= Solution
Emittance reduction has little effect when radiation dominates every source width and divergence. At fixed $\beta=L=2$ m, the strongest condition is $\lambda L/(16\pi^2)\gg\epsilon_x\beta$, or $\boxed{\lambda\gg4.7\times10^{-7}\ \mathrm m}$. The divergence condition only requires $\lambda\gg\epsilon_xL/\beta=3$ nm. There is no sharp “negligible” cutoff without an allowed percentage. If the <optics> were reoptimized to match diffraction instead, the weaker comparison $\lambda/(4\pi)\gg\epsilon_x$ would set the relevant scale; it is not the same fixed-beta problem.
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