Global Coherence of a Quantum-Sourced Einstein–Scalar History

This tenth paper in the Schwarzschild Quantum Gravity series establishes a globally coherent Einstein–real-scalar history from the quantum-sourced, self-consistent gravitational construction developed in Papers I–IX. The investigation begins with exact transport of the physical boundary mass, retarded clock, scalar transmission data, and characteristic area target under changes of temporal anchor. It then proves nonlinear uniqueness for independently constructed retarded exteriors throughout their common equation-certified physical domains. The central mathematical result is a two-arrival characteristic comparison that accommodates distinct clock windows and characteristic flows, using parent-uniform annular estimates and radial continuation to establish full overlap coherence. These results define a Hausdorff physical-past spacetime carrying globally consistent (C^2) metric and real-scalar fields satisfying the nonlinear Einstein equations and covariant scalar wave equation. The complete two-region gravitational action—including Einstein–Hilbert and scalar bulk contributions, the moving material interface, Gibbons–Hawking–York boundary terms, and corner contributions—descends consistently to finite collars of the global history. Equality of the complete action functions also transports the existence and values of the corresponding one-sided first variations under common admissible constitutive and boundary prescriptions. For the selected positive-mass turning-clock sector, the physical-past seed domain terminates at a unique transverse marginal boundary of positive radius, where (2M=R). A sign-flexible characteristic construction establishes a regular local Einstein–scalar exterior across this boundary. Together, these results establish the global coherence, nonlinear field equations, and finite-region variational structure of the quantum-sourced physical history, providing a mathematical foundation for investigating its continuation beyond the first marginal boundary.