Classicality as a Residual-Rigid Macroscopic Phase of Quantum Dynamics

This work formulates classical physics as a stable residual-rigid macroscopic phase of quantum dynamics. The central thesis is not that classical mechanics is an independent fundamental theory, nor that decoherence alone solves the measurement problem, but rather that classical behavior arises conditionally when quantum states belong to a protected small-residual basin around a regular classical stratum. Let ρ denote a quantum state and let Mcl denote a regular classical stratum in the quantum state space. The non-classicality residual is defined as the multi-component map Rcl(ρ) = (Roff(ρ), Rloc(ρ), REhr(ρ), Rdyn(ρ), Rred(ρ)), where the components measure, respectively, off-diagonal coherence in a pointer decomposition, failure of semiclassical localization, Ehrenfest defect, transverse dynamical instability, and lack of redundant environmental records. The associated residual energy is ηcl(ρ) = woff ‖Roff‖² + wloc ‖Rloc‖² + wEhr ‖REhr‖² + wdyn ‖Rdyn‖² + wred ‖Rred‖², with positive weights determined by the macroscopic resolution scale of the physical model. The mathematical structure is residual-quadratic. If R(u*) = 0, η(u) = ‖R(u)‖², and L = DR(u*), then the canonical second variation satisfies D²η(u*)h,h = 2 ‖Lh‖², and the associated stability operator is A = L∗L. Applied to the classicality residual, this gives the transverse stability operator Acl = DRcl∗ DRcl. The principal rigidity statement is conditional: if the linearized residual has a coercive transverse spectral gap and if small residual energy cannot occur far from the classical stratum, then ηcl(ρ) ≤ ε* implies dist(ρ,Mcl)² ≤ C ηcl(ρ). Thus classical physics is interpreted as a protected low-residual basin in quantum state space: a stable macroscopic regime in which decoherence, semiclassical localization, Ehrenfest control, dynamical invariance, and environmental redundancy jointly enforce rigidity of classical behavior. The framework also identifies the main failure modes of classicality: closing of the transverse spectral gap, absence of decoherence, loss of localization, failure of redundant records, tunneling, revivals, mesoscopic superpositions, and concentration-like escape channels. Consequently, the classical regime is not asserted as a universal limit of all quantum states, but as a gapped residual-rigid phase valid under explicit stability and anti-concentration hypotheses.