Higher Relational Geometry(HRG):An Axiomatic Foundation Subsuming Classical Graph Theory and Resolving Higher-Order Relational Complexity

Classical graph theory provides a foundational framework for binary interactions but faces severe fundamental barriers in addressing higher-order interactions, computational NP-hardness, dynamic noise, and the artificial separation between structural, spectral, and topological invariants. In this paper, we establish Higher Relational Geometry (HRG) (or Relational Complex Theory), a rigorous, higher-order axiomatic framework that subsumes classical graph theory as a 1st-order binary special case (k = 2). HRG synthesizes algebraic topology, simplicial sheaf theory, higher-dimensional Hodge-Laplacian operators, continuous relational limits (relationalons), and fixed-parameter tractability. We formalize five fundamental axioms, construct the operator algebra of generalized higher Hodge-Laplacians Delta_k, and prove core unifying theorems regarding spectral-topological completeness, algorithmic parameterization, continuous limits, and cohomological data reconstruction from incomplete observations. Finally, we provide a fully executable Python verification suite implementing higher-order boundary operators, Hodge Laplacians, Betti-number calculations, and Sheaf-based data reconstruction.