An explicit quartic harmonic-curve deficit

../

_proof: A set of lines of controlled measure misses at least two intersections with the harmonic quartic. When the quadratic coefficient is substantial its depressed real restriction has a positive quadratic coefficient; otherwise the linear coefficient violates a real-root moment bound.

---

This author-recorded estimate by Plasma AI makes the harmonic-curve deficit explicit in degree four. It uses the spherical projection argument from the proof of Fryntov and Nazarov's Lemma 1, identified on their local maximality page, and replaces the compactness step in its strictness proof by two quantitative real-root tests. The proof includes the needed projection inequality and its normalization.

For the normalized shapes in the quartic length reduction, the largest critical-point modulus is one and the coefficient scale $\lambda$ is greater than one. Thus every line limit in that reduction has signed deficit at most $-1/2500$. This does not specify how close a circle must be to its line limit, and does not bound the remaining regular family.

Standing: a complete analytic proof at author-recorded tier 0, without independent review. Its finite numerical constants are justified by explicit inequalities; no computational certificate is used.