Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero

Abstract: An exact zero of a generated harmonic needs two real cancellations; a damped oscillator's leading fundamental-source balance supplies one. Absorber loss leaves a phase deficit of fixed sign, amplitude independent at leading order, closed instead by the orbit's own $5Ω$ motion returning to $3Ω$ at weight $|Δ_1|<sup>4$. Closure is therefore unreachable at weak amplitude: the channel supplying it ranks at $O(F<sup>7)$, not at the leading $O(F<sup>3)$. The fifth harmonic is $1.8\times10<sup>{-3}$ of the relative-coordinate fundamental, yet a balance homotopy weighting it below $0.48$ annihilates the tracked pair in a saddle-node.