Critical values of local quartic length maximizers

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_proof: The length formula from Eremenko and Hayman's local deformation and equality in the holomorphic mean-value inequality force a putative maximizing deformation to be the identity outside its support. Polynomial identity then contradicts the moved critical value.

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The proof is an author-recorded deduction by Plasma AI from the local quasiconformal deformation in the proof of Eremenko and Hayman's Lemma 5. The source states that some global maximizer has unit critical values. Here the equality case in its length formula rules out an off-circle critical value at any local maximizer.

The source deformation, its analytic parameter dependence, and its principal normalization at infinity are external premises. The proof identifies their exact use and reading coverage. No compactness theorem, continuation of a critical value to the circle, cubic inequality, or computational evidence is needed for this deduction.

In the quartic length reduction, this result lets a strict bound for every nonunicritical quartic with unit critical values imply the full equality classification. It does not supply that strict bound. Standing: proved at author-recorded tier 0, without independent review. There is no remaining gap in the stated necessary condition; the quartic length inequality remains an obligation of the linked research sketch.