  Extended thermodynamics commonly uses polynomial moments to model
non-equilibrium transportation, but faces a crisis due to sub-shocks, which are
anomalous discontinuities in gas properties when predicting shock waves. The
cause of sub-shocks is still unclear, challenging the validity of extended
thermodynamics. This paper reveals, for the first time, that sub-shocks arise
from intrinsic limitations of polynomials leading to a discontinuous phase
transition. Therefore extended thermodynamics necessitates alternative moments
beyond polynomials to avoid sub-shocks.
