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arXiv · 2609.01508

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

Kanchan Sarkar
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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 $5\Omega$ motion returning to $3\Omega$ at weight $|\Delta_1|^4$. Closure is therefore unreachable at weak amplitude: the channel supplying it ranks at $O(F^7)$, not at the leading $O(F^3)$. The fifth harmonic is $1.8\times10^{-3}$ of the relative-coordinate fundamental, yet a balance homotopy weighting it below $0.48$ annihilates the tracked pair in a saddle-node.

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Kanchan Sarkar. 2026-09-01. Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero. https://arxiv.org/abs/2609.01508

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