Strong primeness for equivalence relations arising from Zariski dense subgroups

The paper states and proves that for Zariski dense subgroups of k-isotropic almost k-simple algebraic groups over local fields of characteristic 0, the orbit equivalence relation R(Γ ↷ X) is strongly prime, giving a direct, non-inductive argument that culminates in Theorem A and its proof completion (thereby establishing the embedding dichotomy) . By contrast, the candidate solution assumes as an input the very strong primeness property it aims to establish (Step 3), and then uses it to conclude strong primeness via a product-intertwining step—this is circular. Although the model references product-intertwining ideas attributed to Spaas, the paper’s proof of Theorem A does not require assuming strong primeness; it derives it directly using algebraic representations and Popa–Ioana intertwining for equivalence relations .