  We propose a model-independent analysis of near-threshold enhancements using
independent S-matrix poles. In this formulation, we constructed a Jost function
with controllable zeros to ensure that no poles are generated on the physical
Riemann sheet. We show that there is a possibility of misinterpreting the
observed near-threshold signals if one utilized a limited parametrization and
restrict the analysis to only one element of the S-matrix. Specifically, there
is a possibility of the emergence of ambiguous pair of poles which are
singularities of the full S-matrix but may not manifest in one of its elements.
For a more concrete discussion, we focused on an effective two-channel
scattering where the full S-matrix is a $2\times2$ matrix. We apply our method
to the coupled two-channel analysis of the $P_\psi^N(4312)^+$ and found that
the compact pentaquark interpretation cannot be ruled out yet.
