Self-Interacting Extended Scalar Field and Emergent Scale Dynamics in Flat Spacetime

We study a self-interacting scalar field defined in flat spacetime, where the interaction couples the field to its ground state configuration. Using an ansatz for this ground state, the Klein-Gordon equation allows an analytical solution that requires two constraints: one independent of the excitation level and another that depends on it. These constraints organize the accessible configurations and make it possible to describe a process in which the mass evolves until it reaches a stable regime. This leads, at late times, to a residual value whose order of magnitude is similar to that of the lightest neutrinos, although in this framework it is not interpreted as a particle mass but as a geometric remnant of the relaxation process. From these constraints, relations arise between the residual value and the vacuum energy, which in this approach are understood as consequences of the stabilization mechanism itself. In addition, by modeling quantum transitions between adjacent levels of the system, we obtain an effective expansion rate for the scale parameter associated with the field, whose magnitude is compatible with late-time cosmological expansion. The model is presented as an effective framework in flat spacetime, and its scope and limitations are discussed explicitly.