= Solution
Each compact <Fourier mode> obeys the noncompact equation with dispersion relation $\omega^2=\mathbf k^2+m_n^2$, where
$$
\boxed{m_n=\frac{2\pi|n|}R\quad(\hbar=c=1).}
$$
Thus momentum in the extra dimension appears as rest mass to a four-dimensional observer. There is one massless zero mode and equally massive $n,-n$ modes. For a real scalar they can be organized into sine and cosine components rather than independent complex fields.
At equal compact positions all phase factors are one, and the propagator becomes the <Kaluza-Klein tower>
$$
\boxed{G_{\mathrm{ret}}(\Delta w=0)=\frac1R\left[
G_{\mathrm{ret}}^{(4,0)}+2\sum_{n=1}^\infty
G_{\mathrm{ret}}^{(4,m_n)}\right].}
$$
The factor $1/R$ follows from the normalized compact eigenfunctions. It cannot be discarded if the five-dimensional field and source normalizations are retained.
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