Variational Backbone and Regime Closures X: Gravity as a Metric-Cone Readout of the Core Hessian From the R2-selected effective background to the Einstein-form display

Part X formulates metric gravity as a metric-language readout of the effective Hessiangenerated by the Part I VBRC core density. The starting datum is not a primitive Lorentzianmanifold (M, g), not a metric background placed underneath the theory, and not a functionalwhose first input is an independent gravitational field. The starting datum is the retainedeffective pairqeff = (IR, FX), FX = ΣX(II ) ∈ Ran ΣX,together with the Part I core comparison density, written before metric language asecore =β2QR(IR) + α2BF (FX, FX) + η Bint(IR, FX).The objects QR,BF ,Bint are effective comparison data, not yet contractions or differentialoperators supplied by a pre-given metric. In a first-order retained representative one maylater realize QR by BR(D(X)R IR, D(X)R IR), but D(X)R is not ∇gX until the metric-readout gatehas produced gX.The R2 reading selects, when the relevant stability gate holds, an effective backgroundq⋆eff = (I⋆R, F⋆X).The effective HessianH⋆X = D2Ecore(q⋆eff)then carries the principal-symbol and cone data from which the Part X metric representativeis read:gX = Rmetσprin(H⋆X).Only after this metric-readout gate has been passed do expressions such as dvolgX , ∇gX , ∂/gX,R(gX), and Gµν(gX) become available. Thus geometry is not the primitive stage on whichecore is written; it is the readable representative extracted from the stabilized comparisonstructure generated by ecore.After the metric representative is licensed, the same core-generated structure is firstdisplayed in metric language on the retained-effective pair by EmetX gX, IR, FX, with FX ∈Ran ΣX. Only after a Lichnerowicz representative has been declared may one chooserepresentative data ψ0 in a summary fiber and write FX = D(X)I ψ0 = ∂/gX ψ0. The residualequation KXµν = 0 is therefore not a primitive variation of a pre-existing metric object. It isthe metric-representative form of the core-generated stationarity/readout. The Lichnerowicz,retained-Schur, and summary-side dual-Schur constructions are then treated as readout viewsof the same Hessian, not as independent source sectors; the legacy label VclSch denotes thissummary-side dual-Schur readout, not a physical closed sector.Under the locked density subcase |ψ0|2 = ρ0, the Lichnerowicz readout yields the Einstein–Hilbert principal display with induced coefficient γE = αρ0/4. The Einstein-form expressionγE Gµν(gX) = Teff,(v)µν + RE,(v)µνis a GR-locked comparison display of the metric representative, not the starting law of thePart. The paper therefore makes no claim that VBRC begins with a metric background,no claim that an Einstein equation is inserted as an independent axiom, and no claim thatunread-sector content appears as raw matter. Unread content affects the metric displayonly through protocol-licensed summaries and their derived Lichnerowicz/Schur/readoutfootprints.