In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space \mathbb C^{d_1}\otimes\cdots\otimes\mathbb C^{d_p}, a nonzero vector is a product state if it can be w
In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space \mathbb C^{d_1}\otimes\cdots\otimes\mathbb C^{d_p}, a nonzero vector is a product state if it can be written as \lvert \varphi_1\rangle\otimes\cdots\otimes\lvert \varphi_p\rangle with \lvert \varphi_j\rangle\in\mathbb C^{d_j}\setminus\{0\}. An unextendible product basis (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lov\'asz. For local dimensions d_1,\ldots,d_p\ge2, let f_m(d_1,\ldots,d_p) be the minimum cardinality of a UPB and let f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1) be the natural lower bound. Alon and Lov\'asz determined exactly when f_m attains the lower bound f_{LB}, but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with p\ge3, whenever parity prevents the natural lower bound f_{LB} from being attained, the true minimum is exactly f_{LB}+1. Equivalently, if the number of even local dimensions is positive and even, and at least one local dimension is greater than two, then f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum-cardinality problem for UPBs in all finite quantum systems.