Splitting-and-Branching 3D Generation: A Recursive Structural Growth Framework for Three-Dimensional Shape Synthesis

Most three-dimensional (3D) generative methods construct shapes by learning distributions over geometric representations, optimizing implicit fields, synthesizing point sets, or reconstructing scenes from observations. This paper proposes Splitting-and-Branching 3D Generation (SB3G), a structural generation framework in which a 3D object is produced through successive splitting events, branch creation, substructure inheritance, and recursive geometric expansion. Rather than treating a complete shape as the primary generation unit, SB3G treats structural components and their relationships as evolving entities. Given an initial seed structure, a generation controller determines whether to preserve, split, branch, terminate, or refine individual components. Newly created components inherit selected geometric and semantic attributes from their parent structures while introducing controlled variations. A topology-aware constraint mechanism regulates connectivity, intersection, collision, component size, and geometric continuity. The framework is designed to accommodate tree-like structures, crack networks, branching vessels, porous structures, and recursively organized fractal geometries. We formulate SB3G as a variable-cardinality structural growth process, define its state representation, transition operators, objective function, and termination criteria, and outline a practical implementation and evaluation protocol. SB3G is a proposed methodological framework rather than an experimentally validated model. Its potential contribution lies in unifying recursive structural branching and topology-aware geometric growth within a single generation formulation, while its novelty and effectiveness remain subject to comparison with existing procedural, graph-based, fracture-generation, and neural 3D modeling methods.