Discrete Complexity Geometry of Computational Universes:\\ Metrics, Volume Growth, and Local Curvature on Configuration Graphs

In the previous work, we axiomatized the ``computational universe'' as a quadruple U_{comp} = (X,T,C,I) with configuration space, one-step update relation, single-step cost, and information quality function. Building on this foundation, we develop a ``discrete complexity geometry'' framework that uses purely discrete graph-theoretic and metric structures to characterize time complexity of computational processes, ``geometric difficulty'' of problems, and the structure of reachable domains and horizons under finite resources. First, we associate each computational universe with a weighted directed graph G_{comp} = (X,E,w), where edge set E = T and edge weights w(x,y) = C(x,y). Under appropriate finiteness and generalized reversibility assumptions, this structure induces a generalized metric d:X\times X \to [0,\infty], whose shortest path values are equivalent to a physicalized version of discrete time complexity. We define complexity balls B_T(x_0) = \{ x : d(x_0,x)\le T \} and complexity volume growth functions V_{x_0}(T) = |B_T(x_0)|, introducing a ``complexity dimension'' \dim_{comp}(x_0) measuring growth order of complexity near a given starting point. Second, we introduce a discrete Ricci curvature \kappa(x,y) based on transition probabilities and first-order Wasserstein distance on weighted graphs, qualitatively characterizing ``divergence'' or ``contraction'' tendencies of complex paths in local regions. We prove that under natural assumptions, negative curvature regions correspond to exponential volume growth of complexity balls, while non-negative curvature regions correspond to polynomial or sub-exponential growth, establishing a qualitative connection between curvature and problem difficulty. Third, we view the family of reachable domains \{ B_T(x_0) \}_{T>0} as ``complexity horizons'' evolving with resource budget T, characterizing complexity phase transitions through simple homological and connectivity indicators: when T crosses certain critical values, the number of connected components, fundamental group, or first homology group of reachable domains undergoes mutations, corresponding to algorithms ``suddenly opening new routes'' in complexity geometry. Finally, assuming the computational universe arises from a physical implementation controlled by a unified time scale \kappa(\omega), we discuss how a family of complexity graphs converges to a Riemannian manifold (M,G) under mesh refinement limits, such that discrete complexity distance d approximates geodesic distance induced by G at large scales, providing several rigorous convergence theorems in low-dimensional cases. As the second work in the ``Computational Universe Theory'' series, this paper provides the first bridge from fully discrete computational structures to geometrized complexity, laying foundations for subsequent work unifying information geometry, time scales, and category equivalence between physical and computational universes.