Lewis-labeled graphs: curly arrows and fishhooks as executable electron transfers

Abstract

The curly-arrow formalism is the lingua franca of organic reaction mechanisms, but it is not executable. Molecular graphs used in rule-based modeling encode atomic connectivity while omitting lone pairs, radical electrons, and distinct sigma and pi components. Topological matching therefore cannot determine whether a reactive center holds the electrons a step consumes, while matching on derived formal charge can reject centers that do hold them. We introduce the Lewis-labeled graph (LLG), whose atom labels carry lone-pair and radical populations and whose bond labels separate sigma- and pi-bond occupancies. Bond order, formal charge, and valence electron inventory are derived from these fields. Chemical transformations become resource-constrained double-pushout rules, and curly arrows and coupled fishhooks become locus-sorted electron transfers committed atomically from a common pre-state. For a specified event group, we prove that its execution, application of its induced rule, and the corresponding integral occupancy update are equivalent. Every admissible event conserves valence electrons and net formal charge, and bond-centered fishhook coupling follows from integrality rather than drawing convention. Bidirectional replay recovers all 39,732 mapped reference endpoints, while LLG admits 96 fewer forward and 818 fewer inverse unique outcomes than conventional atom-bond rules. Ten radical records require manual annotation corrections. After review, transition construction succeeds for 101,313 of 101,314 records. The remaining case requires an endpoint atom-map correction rather than an arrow edit. Strict replay accepts all 160 reviewed steps and rejects all 1,120 controlled corruptions. These results establish a common executable state space for molecular graphs, reaction rules, and electron-flow annotations, prior to questions of kinetic or thermodynamic feasibility.