Variational Backbone and Regime Closures VII: Elimination Duality and Structural Invariants From the fixed core density to route-invariant Schur data

VBRC treats Schur duality as an invariant test of the same summary interface. Underthe Internal Invisibility Principle (IIP), neither elimination route is a new ontology; bothare admissible Schur reductions of one retained-effective block Hessian built on the Part Istate space, and Part VII identifies which determinants, indices, criticalities, and stabilitydiagnostics survive the change of route.Part VII asks what remains invariant when both Schur elimination routes are allowed forthe same two-sector structure generated from the fixed core densityecore(IR, F) = β2|DRIR|2 +α2|F|2 + η⟨ΓIR, F⟩, F = ΣH,P (II ),where the structurally fixed datum is the licensed summary-entry grammar F = ΣH,P (II ),with F = DI II a declared first-order representative realization, and Γ is a declaredrepresentative-level typed interface with fixed normalization. The block Hessian is taken onthe retained-effective tangent variablesu = δIR, v = δF ∈ TF Ran ΣH,P ,rather than on raw unread variations δII . Stationary linearization givesH =A B∗B C .Elimination of the summary coordinate v gives the retained-side Schur operator Sop(H) =A − B∗C−1B, while elimination of the retained coordinate u gives the summary-side dualSchur operator Scl(H) = C − BA−1B∗. These reduced operators need not coincide andneed not act on the same space. Thus the two Schur routes compare reductions of oneretained-effective Hessian, not two ontologies of the unread side.The structural claim is narrower. The retained-side route preserves the retained Laplaceprincipal class, while the dual route imports the corresponding A−1-controlled |k|−2responsekernel into the summary-side readout. The shared global spectral package consists ofcriticality and determinant/index data:0 ∈ spec(Sop(H)) ⇐⇒ 0 ∈ spec(Scl(H)) ⇐⇒ 0 ∈ spec(H),and, on a noncritical branch or in a declared relative/regularized convention,LogDet(H) = LogDet(C) + LogDet(Sop(H)) = LogDet(A) + LogDet(Scl(H)),together with the Schur inertia/index identity. The quantity κ = η/√αβ is treated separatelyas a dimensionless classification ratio of the normalized two-sector quadratic representative,not as a new primitive coupling. Thus the Part V retained-side mass/gap diagnostic and thePart VI summary-side response diagnostic are two Schur reductions of the same ecore-inducedretained-effective block-Hessian geometry, not independent constructions.