This paper establishes a rigorous multi-layer classification system for discrete mathematics, extending the axiomatic layered paradigm previously developed for quantum mechanics and the Standard Model. The system comprises eight layers (Level 0: Meta-Logic and Universe Axioms; Level 1: Kingdom; Level 2: Phylum; Level 3: Class; Level 4: Order; Level 5: Family; Level 6: Genus; Level 7: Species). Each layer is equipped with explicit axioms (totaling 103), compatibility conditions (D1–D9), and fundamental theorems with complete proofs (at least 4 steps for standard theorems, at least 8 steps for core theorems). We embed all major branches of discrete mathematics—graph theory, hypergraph theory, matroid theory, group theory, lattice theory, order theory, combinatorial design, formal languages, automata theory, computational complexity, random combinatorics, algebraic combinatorics, and topological combinatorics—into the classification. We further identify 53 gaps (7 per layer for Levels 1–7, plus 4 meta-predictions for Level 0), each elevated to a predictive branch with full axiom systems, main theorems, and rigorous proofs (at least 14 steps per prediction). We prove strong rigidity (20 steps), NP-completeness and #P-completeness of the classification existence problem (20 and 18 steps, respectively), and construct non-standard finite structures via a compactness argument (24 steps). All earlier conjectures and open problems have been resolved and promoted to theorems. The system is shown to be extensible, admits an ∞-categorical lift, and establishes a unified bridge to the classification systems of quantum mechanics and the Standard Model. No foundational step is omitted; all proof steps are explicitly enumerated. The system relies on a small number of classical foundational theorems (Tutte’s planarity criterion, Witt’s uniqueness theorem for Steiner systems, Gromov’s polynomial-growth theorem, the PCP theorem, the Baker–Gill–Solovay relativization theorem, the Grohe–Marx WL completeness theorem, Tutte’s excluded-minor characterization of graphic matroids, the Adiprasito–Huh–Katz hard Lefschetz theorem for matroids, Jensen’s diamond principle and the Solovay–Tennenbaum theorem for Suslin tree independence, and the Kleitman–Golden theorem on the minimum number of spanning trees in cubic graphs).
shifa liu (Wed,) studied this question.