Pure Yang–Mills in the SCEQC Framework: The Mass Gap as the Poincaré Constant of the SP Vacuum Density

This work advances the Scretching–Planck (SP) framework toward the Yang–Mills mass-gap problem (Pure Yang–Mills in the SCEQC Framework) by establishing an exact identity between the finite-lattice mass gap and the Poincaré spectral constant. It proves a positive mass gap on every finite lattice, connects the SP ground-state density to the Wilson path integral, and incorporates rigorous strong-coupling results. SU(2) lattice Monte Carlo simulations provide supporting numerical evidence consistent with dimensional transmutation. The analysis reduces the remaining continuum mass-gap challenge to a single unproven condition: a uniform positive lower bound on the spectral gap along the renormalisation-group trajectory. These results clarify what is rigorously established, what is numerically supported, and what remains to be proved.