Three XYZ Directions Read from Complete Pairwise Relational Waves: Growth of Internal Relational Directions and Saturation of Spatial-Direction Readout in Closed AB, ABC, and ABCD Systems

This preprint extends earlier closed two-body AB phase-system experiments, in which the only positional-phase relation was AB and position and acceleration-like readouts were therefore one-dimensional, to complete pairwise relation networks for ABC and ABCD systems. Each pairwise relation is treated as a physical complex relational wave; no observer-only C or D channel, per-step state normalization, observation damping, or absolute background axis is used. A real skew-symmetric generator is constructed from shared endpoints and phase differences and evolved by a Cayley orthogonal update, preserving both the quadratic closure sum_e X_e^2=R^2 and the absolute-square sum. Across 32 trials and 720 steps for each configuration, AB had one stationary relation and generator rank 0. ABC had three active relations and generator rank 2 with a one-dimensional kernel: the result is read as two phase-readable XY axes and a Z normal determined by the XY plane. ABCD had six active relations and generator rank 6, decomposing into three independent rotation planes: the result is read as the three uniquely readable XYZ spatial directions, while the other three internal directions are not uniquely oriented. The paper distinguishes the number of internal relational directions from the number of uniquely readable spatial directions and interprets additional relations beyond four bodies as increasing non-unique internal directions rather than adding new spatial axes beyond XYZ.