Local Relational Signatures in Vertex-Transitive Graphs, with Applications to Regular Convex 4-Polytopes

This paper develops a purely combinatorial, local-to-global framework for analyzing finite, connected, vertex-transitive graphs equipped with bounded relational data. We introduce the notion of a local relational signature, consisting of vertex degree, the isomorphism type of the vertex–figure adjacency graph, prescribed overlap conditions between adjacent neighborhoods, and a finite set of forbidden configurations supported at bounded radius. We show that, for each admissible signature, there exists a unique global realization up to isomorphism. Applying this framework, we recover exactly the six regular convex four-dimensional polytopes as the only graphs compatible with these local constraints, identified through their 1-skeleta. The 24-cell is distinguished combinatorially as the unique case whose local structure is rigid already at the level of the vertex figure, without requiring additional radius-2 refinement. All arguments are purely local and combinatorial and avoid geometric realization, metric structure, and Coxeter-theoretic classification.