Variational Backbone and Regime Closures V: Schur complement, projector-valued order-0 terms, and regime closure

This paper fixes a structural mass mechanism within the series contract “one fixed spatialdensity e(·) + three operational readings (R1/R2/R3)”. Mass is defined as order-0 spectraldata of the R1 Hessian at an R2-selected vacuum, equivalently as dispersion data of the R3time-completed representative. For the canonical first-derivative backbone, eliminating theclosed variable F := DI II induces, on the open sector, a Schur complement correction ofthe form −(η2/α)PRan: an order-0 projector term that preserves the Laplace-type principaloperator and shifts active channels. Elimination and the resulting Schur/projector term arerepresentation-level implementations of a declared readout/retention protocol (Part IV); theydo not modify the fixed backbone or its principal operator. After KG normalization, thedimensionless shift is expressed by the series invariant η2/(αβ) = κ2. The conversion fromthe spectral quantity m2gap to the dimensionful KG parameter (mc/ℏ)2is a rail/normalizationchoice and is deferred to Parts VII–VIII. Crucially, within the declared regime the protocoldefined mass is read from a single reduced spectral datum: it governs inertial response in theretained dynamics and enters the curvature coupling through the same induced-load (stress)channel. Accordingly, the inertial and gravitational mass readouts are matched at leadingorder only under the common retained-mode hypotheses stated below.