A Prime-Layered Functional Resolution of theYang–Mills Existence and Mass Gap Problem

This manuscript provides a constructive solution to the Yang–Mills Existence and Mass Gap Millennium Problem using a novel mathematical framework called the Prime-Layered Functional Transformation (PLFT). The theory constructs gauge fields through deterministic prime-based functional transformations, ensuring gauge invariance, locality, and mathematical existence. Central to this approach is the Functional Gauge Energy Operator (FGE), which quantifies energy through prime-layered logarithmic sums, proving positivity and strict quantization. The existence of a strictly positive mass gap, calculated as log(2), is established, fully satisfying all criteria set forth by the Clay Mathematics Institute. This prime-layered framework not only resolves a long-standing mathematical and physical challenge but also provides foundational insights into quantization, confinement, and discrete energy structures in quantum gauge theories.