact coordinates cover centered monic quartics with unit critical values, including point and value collisions; the unicritical length comparison is also proved.

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Each claim states its hypotheses, conclusion and evidence standing. The cubic lemniscate claim combines a written reduction with a ball-arithmetic computation whose coverage of the parameter range is checked by its evidence program. The quartic claims treat degree four: coordinates, comparisons, variation formulas, deficits and obstructions for the length of quartic lemniscates.