  This is the first installment of a series of three papers in which we
describe a method to determine higher-point correlation functions in one-loop
open-superstring amplitudes from first principles. In this first part, we
exploit the synergy between the cohomological features of pure-spinor
superspace and the pure-spinor zero-mode integration rules of the one-loop
amplitude prescription. This leads to the study of a rich variety of
multiparticle superfields which are local, have covariant BRST variations, and
are compatible with the particularities of the pure-spinor amplitude
prescription. Several objects related to these superfields, such as their
non-local counterparts and the so-called BRST pseudo-invariants, are thoroughly
reviewed and put into new light. Their properties will turn out to be
mysteriously connected to products of one-loop worldsheet functions in packages
dubbed "generalized elliptic integrands", whose prominence will be seen in the
later parts of this series of papers.
