The AbuGhuwaleh Hyperplane-Residue Calculus for Multidimensional Improper Integrals: Tensor Spectra, Simplex Reduction, and Monomial Pole Geometry

This paper develops a multidimensional sequel to the author's hyper-residue program for improper integrals. The central shift is geometric: instead of a single Mellin variable with poles at isolated points, we build a several-variable spectral-Mellin calculus whose singular set consists of coordinate planes and genuine affine hyperplanes. The first part of the paper proves an AbuGhuwaleh tensor spectral solver for Laplace-orbit kernels on the positive orthant and derives a rectangular meromorphic continuation theorem for gapped product spectra. The second part proves an AbuGhuwaleh simplex-radial solver: when the kernel depends on a weighted simplex radius ρ_λ(x)=λ₁x₁+⋯+λdxd, the d-dimensional improper integral collapses exactly to a one-dimensional spectral symbol evaluated at the combined weight Σ(z)+τ, thereby producing affine hyperplane poles Σ(z)+τ=-n. The third part pushes the construction further by monomial changes of variables. This yields oblique linear-pole geometries of the form (A-Tz)ⱼ=-n and A⁻¹,z+τ=-n, together with explicit hyperplane residues and regular values. We also prove oscillatory pole-collapse, moment-cancellation, frequency-transport, scaling, and logarithmic differentiation theorems. Benchmark examples include algebraic radial kernels, one-atom hyperplane models, cancellation of the first affine pole, tensor-product Beta families, and a genuinely oblique two-variable monomial family. The resulting framework is substantially stronger than a one-dimensional gapped theory because it creates an explicit multidimensional residue geometry for improper integrals while remaining completely spectral and constructive.