Rigorous Constructive Framework for a Mass Gap in Elastic Quantum Models (Part I: Flat DEM/QEM)

We present a rigorous constructive framework for demonstrating the existence of a mass gap in Elastic Quantum Models (EQM), formulated on flat spacetime. The model couples four–dimensional Yang–Mills fields with a quantum–elastic scalar sector that ensures rigidity. By verifying the Osterwalder–Schrader axioms, we reconstruct a Hilbert space with a positive self–adjoint Hamiltonian. Uniform exponential clustering of gauge–invariant correlators implies a spectral gap: Spec(H)={0}∪[mgap,∞),mgap>0.\mathrm{Spec}(H) = \{0\} \cup [m_{\rm gap}, \infty), \quad m_{\rm gap} > 0.Spec(H)={0}∪[mgap,∞),mgap>0. The analysis is performed with lattice regularization, reflection positivity, cluster expansion, and renormalization group arguments, leading to non–trivial dynamics beyond free field theory. This work establishes a mathematically controlled proof of a mass gap in EQM, providing a new constructive approach linked to Yang–Mills theories and related to the Millennium Prize Problem.