We develop a unified meromorphic profile theory for Abu Ghuwaleh master theorems over rational kernels. The analytic branch initiated in the 2022 master-theorem papers is extended in a single framework that simultaneously treats interior pullback poles, boundary collisions, finite-part contributions on the real axis, and general symmetric analytic profiles on the closed upper half-plane. The foundation is an upper-half-plane finite-part contour theorem with an explicit correction term at infinity together with a local divisor-transport operator for arbitrary pole order of the meromorphic factor and arbitrary multiplicity of the pullback equation. For symmetric admissible profiles u (z) =+ (z) the resulting whole-line master theorem decomposes the transform into four contributions: kernel residues in the upper half-plane, interior pullback residues, boundary half-residues, and the coefficient at infinity. Even and odd kernels yield half-line symmetric and antisymmetric master formulas, while logarithmic derivatives produce weighted pullback counting identities. The exponential specialization recovers the periodic lattice branch in fully explicit form, including interior meromorphic corrections, a universal boundary transport law for higher-order poles, and a radial half-jump theorem describing boundary collision. The Blaschke specialization supplies finite-valence sampling theorems and critical derivative-sampling formulas. Explicit applications are obtained for Cauchy kernels, higher resolvent hierarchies, cotangent and cosecant-square finite-part transforms, product-Cauchy kernels, and single- and power-Blaschke profiles. The paper provides one coherent master-theorem package containing the analytic, interior-meromorphic, boundary-singular, and symmetric-profile branches within a single residue-transport formalism.
Mohammad Abu-Ghuwaleh (Wed,) studied this question.