The Nycloid and Non-Repetition Curvature Limit (NRCL): A Geometric Support Theory for MED Deformation and the Brachistochrone–AMAR Curve

This paper presents the discovery of a new geometric extremal principle—the Naik’s Non-Repetition Curvature Limit (NRCL)—derived from the author’s search for a curve that maximizes downward curvature while maintaining strict monotonicity in both coordinates and avoiding all repetition of state. This requirement originates from the author’s MED (Mass–Energy Deformation) Theory, where deformation along the MED axis must remain irreversible, monotonic, and coordinate injective.

From these conditions, a curvature inequality is derived without reference to any classical curves. Solving this extremal condition yields a curvature law of the form _κ(∆y) ∝ _{√(1/ ∆y)}. Only afterwards is it recognized that the resulting extremal curve matches a descending half-cycloid. This motivates the introduction of the Nycloid—a cycloid reinterpreted as the maximum-curvature monotonic descent curve consistent with NRCL.

Finally, the Nycloid curvature law is inserted into the MED deformation framework of the AMAR Breathing Universe Theory, providing the geometric saturation limit for deformation in the companion paper “Brachistochrone-AMAR Gravity: Using Newton's Cycloid for Mass–Energy Density Deformation and Testing Against Real Galactic Rotation Curves”.