Coherence–Flux Conservation with an Internal Parameter

I introduce a generalized conservation–balance framework for scalar fields by extending the standard continuity equation with an internal variable T representing a hidden structural degree of freedom. For the Lagrangian density L(phi, d_t phi, grad phi, d_T phi) = 1/2 (d_t phi)^2 − c^2/2 |grad phi|^2 − alpha/2 (d_T phi)^2 − V(phi), the associated variational structure yields an extended continuity equation that reduces to an exact conservation law only in the shift-invariant case V′(phi) = 0. For a general potential V(phi), the symmetry is explicitly broken and the equation becomes a balance law with a source term d_t J0 + div J + d_T Js = − V′(phi), where J0 = d_t phi is the canonical momentum density, J = c^2 grad phi the spatial flux, and Js = alpha d_T phi the structural flux. This formulation unifies apparently non-conservative dynamics in observable spatial domains by interpreting dissipation-like behaviour as flux redistribution along the internal direction T, rather than as a violation of conservation. I establish well-posedness of the resulting wave equation posed on an extended spatial domain (x, T), derive the corresponding extended-domain balance relation, and show how the structural flux accounts for apparent mass or energy leakage, amplitude attenuation, or effective damping without violating the global balance. Numerical experiments based on finite-difference schemes confirm that, in the shift-invariant case, the extended integral invariant is preserved within a relative error below approximately 2.5% under perturbations and mesh refinements. This variational extension provides a compact and physically motivated framework for modeling apparent non-conservation phenomena in systems with hidden internal variables, including wave propagation in complex media, transport in composite or microstructured materials, and effective models in optics or mathematical biology.