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 unreadsector pointwise. It reads only the licensed summaryF = ΣH,P (II ).Part VI takes this summary entry as given and asks the dual question to Part V: whatclosed-summary response operator is produced when the retained/open variation is eliminatedfrom 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 thelicensed 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 fromecore =β2|DRIR|2 +α2|F|2 + η⟨IR, F⟩, F = ΣH,P (II ),written as a functional Eg⋆VIIR, F of the retained-summary pair (IR, F), we linearize aroundan upstream-selected background representative (g⋆, I⋆R, F⋆). The resulting quadratic formin open/closed-summary variations (u, v) is Schur-reduced by eliminating the open variationu, givingLcl,eff = C − B∗A−1B.In the canonical Laplace-type representative,Lcl,eff = αI − η2PRan(βD∗RDR + M⋆R)−1PRan.Thus the unread/closed-summary channel is not inserted as an arbitrary dark component; itreceives a calculable resolvent-shaped response produced by the same variational core.Source-like, background-like, DM-like, and DE-like language appears only after additionaldownstream readout gates are declared. Before such gates, the generic output of Part VIis one mixed closed-summary response operator. The source/background split is a readoutdecomposition of this operator, not an ontological splitting into independent fluids, andPart VI does not claim a completed dark-sector phenomenology, a unique halo profile, or aunique equation of state.