Yang-Mills Mass Gap and Confinement in ROG-H

The Yang–Mills existence and mass gap problem asks for a rigorous non-abelian gauge theory on continuous spacetime ℝ⁴, and a proof that the energy spectrum has a strictly positive lower bound. This paper states the structural content of the problem in the discrete, relational, quaternionic geometry of Pure Temporal Geometry (PTG-ℍ). The domain is a discrete relational graph H = (V_q, E_q) with state space ℍ. A quark is a node of the graph. A gluon is an edge, carrying the Chrono-Elastic Wave as its internal state. Flavor is set by the Quark Assignment Principle: the sign of the Expectation axis Φ₃. Four structural results are proved. Existence requires at least two nodes joined by an edge. The mass gap is the torsion coupling κ_g = γδ, strictly positive under Axiom G1. Confinement follows because an isolated node has no temporal geometry. Hadronization is a JUMP-induced bifurcation of the graph when the accumulated torsion exceeds the critical threshold Φ_critical. The four-dimensional Horizon operator fails to invert the Heisenberg torsion, so the gluon cycle closes through the Sequence gauge shift, giving net gauge restoration. The same structure yields the photon, Z boson, and W boson as degenerated projections of the master gluon cycle.