From Approximation to Local Closure: A Qualitative Reduction Principle for Newtonian and Relativistic Dynamics

Approximation language is indispensable in relating Newtonian and relativistic dynamics, but by itself it does not specify which local questions remain closed after reduction. This paper develops a qualitative local-closure principle that refines standard approximation language into an explicit criterion for legitimate reduction between nearby finite-dimensional dynamical descriptions. The motivating case is the relation between Newtonian and special-relativistic particle dynamics in a local low-velocity regime, but the main contribution is structural rather than model-specific. A new qualitative distinction is introduced by separating local predictive agreement, descriptor compatibility, and question-closure. Small trajectory error alone is proved insufficient for arbitrary question transfer. Legitimate reduction is then shown to require a reduced descriptor family on which the target question descends with controlled error. A concrete local theorem is established for Newtonian and relativistic dynamics under explicit low-velocity, regularity, and finite-time assumptions. The result shows that Newtonian dynamics is not merely a rough approximation in the relevant regime, but a question-limited reduced description whose legitimacy depends on which local answers remain closed. The result provides a reusable structural criterion for legitimate reduction in finite-dimensional local dynamics.