This paper investigates the fundamental consistency of classical mechanics when applied to the one-body problem with a changing radial constraint, demonstrating a logical paradox inherent in the standard model. It is mathematically established that the centripetal force acts exclusively along the radius vector, meaning the component of force perpendicular to the momentum vector (F⊥ ) is zero. However, the Conservation of Angular Momentum (CoAM), defined as L=rP⊥ =constant, mandates that if the radius (r) changes, the perpendicular momentum (P⊥ ) must instantaneously change to compensate. Since Newton’s Second Law dictates that any change in momentum must be caused by a net force, the required change in P⊥ without a corresponding perpendicular force (F⊥ =0) constitutes a fundamental axiomatic contradiction. This paper concludes that CoAM is mathematically inconsistent in any variable-radius system, demanding a revision of the conservation principle.