  Weinberg recently pointed out a flaw in the standard argument that large
$N_c$ QCD with color-fundamental quarks [QCD(F)] cannot yield narrow tetraquark
states. In particular, he observed that the argument does not rule out narrow
tetraquarks associated with the leading-order connected diagrams; such
tetraquarks would have a width scaling as $N_c^{-1}$. It is shown here,
however, that while the standard analysis of tetraquarks does not rule them
out, a more thorough analysis rules out quantum-number exotic tetraquarks
associated with the leading-order connected diagrams. This analysis is based
entirely on conventional assumptions used in large $N_c$ physics applied to the
analytic properties of meson-meson scattering. Our result implies that one of
three possibilities must be true: i) quantum-number exotic tetraquarks do not
exist at large $N_c$; ii) quantum-number exotic tetraquarks exist, but are
associated with subleading connected diagrams and have anomalously small widths
that scale as $N_c^{-2}$ or smaller; or iii) the conventional assumptions used
in large $N_c$ analysis are inadequate.
