Mathematical Theory of Molecular Generation (MTMG): From Abstract Topology to Physical Chemical Structures

This paper introduces the Mathematical Theory of Molecular Generation (MTMG), a rigorous foundational framework that models molecular structures as a deterministic projection from abstract topological and algebraic spaces to physical molecular manifolds. Abandoning empirical and data-driven heuristics, MTMG conceptualizes chemical generation as an exact mapping guided by universal mathematical laws, akin to architectural generation from geometry. By embedding molecular graphs into three-dimensional Euclidean space and enforcing strict algebraic invariants, valence constraints, and spectral graph properties, MTMG guarantees the physical stability and chemical validity of generated entities ab initio. We formalize a comprehensive suite of generation operators—including expansion, fusion, symmetry multiplication, and topological surgery—and prove the completeness and convergence of the framework over chemical space. This work transforms molecular science from an empirical discovery discipline into an exact mathematical science.