Non-Hermitian Twisting Theory under the open boundary condition

The non-Hermitian skin effect (NHSE) is a hallmark of non-Hermitian system, yet its generalized Brillouin zone (GBZ) description is restricted to periodic systems. We develop a site-resolved theory via a local scaling transformation (LST), introducing local twisting $T_n$ to quantify metric operator $\xi$ nontriviality. This elucidates the NHSE's origin and uncovers the generalized multiple-channel skin effect (MCSE). Exploiting $T_n$'s translational independence, we define the Zahlen-Brillouin Zone (ZBZ), extending non-Hermitian band theory to nonperiodic and disordered lattices. Furthermore, we unify the $\xi$ with GBZ Riemannian geometry, establishing the metric and state correspondence (MSC) as the principle for real-space localization. With a global skin index $\mathbf\Gamma$ for phase transitions, our results provide a universal paradigm for non-Hermitian physics in both crystalline and amorphous media.