This supplement extends the MUT v5.8 scalar-impedance framework to the problem of temporal directionality. It proposes that the Arrow of Time is not a fundamental asymmetry, but a deterministic consequence of the global decay of the Background Scalar Field frequency � under cosmological scaling.Using the relation �, temporal flow is derived directly from scalar field evolution. As the cosmic scaling factor � increases, the master frequency relation � predicts a monotonic reduction in field frequency. Macroscopic irreversibility emerges from this global frequency gradient rather than from statistical postulates alone.Thermodynamic entropy is reinterpreted as phase decoherence within the scalar manifold, expressed as �. This provides a deterministic bridge between microscopic reversible laws and macroscopic temporal asymmetry.This work integrates cosmological expansion, entropy growth, and temporal directionality into a unified scalar-field framework within the MUT v5.8 series.The version number (v5.8.12) reflects internal theoretical development order.