Topological Condensed-Matter and Solid-State Physics: A Three-Axiom Reconstruction of Many-Body Matter, Crystals, Collective Excitations, Correlations, Topological Phases, and Transport

Abstract

We develop a conditionally closed theoretical architecture for condensed-matter and solid-state physics from three physical axioms: microscopic ontic definiteness and finite localization, the physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. Electrons, nuclei, atoms, and molecular units enter as finite topological carriers, whereas crystals, energy bands, phonons, quasiparticles, order parameters, defects, and topological phases are derived as spectral, variational, statistical, or homotopy properties of the resulting many-body system. We construct a sorted first-order language, a model-theoretic conservative extension of the underlying physical theory, and a constrained Legendre descent from a unified matter–substrate action to a self-adjoint, lower-bounded many-body Hamiltonian. A signed ownership ledger prevents double counting among self-energy, screening, correlation, electron–phonon, electromagnetic, historical-memory, and environmental contributions. For neutral stable branches, we establish a thermodynamic limit under translation covariance, two-sided almost-additivity, controlled Coulomb surface terms, and passive auxiliary realizations of causal memory. The framework derives crystal symmetry, Bloch and Wannier representations, lattice dynamics, phonon quantization, electronic bands, semiconductor statistics, Green functions, conserving Kadanoff–Baym dynamics, strong-correlation reductions, magnetism, polarons, superconductivity, disorder, localization, transport, critical phenomena, dielectric response, defects, interfaces, and low-dimensional systems. Conventional Bloch, tight-binding, nearly-free-electron, Debye, Landau, Fermi-liquid, Hubbard, Heisenberg, BCS/BdG, Boltzmann, and Kubo theories are recovered only as controlled limits with explicit validity domains and failure conditions. Fundamental-carrier topology, band topology, and interacting many-body topology are distinguished rigorously. Stability criteria are given for spectral and mobility gaps, Green-function invariants, Floquet and non-Hermitian phases, completely positive open dynamics, Liouvillian steady-state bundles, and finite-memory auxiliary systems. We also specify when thermodynamic, zero-frequency, zero-wave-number, long-time, symmetry-breaking, and regulator limits commute and when their ordering defines distinct physical observables. A terminal completion theorem identifies the conditions for pure-theory closure: semantic conservativity, action traceability, self-adjoint lower-bounded dynamics, thermodynamic-limit control, causality, positivity, conservation laws, controlled standard limits, valid topological domains, and falsifiable observable maps. This theoretical closure does not imply that the spectrum, phase diagram, or transport coefficients of any particular material have been calculated or experimentally confirmed; such tasks belong to separate material-application programs. Keywords **global realism; topological condensed matter; solid-state physics; many-body theory; causal memory; historical disturbance field; Green functions; strong correlations; superconductivity; topological phases; quantum transport; thermodynamic limit; open quantum systems; Liouvillian topology; controlled effective theories**