The UD Origin of Thermodynamic Irreversibility: From Field Equations to Entropy Evolution

We derive the irreversible increase of the spatial-to-material ratio from the UD field equations. The conditional asymmetry of the $U$ and $D$ attributes—$U$ expands unconditionally while $D$ condenses only above the threshold $\alpha$—is shown to imply $\dot U_u/U_u > \dot D_d/D_d$ on global scales. From this inequality and the global constraint $E = C$, we prove strictly that the global entropy $\Xi \equiv (U_u + U_d)/(D_d + D_u)$ satisfies $\dot\Xi > 0$ in the time-averaged sense. The correspondence between the UD entropy $\Xi$ and the classical thermodynamic entropy $S$ is established: the two are proportional, $S = k_B \mathcal{N} \ln\Xi$, with the proportionality constant fixed by the ideal gas limit. For local isolated systems, the same field equations force the entropy to converge to unity, with the approach direction determined by the initial $E/C$ ratio. The ideal gas law and the equipartition theorem are derived from UD momentum transfer, giving the temperature calibration $T_{\rm th}=T_0\cdot E/C$ with $T_0=2mc^{2}/3k_{B}$. For a multi-species gas at thermal equilibrium, heavier particles have smaller $E/C$. Apparent local entropy decrease during structure formation is quantified. This work supplements classical thermodynamics with a first-principles field-theoretic foundation. All results follow from the UD axioms without adjustable parameters.