Updating direct methods III. Reduction of structural complexity when first-rank semi-invariants are estimated via the Patterson map

A new theory for the probabilistic estimation of first-rank one-phase semi-invariants is presented. In this approach, atomic positions are treated as primitive random variables but are constrained by the a priori knowledge of interatomic vectors. This information is always available, thus allowing the new technique to be considered an ab initio probabilistic method conditioned by the knowledge of the Patterson map. The theoretical foundation for the estimation of triplet invariants was outlined in the first paper of this series [Giacovazzo (2019). Acta Cryst. A75, 142–157]. Subsequent experimental tests, shown in the second paper of this series [Burla et al. (2024). J. Appl. Cryst.57, 1011–1022], have demonstrated the significant superiority of this new approach over existing methods. The improvements were so notable that it has been suggested this technique could be valuable for the ab initio solution of macromolecular structures. This work expands the probabilistic approach to include the estimation of first-rank one-phase semi-invariants, The hope is that they can contribute to the ab initio solution of macromolecular structures. Only in this way can one-phase semi-invariants go from being a historical curiosity to an effective tool for solving macromolecular structures.