Derivations for the MPS overlap formulas of rational spin chains

We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in ${\mathfrak{g}\mathfrak{l}}_N$ symmetric spin chains. This formula expresses the normalized overlap as a product of a MPS-independent Gaudin-determinant ratio and a MPS-dependent scalar factor constructed from eigenvalues of commuting operators, defined via the $K$-matrix associated with the MPS. Our proof is fully representation-independent and relies solely on algebraic Bethe Ansatz techniques and the $KT$-relation. We also propose a generalization of the overlap formula to ${\mathfrak{s}\mathfrak{o}}_N$ and ${\mathfrak{s}\mathfrak{p}}_N$ spin chains, supported by algebra embeddings and low-rank isomorphisms. These results significantly broaden the class of integrable initial states for which exact overlap formulas are available, with implications for quantum quenches and defect CFTs.