  We investigate the possibility of secondary production as the source of
cosmic ray (CR) positrons, taking into account observed electron and antiproton
spectra and allowing that the reported steepening in the AMS-02 positron
spectrum may be due to radiative losses. Using a simple argument based on the
antiproton spectrum, we show that positrons cannot be purely secondary. Namely,
the antiproton to positron ratio requires that positrons be uncooled up to a
few hundred GeV. This requirement implies that CR protons escape from the
Galaxy too quickly to reproduce the observed antiproton flux. We further show
that this result rules out also more complex secondary production scenarios
such as reacceleration and the Nested Leaky Box. Thus, we conclude that while
antiprotons can be produced exclusively in secondary interactions, a primary
source of CR positrons -- be it dark matter or pulsars -- is required to
explain their spectrum.
