Syzygy Transformations of Cluster Characters and Geometric Twists

We use the full multiplication formula to study syzygy and suspension of cluster characters. In stably 2-Calabi--Yau Frobenius categories, syzygy and cosyzygy induce inverse automorphisms of localized character algebras; in Hom-finite 2-Calabi--Yau triangulated categories, the same uniqueness principle transports all cluster characters under suspension. The latter gives our Auslander--Reiten $F$-polynomial identity. No mutation reachability is required. We extend positroid twist identities to all objects, recover the Grassmannian and unipotent-cell formulas, identify partition-function algebras before boundary localization, and derive tropical covariance when the relevant tropical coordinate map is a homeomorphism.