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 \[ \mathcal M_X[K;\psi] = \int_0^\infty K(t)\psi(Xt)\,\frac{dt}{t}, \] Mellin inversion gives the exact factorization \[ \mathcal M_X[K;\psi] = \frac{1}{2\pi i} \int_{(c)} \widehat K(s)\widehat\psi(-s)X^s\,ds. \] Thus a pole of order \(m\) of \(\widehat K\) at \(s=s_0\) is annihilated exactly when \(\widehat\psi\) has a zero of order at least \(m\) at \(-s_0\). We construct compact-support-preserving neutralizers using the logarithmic differential operator \(\mathcal D=t\,d/dt\), 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 \(X^{1/4}\) under explicit boundary growth hypotheses. Reciprocal-zeta singularities then identify \(X^{1/8}\) 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 \[ \delta_n(\mathcal K;0) = \inf_{\substack{\deg Q\le n\\Q(0)=1}} \sup_{z\in\mathcal K}|Q(z)|, \] classical Bernstein--Walsh--Siciak theory gives \[ \lim_{n\to\infty} \delta_n(\mathcal K;0)^{1/n} = e^{-V_{\mathcal K}(0)}. \] We realize the corresponding extremal polynomials as compact-support-preserving Mellin differential filters \(Q(-\mathcal D)\), 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.