Thermal freeze-out and collective signatures in $$pp$$ collisions with a non-extensive statistical study on energy and pseudorapidity systematics

This paper presents a comprehensive analysis of the double-differential [Formula: see text] distributions of charged particles across twelve pseudorapidity regions ([Formula: see text], bin width [Formula: see text]) in [Formula: see text] collisions at [Formula: see text], 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 [Formula: see text] ranges (validated by Data/Fit panels). The extracted parameters including kinetic freeze-out temperature ([Formula: see text]), mean transverse flow velocity ([Formula: see text]), non-extensivity parameter ([Formula: see text]), and mean transverse momentum ([Formula: see text]) reveal systematic dependencies on η and [Formula: see text]. Both [Formula: see text], [Formula: see text], and [Formula: see text] decrease with [Formula: see text], attributed to reduced energy deposition and weaker thermalization in fragmentation-dominated high-[Formula: see text] regions. These parameters also exhibit marked growth with [Formula: see text], reflecting collision-energy-driven dynamics. In contrast, [Formula: see text] increases with [Formula: see text], signaling greater deviation from thermal equilibrium (Boltzmann limit [Formula: see text]) at larger [Formula: see text], while mid-η particles ([Formula: see text]) approach near-equilibrium behavior. Parameter correlations, particularly between [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], 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 [Formula: see text] interactions.