Effective Manifold Theory: A Transition-Atlas Framework for Native Generators, Geometric Charge, and Path Action

This preprint introduces Effective Manifold Theory (EMT) and Projective/Preparatory EMT (P-EMT) as a transition-atlas framework for representing dynamical systems through native generators, event sections, geometric transition charge, effective barriers, and accumulated path action. The central claim is not that EMT replaces existing dynamical systems, thermodynamics, or field-theoretic methods, but that it provides a representation layer that identifies whether a transition is dominated by native physical coordinates, instantaneous geometric charge, threshold-current behavior, accumulated geometric action, or memory/preparation effects. The manuscript develops the EMT lift for discrete and continuous observables, defines geometric charge χQ, introduces a photoelectric-like threshold law J_out = κ(χQ − Φ)_+, discusses accumulated EMT action A_EMT as a non-arbitrary P-EMT memory coordinate, and relates the framework to Poincaré sections, Hodge decomposition, MaxEnt/MaxCal, stochastic thermodynamics, effective-field reasoning, and Maxwell-like field analogies on transition atlases. Computational benchmarks on bee motion, Lorenz dynamics, three-body dynamics, and diffusion-like systems are included as preliminary evidence that EMT functions as a transition-mechanism diagnostic rather than a universal predictor. This is a theoretical and computational preprint. The claims about Maxwell-like fields, QFT analogies, and tensorial P-EMT are presented as effective or future directions, not as established equivalences to electromagnetism, quantum field theory, or fundamental physics.