Stability–Rigidity with Cascade

We establish a stability–rigidity framework in which structural persistence is maintained under controlled perturbative cascades. The core result demonstrates that rigid behavior can be recovered through stability amplification even when initial configurations are only weakly constrained. We introduce cascade-tightening operators that iteratively refine local constraints to global rigidity, yielding metastable plateaus and collapse thresholds. These refined mechanisms complement prior metastable barrier and torsional tightening frameworks, and collectively broaden the toolkit for analyzing rigidity emergence in complex mathematical and computational landscapes. This manuscript is the foundational theorem note in the Stability–Rigidity program, establishing the cascade-tightening mechanism used in later constructions. All subsequent papers referring to cascade-tightening or cascade-amplification should cite this version.