Loewner Extrema and Rank-One Certification Gaps for Cumulative-Tail Matrix Response Bodies

We give an exact spectral classification in every finite real matrix dimension for thecompact response body En(S) = {G[K] : K ∈ An(S)}. It has least and greatestelements in Loewner order if and only if S has at most one distinct eigenvalue in(aℓ, bℓ). On exactly these spectral strata, En(S) is the completely filled matrix interval[g−(S), g+(S)]⪯ and is characterized by its directional quadratic-form bounds. Iftwo distinct interior eigenvalues are present, exact moment-preserving mixed-blockperturbations destroy both Loewner extrema.For n = 2 we then compare the response body with the intersection of all exact rankonedirectional slabs. Near every interior scalar moment we construct an explicit openoperator-norm ball that satisfies every such exact lower and upper rank-one inequalitybut lies outside the convex positive-semidefinite upper image US = conv E2(S) +Sym+(2). A positive-definite load gives the separating inequality. The proof combinesan exact collision normal, a global finite-scale primal–dual localization, compactrecovery, and a uniform rank-one completion. We further identify the intrinsic positiveloadcompatibility defect with a directed operator-norm excess and prove that boththe one-sided and exact two-sided certification gaps are of sharp order Θ(dist(S,RI)2)near the scalar locus. A local Lipschitz estimate for S 7→ En(S) gives quantitativestability. Finally, a normal-incidence anisotropic Maxwell layer is shown to realize Gas its cubic low-frequency input-admittance coefficient; the electromagnetic model isan application of the preceding analysis, not an additional hypothesis in the abstracttheorems.