Gradient and Rotational Structure of the Temporal Field: A Self-Contained Dynamical Decomposition in Time–Scalar Field Theory

We present a self-contained formulation of Time–Scalar Field Theory (TSFT) in which a scalar temporal field Θ(xμ) admits a natural decomposition into gradient and rotational components. This construction introduces two derived structures: a gradient sector associated with temporal variation and a rotational sector associated with temporal vorticity. The resulting framework provides a complete dynamical description of scalar temporal propagation, including conservation laws, wave dynamics, and phase structure. We show that this decomposition yields a mathematically consistent field theory without requiring additional postulates beyond the existence of a scalar temporal field. In appropriate limits, wave-like behavior emerges naturally, and complex phase structure arises from coupled real scalar components. This provides a pathway toward quantum-like dynamics as an emergent property of temporal field behavior. The resulting formulation is self-contained, covariant, and compatible with standard scalar field dynamics. This work establishes a foundational extension of Time–Scalar Field Theory and provides a mathematically consistent framework for further exploration of emergent geometry, wave propagation, and coherence structure.