Clay Walls for Yang--Mills: a Logical Map of the Infrared and Ultraviolet Barriers

This work presents a structured analysis of the two logically independent mathematical barriers underlying the Yang–Mills mass gap problem, one of the Clay Millennium Problems. The first barrier, Wall 1, concerns the existence of a uniform spectral gap for the lattice Yang–Mills Hamiltonian in the thermodynamic limit at fixed lattice spacing. The second barrier, Wall 2, addresses the preservation of this gap in the continuum limit, where the vanishing of the Casimir gap creates a fundamental ultraviolet obstruction. The paper systematizes five classical formulations of Wall 1 (spectral form, vacuum overlap, no‑dilute condition, Poincaré inequality, effective potential), identifies the shared structural difficulty (spreading of minimizing sequences), and analyzes the 1‑link model as a reference case with uniform Bakry–Émery curvature. We also discuss gap stability under RelBound perturbations and the role of gradient flow in the UV regime. The result is a logical map of the infrared and ultraviolet obstacles that organizes classical physical and mathematical intuitions without claiming to resolve the Clay problem.