We develop a spectral and computational framework for probability and dynamics within the theory of Universal Modular Dynamics (UMD), where physical structure is encoded in a transition operator rather than in a predefined spacetime background. Starting from a discrete set of states with transition probabilities, we introduce an information-theoretic distance based on the logarithm of transition weights and show that this structure induces an emergent geometry. In the continuum limit, this construction yields a Riemannian metric defined directly from the local behavior of the transition operator. We establish a distinction between structural probability and dynamical probability, and introduce the concept of dynamic drift, which quantifies their deviation. A rigorous bound is derived showing that the magnitude of this drift is controlled by the spectral gap of the transition matrix. This leads to a natural renormalization group (RG) interpretation, where the subleading eigenvalue governs flow toward criticality. We further demonstrate that UMD exhibits a well-defined phase structure characterized by spectral properties, including critical regimes with universal scaling behavior. Finitesize scaling and universality classes are identified, and the role of information retention is clarified. Extending the geometric construction, we define curvature directly from non-factorization of transitions and show that an effective gravitational dynamics emerges. A variational principle is formulated, leading to field equations in tensor form, where curvature is determined by information density and its gradients. We prove that the framework is generally covariant and establish a rigorous continuum limit connecting discrete transition dynamics to smooth manifolds. Consistency tests confirm the recovery of flat space, weak-field behavior, diffusion dynamics, and critical scaling. These results suggest that geometry, dynamics, and gravitational behavior can be understood as emergent phenomena arising from the spectral structure of probabilistic transitions, providing a unified perspective on probability, information, and physical law.