Coherent soft-hard evolution with event activity in pp collisions at s=0.9, 2.76 and 7 TeV

Multiplicity-differential transverse-momentum spectra of Formula: see text, Formula: see text, Formula: see text, and Formula: see text produced in Formula: see text collisions at Formula: see text, Formula: see text, and Formula: see text TeV are analyzed in event-activity classes characterized by the mean charged-particle multiplicity Formula: see text. The spectra are fitted with two widely used phenomenological descriptions: (i) a thermodynamically consistent Tsallis form and (ii) a Hagedorn parameterization supplemented by an effective transverse-boost component. Both functions reproduce the measured Formula: see text distributions across energies and multiplicity bins with stable fit quality, with the largest deviations confined to the edges of the fitted ranges where acceptance effects and hard-scattering contributions are most influential. When presented versus Formula: see text, the extracted parameters show clear and systematic trends. The effective temperature scales (Formula: see text in the Tsallis description and Formula: see text in the Hagedorn-based description) decrease with increasing event activity and exhibit mass ordering, while the high-Formula: see text tails harden with multiplicity: the Tsallis nonextensivity parameter Formula: see text increases and the Hagedorn power index Formula: see text decreases. Derived quantities such as Formula: see text, Formula: see text, Formula: see text, Formula: see text, and the characteristic momentum scales Formula: see text and Formula: see text provide compact measures of the soft-to-hard transition as event activity grows. The corresponding HagedornFormula: see textflow boost variables Formula: see text, Formula: see text, and Formula: see text, together with tail-related exponents, furnish complementary characterizations of the effective transverse-boost and power-law behavior. The extracted transverse-boost parameter Formula: see text (and its related variables) increases mildly with Formula: see text, indicating a stronger effective transverse expansion in higher-multiplicity events within the adopted parameterization. These multiplicity- and energy-resolved results provide a concise set of spectral parameters that can be used to benchmark phenomenological descriptions and to serve as quantitative constraints for event-generator descriptions of small-system particle production.