Relative Field Localization (RFL): Extended Conserved Charge and Q-ball Localization — A Covariant Scalar-Field Framework

The Relative Field Localization (RFL) framework is built around two central physical ideas: a complex field carries a conserved, spatially extended U(1) charge, and an interaction with the environment may drive this extended configuration toward a localized, finite-energy Q-ball-like state. This work develops a covariant scalar-field foundation for RFL. It derives the Noether current, invariant charge, stress-energy tensor, nonrelativistic positive-density limit, stationary Q-ball equation, existence conditions, energy and charge functionals, and virial identity. The study also presents numerical solutions for flat-space Q-ball configurations, examines their branch structure and low-lying perturbation spectra, and provides a self-gravitating spherical background for comparison with the scalar-field sector used in Quantum Gravity Theory (QGT). The localization-time expression is corrected to retain the finite-radius factor (a^{-3}). Thermal emission and absorption channels are distinguished, and the relationship between Lindblad dephasing, conditioned quantum trajectories, and physical Q-ball formation is clarified. A Born-compatible localization instrument is presented as a mathematical ansatz, while its microscopic derivation from the system–environment interaction remains an open problem. The framework preserves the central RFL proposal of extended conserved charge and interaction-triggered Q-ball localization, while distinguishing mathematical identities, conditional results, numerical findings, and unresolved physical hypotheses.