This paper presents a comprehensive analysis of the double-differential \(p_T\) distributions of charged particles across twelve pseudorapidity regions ( \(|\eta | < 2.4\) , bin width \(\Delta \eta = 0.2\) ) in \(pp\) collisions at \(\sqrt{s} = 0.9\) , 2.36, and 7 TeV. Using a modified Tsallis function incorporating an effective transverse flow velocity, we achieve excellent agreement between the model and experimental data, with consistent fit quality across all \(p_T\) ranges (validated by Data/Fit panels). The extracted parameters including kinetic freeze-out temperature ( \(T_0\) ), mean transverse flow velocity ( \(\langle \beta _T\rangle\) ), non-extensivity parameter ( \(q\) ), and mean transverse momentum ( \(\langle p_T \rangle\) ) reveal systematic dependencies on \(\eta\) and \(\sqrt{s}\) . Both \(T_0\) , \(\langle \beta _T\rangle\) , and \(\langle p_T \rangle\) decrease with \(|\eta |\) , attributed to reduced energy deposition and weaker thermalization in fragmentation-dominated high- \(|\eta |\) regions. These parameters also exhibit marked growth with \(\sqrt{s}\) , reflecting collision-energy-driven dynamics. In contrast, \(q\) increases with \(|\eta |\) , signaling greater deviation from thermal equilibrium (Boltzmann limit \(q \rightarrow 1\) ) at larger \(|\eta |\) , while mid- \(\eta\) particles ( \(|\eta | \approx 0\) ) approach near-equilibrium behavior. Parameter correlations, particularly between \(T_0\) , \(\langle \beta _T\rangle\) , \(q\) , and \(\langle p_T \rangle\) , highlight interplay between thermal and collective-like dynamics. The results provide critical insights into the interplay of thermalization, effective flow, and non-equilibrium effects in small collision systems, advancing the understanding of bulk hadronic matter in high-energy \(pp\) interactions.
