Co-evolving Probability and Geometry through Unbalanced Transport

We propose a geometric formulation of generative renormalization in which probability and geometry evolve jointly across scale. At scale $z$, the state is a pair $(\rho_z,g_z)$ consisting of a positive measure and a Riemannian metric. We model $\rho_z$ using Wasserstein-Fisher-Rao (WFR) transport, allowing both displacement and birth-death dynamics, while $g_z$ evolves through a coupled geometric flow. Interpreting $z$ as an emergent radial coordinate yields the bulk metric $ds^2=N(z,x)^2dz^2+g_{ij}(z,x)dx^idx^j$. We formulate the coupled dynamics variationally and characterize scale-invariant solutions that induce warped bulk geometries. The framework provides a transport-theoretic route from renormalization to emergent geometry in which the manifold co-evolves with the generative process.