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 Hagedorn[Formula: see text]flow 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.