On Meta-Structure: A Mathematical Meta-Language of Objects and Relations

This paper proposes the meta-structure as a candidate mathematical metalanguage. At its core is an ordered pair L = (V, E), where V is a collection of objects and E a collection of relations on those objects. Objects and relations are taken as primitives; neither a membership relation nor composability is presupposed. From this minimal starting point, we generate the fundamental structures of graph theory layer by layer through two operations: rule-based constraints and structural expansions. A single meta-level axiom is introduced, the Graph Transfer Axiom. It asserts that if two structures admit graph representations that are isomorphic, then propositions about one can be transformed into propositions about the other. The axiom thereby turns cross-domain knowledge transfer from an analogy reliant on intuition into a mathematically guaranteed operation. The work is organized into five parts. Part I defines the meta-structure and its basic operations. Part II provides equivalent reformulations of five major founda-tional theories, ZFC set theory, Peano arithmetic, Hilbertian geometry, category theory, and homotopy type theory, showing that the expressive power of the meta-structure captures a common core shared by foundations ranging from the classical to the cutting-edge. Part III reconstructs the fundamental structures of graph theory from the meta-structure through the alternating use of rule-based constraints and structural expansions. Part IV takes six representative fields, group theory, number theory, linear algebra, analysis, probability theory, and topology, and re-formulates their core concepts and fundamental theorems in the language of graphs. A dedicated chapter juxtaposes the translated theorems across these fields, bringing to light the cross-disciplinary structural unity revealed by the meta-structure framework. Part V summarizes the characteristic features of the meta-structure approach and points to future directions, including dynamic meta-structures and the broader prospects of cross-disciplinary knowledge transfer. The work strategy is as follows. The equivalent reformulations of the five foun-dational theories in Part II establish the expressive foundation of the meta-structure framework. All mathematical branches are ultimately built upon these axiomatic systems; therefore, once the equivalent translation of the foundational systems is established , any mathematical theory built upon them can in principle be equivalently reformulated in the meta-structure language. The translations of the graph-theoretic spectrum and the six representative fields in Parts III and IV are demonstrations of the translation patterns rather than exhaustive verifications. In each field, basic definitions , axioms, and a selection of core theorems are given complete graph-theoretic translations to exhibit the concrete patterns of translation. Once these patterns have been exhibited, the translation of the remaining theorems in that field is a pat-2 3 terned extension. The six fields span discrete and continuous mathematics, algebraic and order structures, and deterministic and stochastic systems, thereby exhibiting the consistency and operability of the translation patterns across diverse types of mathematical thought. Taken together, the work demonstrates the dual expressive capacity of the meta-structure as a mathematical metalanguage: at the abstract level, it captures structural correspondences across mathematical fields; at the concrete level, it supplies graph-theoretic constructions that are directly operable. The Graph Transfer Axiom serves as a foundation for cross-disciplinary knowledge transfer.