Möbius-Coordinate Map Theorem for the Poincaré Disk

We develop an explicit scalar coordinate treatment of the known orthogonal Möbius gyrodecomposition of the Poincaré disk. The Möbius real and imaginary parts express each point uniquely as an ordered Möbius sum of components on the Euclidean orthogonal real and imaginary diameters. The Möbius-Coordinate Map Theorem identifies the disk diffeomorphically with (−1,1)2, and the coordinate parameters are bounded reparametrizations of Lobachevsky coordinates in a specified convention. We describe the associated equivalence classes: constant Möbius real part gives perpendicular hyperbolic lines, whereas nonzero constant Möbius imaginary part gives curves of constant signed distance from the real diameter. Rotational covariance supplies the formulas along arbitrary diameters. As a concrete application, the coordinates give the perpendicular foot and distance to a diameter directly from a disk point. The contribution is this explicit two-dimensional coordinate realization and its geometric interpretation, rather than a new general orthogonal decomposition theorem.