= Solution
Normalize a point source by $\int\rho\,d^3x\,dz=q_1$ on the compact spatial direction. The periodic delta has the <Fourier series>
$$
\delta_{2\pi R}(z)=\frac1{2\pi R}\sum_{n\in\mathbb Z}e^{inz/R}.
$$
Writing the potential as $\phi(r,z)=\sum_n\phi_n(r)e^{inz/R}$ reduces the <Poisson equation> to
$$
\left(\nabla_3^2-\frac{n^2}{R^2}\right)\phi_n=-2q_1\delta^3(\mathbf x).
$$
Away from the source, the decaying radial solution is proportional to $e^{-|n|r/R}/r$. Its distributional normalization follows from integrating the Laplacian over a small sphere: the radial flux of $1/r$ is $-4\pi$, so
$$
\phi_n(r)=\frac{q_1}{2\pi r}e^{-|n|r/R}.
$$
This is the contribution of the $n$th <Kaluza-Klein mode>, with inverse range $|n|/R$. At equal compact coordinate, the sum is a geometric series,
$$
\phi(r,0)=\frac{q_1}{2\pi r}\left(1+2\sum_{n=1}^\infty e^{-nr/R}\right)
=\frac{q_1}{2\pi r}\coth\frac{r}{2R}.
$$
Therefore the pair interaction energy, without the individual self-energies, is
$$
\boxed{U(r)=\frac{q_1q_2}{2\pi r}\coth\frac{r}{2R}.}
$$
Its normalization is fixed by the stated $-4\pi R\rho$ source, rather than by importing another convention for <Coulomb's law>. More generally summing the phases gives the <periodic electrostatic Green function with one compact dimension>,
$$
\phi(r,z)=\frac{q_1}{2\pi r}\frac{\sinh(r/R)}{\cosh(r/R)-\cos(z/R)}.
$$
An independent check uses the <method of images> in four spatial dimensions. Since $\nabla_4^2s^{-2}=-4\pi^2\delta^4$, its periodic image sum is $Rq_1\sum_n[r^2+(z+2\pi nR)^2]^{-1}/\pi$, with the same normalization and short-distance singularity.
For $r\gg R$, only the zero <Kaluza-Klein mode> survives:
$$
\boxed{U(r)=\frac{q_1q_2}{2\pi r}\left[1+2e^{-r/R}+O(e^{-2r/R})\right].}
$$
For $r\ll R$, $\coth x=x^{-1}+x/3+O(x^3)$ gives
$$
\boxed{U(r)=\frac{q_1q_2R}{\pi r^2}+\frac{q_1q_2}{12\pi R}+O(r^2/R^3).}
$$
The leading potential thus changes from inverse distance to inverse squared distance. Differentiating gives the signed outward force
$$
F_r=\frac{q_1q_2}{2\pi}\left[\frac{\coth(r/(2R))}{r^2}
+\frac{\operatorname{csch}^2(r/(2R))}{2Rr}\right].
$$
It scales as $r^{-2}$ at large distance and as $2q_1q_2R/(\pi r^3)$ at small distance. \b[A precision inverse-square-law test over separations approaching $R$ would reveal the extra dimension] through this change of force law or its exponential corrections. The model assumes localized sources at equal compact coordinate and an electromagnetic field able to explore the circle; a field confined to ordinary spacetime would not show this effect. Finite source size and ordinary screening must be controlled when comparing a real force measurement with this point-source prediction.
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