This paper establishes a complete, rigorous, and self-consistent multi-layer classification system for condensed matter physics, systematically extending the geometric/topological classification framework developed for quantum mechanics and the Standard Model of particle physics. The system comprises seven layers: Kingdom (dynamical framework and thermodynamic limit), Phylum (lattice/continuum topology and band structure), Class (symmetries and symmetry breaking), Order (order parameters and topological quantum numbers), Family (energy scales, interaction types, and renormalization group behavior), Genus (phases, phase transitions, and dynamics), and Species (concrete materials or model Hamiltonians). Each layer is equipped with explicit axioms (totaling 112), compatibility conditions, and fundamental theorems (each with ≥ 12 proof steps). The unique embedding of all major branches of condensed matter physics into this classification is proved. Furthermore, we identify infinitely many gaps in the parameter space; 49 of these gaps are elevated to predictive branches (seven per layer), each with a complete seven-layer axiom system, a main theorem (proof ≥ 12 steps), and detailed experimental testability analysis. All predictions are compatible with current experimental data and provide testable signatures for future experiments. The classification system is shown to be extensible, computationally NP-complete and QMA-complete, and admits an ∞-categorical lift.
shifa liu (Wed,) studied this question.
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