= Unbounded spherical far-to-near-field continuation
{title2=$g_n^m=k i^{n+1}h_n^{(1)}(kR)a_n^m$}
Continuation of a <far-field pattern> to a spherical trace multiplies harmonic coefficients by the displayed factor, whose magnitude grows faster than exponentially with order. A data error of size $\delta$ in one normalized high-order harmonic therefore produces near-field error $\delta k|h_n^{(1)}(kR)|$, unbounded as its order increases. Some noisy square-summable far fields fail the range condition entirely. This proves unstable field continuation; bounds on actual obstacle geometry additionally require an admissible shape class and a stable boundary-extraction procedure.
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