[en] The experimental detection of the sharp lines of the (e⁺e^-) Puzzle is viewed as a struggle against Doppler broadening. Gedanken experiments which are realistic in zeroth order of detail are analyzed to show that the ORANGE and EPOS/I geometries select narrower slices of a Doppler broadened line than spherically inclusive (APEX and EPOS/II -like) apparati. Roughly speaking, the latter require event-by-event Doppler reconstruction simply to regain an even footing with the former. This suggests that APEX' or EPOS/II's coincident pair distributions must be statistically superior to those of EPOS/I or ORANGE in order to support a comparable inference about sharp structure. Under such circumstances, independent alternative data is invaluable. Therefore, a corroboration of Sakai's 330.1 keV (< 3 keV wide) electron line in few MeV e⁺ or e^- bombardments of U and Th targets could prove crucial