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

We provide a corridor-level closure of the R3 (conservative / time-completed) reading at the level of principal symbols under the density-fixed rule. (The geometric/dissipative metric closure is established in Part II.) This paper does not derive the full coupled theory from E[g, IR, II ]; it classifies the Laplace-backbone-compatible conservative completions and correspondence gates that force the KG→Sch→HJ corridor. Concretely, (i) any local linear second-order time completion preserving the Laplace backbone yields a Klein–Gordon-type principal part; (ii) a rest-mass phase rotation gives an exact 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 Schrödinger dynamics; (iv) in the local-potential presentation, WKB yields exact continuity/phase equations and formally reduces to Hamilton–Jacobi as ℏ → 0. As a design constraint in the 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 term is locked to be quadratic; “HJ-only modification” is excluded unless the operator class is changed.