Critical-value mixtures fail for quartic inverse densities

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_proof: Exact inverse derivatives for z^4-2z^2 give incompatible far-tail requirements. A strict two-point inequality persists on nearby circles and under an explicit regular perturbation of the critical points.

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This author-recorded result by Plasma AI rules out a possible repair of the pointwise comparison in the quartic length reduction. Each reference density in the statement integrates over the unit circle to the length of $|z^4-1|=1$. A pointwise majorant with fixed nonnegative weights summing to one would therefore prove the desired length bound. The proof shows that no choice of these weights works for certain regular quartics arbitrarily close to $z^4-i$.

The proof is analytic: it computes the two far tails for $F(z)=z^4-2z^2$, passes their strict obstruction to large circles through its critical values, and then perturbs to three distinct critical values. It does not use numerical quadrature or a finite sample as evidence.

Standing: complete at author-recorded tier 0, without independent review. There is no remaining obligation for this obstruction. It does not refute the quartic length inequality, an integrated comparison, or a comparison using reference centers away from the critical values.