Topological–Historical Wave Dynamics: A Complete Global-Realist Theory of Physical Waves, Memory, Spectra, Conservation Laws, Computation, and Experimental Tests

Abstract

We establish a logically closed, mathematically formulated, computationally executable, and experimentally falsifiable theory of wave dynamics within the global-realist framework. The construction inherits exactly three physical axioms: microscopic ontic definiteness and finite localization, the physical reality of the spacetime–vacuum substrate, and persistent causal source–response coupling. A physical wave is defined as a propagating or mode-transferring retarded disturbance generated by an actual finite source history, rather than as an independent substance or a universally ontic wave function. A unified causal chain is developed from finite-core source functionals and local, nonlocal, or hereditary medium response to constraint-compatible retarded Green operators, physical modes, energy–momentum and wave-action transport, scattering, interference, diffraction, nonlinear conversion, stochastic propagation, source reaction, and detector records. The theory treats strong, weak, and distributional solutions; hyperbolic, parabolic, dispersive, and mixed operators; gauge constraints; direct and tail propagation; spectral poles, branch cuts, continua, quasinormal modes, bound states in the continuum, and non-Hermitian representations. Controlled reductions include WKB rays, diffusion limits, transport equations, envelope models, solitons, shock laws, and classical branch theories with explicit error bounds and limit-order requirements. Scalar, electromagnetic, gravitational, acoustic, elastic, surface, internal, plasma, lattice, thermal, nuclear, matter, active, nonreciprocal, chiral, gyrotropic, and time-varying waves are internalized using unified notation while retaining their distinct sources, constraints, polarizations, constitutive structures, boundaries, nonlinearities, and observables. Executable regression suites verify smooth propagation, entropy shocks, passive memory, constraint preservation, dispersive transport, matched layers, resonance poles, quasi-BIC opening, spectral-pollution rejection, and an infinite-domain resonance computed independently by exact Dirichlet-to-Neumann reduction and exterior complex scaling. The resulting architecture distinguishes structural derivation, numerical verification, parameter calibration, and held-out experimental validation, thereby providing explicit rejection criteria for every admitted wave model. **Keywords** Global realism; wave dynamics; causal Green functions; finite-core sources; historical memory; wave propagation; dispersion; wave action; scattering; nonlinear waves; stochastic media; quasinormal modes; bound states in the continuum; topological waves; numerical verification; experimental falsifiability.