  A parametrized mathematical model is required to extract the information
carried by transverse momentum $p_t$ spectra from high-energy nuclear
collisions and subject it to physical interpretation in terms of possible
hadron production mechanisms. The importance of proper model construction and
implementation has increased with the emergence of claims for ``collectivity''
(flows) associated with small collision systems (e.g. $p$-$p$ and $p$-Pb). A
two-element spectrum model, denoted herein as the Bylinkin model, includes an
exponential element and a ``power-law'' element interpreted by the authors to
represent emission from a thermalized source and from jet production
respectively. Application of the Bylinkin model to various collision systems
has led to conclusions about achievement of thermalization and other
characteristics of nuclear dynamics. In connection with the Bylinkin model
there has emerged theoretical conjecture that the thermalization mechanism
signaled by the exponential element is hard processes interacting with quantum
entanglement within projectile protons. Predating the Bylinkin model is a
two-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 from
which a systematic description of high-energy nuclear collisions has emerged.
The Bylinkin model can be seen as a limiting case of the TCM model functions
that is unsuited to represent underlying production mechanisms. The present
study provides detailed comparisons of the two models for a variety of
situations and contrasts two very different data interpretations that result.
