= Solution
At distances and elapsed times much smaller than $R$, propagation probes the local covering space and resolves the compact direction. Formally $R^{-1}\sum_n$ becomes $\int dk_w/(2\pi)$ when many nearby compact momenta participate, giving the massless five-dimensional <retarded Green function>. Equivalently, the compact <Green function> is a sum of covering-space images at $\Delta w+jR$. Before a signal can travel to any nonzero winding image, only the direct image contributes. \b[Short-scale physics is five-dimensional.]
At low frequency and long wavelength, $|\omega|,|\mathbf k|\ll2\pi/R$, massive modes cannot be excited as propagating particles, and the massless zero mode dominates at long noncompact range. Then $G\simeq R^{-1}G^{(4,0)}$, with higher-mode corrections and the corresponding effective source normalization. \b[Low-energy long-scale physics is four-dimensional.]
A precise static illustration avoids ambiguities of pointlike retarded signals. At equal compact positions and zero frequency, the <Yukawa potential> sum is
$$
G_{\mathrm{stat}}(r,0)=\frac1{4\pi Rr}\left[1+2\sum_{n=1}^\infty e^{-2\pi nr/R}\right]
=\frac1{4\pi Rr}\coth\left(\frac{\pi r}R\right).
$$
This <static Green function across a circle compactification> has limits
$$
\boxed{G_{\mathrm{stat}}\sim\frac1{4\pi^2r^2}\quad(r\ll R),\qquad
G_{\mathrm{stat}}\sim\frac1{4\pi Rr}\quad(r\gg R).}
$$
They are the Coulomb kernels in four and three spatial dimensions respectively, with exponentially small massive corrections at large $r$.
For an unsmeared point-source retarded distribution, coordinate separations alone are not enough to impose a low-energy approximation. A high-frequency pulse can still excite massive modes and resolve winding signals at arbitrarily long travel distance/time; its image fronts satisfy $\Delta t^2=r^2+j^2R^2$. Thus <large separation does not impose a Kaluza-Klein low-energy limit>. The usual large-separation four-dimensional statement means low-energy sources or coarse-grained measurements, or a static long-distance limit. It is not a universal pointwise limit erasing all massive oscillations from a sharply resolved retarded pulse.
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