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

This paper treats Schur duality in VBRC as an invariant test of one retained-summary interface. Under the Internal Invisibility Principle, the two Schur elimination routes are not two ontologies. They are two admissible reductions of the same retained-effective block Hessian, where unread content enters only through a licensed summary channel. Eliminating the summary variation gives the retained-side Schur operator used in Part V. Eliminating the retained variation gives the summary-side dual-Schur operator used in Part VI. These reduced operators need not be equal and need not act on the same space. The invariant content lies instead in the shared spectral diagnostics: criticality, determinant identities, and inertia/index data, interpreted on noncritical branches or under declared relative regularization. Part VII therefore shows that the retained-side mass or gap diagnostic and the summary-side response diagnostic are dual reductions of the same core-induced block-Hessian geometry. The result is a Schur-invariant consistency layer for the VBRC summary-interface calculus, not a new hidden-sector ontology.