A Calculable and Falsifiable Three-Axiom Framework for Nonlinear Systems: Mechanics, Memory, Chaos, Continua, Networks, and Corrected Gravitation

We develop a self-contained mathematical framework for nonlinear physical systems based on three physical axioms: microscopic ontic definiteness with finite localization, the physical reality of the spacetime–vacuum substrate, and persistent causal delayed source–response coupling. Classical theories are reconstructed as controlled reductions rather than introduced as additional ontological principles. The framework covers constrained Hamiltonian and Lagrangian mechanics, perturbation theory, slow–fast dynamics, bifurcations, deterministic chaos, statistical reduction, nonlinear continua, fracture, fluids, turbulence, pattern formation, adaptive networks, and corrected gravitation. For every reduction, we specify the state space, closure assumptions, nondimensional parameters, neglected finite-core, historical, correlation, and boundary residuals, quantitative error measures, and rejection criteria. General relativity is recovered only in a declared local, short-memory, low-frequency, and point-source limit. Probability, entropy, information, networks, and emergent structures are treated as effective descriptions or observational representations, not fundamental entities. Conditional theorems, counterexamples, conservation ledgers, numerical benchmarks, and falsification interfaces distinguish rigorous results from assumptions and computed evidence. The resulting construction establishes a calculable and experimentally testable unified framework; it does not claim universal empirical confirmation or promote numerical agreement to the status of physical proof. **Keywords** Nonlinear systems; dynamical systems; geometric mechanics; delayed response; memory effects; bifurcation theory; deterministic chaos; singular perturbation; continuum mechanics; turbulence; pattern formation; adaptive networks; statistical reduction; corrected gravitation; controlled limits; numerical benchmarks; falsifiability.