A-3_Consistency Conditions for Velocity-Dependent Scales and the Uniqueness of the Lorentz Factor

This version updates the manuscript to the finalized English edition prepared for submission to General Relativity and Gravitation (GRG). The structure, arguments, and mathematical derivations remain consistent with the previously released Japanese version, but the following improvements have been made: Full English translation with polished academic style suitable for journal submission Clarification of the general consistency condition for velocity-dependent scales Refined derivation showing the uniqueness of the Lorentz factor Improved explanations regarding consistency with electromagnetism, empirical observations, and special relativity Expanded appendices providing mathematical background (hyperbolic identities, velocity composition, alternative formulations, differential equation approach) Minor corrections to notation and references Overall enhancement of readability, precision, and transparency This update does not alter the scientific claims or conclusions of the original version. The initial Zenodo release continues to serve as the priority timestamp for the research.