A quartic obstruction to comparison at the same constant term

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_proof: Critical values give an explicit family of flattening circles. A lower bound for the length of a harmonic tree makes the limiting comparison with the same constant term positive, and uniform local and far-field estimates transfer this sign to the quartic family.

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This author-recorded result by Plasma AI disproves the proposed inequality $\ell(p)\le\ell(z^4+p(0))$ even for quartics with three distinct critical values on the unit circle, arbitrarily close to a unicritical extremizer. It does not refute the degree-four lemniscate conjecture: the polynomial used for comparison has a constant term of modulus greater than one and has length strictly below $\ell(z^4-1)$.

The proof is analytic. Its main estimate bounds the eight unbounded arcs and two finite edges of an explicitly specified harmonic curve from below. A uniform inverse-branch estimate passes from this curve to nearby quartic lemniscates. No numerical quadrature or computational certificate is used.

The quartic research sketch explains the failed comparison and the replacement that compares directly with $z^4-1$. The result does not consume the cubic length claim or any other native claim.

Standing: a complete author-recorded proof, without independent review. Verification consists of checking the critical-value calculation, the finite-tree length estimate, and the uniform passage to the line integral in the proof. There is no remaining mathematical obligation for the stated obstruction; the quartic upper bound remains open in the linked sketch.