Non-Hermitian skin effect (NHSE) is a phenomenon in non-Hermitian systems where eigenstates accumulate at system boundaries. While well understood in one dimension via non-Bloch band theory, higher dimensions face challenges due to diverse open boundary conditions and lattice geometries. This work presents a geometry-adaptive non-Bloch band theory in arbitrary dimensions using spectral potential. It accurately determines energy spectra, density of states, and generalized Brillouin zone (GBZ) in the thermodynamic limit (TDL), revealing geometric dependencies. NHSE is classified into critical and non-reciprocal types using net winding numbers. Critical NHSE hosts scale-free skin modes, while non-reciprocal NHSE includes normal/anomalous corner modes, boundary modes, or scale-free modes. The non-convergence and instability of non-Bloch spectra in the presence of scale-free modes are attributed to the non-exchangeability of the zero-perturbation limit and TDL. The instability drives energy spectra towards Amoeba spectra in the critical case. The findings provide a unified non-Bloch band theory governing energy spectra, density of states, and GBZ in the TDL, offering a comprehensive understanding of NHSE in arbitrary dimensions. The work addresses key challenges in higher-dimensional NHSE, including spectral convergence, GBZ determination, and classification. It demonstrates that non-Bloch spectra and NHSE are geometry-dependent, and introduces a framework for analyzing NHSE in various lattice geometries. The results highlight the importance of geometric information in formulating non-Bloch band theory and provide insights into the stability and convergence of non-Bloch spectra. The study also reveals the relationship between non-Bloch spectra, Amoeba spectra, and perturbed spectra, establishing a theoretical foundation for understanding NHSE in arbitrary dimensions.Non-Hermitian skin effect (NHSE) is a phenomenon in non-Hermitian systems where eigenstates accumulate at system boundaries. While well understood in one dimension via non-Bloch band theory, higher dimensions face challenges due to diverse open boundary conditions and lattice geometries. This work presents a geometry-adaptive non-Bloch band theory in arbitrary dimensions using spectral potential. It accurately determines energy spectra, density of states, and generalized Brillouin zone (GBZ) in the thermodynamic limit (TDL), revealing geometric dependencies. NHSE is classified into critical and non-reciprocal types using net winding numbers. Critical NHSE hosts scale-free skin modes, while non-reciprocal NHSE includes normal/anomalous corner modes, boundary modes, or scale-free modes. The non-convergence and instability of non-Bloch spectra in the presence of scale-free modes are attributed to the non-exchangeability of the zero-perturbation limit and TDL. The instability drives energy spectra towards Amoeba spectra in the critical case. The findings provide a unified non-Bloch band theory governing energy spectra, density of states, and GBZ in the TDL, offering a comprehensive understanding of NHSE in arbitrary dimensions. The work addresses key challenges in higher-dimensional NHSE, including spectral convergence, GBZ determination, and classification. It demonstrates that non-Bloch spectra and NHSE are geometry-dependent, and introduces a framework for analyzing NHSE in various lattice geometries. The results highlight the importance of geometric information in formulating non-Bloch band theory and provide insights into the stability and convergence of non-Bloch spectra. The study also reveals the relationship between non-Bloch spectra, Amoeba spectra, and perturbed spectra, establishing a theoretical foundation for understanding NHSE in arbitrary dimensions.