WARP Graphs: A Worldline Algebra for Recursive Provenance

Many computational systems are best described as graphs of graphs of graphs. This paper introduces WARP graphs (Worldline Algebra for Recursive Provenance), a minimal categorical structure that captures nested graph-shaped state. Formally, a WARP graph is a finite directed multigraph whose vertices and edges carry attached WARP graphs, yielding a finite well-founded hierarchy.

The “worldline” terminology anticipates the deterministic evolution of such states developed in subsequent papers; this work focuses on the static state object. WARP graphs unify hierarchical system structure, syntax, control flow, provenance, and traces into a single algebraic object.

The paper gives an inductive definition, an initial-algebra characterisation, a category of morphisms, and embeddings of ordinary graphs and hypergraphs. It forms the structural foundation for the AIΩN Foundations Series, enabling deterministic rewriting, holographic provenance, and observer geometry in later developments.