Major Conjectures and Cosmological Tests O: The Cosmological Falsifiability of the Yang–Mills Existence and Mass Gap Problem

This paper addresses the Yang–Mills existence and mass gap, one of the Millennium Prize Problems, by proposing an indirect test based on cosmological observations. We hypothesize that themathematical structure of the SU(3) gauge field with a unique vacuum and a positive mass gap ∆was realized in the early universe via a first-order phase transition, leaving characteristic imprintson the power spectrum of the primordial gravitational wave background. Under the axioms of information conservation and computability, we construct a computational tree model linking the massgap ∆ to the computational depth s, and derive that the gravitational wave background exhibits apeak frequency f∗ in the µHz–mHz band, a peak amplitude Ω∗ ≈ 3.2 × 10−13, and a relative amplitude of about 8.5% periodic modulation. Through numerical simulations, we demonstrate thatnext-generation space-based gravitational wave detectors (such as LISA, BBO, µAres) have the capability to detect such a signal with high significance. If future observations fail to find this signal, itwill impose strong constraints on our proposed physical realization model. This work translates, forthe first time, the Yang–Mills mass gap problem into an observable cosmological test, opening a newavenue for understanding the relationship between mathematical structures and physical reality.