ABSTRACT This paper rigorously addresses the consumed capacity underestimation caused by division-based methods in graph theory and polyhedral geometry. Graph-theoretically, such underestimation occurs because subgraphs in shorter sections approximate their combined graph. Our polyhedral study suggests that the capacity analysis is computationally ‘easy’ because its linear programme in the complete section has an orthogonal projection on the Euclidean plane containing five proper faces, one being the objective-defined one-dimensional optimal facet. Geometrically, we conclude that the polyhedra of the linear programmes in shorter and complete sections have parallel facets, with the former nested within the latter. Consequently, the optimal values of the former cannot exceed those of the latter. With these theoretical insights, we derive a train operation labelling algorithm that takes proven linear time. Beyond enabling faster capacity computation, the findings deepen our understanding of the connection between infrastructure capacity utilisation and timetable structure, ultimately leading to more efficient railway optimisation.