Momentum Identity Theorem (MIT): Conservation of Momentum Does Not Require Conservation of Momentum Identity

The Momentum Identity Theorem (MIT) introduces a conceptual distinction between the quantity of momentum and the dynamical identity of momentum within the framework of classical mechanics. While the conservation of momentum is one of the most robust principles in physics, the notation p=mvp = mvp=mv does not explicitly distinguish between momentum sustained by active constraints within non-inertial systems and momentum that persists autonomously under inertial motion. This work argues that two equal quantities of momentum need not possess identical dynamical identities. The theorem proposes that transitions from constrained motion to autonomous motion may preserve the numerical value of momentum without constituting additional momentum transfer events. Consequently, the emergence of autonomous momentum from restricted momentum should not automatically be interpreted as an independent event of action-reaction unless an additional impulse can be demonstrated. The MIT does not modify Newtonian mechanics, Noether's theorem, or the principle of momentum conservation. Instead, it formalizes an operational distinction that appears implicit within classical mechanics and may clarify conceptual ambiguities in systems involving transitions between non-inertial and inertial regimes. Although motivated by discussions arising within the broader framework of Momentum Engineering, the theorem is presented independently of any specific technological implementation. Its purpose is to contribute a precise conceptual refinement to the description of momentum and to stimulate further theoretical and experimental investigation. Keywords Momentum Identity Theorem MIT Momentum Identity Momentum Conservation Classical Mechanics Non-Inertial Systems Inertial Frames Constrained Motion Autonomous Motion Restricted Momentum Dynamical Identity Momentum Transition Action and Reaction Impulse Rotational Dynamics Reference Frames Conceptual Physics Momentum Engineering Theoretical Physics Foundations of Mechanics