A Calculus of Mellin-Residue Neutralization: Finite-Rank Scattering Geometry and Potential-Theoretic Spectral Filters

We develop a Mellin-residue calculus for removing finite-dimensional polar obstructions from multiplicatively localized observables. For \[ M_X[K;ψ] = ∫_0^∞ K(t)ψ(Xt)\,dt/t, \] Mellin inversion gives the exact factorization \[ M_X[K;ψ] = 1/2π i ∫(c) K(s)ψ(-s)X^s\,ds. \] Thus a pole of order \(m\) of \( K\) at \(s=s_0\) is annihilated exactly when \(ψ\) has a zero of order at least \(m\) at \(-s_0\). We construct compact-support-preserving neutralizers using the logarithmic differential operator \( D=t\,d/dt\), and prove a corresponding minimal differential-order statement. The residue calculus is then formulated intrinsically in terms of finite families of residue functionals. An explicit finite-rank idempotent projects onto their common kernel, and the correction rank is shown to be minimal. In the automorphic setting, this leads to an effective-residue-rank formulation for Eisenstein scattering: the number of neutralization conditions is governed by the rank of the effective scattering residue map rather than by the number of cusps. At the standard Eisenstein pole \(w=1\), the residual obstruction is rank one under the usual normalization. Exact residue removal is separated sharply from analytic continuation. A contour-depth theorem shows that the remainder exponent is determined by the region to which the neutralized Mellin integrand can genuinely be continued. For an abstract Rankin--Selberg heat factorization, neutralization removes the \(u=1\) polar contribution exactly and yields the post-neutralization scale \(X1/4\) under explicit boundary growth hypotheses. Reciprocal-zeta singularities then identify \(X1/8\) as the subsequent RH-sensitive scale, without claiming an unconditional gain to that exponent. Finally, finite exact annihilation is extended to uncertain compact spectral sets. For \[ δ_n( K;0) = inf_{{ Q≤ n\(0)=1}} z∈ K|Q(z)|, \] classical Bernstein--Walsh--Siciak theory gives \[ limn→∞ δ_n( K;0)1/n = e^{-VK(0)}. \] We realize the corresponding extremal polynomials as compact-support-preserving Mellin differential filters \(Q(- D)\), and combine them with exact residue neutralization. Exact Chebyshev and disk models, a polynomial-hull no-go regime, and a moving-pole exponent law are also established. The resulting framework unifies exact Mellin residue annihilation, finite-rank scattering geometry, quantitative contour transfer, and potential-theoretically optimal spectral filtering in a single calculus.