  In the spirit of Koml\'os's theorem, we investigate when a sequence of
semimartingales satisfying a boundedness condition admits forward convex
combinations converging in the \'Emery topology. A dichotomy into two classes
of universal counterexamples to a Koml\'os-type theorem is established,
mirroring a result of M\'emin. As an application of this classification, we
obtain a remarkable Koml\'os-type theorem for semimartingales, with convergence
in a $\sigma$-localized$\unicode{x2014}$in the sense of
Kallsen$\unicode{x2014}$version of the \'Emery topology. This provides the
first general convex compactness principle for semimartingales in the
literature for which the limit $X$ is necessarily a semimartingale, solving a
serious gap in previous results that limited the possible wealth processes
vis-\`a-vis $X$ to those arising from predictable finite variation integrands,
a severe restriction in applications to mathematical finance.
