The annihilation of cosmic positrons ($e^+$) with electrons in the interstellar medium (ISM) results in the strongest persistent gamma-ray line signal in the sky. For 50 years, this 511 keV emission has puzzled observers and theoreticians. A key issue for understanding $e^+$-astrophysics is found in cosmic-ray propagation, especially at low kinetic energies (< 10 MeV). We want to shed light on how $e^+$s propagate and the resulting morphology of the emission. We approach this positron puzzle by inferring kinematic information of the 511 keV line in the inner radian of the Galaxy. This constrains propagation scenarios and source populations. By dissecting the 511 keV emission as measured with INTEGRAL/SPI, we derive spectra for individual regions in the sky. The centroid energies are converted into Doppler-shifts, representing the line-of-sight velocity along different longitudes. This results in a longitude-velocity diagram of $e^+$-annihilation. We also determine Doppler-broadenings to study annihilation conditions as they vary across the Galaxy. We find line-of-sight velocities in the 511 keV line that are consistent with zero, as well as with galactic rotation from CO measurements, and measurements of radioactive Al-26. The velocity gradient in the inner 60 deg is determined to be $4pm6$ km/s/deg. The 511 keV line width is constant as a function of longitude at $2.43pm0.14$ keV. The positronium fraction is found to be 1.0 along the galactic plane. The weak signals in the disk leave open the question whether $e^+$-annihilation is associated with the high velocities seen in Al-26 or rather with ordinarily rotating components of the Galaxys ISM. We confirm previous results that $e^+$s are slowed down to the 10 eV energy scale before annihilation, and constrain bulk Doppler-broadening contributions to <1.25 keV. Consequently, the true annihilation conditions remain unclear.
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