Topological Constraints in Evolutionary Matter Design: A Graph-Theoretic Framework for Genus-Controlled Self-Assembly

A theoretical framework for controlling the topological genus of self-assembled molecular structures through geometric constraints in evolutionary optimization. The approach combines graph Laplacian methods with curvature-based penalty functions to stabilize non-trivial topologies (genus g ≥ 1) that would otherwise collapse to minimal-energy spherical configurations. We demonstrate that toroidal structures (g=1) can be reliably obtained by incorporating an approximate Betti number \beta_1 in the fitness function alongside a "phantom field" potential that biases hydrophobic collapse toward annular rather than spherical attractors. The framework achieves fitness scores within 0.4% of theoretical optima. Rigorous topological validation via winding number analysis (w=0.98 ± 0.02) and persistence scanning confirms the robustness of the genus-1 topology across length scales. We characterize the resulting structures using discrete Ricci curvature, confirming the expected negative curvature in the interior region (mean κ ≈ -1.4) that maintains pore patency.