Variational Backbone and Regime Closures V: Core-Density Schur Reduction and the Order-Zero Spectral Mass Shift

VBRC treats mass-like gaps as summary-interface footprints rather than primitivecoefficients. Part V computes the Schur-reduced retained operator generated by the fixedcore densityecore =β2|DRIR|2 +α2|F|2 + η⟨IR, F⟩, F = ΣV (II ).The calculation is performed on the retained-effective variables (IR, F); the unread coordinateII appears only through the representative pullback F = D(V )III . At an R2-selectedeffective background q⋆eff = (g, I⋆R, F⋆), the quadratic expansion and Schur elimination ofv ∈ SV := TF ⋆ Ran(ΣV ) giveLeff(q⋆) = βD∗RDR + V′′⋆ −η2αPSV.In the first-order representative, PSV = PRan(DI ). The Schur term is an order-0, projectorvalued spectral footprint, not itself a positive scalar mass coefficient. The mass/gap datumism2gap(q⋆) = λ+minLeff(q⋆) HphysR.R3 provides only the dispersion checkc−2∂2t u + Leffu = 0, ω2j = c2λj .Thus Part V is a calculation of a summary-mediated spectral gap, not a theory of primitivemass insertion or a unique hidden-sector ontology.