A Theoretical Definition of Tensor-Based Cosmic Structure and Observational Alignment of Universal Axes

The supplementary note titled “Application of the Tensor-Induced Flat Rotation Equation to the Milky Way and the Andromeda Galaxy” examines whether the observed flat rotational structures of spiral galaxies can be reproduced through spatial tensor alignment and vector transformation without introducing additional mass components such as dark matter. The study applies the same structural equation framework to both the Milky Way and the Andromeda Galaxy in order to investigate the possibility of a unified rotational mechanism. The supplementary note titled “Pre-Registered Predictions and Verification Plan for Cosmic Anisotropic Structure” presents a pre-analysis prediction framework for large-scale cosmic anisotropy. The document defines, prior to observational analysis, the predicted statistical symmetry relationships between galactic rotation directions, black hole jet orientations, polarization distributions, and the proposed T1 axis. The framework is intended to minimize post-hoc interpretation by fixing theoretical predictions and verification procedures before statistical analysis using publicly available astronomical datasets. Together, these supplementary studies extend the present theoretical framework toward quantitative verification of galactic rotational structure and large-scale directional anisotropy within the observable universe.