Gauge Field Rigidity on Hyperkähler Space: The Exact Global Resolution of the Yang-Mills Mass Gap Problem

Abstract The Yang-Mills existence and mass gap problem is one of the most fundamental unresolved challenges in mathematical physics. Historically, attempts to establish a mathematically rigorous quantum Yang-Mills theory on flat Euclidean space R4 have been stymied byultraviolet divergences, infrared instabilities, and the non-physical nature of continuous perturbation schemes. This paper presents an exact, non-perturbative global resolution of theYang-Mills Mass Gap Problem.We abandon flat coordinates and formulate the gauge field on a four-dimensional Ricciflat Hyperkähler manifold M4. We lift the classical connection 1-form A into a quaternionicoperatorconnectionAactingonthenon-commutativeCliffordbundle,whosestabilityisgoverned by the Reflexive Topos Logic (RST) established in our preceding papers. We demonstratethattheself-consistent, nilpotentboundaryconditionofthediscreteexteriorderivatived2discrete ≡ 0 forces any zero-mass excitation states(representing masslessgluons)togenerateanasymmetric, divergent algebraic shear tensorshear on the Sp(1) gauge bundle. To preventthe collapse of global consistency, the background geometry of the Hyperkähler space generates an infinite gauge restoration force, locking the minimum excitation energy away fromzero. This proves the existence of a strictly positive mass gap ∆ > 0 as a direct consequenceof topological gauge rigidity.