Friedrichs-Selected Schwarzschild Quantum Matter

The first two papers in this series establish a continuous program for a single real massless scalar field in the Schwarzschild interior. Paper I derives its action-owned radial operator and selects the no-log Friedrichs realization. Paper II carries the same field into a continuum Fock theory and a separately derived physical Cauchy theory with an all-mode self-adjoint Hamiltonian and norm-preserving physical clock.

The present paper constructs the local quantum matter carried by that theory. Starting from the original four-dimensional real scalar action, it derives the Hilbert stress, real bilinear polarization, exact Hermitian complexification, and norm-weighted reference subtraction, defining a carrier-independent stress law with single-real-field provenance across both quantum realizations.

On the physical Cauchy Fock carrier, finite signed-mode banks define literal real-field quadratics containing all ladder sectors. The local stress retains normal and anomalous components; the real Bogoliubov law induces a local mixing jet that is proved equal to the geometric mode gradient. The theory establishes nonzero anomalous coherence and explicit nontrivial radial stress.

Pointwise all-mode matter is constructed on a dense transported regular class. Directed finite-bank exhaustion converges to an intrinsic all-mode vacuum-relative stress tensor whose covariant conservation follows from the on-shell real mode equations. An independent Friedrichs continuum realization supplies regulator-independent finite-particle matter with action ownership, conservation, and exact number-state scaling. Both carriers instantiate the same original-real-scalar stress law.

Finite spherical states generate explicit quantum sources and a genuine coupled linear Einstein response. Comparison with a separately constructed nonlinear Einstein–scalar branch fixes the paper boundary: the original-geometry quantum source class and the changed-geometry nonlinear source obey distinct conservation laws, opening the changed geometry as the next quantum-theory problem.