Source-Weighted Ward Identity in Discrete Causal Structures

We report a machine-precision verification of a discrete Ward identity on causal graphs,demonstrating that the \emph{source-weighted conservation law} and the \emph{local energy exchange law}form a complete algebraic closure.The results support a causal–energetic duality in which time acts as an active energetic degree of freedom.The framework naturally extends toward multiscale causal domains and cosmological energy exchange phenomena.