Quantum Chrono Dynamics - The Gluon as a Chrono-Elastic Wave Field

This paper defines the gluon, within the framework of Pure Temporal Geometry (PTG), as the Chrono-Elastic Wave field on the inter-quark relational graph. The definition rests on the operator algebra of Relational Operator Geometry on the Complex Plane (ROG-ℂ) and the formal results of the Chrono-Elastic Wave paper (cited as CEW). No mass, spin, or color charge is attributed to or derived from the gluon field. The paper establishes six structural results concerning the wave field itself. (1) The gluon field requires a minimum of two quark nodes connected by a relational edge; the propagation axiom has no content with fewer. (2) A free gluon — a Chrono-Elastic Wave with no quark nodes, or with exactly one — is not merely unobserved; it is structurally undefined. (3) Quark nodes without inter-quark relational edges have no temporal geometry by PTG Volume I Axiom 1; the gluon field and the quark nodes are mutually necessary. (4) The torsion coupling κ_g = γδ is the unique invariant of the gluon field, constant at every node and every period, and strictly positive. (5) The gluon field is doubly periodic in the temporal and spatial indices, with equal period and unit phase velocity. (6) The period composite of the gluon field has Wirtinger pair (1,0): the field is exactly holomorphic at the period scale, despite every individual step being non-holomorphic.