A rigorous, navigable definition of Chemical Space: the Polytope Formalism as a foundation for AI-driven chemical discovery

Chemistry is widely conceived as the exploration of "Chemical Space" – the domain of all possible chemical entities and the transformations that interconvert them – yet this notion has remained largely heuristic, lacking a rigorous, unified mathematical definition. Existing representations, including molecular graphs, quantum-mechanical state descriptions, and cheminformatics embeddings, each capture particular aspects of chemical identity but none exhaustively define chemical entities or organise them by the physical transformations that connect them. As a result, both human-led and machine-learning approaches explore chemical possibility locally, extrapolating from training data without a principled map of the space itself. The Polytope Formalism addresses this foundational gap. It treats chemical identity as a structured, discrete configuration space whose elements correspond to formally definable chemical states, and whose adjacency relations correspond to physical mechanisms. Developed first for stereoisomerism and extended to molecular constitution, it systematically enumerates complete sets of species – traditional isomers together with their interconversion intermediates – guaranteeing that nothing is missing. Critically, these networks map directly onto the topology of the underlying potential-energy surfaces, functioning as a discretised encoding of them. This yields a unified, navigable representation of chemical possibility: a rigorously defined space within which completeness, adjacency, and transformation are exactly characterised. Such a structure offers AI systems something current representations cannot – a principled, physically grounded map to search, rather than an unbounded continuum to approximate. This poster presents a high-level overview of the formalism and its prospects as a foundation for the automated exploration of Chemical Space. 1 P. J. Canfield and M. J. Crossley, Chem. Sci., 2025, 16, 6705–6719. 2 P. J. Canfield and M. J. Crossley, Chem. Sci., 2026, 17, 2102–2118.