Quantum Theory in the Dynamical Sector of Absolute Frame Theory: Modular Reflection Positivity and its Unification with Black-Hole Unitarity

A companion paper reconstructed finite-dimensional quantum theory as the operational shadow of the M--A channel of Absolute Frame Theory (AFT), and closed the reconstruction rigorously in the static sector through a reflection-positivity (RP) theorem about a single Euclidean time. That theorem explicitly left open the genuinely time-dependent sector: the sliding embedding mode, the curved (horizon) case, and the interacting thermodynamic limit. We resolve the structure of that open frontier. First, the time-dependent residue splits cleanly: uniform sliding is a removable Euclidean isometry---a translation, not a boost, carrying the vacuum and not a thermal state---so the static theorem applies verbatim in the comoving frame; only accelerated sliding, which possesses a horizon, is genuinely new. Second, for the sector with a horizon the correct tool is modular: by the Tomita--Takesaki theory together with the Bisognano--Wichmann and Sewell theorems, the modular flow is the horizon boost, the modular conjugation J is the geometric reflection, and the state is Kubo--Martin--Schwinger thermal---precisely the modular backbone already established for black-hole unitarity in a companion work. We show that the dynamical quantum-theory reconstruction and black-hole unitarity are then one and the same modular object: replacing the static reflection by the modular conjugation J transports all four Osterwalder--Schrader outputs (a physical Hilbert space, a positive Hamiltonian, unitary evolution, and a single operator-level imaginary unit yielding local tomography), and the inertial limit of vanishing surface gravity recovers the static theorem as the zero-temperature face of the construction. Both reductions rest on a single shared hypothesis---that the constrained embedding field theory satisfies the algebraic quantum field theory axioms. We then establish that the defining Karush--Kuhn--Tucker active-set constraint of the horizon introduces no AFT-specific obstruction to those axioms: the constraint functional is strictly quasi-local, real, and bounded below, the active-set (inequality) character preserves reflection positivity, and the only intrinsic subtlety---a Planck-scale collar correction---is the action-quantization envelope of the theory, not a spacetime granularity. The shared hypothesis thereby reduces to the generic constructive-field-theory caveat shared by every interacting four-dimensional theory. We tag each statement as derived, reduced to a named condition, or open, and we close by listing the frontiers that remain: the non-adiabatic Planckian endpoint of evaporation, the constructive limit, and the conceptual roots inherited from the static reconstruction.