Variational Backbone and Regime Closures IX: Classical mechanics as a licensed readout in the R3 regime The Hamilton–Jacobi validity domain and the Newtonian readout instance

This paper treats classical mechanics as a fixed-ℏ validity regime of the backbone rather
than as a separate primitive theory or a mere formal ℏ → 0 limit. With ℏ fixed at the
backbone level, it appears as a licensed Hamilton–Jacobi/trajectory readout of the R3
dynamics on regions where a semiclassical gate is objectively satisfied. For concreteness we
use the Schrödinger-form R3 representative as an 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) single-phase representability Ψ = AeiS/ℏ
, (ii) a non-nodal
interior A ≥ a0 > 0, and (iii) residual smallness quantified by the indicator Ξ = |RS|/ΛHJ,
where RS := ∂tS + H(x, ∇S, t) and ΛHJ := |∂tS| + |H(x, ∇S, t)| + δ0. Our main theorem is
conditional on the declared semiclassical realization regime: on any regime-valid region, the
phase is HJ-dominated with controlled residual and the amplitude obeys an R3-consistent
transport law with a controlled remainder. While computationally equivalent to standard
WKB hierarchies for the same anchor class (Part III; cf. [3]), the status differs: classicality is
governed by regime membership at fixed ℏ, yielding explicit validity and breakdown criteria
and a backbone-fixed correction hierarchy. Finally, Newtonian mechanics is formulated as an
explicit readout instance applied to regime data and is licensed only while regime conditions
persist along the instance evolution; under an additional macroscopic residual-regularity
gate (G–HJ4) we obtain a force-level accuracy bound controlling ∇RS at O(εΞ) within the
declared regime.