The Li Equation Mathematical Formalization, Cosmological Implications and Phenomenological Models

This third part of our comprehensive study of the Li Equation $k_B T_P \cdot t_P = \hbar$ develops its mathematical formalization within an operator-algebraic framework, explores its profound implications for cosmology and the very early universe, and constructs specific phenomenological models yielding testable predictions. We begin by embedding the equation in a generalized algebraic structure where time and temperature arise as conjugate variables in a thermodynamic phase space for quantum geometry, leading to a proposed $\widehat{T}$-$\hat{t}$ commutator. We then apply this framework to cosmology, showing how the Li Equation naturally suggests a non-singular Big Bang with an initial Planckian state characterized by $T_{\text{init}} \sim T_P$ and $t_{\text{init}} \sim t_P$, and derive modifications to the inflationary potential and the primordial power spectrum. Finally, we build detailed phenomenological models for vacuum dispersion, spacetime-induced decoherence, and modifications to black hole radiation, extracting quantitative predictions for observations ranging from gamma-ray astronomy to laboratory-based optomechanical systems. This part transforms the Li Equation from a symbolic relation into a concrete theoretical tool with direct relevance to observable physics.