Variational Backbone and Regime Closures II: R1 and R2 Laws on the Effective State Space Effective law generation (IR, F) before representative pullback

Part II computes the R1 and R2 laws generated by applying the Part I readings to thecore densityecore =β2|DRIR|2 +α2|F|2 + η ⟨IR, F⟩, F = ΣH,P (II ), α, β > 0.Following the constitutional setting of Part I, the primary variables are (IR, F) on theretained-effective state space C(H,P)eff = C(P)R × Ran ΣH,P ; the first-order expression F = DI IIis a representative pullback, and equations involving II are representative lifts of the effectivelaws, not primary dynamics of an unread interior.The primary R1 output is the effective stationarity systemβ D∗RDRIR + η F = 0,ZM⟨αF + ηIR, δF⟩ dvol = 0 ∀ δF ∈ TF Ran Σ.When the summary range is the full retained section space, the constrained condition reducesto the algebraic relation αF + ηIR = 0; Schur elimination then gives the retained lawβD∗RDRIR − (η2/α)IR = 0. In the first-order representative F = DI II , the constrainedcondition becomes D∗I(αF + ηIR) = 0, and a closed-range projector PI = DI (D∗IDI )−1D∗Iyields the projected Schur form βD∗RDRIR − (η2/α)PI IR = 0.The primary R2 output is the effective summary flow∂τ IR = −β D∗RDRIR − η F, ∂τF = −MF (α F + η IR),where MF is the declared positive summary mobility on Ran Σ. In the first-order representative induced by the L2 metric on unread representatives, MF = DID∗I; a representative lift∂τ II = −D∗I(αF +ηIR) is consistent but not the primary R2 law. The flow carries a Lyapunovidentity dEcore/dτ ≤ 0, and its fixed points coincide with the R1 effective stationarity states.A settled background is therefore an R2 fixed-point readout, not primitive data of ecore.The unread side enters the retained law through summary-footprint channels: the licensedsummary F, the constrained summary-side condition, and the Schur-effective operator(algebraic −η2/α in the range-saturated regime, or projected −(η2/α)PI in the first-orderrepresentative when the range is a proper subspace). Metric, Einstein–Hilbert, and Ricci-typestructures are not new law-spaces opened by gates; they are restricted branch-readouts ofthis already generated R1/R2 law-space. The metric branch is treated only as a restrictedreadout: with DI = ∂/g and a settled summary representative ψ0, the term α2|F|2carriesthe Lichnerowicz footprint R|ψ0|2, which becomes an EH term only under constant-densitylocking |ψ0|2 = ρ0.