Constraint geometry and reachability in natural product biosynthesis

Biochemical systems occupy restricted regions of admissible state space defined by coupled metabolic, enzymatic, regulatory, and ecological constraints. Behaviour in these regions is controlled by mechanisms, but at the boundaries, where admissibility changes, systems reorganise suddenly. Here I develop a boundary-centred formalism for natural product biosynthesis in which biosynthetic gene clusters (BGCs) are treated as bounded subsystems whose reachable transformations form a finite region Ω generated by a constraint field C . This framework, referred to here as Boundary Theory (BT), treats biosynthetic systems as occupying bounded admissible regions whose geometry governs reachability and transition behaviour. The boundary (∂Ω) separates trivially reachable interior transformations from those requiring external justification, and perturbations deform ∂Ω to make new regions Ω′ accessible. Boundary curvature governs transition difficulty: steep regions require strong justification, whereas shallow regions permit easier traversal. This framework provides a geometric explanation for biosynthetic silence, conditional activation, and the failure of purely mechanistic interventions, offering a structural basis for predicting when and how BGCs reorganise under altered constraint regimes. This geometric framework is illustrated across major biosynthetic classes including polyketides, terpenoids, alkaloids, and NRPS/RiPP pathways, showing how class-specific constraint regimes shape Ω, ∂Ω, and the conditions under which new regions Ω′ become reachable.