Mathematically Rigorous Proof of Toponium Existence in the TEBAC 9D$^+$ Framework

This work presents a mathematically rigorous and phenomenologically grounded analysis of toponium formation based on the novel TEBAC 9D$^+$ framework. We establish necessary and sufficient conditions for the existence of toponium as a quasi-bound state near the top–antitop production threshold. The approach unifies analytic derivations, non-relativistic QCD corrections, and topological effects, and directly confronts theoretical predictions with recent experimental data from the LHC. All key parameter definitions, full numerical results, over 50 scientific illustrations, and the complete Python codebase used in the analysis are included. This resource aims to provide a comprehensive reference for both theoretical developments and future experimental studies on heavy quarkonium systems.