Global Stability of the Hybrid Balaban–Brydges Flow for a Magnetically Charged Gas: Connecting the UV Lattice Gauge Theory to the IR Semiclassical Mass Gap

AbstractThis paper details the rigorous construction of the Hybrid Renormalization Group flow utilized in the Constructive Gauge Theory Program (CGTP). We demonstrate the ultraviolet stability of the theory employing Balaban’s Block Axial Gauge, which preserves locality while controlling gauge invariance during the block‑spin transformations. Crucially, we prove a “Matching Lemma” which guarantees that the effective action at the transition scale satisfies the Kotecký–Preiss convergence condition required for the infrared Phase II. Finally, utilizing a polymer expansion for the bion gas, we demonstrate that the effective mass parameter follows a dynamical flow with a strictly positive global attractor, ensuring the existence of the mass gap in the continuum limit.