arXiv:2607.23487v2 Announce Type: replace Abstract: This paper establishes a comprehensive theoretical framework for the continuous-to-discrete modeling of spatially stationary holographic MIMO (HMIMO) channels utilizing the Nystrom method with Gauss-Legendre quadrature (NGLQ). Starting with an operator-theoretic analysis of the NGLQ method, we prove that its quadrature error exhibits a super-exponential decay. Furthermore, we derive a spatial sampling threshold, termed computational degrees of freedom (cDoF), which reveals a {\pi}/2 oversampling penalty over the physical DoF for 1D arrays, compounding to a 68% computational redundancy for 2D separable grids. To address the ill-conditioning of the eigenvalue decomposition (EVD) problem inherent to the Nystrom discretization, we invoke the multidimensional Szego-Widom asymptotic expansion. This analysis yields a physically grounded semi-analytical expression for the effective DoF (eDoF) of 2D rectangular apertures, capturing the anisotropic boundary truncation effects to guide partial EVD and reduce computational complexity. Numerical evaluations confirm the tightness of the cDoF threshold under worst-case end-fire conditions. Moreover, simulations utilizing closed-form kernels for isotropic scattering verify that the derived eDoF acts as an accurate asymptotic approximation. Finally, by deploying the exact non-uniform discrete Fourier transform to eliminate interpolation error floors, we demonstrate spectral convergence down to the machine-precision level for non-isotropic scattering environments.