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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 $\int d{k}_w/(2\pi )$ 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. Short-scale physics is five-dimensional.

At low frequency and long wavelength, $|\omega |,|\mathbf{k}|\ll 2\pi /R$, 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. 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 isThis static Green function across a circle compactification has limitsThey 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 $\Delta t^2=r^2+j^2{R}^2$. 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.