Variational Backbone and Regime Closures VI: Closed Curvature from Open Elimination Dual Schur complement, Green-operator kernels, and channel selectivity

Under the series contract “one fixed spatial density e(·) + three operational readings(R1/R2/R3)”, Part V [5] extracted open-sector mass data by eliminating the closed variableG = DI II in the vacuum Hessian, yielding an order-0 projector-valued Schur correctionon the open operator. This Part VI records the dual elimination: for the same quadraticbackbone, we eliminate the open variation and obtain a closed-curvature effective operator onHcl of the form C − B∗A−1B. In the canonical first-derivative representative, this becomesLcl,eff = αI − η2 PRan (−β∆ + V′′⋆)−1 PRan,which is coercive up to a smoothing (order −2) nonlocal kernel. On closed manifolds thecorrection is compact, so the closed spectrum is a discrete shift from α with accumulationat α. We record the required invertibility gate on the physical open subspace, the resultingspectral and topological features in the de Rham card, and a basis-invariant finite-modeextraction protocol for the closed sector.