Quarternionic Relational Operator Geometry

This paper gives the quaternionic realisation of Relational Operator Geometry (ROG). The quaternion q = Δ + Φ₁i + Φ₂j + Φ₃k is the state space that carries the three-part structure of time: Memory (Φ₁), Sequence (Φ₂), and Expectation (Φ₃), together with the Now (Δ) as the active present. The five-dimensional operator manifold GL (2, ℝ) × U (1) acts on this four-dimensional state space. SINK writes the Now into Memory. PUMP reads Expectation into the Now. These are different axes on each side, so SINK and PUMP do not close on a shared pair the way they do in the two-dimensional theory. Direct computation gives the torsion commutator: Ŝ, P̂ (Δ, Φ₁, Φ₂, Φ₃) = (0, γδΦ₃, 0, 0) This is a single directed channel: Expectation into Memory. It touches neither the Now nor Sequence, and it does not touch Expectation itself. SINK, PUMP, and their commutator close as a Heisenberg algebra, not as sl (2, ℝ). This paper gives the complete operator actions, the Heisenberg structure of the torsion, the fixed points of each operator, JUMP as a unitary quaternion rotation, and the correct reduction of the wave case: a one-way relay rather than a closed oscillator, consistent with the horizon-closure results proved elsewhere in this series. All results concern time only. No spatial claim is made. The Chrono-Elastic Cycle and the Chrono-Elastic Wave close the SINK–PUMP torsion on the complex plane ℂ using the Horizon operator Ĥ, complex conjugation. In the quaternionic state space ℍ, Memory and Expectation are separate axes, and the torsion is a directed transfer between them. The extended Horizon operator Ĥ_ℍ commutes with this torsion instead of inverting it, and closure by reflection fails. We show that closure is instead supplied by a gauge shift 𝒢 carried by the Sequence axis Φ₂, which reassigns the carrier's identity between Giver and Receiver without changing the numeric state. We state the Heisenberg Conjugation Identity, which replaces the Horizon Conjugation Identity, and use it to prove that the six-step Heisenberg Cycle closes exactly at a single node, and that the same six steps, propagated along the relational graph, give a doubly periodic Heisenberg Wave with net gauge restoration. The result extends both papers without altering any result already proved on ℂ.