Vacancy order beyond complete local coordination in perovskites

Abstract In oxygen-deficient perovskites, even the complete site-resolved coordination fields of both cation sublattices need not determine the oxygen-vacancy arrangement. Here we solve this information gap exactly on the perovskite parent network. A joint map from oxygen occupation to A- and B-site coordination gives the full compatibility conditions and separates finite-support coordination-preserving variations from additional periodic freedoms. At ABO_2.5 composition, differences between explicit binary orders span this nonlocal sector, and occupancy capacity makes a large exactly characterized subset rigid against every finite-support rearrangement preserving both coordination fields. A CaMnO_2.5 vacancy-order microdomain identified by electron diffraction and HREM maps exactly onto the minimal member of this rigid sector. A general domain-stitching identity then shows how changes between same-coordination orders reappear as interface defects. In a fully specified exchange channel, the optimal peak joint defect measure grows as 2L+2 although separately optimized A- and B-site peaks remain constant. Experimental bulk structures, vacancy-order domains and homologous intergrowths place the exact results within the broader coordination fibre.