Subdivision Geometry (Renji Geometry): The Discrete Generative Structure of Continuous Systems

Abstract (Revised English Version with Renji Geometry) This paper introduces Subdivision Geometry (Renji Geometry) as a new geometric framework for describing the discrete generative structures underlying continuous phenomena. Renji Geometry is built upon a triad of concepts: generative units, which represent the minimal discrete components of locally continuous structures; subdivision operations, which describe how these units combine and branch; and the finite subdivision property, which structurally prohibits infinite refinement. Within this framework, we show that diverse continuous phenomena—including vortex formation, energy concentration, singularity development, convection-cell generation, and solar granulation—can be understood as finite-depth branching trees. We formalize the foundational concepts of Renji Geometry and present the main theorems of the theory, including the finite classification of branching structures, finite generation of local structures, discreteness of eigenmodes, and structural exclusion of infinite-depth singularities. We further demonstrate applications across analysis, geometry, physics, and computation: the structure of blow-up in the Navier–Stokes and Euler equations, energy concentration in Yang–Mills theory, singularity models in Ricci flow, generative lower bounds in computational complexity, compression of solar granulation patterns, and generalized convection and vortex generation. These results reveal the universality of discrete generative structure underlying a wide range of continuous phenomena.