Variational Backbone and Regime Closures III: Conservative & Semiclassical Closure Density-fixed Laplace backbone and the R3 regime: KG-type principal part, exact KG→Sch identity, NR/WKB gates, and semiclassical HJ locking

We provide a regime-level closure of the R3 (conservative / time-completed) readingat the level of principal symbols under the density-fixed rule. (The geometric/dissipativemetric closure is established in Part II.) This paper does not derive the full coupled theoryfrom E[g, IR, II ]; it classifies the Laplace-backbone-compatible conservative completions andcorrespondence gates that force the KG→Sch→HJ regime.Concretely, (i) any local linear second-order time completion preserving the Laplacebackbone yields a Klein–Gordon-type principal part; (ii) a rest-mass phase rotation gives anexact KG→Sch identity with explicit relativistic correction; (iii) under a quantitative slowenvelope gate ε = ℏω/(mc2) ≪ 1 on a fixed time window, the identity reduces to Schrödingerdynamics; (iv) in the local-potential presentation, WKB yields exact continuity/phaseequations and formally reduces to Hamilton–Jacobi as ℏ → 0. As a design constraint inthe series, within a fixed linear scalar, local, second-order Laplace-type Schrödinger class(up to lower-order perturbations, possibly order-0 operators), the semiclassical kinetic termis locked to be quadratic; “HJ-only modification” is excluded unless the operator class ischanged.