Relative Field Localization (RFL): Microscopic Jump-Operator Derivation and Open-System Closure

The third closure item of Relative Field Localization (RFL) concerns the microscopic derivation of jump operators from the system-environment interaction, the preservation of the global U(1) charge, and the ability of the resulting open dynamics to populate finite-energy Q-ball configurations. This report develops the eight requested components in a common notation. Starting from the density-coupling Hamiltonian, the interaction is decomposed into Bohr-frequency components and yields distinct emission and absorption operators. The Born-Markov reduction, secular conditions, and positivity of the transition-rate matrix are stated explicitly. Charge preservation follows from the neutrality of the Noether current. The existence interval and finite-energy conditions for the stationary Q-ball are checked, and a direct transition is contrasted with a sequence of adjacent-mode transitions. A finite two-mode, fixed-charge Fock-space projection is used to test an actual intermediate-state network. Trace preservation, complete positivity within the derived Lindblad regime, charge conservation, detailed balance, recoil, and numerical convergence are audited with machine-level precision.