Recollement Defects and MacPherson–Vilonen Boundary Residuals

We study the adjunction-defect calculus underlying MacPherson--Vilonen gluing. For an open--closed decomposition \(X=U\sqcup Z\), recollement gives adjoint triples \(j_!\dashv j^*\dashv j_*\) and \(i^*\dashv i_*\dashv i^!\). We package this data as a MacPherson--Vilonen adjunction package and compute the twist--cotwist defects of its four constituent adjunctions. The nontrivial open-adjunction defects are \(T_{D^b_c(X)}(j_!\dashv j^*)\simeq i_*i^*(-)[-1]\) and \(T_{D^b_c(X)}(j^*\dashv j_*)\simeq i_*i^!(-)[1]\). By contrast, full-faithfulness in genuine recollement forces several open and closed defects to vanish, so imposing sphericality on all four raw recollement adjunctions is degenerate. Thus compatible sphericalization is formulated only conditionally, requiring shared-object data and transportability of compatibility morphisms. The main unconditional result identifies the boundary residual \(\operatorname{BRes}_{Z}(M):=\operatorname{Cone}(j_!j^*M\to j_*j^*M)\) as an extension \(i_*i^*M\to \operatorname{BRes}_{Z}(M)\to i_*i^!M[1]\to\). Iterating over closed-stratum filtrations yields residual towers, with filtered--graded, duality, motivic, and schober-theoretic outlooks.