Dynamical Closure of Time–Scalar Field Theory: An Action-Based Framework with Emergent Geometry and Coherence Structures

We develop a dynamical formulation of Time–Scalar Field Theory (TSFT) in which a scalar temporal field Θ(xμ) governs both physical structure and emergent geometry. The theory is defined by an action on an auxiliary background manifold, avoiding circular dependence between field dynamics and the induced metric. We show that the scalar field admits stable, localized solutions corresponding to coherence structures, which arise as extrema of the potential and remain stable under perturbations. The physical spacetime metric is constructed as a conformal-disformal functional of the scalar field, consistent with prior classification results for scalar-induced geometries. We derive the field equations from the action and analyze the resulting dynamics, including perturbative stability and wave-like propagation. In the weak-field limit, we demonstrate that the induced metric reproduces the Newtonian potential and yields the Parametrized Post-Newtonian parameter γ = 1 at leading order, with disformal contributions entering at higher order. This framework provides a non-circular pathway from scalar field dynamics to emergent geometry and stable localized structure. Within this formulation, both spacetime curvature and particlelike behavior arise from a common underlying temporal field. While the functional form of the metric coefficients remains to be determined, the results establish a consistent foundation for further development of scalar-temporal approaches to gravity and quantum-like structure.