A Self-Consistent Einstein–Scalar Quantum Sector

A Self-Consistent Einstein–Scalar Quantum Sector: Friedrichs selection, native quantum evolution, and exact gravitational matter is the fourth paper in a machine-checked construction of real scalar quantum dynamics on Schwarzschild-derived and self-consistent nonlinear geometries. The paper constructs a nonlinear Einstein–scalar sector in which the geometry supplies the native real scalar action, terminal coordinate, radial operator, self-adjoint quantum law, Cauchy evolution, Fock carrier, and physical Hamiltonian, while quantum matter constructed from that same geometry supplies its exact Einstein source. Finite ground-state action selects the no-log Friedrichs realization of the native radial operator. The resulting native quantum theory supports global signed Cauchy modes, Bogoliubov evolution, a self-adjoint all-mode Hamiltonian, strong physical-time unitary transport, vacuum-relative local stress, and conserved spacetime quantum matter. A genuine coherent state on the completed Fock carrier provides the most transparent source realization. Its full normal and anomalous field pairing reduces exactly to the classical quadratic stress of its real coherent mean field, and the resulting vacuum-relative quantum stress satisfies the nonlinear Einstein equation with the same physical coupling. The paper also gives a one-occupation exact source construction, identifies the native particle spectrum through (N_k=|\\beta_k|^2), records the ultraviolet occupation bound, and specifies the bridge between the paper’s vacuum-relative stress and conventional locally covariant renormalized stress. The physical comparison surface includes geometry, effective potential, mode occupation, stress, and scalar-curvature observables relative to a matched Schwarzschild reference. At the terminal endpoint, scalar curvature diverges and excludes regular (C^2) metric extension in the formalized class. Independently, the physical coherent-state orbit has no strong endpoint limit in the original Fock representation, while unitary evolution remains valid throughout every strictly positive native time. The resulting sector closes the physical geometry–matter loop [g_{\\rm NL}\\longrightarrow\\mathcal Q(g_{\\rm NL})\\longrightarrow\\Delta T_{\\mu\\nu}^{\\rm quantum}\\longrightarrowg_{\\rm NL},] providing the physical-sector foundation for the continuing series, where the same Einstein–scalar action is subsequently connected to its functional gauge and master-action structure. All principal mathematical claims are formalized and machine checked in Lean 4.