Variational Backbone and Regime Closures IX: Classical Mechanics as an R3 Regime Sector The Hamilton–Jacobi regime and the Newtonian readout instance

Classical mechanics is often presented as the ℏ → 0 limit of quantum theory. Wereassign its status with ℏ fixed at the backbone level: classicality is sectorial under the R3conservative reading, appearing as a Hamilton–Jacobi (HJ) regime on regions where a declaredsemiclassical gate holds. For concreteness we use the Schrödinger-form R3 representative asan anchor (the Part III anchor class); KG-type completions are deferred to later extensions.On a space–time domain U ⊂ M × R, we define an R3–HJ regime sector by (i) singlephase representability Ψ = AeiS/ℏ, (ii) a non-nodal interior A ≥ a0 > 0, and (iii) residualsmallness quantified by the indicator Ξ = |RS|/ΛHJ, where RS := ∂tS + H(x, ∇S, t) andΛHJ := |∂tS| + |H(x, ∇S, t)| + δ0. Our main theorem states that on any regime-valid regionthe phase is HJ-dominated with controlled residual and the amplitude obeys an R3-consistenttransport law (with controlled remainder). While computationally equivalent to standardWKB hierarchies (Part III; cf. [3]), the status differs: classicality is governed by regimemembership at fixed ℏ, yielding explicit validity and breakdown criteria and a backbone-fixedcorrection hierarchy. Finally, Newtonian mechanics is formulated as an explicit readoutinstance applied to regime data and is licensed only while regime conditions persist alongthe instance evolution; under an additional macroscopic residual-regularity gate (G–HJ4) weobtain a force-level accuracy bound controlling ∇RS at O(εΞ).