  Most high-energy constructions that realise a phase of cosmic inflation
contain many degrees of freedom. Yet, cosmological observations are all
consistent with single-field embeddings. We show how volume selection effects
explain this apparent paradox. Because of quantum diffusion, different regions
of space inflate by different amounts. In regions that inflate most, and
eventually dominate the volume of the universe, a generic mechanism is unveiled
that diverts the inflationary dynamics towards single-field attractors. The
formalism of constrained stochastic inflation is developed to this end.
