Topological Analytical Mechanics: The Global-Realist Replacement of Point-Particle, Memoryless Mechanics by Finite-Core, Sector-Resolved, and Historically Coupled Dynamics

Abstract

We reconstruct analytical mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. Material bodies are represented not as structureless points but as finite topological excitations with actual histories; forces are projections of conservative interaction, finite-core contact, and retarded substrate response; and the local mechanical state is enlarged by sector and historical variables whenever memory remains dynamically active. Classical analytical mechanics is recovered as the controlled zero-topology, zero-memory, point-core limit of this enlarged theory. From a common parent action we derive the collective mass metric, topological connection, finite-core potential, causal memory kernel, Euler–Lagrange and Hamilton equations, regular and constrained Legendre transforms, contact and port-Hamiltonian structures, Noether balances, stability and bifurcation criteria, many-body and continuum reductions, relativistic and field formulations, stochastic and discrete mechanics, optimal control, and the canonical quantization interface. Exact elimination of stable historical variables gives the equivalent retarded integro-differential dynamics. Classical rigid-body, Keplerian, collision, contact, elastic, fluid, statistical, and semiclassical equations are recovered on their declared asymptotic domains. The manuscript contains no auxiliary physical postulates. Twenty-seven former auxiliary axioms are reclassified as derived conditions and assigned explicit traceability certificates. Logical, topological, causal, variational, conservation, and gauge conditions follow from the three axioms together with standard mathematics; positivity, coercivity, reciprocity, and regularity conditions define admissible realization classes. These proofs establish structural derivability and mathematical consistency, not numerical completion. Finite-core profiles, response spectra, Green operators, collision matrices, sharp constants, and uncertainty bounds still require extensive computation, controlled estimation, and independent experiment. The principal achievement is the axiomatic unification and reconstruction of analytical mechanics. Thermodynamics, optics, fluid mechanics, particle physics, cosmology, and astrophysics retain separate full theories and enter here only through their common mechanical interfaces. The resulting framework replaces mutually independent idealizations with a finite-core, sector-resolved, causal, and experimentally falsifiable mechanics. **Keywords** Topological analytical mechanics; global realism; finite-core mechanics; historical disturbance field; causal memory; topological sectors; variational principles; Hamiltonian mechanics; Noether theorem; nonholonomic mechanics; classical-limit recovery.