A Universal Obstruction to Simulating ( 𝑥 , 𝑦 , 𝑧 , 𝑥 + 𝑦 + 𝑧 ) (x,y,z,x+y+z) in Near-Extremal 3-AP-Free Sets

This record contains a preprint entitled “Digit Rigidity and Container Obstructions for Three-Term Arithmetic Progression–Free Sets.”The paper develops a digit-based combinatorial framework for three-term arithmetic progression–free sets, encoding arithmetic relations with carries via a finite 3-uniform hypergraph. Using standard hypergraph container theorems, it establishes a digit-slice compression principle that forces structural rigidity in near-extremal configurations. This rigidity activates a canonical obstruction showing that the linear form (x,y,z,x+y+z)(x,y,z,x+y+z)(x,y,z,x+y+z) cannot be simulated except with exponentially suppressed frequency. The argument is finite, combinatorial, and unconditional.