Second variation of quartic lemniscate length

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_proof: Polynomial division constructs a holomorphic motion before a real normalization puts the critical values back on the unit circle. Its velocity and acceleration give the length derivatives and, by the chain rule, the Hessian in angular-gap coordinates.

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This author-recorded result by Plasma AI identifies the precise second variation in the quartic length reduction. The proof constructs the local family by a polynomial differential equation. It treats the inverse-branch singularities at the critical values before differentiating the length integral.

The formulas do not establish a sign. In particular the inequality $\operatorname{Var}_{\mu}(Q)\ge1$, which would make $\log\ell$ subharmonic under this motion, remains unproved. The angular-gap criterion is stronger and remains unproved as well. At a stationary member the weaker inequality $\mathbb E|Q|^2>1$ would exclude a local maximum.

Standing: a complete analytic derivation at author-recorded tier 0, without independent review. No numerical evidence is needed for the identities. The exploratory checks linked from the research sketch test their implementation and do not confer mathematical standing.