Non-trivial Vacuum Topology and the Yang-Mills Mass Gap: A Research Programme Based on a Scalar Condensate Model

Within the set-theoretic framework of \mathrm{ZF}^{+} with the axiom of countable choice, we propose a scalar-field effective scheme for studying the mass gap problem in SU(3) Yang-Mills theory on four-dimensional globally hyperbolic Lorentzian manifolds. The central idea is to map the vacuum gluon condensate of the gauge field to a quantum theory of a massive scalar field. We present the following partial progress: (1) In non-trivial topological sectors, using the Bogomol'nyi inequality and estimates of moduli space volumes, we provide a framework for arguing that the gluon condensate is positive; (2) Under the assumption of a positive gluon condensate, through the effective mass relation \mu^2(\rho) = \kappa \rho^{\alpha} (\alpha > 0), we reduce the problem to establishing a spectral lower bound for the massive scalar field; (3) In the globally hyperbolic setting, using the algebraic quantum field theory framework, we quantise the scalar condensate mode and prove that its Hamiltonian spectrum has a strictly positive lower bound \mu_0 > 0. The conclusions of this paper depend on a core assumption—the strict positivity of the gluon condensate (\rho_{\mathrm{vac}} > 0). We provide a preliminary derivation of this assumption based on topological arguments. Under this assumption, we establish the logical chain “non-trivial vacuum topology \Rightarrow positive gluon condensate \Rightarrow positive mass gap”, providing a constructible framework for the Yang-Mills mass gap problem.