A wave, in the sense developed here, is a field on a discrete relational chain that is periodic in both the temporal index and the spatial index, generated at each node by the same finite, non-trivial, identity-preserving operator sequence, offset node to node by unit relational latency. This paper constructs such a field — the Chrono-Elastic Wave — from two strictly irreversible shear operators (SINK, PUMP) and one sign-reflecting operator (the horizon operator), realised throughout in the complex plane ℂ via the Wirtinger calculus of Relational Operator Geometry on the Complex Plane (ROG-ℂ). Four results are established. First, the horizon operator is shown to coincide exactly with complex conjugation, with Wirtinger pair (0,1). Second, the commutator of SINK and PUMP — the torsion generator — is shown to equal the horizon operator itself, up to the scalar −γδ; horizon conjugation is therefore not an auxiliary device but the torsion generator applied directly. Third, this identity is used to prove that horizon-conjugating the reverse-order SINK–PUMP product yields exactly the group inverse of the forward-order product, both as real 2×2 matrices and independently as Wirtinger pairs, giving closure of a finite local operator sequence: the disturbance returns to its exact initial value after one period. Fourth, a propagation axiom extends this local periodicity to a field on the integer chain, and the resulting field is proved to be periodic in the spatial index with the same period as the temporal index, at the maximal admissible propagation speed of one relational tick per step. The net Wirtinger pair of one complete period is shown to be (1,0): exactly holomorphic, despite every individual step in the period being non-holomorphic. All claims are stated as definitions, axioms, theorems, and proofs.
Isong Otto Beseka (Fri,) studied this question.