  In this final part of a series of three papers, we will assemble
supersymmetric expressions for one-loop correlators in pure-spinor superspace
that are BRST invariant, local, and single valued. A key driving force in this
construction is the generalization of a so far unnoticed property at tree
level; the correlators have the symmetry structure akin to Lie polynomials.
One-loop correlators up to seven points are presented in a variety of
representations manifesting different subsets of their defining properties.
These expressions are related via identities obeyed by the kinematic
superfields and worldsheet functions spelled out in the first two parts of this
series and reflecting a duality between the two kinds of ingredients.
  Interestingly, the expression for the eight-point correlator following from
our method seems to capture correctly all the dependence on the worldsheet
punctures but leaves undetermined the coefficient of the holomorphic Eisenstein
series ${\rm G}_4$. By virtue of chiral splitting, closed-string correlators
follow from the double copy of the open-string results.
