Hadronic Mass Generation through Three-Dimensional Magnetic Locking: A Discrete Resolution for Yang—Mills Theory

The Yang-Mills mass gap problem asks for a rigorous proof, within the axiomatic framework of constructive quantum field theory, that quantum Yang-Mills theory in four-dimensional Minkowski space exists and exhibits a strictly positive mass gap ∆ > 0. We argue that the persistent failure of the program is the consequence of a categorical error: the continuum functional integral that the program seeks to construct sums over a configuration space whose elements have no ontological referent. We axiomatize an Endpoint-Only Ontology in which physical content is exhausted by interaction events and conditional probability distributions over them, with no intermediate trajectories. Within this ontology, we formalize a mechanism for hadronic rest mass generation based on three-dimensional magnetic locking of discrete photonic packets, derive linear quark confinement and the deconfinement transition, identify the three colour charges of QCD with the three orthogonal rotation axes of the lock structure, and show that SU(3) emerges as a mathematical consequence of antisymmetry combined with three-dimensional rotational closure. The mass gap is a one-line corollary: stable rotational closure in three dimensions requires at least three coupled locks, each carrying positive lock energy. We prove a non-existence theorem for the continuum limit of confining non-Abelian gauge theory: under the Endpoint-Only Axiom, no probability measure on the space of smooth gauge connections satisfies the Osterwalder-Schrader axioms while reproducing the conditional probability structure required by confinement and a positive mass gap. The argument employs the elementary fact that a tempered distribution invariant under continuous translations cannot have countable support unless it is zero. The Higgs mechanism is preserved as the parameterization of bare elementary-fermion mass; the lock mechanism accounts for the dominant 99% of hadronic rest mass that the Higgs alone does not explain. Eight falsifiable predictions are provided. The Clay problem, on the present analysis, is dissolved: the object whose existence is to be proved does not exist as a continuous field theory, and the mass gap exists trivially in the discrete description that does.