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 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 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.