In this manuscript, we introduce basic \(\mathbb{F}^{\mathscr{B}}\) -contractions as a new class through Hardy-Roger, Reich, and Ćirić contractions to prove the existence and uniqueness of fixed point results. This framework includes concrete examples meeting fundamental \(\mathbb{F}^{\mathscr{B}}\) -contraction criteria while failing to comply with the requirements of established \(\mathbb{F}\) -contraction standards. As an application of our main results, we employ the Atangana-Baleanu fractional derivative to analyze the existence and uniqueness criteria of solutions for the fractional-order Lorenz model. Our study analyzes the fractional-order Lorenz model by fixing the fractional order, initial conditions, and two parameters while varying the others until the butterfly effect is observed. This research demonstrates the adaptability of basic \(\mathbb{F}^{\mathscr{B}}\) -contractions in nonlinear dynamics and fractional-order systems, while highlighting the criteria for their existence.
