Master Integral Theorems over Rational Kernels: Admissible Profiles, Fractional Transforms, Semigroup Evolutions, and Arithmetic Applications

We develop a general theory of master integrals generated by analytic compositions over rational kernels. The exponential profile e^ z is replaced by an admissible analytic profile on the upper half-plane, leading to a residue theorem with an explicit correction term at infinity. For the exponential profile this yields unified whole-line and half-line master theorems, a Hadamard finite-part treatment of real poles, universal polynomial-kernel formulas for P/Qʳ, tensor-product extensions, and finite-jet reconstruction from repeated Cauchy poles. A fractional branch is obtained for Cauchy and product-Cauchy kernels, where the master transforms are expressed through modified Bessel functions. A coefficient-multiplier transfer principle produces exact integral identities for Poisson, heat, wave, Schr\"odinger, and modified Helmholtz evolutions. On the q-lattice, a Jackson q-Beta master theorem is established with a sharp classical limit and basic-hypergeometric applications. The Cauchy kernel further yields exact representations for Lambert terms, Eisenstein series, the Dedekind eta function, and Jacobi theta functions. Exact application families include multiscale Cauchy quadratures, logarithmic and polylogarithmic identities, Mittag--Leffler sampling formulas, higher-order damped-node formulas, harmonic and wave Poisson-kernel evaluations, rational-profile identities, and benchmark formulas of Cauchy type.