From Tara to Mvog: A theory of deformable connected geometric evolution

This paper introduces an axiomatic theory of Mvogs, rooted, generation-indexed geometric structures that evolve through the reproduction, placement, and attachment of three-dimensional cells. Every Mvog originates from a distinguished ancestral cell, called Tara, whose descendants are generated through a prescribed birth law. Reproductively active descendants may subsequently assume the role of Tara relative to their own offspring, thereby producing nested genealogical families. The evolution is governed by a Growth Operator defined as the composition of birth, spatial placement, and geometric connection. A Mvog is surface-connected when every pair of cells can be joined by a finite chain of nonempty direct boundary interfaces. When the cellular support is not surface-connected, the Mvog is classified as branch-mediated if the inclusion of prescribed three-dimensional branches produces a connected augmented structure, and as completely disconnected otherwise. The framework is developed within the Trinition geometry induced by the positive-definite metric tensor [Formula: see text], where [Formula: see text] controls anisotropic deformation while preserving the intrinsic genealogical and attachment structures. The principal results establish recursive existence and uniqueness relative to prescribed admissible deterministic data, preservation of surface connectedness under explicit attachment hypotheses, population.volume growth laws, evolution of the total interface measure, and finite termination under a uniform lower-volume condition. The [Formula: see text]-connectivity ratio [Formula: see text] and the interface measure [Formula: see text] are introduced to quantify global cellular connectedness and the geometric magnitude of positive-area attachments. Detailed comparisons with iterated function systems, cell complexes, body packings, contact graphs, branching processes, cellular automata, stochastic aggregation, continuum percolation, and geometric-flow methods clarify that these established frameworks retain only particular projections or specializations of the complete Mvog state. Numerical constructions involving completely disconnected populations, branch-mediated organizations, strongly surface-connected assemblies, double-helical and anthropomorphic forms, and Mvogs supported by Cesáro–Koch and Sierpiński geometries illustrate the scope of the theory. The resulting framework provides a mathematical foundation for studying geometry generated through reproduction and attachment rather than through contraction, lattice updating, or continuous front propagation.