Friedrichs-Selected Schwarzschild Quantum Matter

Friedrichs-Selected Schwarzschild Quantum Matter develops the local matter theory of the same single real massless scalar field whose terminal operator and physical quantum dynamics were established in Papers 1 and 2 of this series. Starting from the original four-dimensional scalar action, the paper derives a shared real-scalar stress law, its Hilbert metric variation, its complex polarization, and the corresponding vacuum-relative quantum stress on the physical Schwarzschild Fock carrier. The physical-time theory is carried into local spacetime matter through the actual unitary clock. The resulting stress retains the complete normal and anomalous quadratic sectors generated by real-scalar Bogoliubov evolution. A dense dynamically transported regular class supports all-mode local stress, arbitrary conjugation-closed bank removal, and covariant conservation. The paper also preserves the independently constructed Friedrichs-continuum matter lane, where a dense regular finite-particle class admits presentation-independent, regulator-independent vacuum-relative stress and exact number-state scaling. Both constructions realize the same original-action real-scalar stress law while retaining their independently constructed quantum carriers. For spherically symmetric finite zero-mode states, the quantum source is coupled directly to the Schwarzschild Einstein tensor at first order, yielding an explicit machine-checked linear gravitational response. The analysis then compares this original-geometry quantum matter with the separately constructed nonlinear Einstein–real-scalar branch. Their full stresses agree at the common anchor with an explicitly derived normalization, while their global source laws separate through the geometry-dependent conservation structure. This identifies a precise mathematical boundary between original-geometry quantum matter and the changed-geometry quantum theory required for nonlinear continuation. All principal claims are formalized in Lean 4. The capstone theorem, Q12_realScalarQuantumMatter_owned_by_originalAction, packages original-action provenance, physical-clock stress covariance, Bogoliubov coherence, dense all-mode regularity, conservation, the independent continuum matter authority, spherical quantum sources, linear Einstein response, and the nonlinear source adjudication.