The Dynamic Formulation of Particle Decay Probability: Exposing the Emission Retardation Mechanism and the Absence of Geometric Time Dilation in Linear Translation

In contemporary physics, the observed prolongation of a particle's lifetime during straight-line propagation is fundamentally misinterpreted as geometric time dilation within a four-dimensional spacetime continuum. Within the framework of the Unified Object Project, this paper refutes the existence of autonomous kinematic time dilation in linear translation. We establish that particle decay and radiative emission share an identical physical nature, both governed by the dynamic rupture of structural energy bonds against the static background field ($k=1$). By introducing the concepts of the proper time tail oscillation and the critical energy dispersion diameter ($D_{90}$), we derive a generalized analytical equation for the dynamic decay probability: $W = (mc^2 + T - V)/(m_{\text{new}}c^2)$. This formulation proves that the observed temporal delays in linear baselines—such as the Ives-Stilwell and Frisch-Smith experiments—are purely energetic phenomena, where the relative kinetic energy alters the internal structural transition rate rather than the geometry of time. Furthermore, this dynamic model successfully resolves subatomic stability anomalies, specifically explaining the proton-neutron decay dichotomy based on localized potential wells.