The paper addresses the exceptional behavior of non-Hermitian systems in relation to the bulk-boundary correspondence, a principle that typically predicts the appearance of boundary states in topological insulators. In non-Hermitian systems, this correspondence can fail, leading to the (dis)appearance of boundary states at parameter values far from those causing gap closings in periodic systems. The authors introduce a comprehensive framework based on biorthogonal quantum mechanics to explain these discrepancies. They define a biorthogonal polarization \( P \) that accurately predicts the occurrence of boundary and domain-wall modes in non-Hermitian systems, even in the presence of the non-Hermitian skin effect. This polarization is formulated directly for systems with open boundaries and exhibits a quantized jump at points where the number and localization properties of boundary modes change. The authors illustrate their theory by deriving phase diagrams for several microscopic open boundary models, including non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators. They show that the biorthogonal polarization correctly predicts the (dis)appearance of boundary modes and the critical points where the bulk gap closes, providing a unified explanation for various numerical observations.The paper addresses the exceptional behavior of non-Hermitian systems in relation to the bulk-boundary correspondence, a principle that typically predicts the appearance of boundary states in topological insulators. In non-Hermitian systems, this correspondence can fail, leading to the (dis)appearance of boundary states at parameter values far from those causing gap closings in periodic systems. The authors introduce a comprehensive framework based on biorthogonal quantum mechanics to explain these discrepancies. They define a biorthogonal polarization \( P \) that accurately predicts the occurrence of boundary and domain-wall modes in non-Hermitian systems, even in the presence of the non-Hermitian skin effect. This polarization is formulated directly for systems with open boundaries and exhibits a quantized jump at points where the number and localization properties of boundary modes change. The authors illustrate their theory by deriving phase diagrams for several microscopic open boundary models, including non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators. They show that the biorthogonal polarization correctly predicts the (dis)appearance of boundary modes and the critical points where the bulk gap closes, providing a unified explanation for various numerical observations.