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A parametrized mathematical model is required to extract the informationcarried by transverse momentum $p_t$ spectra from high-energy nuclearcollisions and subject it to physical interpretation in terms of possiblehadron production mechanisms. The importance of proper model construction andimplementation has increased with the emergence of claims for ``collectivity''(flows) associated with small collision systems (e.g. $p$-$p$ and $p$-Pb). Atwo-element spectrum model, denoted herein as the Bylinkin model, includes anexponential element and a ``power-law'' element interpreted by the authors torepresent emission from a thermalized source and from jet productionrespectively. Application of the Bylinkin model to various collision systemshas led to conclusions about achievement of thermalization and othercharacteristics of nuclear dynamics. In connection with the Bylinkin modelthere has emerged theoretical conjecture that the thermalization mechanismsignaled by the exponential element is hard processes interacting with quantumentanglement within projectile protons. Predating the Bylinkin model is atwo-component (soft+hard) model (TCM) derived empirically from the evolution of$p$-$p$ spectrum data with event multiplicity as a form of data compression.The TCM has been applied to many collision systems and hadron species fromwhich a systematic description of high-energy nuclear collisions has emerged.The Bylinkin model can be seen as a limiting case of the TCM model functionsthat is unsuited to represent underlying production mechanisms. The presentstudy provides detailed comparisons of the two models for a variety ofsituations and contrasts two very different data interpretations that result.

Comparison of spectrum models as applied to single-particle $\bf p_t$ spectra from high-energy p-p collisions and their physical interpretations