Variational Backbone and Regime Closures II: Geometric/Dissipative Metric Closure EH-leading rigidity, R1–R2 separation, and Ricci–DeTurck normalization modulo diffeomorphisms

We work in a density-fixed multi-reading framework: a single local C2scalar functionalE[g, IR, II ] = RM e(·) dvolg is fixed, and distinct regimes are obtained by changing the reading(R1 stationarity, R2 dissipative metric gradient flow in τ , R3 conservative time completionin t), not by swapping primitives. (The conservative/KG→Sch→HJ corridor is closed inPart III.) This paper closes the geometric/dissipative metric reading (R2) undertwo scoped hypotheses: (i) an EH-leading metric sector (the second-order metric principalsymbol is that of the Einstein–Hilbert term, within the declared equivalence class), and (ii)principal-symbol-safe coupling of non-metric fields via a derivative-free stress form, so thatmatter does not introduce new second-order metric operators. A key structural separation isemphasized: R1 identifies the metric Euler–Lagrange derivative (Einstein tensor structure),while R2 is defined by a well-posed parabolic representative of the L2-descent class obtainedafter (a) removing diffeomorphism degeneracy by the DeTurck trick and (b) fixing thescale/volume mode by a standard normalization (e.g. volume normalization using averagescalar curvature). In this representative form the principal symbol is Ricci/Laplace type;field effects enter as lower-order forcing through the stress tensor.