This paper presents a constructive, geometrically grounded solution to the Yang–Mills existence and mass gap problem using the Expansion–Compaction–Torsion (ECT) framework. By representing quarks as fractional torsion loops and gauge bosons as pure torsion threads, we derive confinement, mass generation, and the non-Abelian Lie algebra structure directly from geometric principles. The model embeds SU(3) and SU(2) gauge symmetries as emergent properties of torsional shell dynamics, offering a novel topological foundation for gauge theory. A discrete mass gap arises naturally from compaction-induced energy quantization, fulfilling the conditions of the Clay Millennium Prize Problem. This work builds upon previous publications unifying geometry, topology, and physics through the ECT model, including a constructive proof of the Poincaré Conjecture. The findings provide a unified field-theoretic framework linking particle physics to fundamental geometric flows in spacetime.