A Self-Consistent Einstein–Scalar Quantum Sector

The first three papers in this series derive a selected real-scalar law on Schwarzschild geometry, construct its physical quantum evolution, and promote that evolution to local conserved quantum matter. This fourth paper closes that physical-sector cycle on an independently constructed nonlinear Einstein–real-scalar geometry.

The nonlinear geometry determines its own terminal coordinate, real scalar action, and full radial potential. Its curvature endpoint is Hardy-critical, with inner limit-circle and outer limit-point classification, deficiency indices (1,1), physical endpoint coefficients (A,B), and a complete self-adjoint projective family. The native ground-state form selects the no-log law, and compact-core form closure gives the Friedrichs realization of the nonlinear radial operator.

The same action yields global signed Cauchy modes, action-derived Bogoliubov transport, and an all-mode real Fock implementation. On every owned positive compact interval the ultraviolet mixing decays quadratically in frequency, so the anchor-vacuum occupation spectrum has finite total particle number. An independently constructed self-adjoint full-action Hamiltonian generates a two-time physical unitary satisfying the strong Schrödinger equation on the original energy domain.

Every signed mode has C² spacetime regularity and satisfies the native covariant wave equation. Directed observable-bank removal gives a conserved vacuum-relative all-mode core stress with complete normal and anomalous sectors. For admissible coherent amplitudes with nonzero real mean-field flux, the quantum quadratic pairing equals the classical product of the corresponding real mean field, and the resulting branch satisfies the Einstein equation with the same quantum stress source. A distinguished coherent branch and an independently constructed one-occupation branch provide exact source realizations.

Finally, the native scalar curvature diverges to −∞ along the shrinking-radius endpoint, excluding regular C² endpoint extensions in the formalized isometric-open-embedding class. The result is a self-consistent Einstein–scalar quantum sector in which geometry determines the quantum law, the quantum theory contains exact gravitational sources of that same geometry, and the geometry itself identifies the invariant boundary of its regular classical continuation.