Summary of Relativistic Velocity Addition Formulas and Preliminary Monograph (Work in Progress).

Version 2 Update (January 2026) This updated version includes the following major additions: The "Forgotten Fundamental Formula": A new addition to the summary of formulas, providing a strikingly simple expression (basically a "Tangent Addition") for relativistic velocity addition derived from a Cayley-type projection. While implicitly present in the author's earlier geometric research (namely doi:10.5281/zenodo.18322428 (2014) and doi:10.5281/zenodo.18320378 (2015)), its formal explicit statement reveals the full algebraic power of the Möbius framework. Expanded Draft (120 pages): The monographic work has been extended to 120 pages. While the previous version ended with detailed explanations of the conformal action of the group on the sphere of Square Roots of -1 and the analysis of fixed points, this version also includes the beginning of Chapter 4, introducing the most basic concepts to adapt the solid Mathematical Theory to Relativistic Kinematics. Content of this deposit: Summary of Addition Formulas: Now featuring 8 formulations, bridging the gap between Unit Quaternions (Cinematic/Intuitive) and Projected Möbius-Cayley variables (Algebraic/Efficient). Monograph Draft (120 pages): A comprehensive exploration of the group $q \mapsto (aq-b)(bq+a)^{-1}$ and its conformal properties in Relativistic Kinematics. Python Scripts: Functional tools for double-checking the Velocity-Addition Formulas as well as some isomorphisms between the studied quaternionic structures and standard Lorentz transformations [the Python files were initially not intended to be shared, and do remain quite "poorly documented" though]. Disclaimer: This remains a "Working Paper" and a research-in-progress draft. It is published to establish chronological priority on the "Forgotten Fundamental Formula" and the associated non-linear boost formalism.