Variational Backbone and Regime Closures VI: Open Schur Elimination and Closed-Summary Source/Background Readouts A dual-Schur response operator under downstream source/background gates

Under the Internal Invisibility Principle (IIP), the retained law does not read the unread sector pointwise. It reads only the licensed summary F = ΣH,P (unread content). Part VI takes this summary entry as given and asks the dual question to Part V: what closed-summary response operator is produced when the retained/open variation is eliminated from the same two-sector Hessian generated by the fixed core density? The active closed-side variable of this Part is not the raw unread coordinate II . It is the licensed summary variation around a declared background summary F ⋆ , v = δF ∈ S⋆ F , S ⋆ F := TF ⋆ Ran ΣH,P . When the summary representative is linear, we suppress the star and write SF = Ran ΣH,P ; in the first-order representative F = DI II , this becomes SF = Ran(DI ). Starting from e (P) core = β 2 ∥DRIR∥ 2 P + α 2 ∥F∥ 2 P + η⟨IR, F⟩P , F = ΣH,P (unread content), written as a functional E g⋆ VI[IR, F] of the retained-summary pair (IR, F), we linearize around an upstream-selected background representative (g⋆, I⋆ R, F⋆ ). The resulting quadratic form in open/closed-summary variations (u, v) is Schur-reduced by eliminating the open variation u, giving Lcl,eff = C − B ∗A −1B. In the canonical Laplace-type representative, Lcl,eff = αI − η 2PRan(βD∗ RDR + M⋆ R) −1PRan. Thus the unread/closed-summary channel is not inserted as an arbitrary dark component; it receives a calculable resolvent-shaped response produced by the same variational core. Source-like, background-like, DM-like, and DE-like language appears only after additional downstream readout gates are declared. Before such gates, the generic output of Part VI is one mixed closed-summary response operator. The source/background split is a readout decomposition of this operator, not an ontological splitting into independent fluids, and Part VI does not claim a completed dark-sector phenomenology, a unique halo profile, or a unique equation of state.