Relational Operator Geometry on the Complex Plane

Relational Operator Geometry (ROG), established in the original paper, studies the 5-parameter operator algebra acting on the 2-dimensional real state space V = ℝ². This paper realises V as the complex plane ℂ (by setting z = α + iβ), translates all five ROG operators into the universal form of ℝ-linear maps z → az + bz̅, and develops the additional mathematical structure that the complex plane affords. The central result is that the fundamental torsion commutator [Ŝ(γ), P̂(δ)] = γδ·diag(−1,1) in the ℝ² formulation becomes z → −γδz̅ in the ℂ formulation: pure complex conjugation (scaled). The torsion of the two irreversible operators is precisely anti-holomorphic — it lives entirely in the ∂/∂z̅ component of the Wirtinger calculus. We develop: (1) the complete operator translations with Wirtinger coefficient pairs (a,b); (2) the holomorphicity filtration (finer and more natural than the reversibility filtration of ROG-ℝ²); (3) the fixed-point theorem for ℝ-linear maps on ℂ, which admits lines of fixed points when |b| = |a−1|, invisible in ℝ²; (4) the N-sector generalisation of the 4-grid to continuous argument stratification; (5) the unit circle as a natural JUMP-invariant set; (6) the Möbius extension as the projective super-algebra of ROG-ℂ; and (7) the winding number as a new topological invariant of discrete state trajectories.