A Minimal Extension of Newtonian Dynamics and the Emergence of Geometric Consistency

We present a minimal geometric extension of Newtonian dynamics that provides a pedagogical ladder from classical mechanics to special relativity and beyond. The formulationintroduces a small number of additional curvature degrees of freedom, leading naturally toa five-dimensional description while retaining a simple equation of motion.Despite its compact appearance, the resulting ACORN equation encodes invariant structure that forces the hyperbolic relations underlying Lorentz boosts. Full four-dimensionalMinkowski special relativity is recovered as a special case, without introducing observerbased postulates, coordinate transformations, or auxiliary assumptions.The same geometric framework unifies inertial and electromagnetic structure, explainsinvariant particle properties, and provides a natural setting for quantisation through curvature circulation and closure. Fundamental constants and dimensionless ratios emerge asgeometric projections rather than independent inputs.This paper emphasises clarity and accessibility, showing how successive physical theoriesarise as projections of a single geometric structure. Further development of the curvaturespectrum and its implications for particle physics and cosmology are outlined in a companionvolume.