Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling

Continuous-time generative models compress endpoint-conditioned bridges into Markov velocity fields, but no existing diagnostic measures how much information this compression discards. We introduce the \emph{Markovization gap}---the integrated conditional variance of the bridge velocity given the Markov state---which quantifies this loss before any neural network is trained. To make the gap well-defined across model families, we define \emph{Bridge Graphical Models} (BGMs), separating endpoint coupling, bridge law, Markovian projection, and dynamics representation as independent design choices. This decomposition also formalizes Poisson and electrostatic models as field-line bridge kernels with their own caustic gap. Across synthetic, latent, and pixel-space pilots on CIFAR-10 and Fashion-MNIST, a proxy gap estimated in minutes on CPU consistently ranks coupling/bridge choices in the same direction as training loss and FID measured after hours of GPU training.