A Multi-Level Classification System for Algebraic Structures in Mathematical Physics: Axiomatization, Extendability, Periodic Table Predictions, and Complete Solutions to Key Foundational Problems

Based on the previously established seven-level classification system (Kingdom, Phylum, Class, Order, Family, Genus, Species) for algebraic structures, this paper systematically generalizes it to the field of mathematical physics. Algebraic structures in mathematical physics often possess additional structures such as topology, smoothness, metrics, involutions, norms, conformal structures, and quantum deformations, which are incorporated into a unified framework by extending the Species level or introducing a new “Structure” level. This paper first reviews common algebraic structures in mathematical physics and their additional features, then formally defines the extended multi-level classification system and proves its extendability. Subsequently, core objects such as Lie groups, Lie algebras, Poisson algebras, C*-algebras, von Neumann algebras, vertex operator algebras, quantum groups, Yang–Baxter systems, symplectic algebras, and algebraic structures in quantum field theory are embedded into the classification tree, with formal axiom systems and embedding theorems provided. Based on gap analysis, numerous previously unstudied mathematical physics structures (e.g., non-commutative symplectic algebras, C*-algebras with braided structures, fractional Poisson algebras, quantum vertex algebras, supersymmetric vertex algebras, non-commutative tori) are predicted for each level, accompanied by complete axiom systems and existence constructions. All theorems are given rigorous proofs (general theorems at least 4 steps, important theorems at least 8 steps), and all predictions are equipped with concrete models or free algebra constructions. This paper demonstrates how the discovery of mathematical physics branches can be transformed from random exploration to systematic gap-filling according to a “periodic table”. Connections to classical mechanics, quantum mechanics, field theory, string theory, and condensed matter physics are discussed. In particular, this paper systematically solves key problems including the unified embedding of common algebraic structures in mathematical physics, the proof of infinitude of gaps, the non-triviality criterion for free algebras, the single-sorted equivalence for many-sorted algebras, and the transfinite inductive construction for infinite-depth structures, integrating these solved problems into the corresponding sections of the paper.