Relational Operational Geometry - The Pure Mathematics of PTG

We present Relational Operator Geometry (ROG): a new mathematical discipline whose primary object is a 2-dimensional real vector space V = ℝ² equipped with a distinguished 5-parameter operator algebra Ω = {Î, R̂, Ŝ, P̂, Ĵ} ⊂ GL(2,ℝ) × U(1), a sign-stratification of V into four open quadrants by the function σ: V\{0} → {±}², and the discrete dynamics generated by ordinal sequences T: ℕ → V of state vectors evolving under operator composition. The four base operators Î, R̂, Ŝ, P̂ span gl(2,ℝ) and are classified into reversible (diagonal) and irreversible (unipotent shear) subfamilies. The fifth operator Ĵ acts as a rotation-dilation spanning the U(1) fiber. The fundamental algebraic invariant is the torsion commutator [Ŝ(γ), P̂(δ)] = γδ · diag(−1, 1), which is a scalar multiple of the Cartan element h of sl(2,ℝ). This identifies the ROG operator algebra with gl(2,ℝ) = sl(2,ℝ) ⊕ ℝ·I₂, extended by the U(1) rotation. We develop the complete algebraic structure: the Lie bracket table of all generators, the reversibility filtration of GL(2,ℝ), the sign partition as a CW decomposition of ℝ² with Euler characteristic 1, the quadrant transition graph, the orbit structure under each operator subfamily, the fixed-point theorem, and the discrete cocycle structure of operator sequences. We identify ROG’s position within the mathematical landscape and delineate what is genuinely new.